2. Theory and Modelling Although (experimental) evidence is the starting point of our understanding of the

world and the incorruptible guardian of its truth, we would not be able to advance beyond the level of primitive reflection if we were unable to condense our experience into the closed form of theoretical concepts. Concepts and models represent the indispensable spiritual basis of our view of nature and serve as a powerful source of progress and as a route to novel questions and hypotheses which, in their turn, stimulate the performance of further experiments for their justification or falsification.

Which local views might be better suited to allude to the spiritual power of theory and modelling than those of the magnificent churches of L'Aquila? The painting on page 85 shows the Basilica of Collemaggio, this holy place where on the 29th of August 1294 St. Peter of Mottone, the hermit from the mountains near L'Aquila, took his solemn vows and was elected Pope Celestino V. As we have mentioned already, this very special experiment failed and, after only three months, Celestino V resigned. He returned to his mountains to be close to god and died as a hermit. In the Basilica his preserved body, clad in the papal robes, found his last rest. Most likely, the church shown on page 86 does not dispose of such prominent contents. Thus it has become possible that it hosts a theatre today, rather than a congregation.

The topics of theoretical treatment and modelling considered in this chapter include both typical liquids (S. Yashonath's article on ionic conductivity in water) and solids (A. Bunde and P. Heitjans about anomalous transport and diffusion in percolation systems) as well as the region in between, as considered by Kurt Binder on discussing interdiffusion in critical binary mixtures and by Yu Imoto and Takashi Odagaki in their paper on the diffusion on diffusing particles.

An abstract dedicated to the plenary lecture by Luca Cavalli-Sforza deals with the application of diffusion theory to such important social phenomena as re-settlement, immigration and emigration. Let us take the view of L'Aquila's town hall on page 87 as the symbol of completed settlement. And let us enjoy the view from the city of L'Aquila on page 88 over the surrounding land, which provides some impression of the interest of L'Aquila's first settlers to care for a good overview.

84

85

86

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88

Relationship between Ionic Radius and Pressure Dependence of IonicConductivity in Water

Parveen Kumar1, A.K. Shukla1,2 and S. Yashonath1,3

1 Solid State and Structural Chemistry Unit, Indian Institute of Science,

Bangalore-560012, India2 Central Electrochemical Research Institute, Karaikudi-630006 India

3 Center for Condensed Matter Theory, Indian Institute of Science,

Bangalore-560012, India and,

Jawaharlal Nehru Centre for Advanced Scientific Research,

Jakkur, Bangalore-560064, India

Abstract

Experimental measurements of ionic conductivity in water are analysed in order to

obtain insight into the pressure dependence of limiting ionic conductivity of individual

ions (λ0) for ions of differing sizes. Conductivities of individualions,λ0 do not exhibit

the same trend as a function of pressure for all ions. Our analysis suggests that the effect

of pressure on ionic conductivity depends on the temperature. At low temperatures, the

effect of pressure on relatively small ions such as Li+ exhibit an increase in conductivity

with pressure. Intermediate sized ions exhibit an increasein conductivity with increase

in pressure initially and then at still higher pressures, a decrease in ionic conductivity is

observed. Although there are data at low temperatures for ions of large radius, the effect

of increased pressure is expected to lower conductivity with increase in pressure over the

whole range. At higher temperatures, the dependence of conductivity on pressure changes

and these changes are discussed. Divalent ions such as SO2−4 exhibit different trends as a

function of pressure at different temperatures. Both the divalent ions (Ca2+ and SO2−4 ) for

which experimental data exists, exhibit an increase with pressure at lower temperatures. At

slightly higher temperatures, a maximum in conductivity isseen as a function of pressure

over the same range of pressure.

1. Introduction

Among the transport properties, the most accurately and relatively easily measured are

the ionic conductivities. These have been extensively investigated in different polar sol-

vents where different salts readily dissolve. The changes in conductivity with temperature,

pressure, size of the ion, concentration, etc. have been measured. Therefore a large amount

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, S. Yashonath 89

of data exists in the literature for different salts in a widevariety of solvents.

The importance of understanding the conductivity data in different polar solvents can

not be overemphasized. From a fundamental viewpoint it would lead to a capability to

predict and control as well as manipulate the conductivity.Further, a knowledge of the un-

derlying mechanism determining ionic motion can lead to increased understanding which

could lead to an ability to design new materials with better conductivity. The technological

spin-offs of all this development could be quite remarkable. Battery materials with higher

conductivity and lower dissipation are a possibility. Light materials for battery could reduce

the weight of the battery. Batteries under appropriate pressurized condition can probably

perform better. Some of these ideas could also be of importance in fuel cell technology.

The increased understanding can also lead to important advances in biochemistry and ion

conduction across biomembranes and may help unravel the reasons for selectivity observed

for potassium over sodium.

In spite of availability of a large amount of data, our understanding of the ionic con-

ductivity in water or other solvents still remains rudimentary. The reason for this is the

bewilderingly rich variety that the variation in conductivity exhibit as a function of the dif-

ferent conditions such as temperature, pressure, concentration, ion size, etc. It has been

very difficult, if not impossible, even to explain the variation of conductivity with just a

single variable such as ion size or pressure.

Influence of different variables on ionic conductivity havebeen investigated in the liter-

ature both experimentally and theoretically. Among the different variables that have been

studied, the most widely studied is the influence of size dependence on ionic conductivity.

These are discussed in most textbooks [1–5]. Solvents with hydrogen bonds such as water,

methanol, ethanol, etc. as well as a number of non-hydrogen bonded solvents such as ace-

tonitrile and pyridine are seen to exhibit a maximum in ionicconductivity as a function of

the ion radius. This maximum has been seen in all polar solvents. Positively charged ions

(e.g., alkali ions) as well as negatively charged ions (e.g., halide ions) show a maximum in

conductivity suggesting that such a maximum exists irrespective of the sign of the charge

on the ion or nature of the solvent. Thus, this maximum in conductivity is a universal

behaviour of ions in polar solvents.

This maximum is responsible for the breakdown in Walden’s rule which states that

the product of limiting ionic conductivity of a solutionΛ0 with solvent viscosityη0 is a

constant,Λ0η0 = c. It is generally seen that this product goes through a maximum when

plotted as a function of reciprocal of the ion radius. The maximum in Walden product

arises from a similar maximum in conductivity. This breakdown is probably related to any

90

breakdown in Stokes law.

Water being an important and well known solvent and due to itsimportance in many

chemical as well as biological processes, the conductivitymaximum has been most widely

studied in water as compared to other solvents [6]. The availability of measured conduc-

tivity data is extremely valuable, especially to verify thepredictions of theories or calcu-

lations. Further, existence of accurate potentials to model water and interactions between

water and the ion, has led to the detailed molecular dynamicssimulations whose results are

of great importance in relating the macroscopic behaviour with the microscopic properties

and understanding the cause of the many of the macroscopic behaviour.

Early work of Born [7] was responsible for increased interest in study of conductivity

maximum of ions in solution as a function of ionic radius. A number of groups have in-

vestigated the maximum in ionic conductivity in polar solvents [8–12]. These are aimed

at providing a theoretical framework to understand the underlying cause for the observed

size dependent maximum in ionic conductivity. The complexity of these electrolytic solu-

tions has meant that there are completely different theoretical approaches to understand the

maximum in conductivity.

One such theory is the solvent-berg model which put forward the suggestion

that smaller ions are strongly interacting with the nearestneighbour shell of solvent

molecules [13]. This was considered to be particularly trueof cations since these are gen-

erally smaller in size than the corresponding anions and have a higher charge density. The

ion essentially carries this shell of solvent molecules long enough that this leads to a larger

effective diameter which lowers its conductivity to a valuesmaller than the conductivity of

larger ions which have no strongly attached shells of solvent.

Another set of theoretical attempts to reproduce the observed conductivity variation

with ion radius is based on continuum models. Here dielectric friction arising from polar-

ization interaction between the ion field and the solvent is accounted for. Also accounted

for is the hydrodynamic friction arising from the viscosityof the solventη due to the van

der Waals interaction which is relatively short ranged. Born, Fuoss, Boyd, Zwanzig and

Hubbard and Onsager [7, 13–17] attempted to explain the observed maximum in terms of

the slow relaxation of the dielectric medium (solvent), induced by the electric field of the

ion as the ion diffuses. This gives rise to the dielectric friction,ζDF which is given by the

expression (see Zwanzig [17, 18])

ζDF = 3q2i (ǫ0 − ǫ∞)τD/(cr3

i (2ǫ0 + 1)ǫ0) (1)

whereτD is the dielectric relaxation time of the solvent associatedwith the dynamical

91

properties in continuum treatments. Hereǫ0 and ǫ∞ are the static and high frequency

dielectric constants of the solvent.ri and qi are the radius and charge of the ion. The

friction due to shear viscosity of the solvent, the hydrodynamic friction (which may arise

from the short-range interactions) is given by the Stokes law :

ζSR = 4π η ri (2)

for slip boundary condition. Thus, the total friction on theion, ζ is

ζ = ζSR + ζDF (3)

ζSR is higher, the larger the size of the ion. ButζDF has a 1/r3i dependence and is higher

for ions with smaller radius. The result is that at some intermediate size of the ion,ri, the

total friction ζ is lowest when both the termsζSR andζDF are not too large. This explains

the existence of a conductivity maximum. Hubbard and Onsager [16] have improved the

treatment which leads to better agreement with the experimental mobilities.

Although the maximum in ionic conductivity can be reproduced by the continuum treat-

ment for ions carrying a given type of charge – positive or negative – the theory does not

permit distinction between them asζDF depends onq2i . Thus, the theory can not account

for the two different curves in the plot of conductivity–1/ri obtained experimentally for

positive and negative ions and two different maxima [19]. Clearly there is a need for more

refined theories which treat the charge distribution of the solvent explicitly.

Wolynes proposed a microscopic theory to overcome some of the limitations of the

continuum theories. He separated the contribution into those from the hard replusive in-

teractionζHH and soft attractive interactionsζSS . The correlations between the soft and

hard interactions are neglected.ζHH is identified with the hydrodynamic friction. Both

solvent-berg and continuum treatments are limiting cases of this molecular theory. In this

sense, this may be considered to be more general than other theories.

More recently, Bagchi and coworkers [20–22] have extended the molecular theory to

permit self-motion of the ion. This provides a clearer picture of the various physical factors

responsible for the friction on the moving ion. These are based on mode coupling theory

and separate the overall friction into a microscopicζmicro and a hydrodynamic partζhyd :

1

ζ=

1

ζmicro

+1

ζhyd

(4)

Theζmicro has contributions from several terms. Direct binary collisions as well as the

isotropic fluctuations in density lead to friction that are represented respectively byζbinary

92

36

38

40

38

38.5

39

39.5

35

36

37

0 500 1000 1500 2000Pressure (bar)

16

17

18

19 Li+

K+

Cs+

Cl-

λo (

Scm

2 mol

-1)

Figure 1: Variation of conductivity,λ0 with pressure for monovalent ions at -5C. Thedata have been taken from Takisawa et al. [26].

andζdensity. Coupling with polarization fluctuations is responsible for the dielectric fric-

tion ζDF . Thus,

ζmicro = ζbinary + ζdensity + ζDF (5)

ζhyd is the hydrodynamic friction. This can be usually determined from transverse current-

current correlation function. Although all these terms determine the overal friction on the

ion, often some of these terms are less important than others. Thus, for some ions, Bagchi

and coworkers suggest that some of these terms are small and can be neglected. More recent

studies by Bagchi and coworkers have shown the importance ofultrafast solvation. It leads

to a significant reduction in the contribution to friction experienced by the ion [23–25].

Fleming and coworkers [27], Barbara and coworkers [28] and Bagchi and cowork-

ers [29] have shown the relationship between the solvation energy time correlation function

and the dielectric friction. They have shown that both ion solvation dynamics and dielec-

tric friction are influenced by the dynamics of the ion and thesolvent. In other words, the

dynamics that influences the ion solvation dynamics is also responsible for the dielectric

friction. Bagchi and coworkers show that inclusion of the ultrafast mode in the dielec-

tric relaxation is necessary to obtain closer agreement with the experimentally measured

limiting ion conductivityΛ0.

93

72

74

76

78

18

18.5

19

19.5

76

77

78

22

23

48

49

50

29

30

31

32

0 500 1000 1500 2000

39

40

0 500 1000 1500 200040

42

44Li

+

Na+

Rb+

Cs+

Me4 N

+

Et4 N

+

Pr4 N

+

Bu4 N

+

Pressure (bar) Pressure (bar)

λo (

Scm

2 mol

-1)

λo (

Scm

2 mol

-1)

Figure 2: Variation of conducitivity,λ0 with pressure at 25C for monovalent cations. Datataken from Ueno et al. [35].

There have been computer simulation studies on diffusion ofions in water and other

solvents in the past two decades [30–33]. Computer simulations of Rasaiah and cowork-

ers and Lynden Bell and coworkers [32, 33] on ion motion in water have clarified some

aspects of this intriguing problem. Ion–water intermolecular potential was derived by fit-

ting them to solvation energies of ions in embedded water clusters. They have carried out

simulations to study the dependence of mobility on ion radius and charge. Their finding

that both positively and negatively charged ions exhibit a maximum in diffusivity confirms

the well known experimental results on alkali and halide ions which exhibit a maximum

for intermediate sized ions. Simulations suggest that the solvent coordination around the

ions depend crucially on the charge on the ion. However, theyfind no relation between the

solvent coordination and mobility; this supports the view that solvent-berg model does not

provide the required explanation to account for the maximumin conductivity. The precise

size of the ion at which the maximum in mobility is seen also depends on the charge. Their

calculations suggest that the dielectric friction model ismore appropriate for larger ions

while for small ions the solvent-berg model may be more appropriate. They obtained good

agreement with experimental results.

Chandra and coworkers [30] have studied the effect of ion concentration on the hydro-

gen bond dynamics. They find that water molecules participating in five hydrogen bonds

are more mobile as compared to four or fewer hydrogen bonds [31, 34]. They have also

studied the effect of pressure on aqueous solutions. Studies have also been carried out on

non-aqueous solutions.

94

Experimental studies date back to over several decades. Butmore recently, experi-

mental studies of ionic mobility in water, alcohols, acetonitrile and formamide by Kay

and Evans as well as Ueno and coworkers have shown the existence of a maximum in the

Walden’s product [35–40]. Investigations in D2O show that the ratio ofΛ0η0 in D2O to

that in H2O also exhibits a maximum when plotted againstr−1

ion. HereΛ0 is the limiting ion

conductivity of the solution andη0 is the viscosity of the solvent. Ionic mobility of cations

has also been studied in a series of monohydroxy alcohol [35,38]. It is generally observed

that the mobility is lower in these alcohols than found in water. Further, the mobility is still

lower in higher alcohols. Studies of ionic mobility also exist in solvents such as acetoni-

trile and formamide [35, 38, 39]. Both these solvents exhibit ultrafast solvation dynamics.

For acetonitrile, an inertial component with a relaxation time of 70 fs and for formamide

around 100 fs has been reported [41].

Recently, we proposed that the ionic conductivity maximum in polar solvents has its

origin in the Levitation Effect [42, 43]. The latter is an effect that was observed for guests

in zeolites and other porous solids. On increase in the size of the guest, the self diffusivity

decresed initially when the size of the guest was significantly smaller than the size of the

void and neck within the zeolites or other porous solids. However, the size of the guest

was comparable to the size of the neck then, a maximum in self diffusivity was seen. This

maximum has been shown to arise from the mutual cancellationof forces exerted on the

guest by the zeolite leading to lower net force on the guest when its size is comparable

to the size of the neck. The guest then is less confined relative to when it is smaller. A

similar effect leading to a maximum in self diffusivity exists in solutions dominated by van

der Waals interaction as well as in solutions with significant long-range interaction [44–

47]. Thus, it appears that while the previous theoretical frameworks proposed based on

continuum theories as well as microscopic theories providea reason for the size dependent

maximum in conducitivity, they do not even attempt to explain the variation of conductivity

with other variables such as pressure. Here we have collected all the conductivity data as a

function of pressure and analyse them so as to obtain a clear idea of how the conductivities

of ions in water are altered as a function of pressure. Such anunderstanding is necessary

before one can put forward theories to explain the pressure dependence of conductivity for

ions of different sizes.

2. Analysis of Experimental Measurements

Extensive amount of data is available in the literature for ionic conductivity in water

of different salts. There are also several groups who have studied the dependence of ionic

95

60

62

64

66

68

3030.5

3131.5

3232.5

78.5

79

79.5

34

34.5

35

35.5

0 500 1000 1500 2000Pressure (bar)

74

76

78

80

0 500 1000 1500 2000Pressure (bar)

39.5

40

40.5

41

41.5

λo (

Scm

2 mol

-1)

(a) I-

(a) Cl-

(a) Br-

(b) ClO4

-

(b) CH3 CO

2

-

(b) C2 H

5 CO

2

-

(b) C3 H

7 CO

2

-

Figure 3: Variation of conducitivity,λ0 with pressure at 25C for monovalent anions. Datataken from Nakahara and Osugi [48] and Shimizu and Tsuchihashi [49].

conductivity as a function of pressure [26, 35, 49–53]. Usually, the conductivity of the

solution ,Λ, is measured at several concentrations. These are analysedby means of Fuoss-

Onsager equation [35, 51, 54]

Λ = Λ0− S

√

c + Ec log c + Jc

of conductance of unassociated electrolytes. Herec is molar concentration,S andE are

constants which are a function ofΛ0 and the solvent properties viscosity and dielectric

constant.J is a function of ion size taken as an adjustable parameter. From this,Λ0, the

limiting conductivity or conductivity at infinite dilutionof the solution is obtained:

λ0 = T 0+ Λ0 .

Here,T 0+ is the transference number at infinite dilution andλ0 is the limiting ion conduc-

tivity at infinite dilution of the specific ion. Experimentaldetails are not given here but

those interested can find it from the cited references.

An analysis ofλ0 of the specific ion is investigated here since this is a simplequantity.

In contrast,Λ0 is the conductivity of the solution and its value depends on the conductivities

of the cation as well as the anion. Although many studies in the literature reportΛ0 few

studies reportT 0+ or λ0. The available data for analysis is therefore not extensive.

96

0 500 1000 1500 200030

40

50

60

70

80

0 500 1000 1500 200010

15

20

25

30

35

40

45

50

Li+ / H

2 O

Li+ / D

2 O

Na+ / H

2 O

Na+ / D

2 O

Rb+ / H

2 O

Cs+ / H

2 O

Rb+ / D

2 O

Cs+ / D

2 O

Pressure (bar)

λo (

Scm

2 mol

-1)

Me4 N

+ / D

2 O

Me4 N

+ / H

2 O

Et4 N

+ / H

2 O

Et4 N

+ / D

2 O

Bu4 N

+ / D

2 O

Bu4 N

+ / H

2 O

Pr4 N

+ / D

2 O

Pr4 N

+ / H

2 O

Pressure (bar)

λo (

Scm

2 mol

-1)

Figure 4: Variation of conductivity,λ0 with pressure in light and heavy water for cationsof differing sizes. Filled symbols are for light water and open symbols are for heavy water.Data taken from Ueno et al. [35].

Λ0− p plots for monovalent ions of different sizes at -5C: Figure 1 shows a plot

of the variation ofλ0 as a function of the pressure,p, for Li+, K+, Cs+ andCl− over

a pressure range of 1-2000 bars. The measurements have been made at -5C. We note

that forLi+, the conductivity increases with pressure. For intermediate-sized ions at low

temperatures,K+ andCs+, the conductivity increases initially and then subsequently de-

creases with pressure. Thus, a conductivity maximum is seenfor these ions. We could

not find any data for larger ions such tetraalkyl ammonium ions. ForCl− the behaviour is

similar to what is seen forLi+. These data have been taken from Takisawa et al. [26].

A plot of λ0 against pressure is shown for ions of different sizes of ionsfrom the work

of Ueno et al. [35] (see Figure 2). These measurements have been carried out at room

temperature, 298K. Note that the trends seen in the earlier Figure are valid here also : small

ions such asLi+exhibit an increase in conductivity with pressure. Intermediate sized ions

exhibit a maximum in ionic conductivity at some intermediate pressure but larger ions (such

asX4N+, whereX = Me, Et, Pr, Bu) show only a decrease in conductivity with pressure.

The data of Nakahara and Osugi and Shimizu and Tsuchihashi [48, 49] are shown in

Figure 3. This shows a plot ofλ0 versus pressure,p, for monovalent anions at 25C. For

the larger anions such asI−, ClO−

4 andC3H7CO−

2 conductivity decreases with increase

in pressure. But for anions of intermediate size (Br−, Cl−, CH3CO−

2 andC2H5CO−

2 ) a

maximum in conductivity with increase in pressure is seen. This behaviour is what we ob-

serve in case of monavalent cations. It therefore appears that the maximum in conductivity

97

38.8

39.2

39.6

40

72

73

74

72

74

76

78

76

77

78

79

0 1000 2000

14

16

18

20

22

0 1000 2000

30

33

36

39

42

0 1000 200032

36

40

44

0 1000 200034

36

38

40

42

44

46

Li+ Cl

-Cs

+K

+

(a) -5 o C

(a) -5 o C

(a) -5 o C

(a) -5 o C

(a) -10 o C

(a) -10 o C

(a) -10 o C

(a) 0 o C

(a) 0 o C

(a) 0 o C

(a) 0 o C

(b) 25 o C

(d) 25 o C

(b) 25 o C

(c) 25 o C

Pressure (bar)

λo (

Scm

2 mol

-1)

Figure 5: Variation of conductivity,λ0 with pressure for monovalent ions over a range oftemperature. The data have been taken from (a) Takisawa et al. [26] and (b) Nakahara etal. [55] and (c) Nakahara et al. [56] and (d) Ueno et al. [35].

depends on the size of the ion in a way that is independent of the nature of charge carried

by the ion.

Figure 4 shows a plot ofλ0 versus pressure,p, for monovalent cations of different

sizes in light and heavy water. From this figure, we can observe that conductivities in

heavy water show a similar trend as we observe in case of lightwater; only a uniform

lowering of conductivity is seen in heavy water as compared to light water. A reduction in

conductivity in heavy water is attributed to stronger hydrogen bonding in heavy water as

compared to light water. A sluggish solvent structure can lead to reduced mobility of the

ion and not just the solvent.

Temperature dependance of Λ0−p plots : In Figure 5 we show a plot of the variation

of λ0 with pressure over a range of temperatures. At higher temperatures, some changes

are seen in theλ0 -p curves. Firstly, for ions such asLi+ or Cl−, the increasing conductiv-

ity with pressure changes to an increasing and decreasing trend with pressure exhibiting a

maximum at some intermediate pressure. For the intermediate sized ions such asK+ and

Cs+ the trend is seen to remain the same; however, the pressure atwhich the conductivity

is maximum shifts to a lower pressure. For example, in the case K+ the maximum con-

ductivity is seen at a pressure of 1500 bars at -10C. By 25C the pressure at which the

conductivity is maximum shifts to 500 bars. ForCs+ the pressure at which the conductiv-

98

81.4

81.6

81.8

82

112.8

113.1

113.4

113.7

60

60.4

60.8

80.8

81.2

81.6

82

0 400 800 1200Pressure (bar)

47.2

47.6

48

48.4

48.8

0 400 800 1200Pressure (bar)

63.7

64.4

65.1

65.8

Ca2+ SO

4

2-

λo (

Scm

2 mol

-1 )

λo (

Scm

2 mol

-1 )

15o C

15o C

25o C

40o C

40o C

25o C

Figure 6: Variation of conductivity,λ0 with pressure for divalent ions over a range oftemperature. The data have been taken from Nakahara et al. [48].

ity is maximum shifts from 1000 bars at -10C to 200 bars at 25C. These data have been

taken from the references mentioned in the figure caption.

Figure 6 shows a plot ofλ0 versus pressure,p, for divalent ions at different tempera-

tures. This figure shows that divalent ions likeCa2+ andSO2−4 also exhibit an increasing

trend in conductivity as a function of pressure at lower temperature and shows maximum

in conductivity with pressure at higher temperatures. These trends are similar to the trend

seen in the case of monovalent ions.

3. Conclusions

We summarize the different behaviours and the conditions under which these trends are

seen. Irrespective of whether they are cations or anions, ormonovalent or divalent ions, the

following trends are seen at a given relatively low temperature :

(a) small ions exhibit an increase in conductivity with pressure. (b) intermediate sized

ions exhibit a conductivity maximum as a function of pressure. (c) conductivity decreases

monotonically with pressure for large ions.

Similar trends are seen in heavy water,D2O as well.

With increase in temperature, the following changes are seen for ions of different sizes:

(a) small ions : the increase inλ0 is seen to change to a conductivity maximum at

sufficiently high temperatures. (b) intermdiate sized ions: there is shift of the maximum to

99

lower pressures. (c) larger sized ions : no change; only a decreasing trend is seen.

Acknowledgement : We wish to thank Department of Science and Technology, and Coun-

cil of Scientific and Industrial Research, New Delhi for financial support.

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102

Anomalous Transport and Diffusion in Percolation Systems

Armin Bunde,1 Paul Heitjans,2 Sylvio Indris,3 Jan W. Kantelhardt,4 Markus Ulrich1

1 Universität Giessen, Institut für Theoretische Physik, Germany 2 Leibniz Universität Hannover, Institut für Physikalische Chemie, Germany

3 Forschungszentrum Karlsruhe, Institut für Nanotechnologie, Germany 4 Universität Halle, Institut für Physik, Germany

Corresponding author: Armin Bunde Universität Giessen D-35392 Giessen bunde@physik.uni-giessen.de

Abstract Many disordered systems can be modelled by percolation. Applications of this standard model range from amorphous and porous media to composites, branched polymers, gels and complex ionic conductors. In this brief review we give a short introduction to perco-lation theory and describe applications in materials science. We start with the structural properties of percolation clusters and their substructures. Then we turn to their dynamical properties and discuss the way the laws of diffusion and conduction are modified on these structures. Finally, we review applications of the percolation concept for transport in various kinds of heterogeneous ionic conductors. 1. The percolation transition Percolation represents a standard model for a structurally disordered system with a wide range of applications [1-3]. In Sections 1 to 3 we give a brief introduction into percola-tion theory. For brevity, we skip references to most original works here and instead refer to reviews [1] and [2]. In Section 4 we discuss applications on heterogeneous ionic con-ductors. Let us consider a square lattice, where each site is occupied randomly with probability p or is empty with probability 1 (see Fig. 1). Occupied and empty sites may stand for very different physical properties. For illustration, let us assume that the occupied sites are electrical conductors, the empty sites represent insulators, and that electrical current can only flow between nearest-neighbour conductor sites.

− p

At low concentration p, the conductor sites are either isolated or form small clusters of nearest-neighbour sites. Two conductor sites belong to the same cluster if they are con-nected by a path of nearest-neighbour conductor sites, and a current can flow between them. At low p values, the mixture is an insulator, since no conducting path connecting

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, A. Bunde 103

opposite edges of our lattice exists. At large p values, on the other hand, many conducting paths between opposite edges exist, where electrical current can flow, and the mixture is a conductor. At some concentration in between, therefore, a threshold concentration pc must exist where for the first time an electrical current can percolate from one edge to the other. The threshold concentration is called the percolation threshold, or, since it sepa-rates two different phases, the critical concentration.

Fig. 1. Site percolation on the square lattice: The small circles represent the occupied sites for three different concentrations: p = 0.2, 0.59, and 0.8. Nearest-neighbour cluster sites are connected by lines representing the bonds. Filled circles are used for finite clusters, while open circles mark the large infinite cluster. If the occupied sites are superconductors and the empty sites are conductors, pc separates a normal-conducting phase below pc from a superconducting phase above pc. Another example is a mixture of ferromagnets and paramagnets, where the system changes at pc from a paramagnet to a ferromagnet. In contrast to the more common thermal phase transitions, where the transition between two phases occurs at a critical temperature, the percolation transition described here is a geometrical phase transition, which is characterized by the geometric features of large clusters in the neighbourhood of pc. At low values of p only small clusters of occupied sites exist. When the concentration p is increased the average size of the clusters in-creases. At the critical concentration pc a large cluster appears which connects opposite edges of the lattice. We call this cluster the infinite cluster, since its size diverges in the thermodynamic limit. When p is increased further the density of the infinite cluster in-creases, since more and more sites become part of it, and the average size of the finite clusters, which do not belong to the infinite cluster, decreases. At p = 1, trivially, all sites belong to the infinite cluster. The value of pc depends on the details of the lattice and increases, for fixed dimension d of the lattice, with decreasing coordination number z of the lattice. For the triangular lattice, z = 6 and pc = 1/2, for the square lattice, z = 4 and pc ≈ 0.592746, while the hon-eycomb lattice has z = 3 and pc ≈ 0.6962. For fixed z, pc decreases if the dimension d is

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enhanced. In both the triangular lattice and the simple cubic lattice we have z = 6, but pc for the simple cubic lattice is considerably smaller, pc ≈ 0.3116. So far we have considered site percolation, where the sites of a lattice have been occupied randomly. When the sites are all occupied, but the bonds between them are randomly occupied with probability q, we speak of bond percolation. Two occupied bonds belong to the same cluster if they are connected by a path of occupied bonds. The critical con-centration qc of bonds (qc = 1/2 in the square lattice and qc ≈ 0.2488 in the simple cubic lattice) separates a phase of finite clusters of bonds from a phase with an infinite cluster. Perhaps the most common example of bond percolation in physics is a random resistor network, where the metallic wires in a regular network are cut randomly with probability 1 − q. Here qc separates a conductive phase at large q from an insulating phase at low q. A possible application of bond percolation in chemistry is the polymerization process, where small branching molecules can form large molecules by activating more and more bonds between them. If the activation probability q is above the critical concentration, a network of chemical bonds spanning the whole system can be formed, while below qc only macromolecules of finite size can be generated. This process is called a sol-gel tran-sition. An example of this gelation process is the boiling of an egg, which at room tem-perature is liquid and upon heating becomes a more solid-like gel. The most natural example of percolation is continuum percolation, where the positions of the two components of a random mixture are not restricted to the discrete sites of a regu-lar lattice. As a simple example, consider a sheet of conductive material, with circular holes punched randomly in it. The relevant quantity now is the fraction p of remaining conductive material. Compared with site and bond percolation, the critical concentration is further decreased: pc ≈ 0.312 for d = 2, when all circles have the same radius. This picture can easily be generalized to three dimensions, where spherical voids are generated randomly in a cube, and pc ≈ 0.034. Due to its similarity to Swiss cheese, this model is also called the Swiss cheese model. Similar models, where also the size of the spheres can vary, are used to describe sandstone and other porous materials. 2. The fractal structure of percolation clusters near pc The percolation transition is characterized by the geometrical properties of the clusters near pc [1,2]. The probability that a site belongs to the infinite cluster is zero below pc and increases above pc as

( )βc~ ppP −∞ (1)

with β = 5/36 in d = 2 and β ≈ 0.417 in d = 3. The linear size of the finite clusters, below and above pc, is characterized by the correla-tion length ξ. The correlation length is defined as the mean distance between two sites on the same finite cluster and represents the characteristic length scale in percolation. When p approaches pc, ξ increases as

105

, (2) ν||~ξ −− cpp with the same exponent v below and above the threshold (v = 4/3 in d = 2 and v ≈ 0.875 in d = 3). While pc depends explicitly on the type of the lattice, the critical exponents β and ν are universal and depend only on the dimension d of the lattice, but not on the type of the lattice.

Fig. 2. Four successive magnifications of the incipient infinite cluster that forms at the percolation threshold on the square lattice. Three of the panels are magnifications of the center squares marked by black lines. An educational game is to time how long it takes each player to detect by eye which of the 24 possible orderings is the correct one that arranges the four panels in increasing order of magnification.

106

For percolation concentrations near pc and on length scales smaller than the correlation length ξ, both the infinite cluster and the finite clusters are self-similar. I.e., if we cut a small part out of a large cluster, magnify it to the original cluster size and compare it with the original, we cannot tell the difference: both look the same. This feature is illustrated in Fig. 2, where a large cluster at pc is shown in four different magnications. We leave it to the reader to find out what is the original and what are the magnifications.

Fig. 3. The same as Fig. 2 except that now the system is slightly (0.3 %) above the percolation threshold and the panels are not scrambled. The upper left picture shows the original and the other pictures are magnifications of the center squares marked by black lines. The correlation length ξ is approximately equal to the linear size of the third (lower left) picture. When comparing the two lower pictures, the self-similarity at small length scales below ξ is easy to recognize.

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As a consequence of the (non-trivial) self-similarity, the cluster is characterized by a “fractal” dimension, which is smaller than the dimension d of the embedding lattice. The mean mass of the cluster within a circle of radius r increases with r as , (3) ξ,~)( f <<rrrM d

with the fractal dimension df = 91/48 in d = 2 and df ≈ 2.5 in d = 3. Above pc on length scales larger than ξ the infinite cluster can be regarded as a homogeneous system which is composed of many cells of size ξ. Mathematically, this can be summarized as

(4) ⎩⎨⎧

>><<

.ξif,ξif

~)(f

rrrr

rMd

d

Fig. 3 shows a part of the infinite cluster slightly above pc (p = 1.003 pc) on different length scales. At large length scales ( , upper left) the cluster appears homogene-ous, while on lower length scales ( , lower pictures) the cluster is self-similar.

ξ>>rξ<<r

The fractal dimension df can be related to β and ν in the following way. Above pc, the mass M∞ of the infinite cluster in a large lattice of size Ld is proportional to Ld P∞. On the other hand, this mass is also proportional to the number of unit cells of size ξ, (L/ξ)d, multiplied by the mass of each cell, which is proportional to . This yields (with Eqs. (1) and (2))

fξd

( ) ( ) ( ) ( ),~ξξ/~~~ ff ν

cβ

cddddddd ppLLppLPLM −

∞∞ −− (5) and hence, comparing the exponents of )( cpp − ,

.νβ

f −= dd (6)

Since β and ν are universal exponents, df is also universal. A fractal percolation cluster is composed of several fractal substructures, which are de-scribed by other exponents [1,2]. Imagine applying a voltage between two sites at oppo-site edges of a metallic percolation cluster: The backbone of the cluster consists of those sites (or bonds) which carry the electric current. The topological distance between both points (also called chemical distance) is the length of the shortest path on the cluster connecting them. The dangling ends are those parts of the cluster which carry no current and are connected to the backbone by a single site only. The red bonds (or singly con-nected bonds), finally, are those bonds that carry the total current; when they are cut the current flow stops.

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The fractal dimension dB of the backbone (dB BB ≈ 1.64 in d =2 and dB ≈ 1.86 in d = 3) is smaller than the fractal dimension d

B

f of the cluster, reflecting the fact that most of the mass of the cluster is concentrated in the dangling ends. On the average, the topological length of the path between two points on the cluster increases with the Euclidean dis-tance r between them as (dmin~ dr min ≈ 1.13 in d = 2 and dmin ≈ 1.37 in d = 3). The frac-tal dimension of the red bonds dred can be deduced from exact analytical arguments: The mean number of red bonds varies with p as , and the fractal di-mension of the red bonds is therefore d

ν/11cred ξ~)(~ −− ppn

red = 1/ν. It is important for applications that close to the percolation threshold, the exponents are universal and depend neither on the structural details of the lattice (e.g., square or trian-gular) nor on the type of percolation (site, bond, or continuum), but only on the dimen-sion of the lattice. 3. Anomalous Diffusion and Conduction on Percolation Clusters Next we will focus on the dynamical properties of percolation systems, where to each site or bond a physical property such as conductivity is assigned. Due to the fractal nature of the percolation clusters near pc, the physical laws of dynamics are changed essentially and become anomalous. We start with the infinite percolation cluster at the critical con-centration pc. The cluster has loops and dangling ends, and both substructures slow down the motion of a random walker. Due to self-similarity, loops and dangling ends occur on all length scales, and therefore the motion of the random walker is slowed down on all length scales. The time t the walker needs to travel a distance R is no longer, as in regular sys-tems, proportional to R2, but scales as , where is the fractal dimension of the random walk [1, 2]. For the mean square displacement this yields immediately

w~ dRt 2w >d

w2 /2 ( ) ~ .dr t t< > (7)

The fractal dimension dw is approximately equal to 3df /2 [4]. For continuum percolation in d = 3, dw is enhanced: dw ≈ 4.2 [5]. In general, dw cannot be calculated rigorously. Exceptions are linear fractal structures (like self-avoiding walks), where dw = 2df, or loopless structures. Diffusion processes described by Eq. (7) are generally referred to as anomalous diffusion. Above pc, fractal structures occur only within the correlation length ξ.(p). Thus the anomalous diffusion law, Eq. (7), occurs only below the corresponding crossover time

, which decreases proportional to , if p is further increased. Above , on large time scales, the random walker explores large length scales where the cluster

is homogeneous, and follows Fick's law increasing linearly with time t. Thus,

wξ~ξdt wν

c )( dpp −−

ξt

>< )(2 tr

109

( )⎪⎩

⎪⎨⎧

>><<

><.tif,tif

~ξ

ξ/2

2w

tttt

trd

(8)

The diffusion coefficient defined by 2 ( ) 2D r t d= < > t is (approximately) related to the dc conductivity σdc by the Nernst-Einstein equation,

,/σ B2

dc TkDne= (9) where n is the density and e the charge of the diffusing particles. Below pc, there is no current between opposite edges of the system, and σdc = 0. Above pc, σdc increases by a power law

( ) ,~σ μcdc pp − (10)

where the critical exponent μ is (semi)-universal. For percolation on a lattice, μ depends only on d. For continuum percolation (Swiss cheese model) in d = 3, however, μ is en-hanced: μ ≈ 2.38. Combining Eqs. (9) and (10), we can obtain the behaviour of the diffusion coefficient D as a function of p − pc. Since only the particles on the infinite cluster contribute to the dc conductivity, we have (from Eq. (1)) n ~ P∞ ~ (p − pc)β in Eq. (9). This yields D ~ (p − pc)μ − β. Next we use scaling arguments to relate the exponent μ to dw. Above tξ, the mean square displacement < r2(t) > behaves as < r2(t) > ~ (p − pc)μ − β t, where, for t = tξ, we have < r2(t) > ~ ξ2. On the other hand we know that for times below tξ on distances r < tξ

1/dw, < r2(t) > ~ tξ2/dw. Equating both relations we obtain immediately (p − pc)μ − β tξ ~

tξ2/dw. Using tξ ∼ ξdw ~ (p − pc)−νdw (from Eq. (2)) we get the relation between μ and dw,

( ) .ν/βμ2w −+=d (11)

4. Application of the Percolation Concept: Heterogeneous Ionic Conductors Let us now turn to applications of percolation models in materials. A substantial amount of research has concentrated on “dispersed ionic conductors” after the discovery by Liang [6] that insulating fine particles with sizes of the order of 1μm, dispersed in a conductive medium (e. g. Al2O3 in LiI), can lead to a conductivity enhancement [7]. This effect has been found to arise from the formation of a defective, highly conducting layer following the boundaries between the conducting and the insulating phase [8]. Effectively, the sys-tem thus contains three phases. Theoretical studies therefore have focused on suitable three-component impedance network models.

110

Fig. 4. Illustration of the three-component percolation model for dispersed ionic conductors, for different concentrations p of the insulating material. The insulator is represented by the grey area, the ionic conductor by the white area. The bonds can be highly conducting bonds (A bonds, bold lines), normal conducting bonds (B bonds, thin lines), or insulating (C bonds, dashed lines). (a)

, (b) , (c) , and (d) . c'pp < c'pp = c''pp = c''pp >

4.1 Correlated Bond Percolation Model for Dispersed Ionic Conductors Figure 4 shows a two-dimensional illustration of such composites in a discretized model [9,10]. In its simplest version this model is constructed by randomly selecting a fraction p of elementary squares on a square lattice, which represent the insulating phase (grey), while the remaining squares are the conducting phase (white). The distribution of both phases leads to a correlated bond percolation model with three types of bonds and associ-ated bond conductances σα; α = A, B, C; as defined in Fig. 4. For example, bonds in the boundary between conducting and insulating phases correspond to the highly conducting component (A bonds). This is an extension to the standard bond percolation model, where only two kinds of bonds (e.g., conducting and insulating bonds, σA = 1, σB = 0) are considered. The analogous construction for three dimensions is obvious. Clearly, the experimental situation described above requires 0σ;1τσ/σ CBA =>>= . It is natural to assume that σA and σB are thermally activated, such that their ratio τ ∼

increases with decreasing temperature. B

)/exp( BTkEΔ− A remarkable feature of this model is the existence of two threshold concentrations. At

, interface percolation (i.e., percolation of A bonds) sets in, whereas at (normally not accessible by experiment) the system undergoes a con-

ductor-insulator transition. The first critical concentration

c'pp =

cc '1'' ppp −==

097.0'c =p corresponds to the threshold for third-neighbour site percolation on a 3-dimensional lattice. Figure 5 shows the total conductivity obtained by Monte Carlo simulations [9,10], for three different temperatures (corresponding to τ = 10, 30 and 100). Good agreement with the experimental curves [11] is achieved, which show a broad maximum in the conduc-tivity as a function of p in the range between the two thresholds. We like to note that the model also describes successfully the variation of the total conductivity with the size of the dispersed particles [12]. In particular, it was found that as the particle size decreases

111

while the thickness of the highly conducting interfacial layer is fixed, the maximum in the total conductivity as a function of the insulator concentration p shifts to smaller val-ues of p. The observation of conductivity maxima at very low volume fractions close to 0.1 in certain composite electrolytes, however, was interpreted recently by a grain bound-ary mechanism within the bulk of the electrolyte phase [13].

Fig. 5. (a) Normalized conductivity of the LiI-Al2O3 system as a function of the mole fraction p of Al2O3 at different temperatures (after [11]). (b) Normalized conductivity resulting from Monte Carlo simulations of the three-component percolation model, as a function of p, for (circles), 30 (full squares), and 100 (triangles) (after [10]).

10σ/σ 0B

0A =

4.2 Composite Micro- and Nanocrystalline Conductors In the foregoing subsection, we have discussed dispersed ionic conductors that were prepared by melting the ionic conductor and adding the insulator (mainly Al2O3) to it. Next we consider diphase micro- and nanocrystalline materials, which were prepared by mixing the two different powders and pressing them together to a pellet. This way, in contrast to the classic dispersed ionic conductors discussed above, the grain size of both ionic conductor and insulator can be varied over several orders of magnitude. For reviews on nanocrystalline materials see, e.g., [13-16].

112

Fig. 6. Plot of the dc conductivity of microcrystalline (full circles) and nanocrystalline (1-x)Li2O:xBB2O3 composites (open circles) vs volume fraction p (bottom scale) and mole fraction x (top scale) of insulating B2O3, at T = 433 K. The arrows indicate the compositions where the dc conductivities fall below the detection limit. The dashed lines show the dc conductivities obtained from the continuum percolation model discussed in the text (after [18]).

Figure 6 shows the ionic conductivity of micro- and nanocrystalline (1-x)Li2O:xBB2O3 composites for different contents x of insulator B2O3 [17,18]. For pure Li2O, i.e. x = p = 0, the dc conductivity of the microcrystalline and the nanocrystalline samples coincide. When Li2O is successively substituted by B2O3, the two systems behave very different. In the microcrystalline samples, the dc conductivity decreases monotonically with x, while in the nanocrystalline samples, the dc conductivity first increases and reaches a maximum near x = 0.6, where the conductivity is about one order of magnitude larger than that of pure Li2O. Further increase of the insulator content leads to a decrease of the conductiv-ity. At x = 0.95, finally, the conductivity has dropped below the detection limit. As for the composites discussed above, the overall behaviour (including the differences between nano- and microcrystalline samples) can be explained assuming an enhanced conductivity at the interfaces between unlike grains. Even more remarkable than the increase of the dc conductivity with increasing insulator content, however, is the fact that, starting from the pure insulator B2B O3, only a tiny volume fraction of C is needed to obtain a dc conductiv-ity which is considerably higher than the dc conductivity of pure Li2O.

113

Fig. 7. Continuum percolation model with insulating spheres (radius R) dispersed in an ionic con-ductor and a highly conducting interface, after [20]. The figures show the two critical insulator contents where (a) an infinite highly conducting pathway is formed and (b) this pathway is dis-rupted. 4.3 Continuum percolation model The increase of the ionic conductivity at intermediate insulator contents for the nano-crystalline composites clearly shows that the interfaces between the two components are responsible for the conductivity enhancement. To describe the dependence of the dc con-ductivity of the composites on the insulator content p, we first consider a continuum percolation model [19,20], which is sketched in Fig. 7. The insulating particles are repre-sented by spheres with radius R. Around these insulating particles a highly conducting interface with width λ and ionic conductivity σA is created. The remaining volume repre-sents the ionically conducting phase which has a conductivity σB. An enhancement factor τ = σ

B

A/σBB of 100 and an interface thickness λ = 1 nm are assumed. The grain radii R have been determined by transmission electron microscopy and X-ray diffraction [21,22] and are roughly 5 µm for the microcrystalline composites and 10 nm for the nanocrystalline composites. The overall conductivity of the system, calculated by an effective medium approximation [17], is shown in Fig. 6 (dashed curves). A good overall agreement with the experimental data is found. However, the effective coordination numbers z used in this approach as fit parameters to reproduce the experimental results (z = 7 for the micro-system and z = 59 for the nanosystem) can hardly be rationalized.

4.4 Brick-layer type percolation model

In an attempt to understand the experimental results on a more microscopic basis, we next consider a brick-layer type model where both, micro- and nanocrystalline compos-ites are treated on the same footing [23]. In the model, one starts with a cubic box of size L3 that is divided into a large number of small cubes with equal volumes a3. Each of the small cubes is regarded as a grain of the composite. With a given probability p the cubes

114

are supposed to be insulating B2O3 grains. Thus the volume fraction of the Li2O grains is 1 − p. By definition, conducting grains are connected when they have one corner in com-mon. Fig. 8(a) shows the largest cluster of Li2O grains that connects opposite faces of the large box close to the percolation threshold, at p = 0.9. Again a highly conducting inter-face of width λ between insulating and ionically conducting particles is assumed. Next the small cubes in Fig. 8(a), that represent the Li2O grains, are replaced by a bond lattice sketched in Fig. 8(b). The bonds represent the ionic conduction (i) inside the grain (Σ0), (ii) along the interface area (Σ1) and (iii) along the interface edges (Σ2). The length of each bond is a/2. The cross section each bond represents is (a - λ)2 for the Σ0 bonds, (a - λ)λ for the Σ1 bonds and λ2 for the Σ2 bonds. Having this in mind, the conductance of each bond can be calculated easily. It is assumed that (i) in the bulk of the insulating BB2O3 grains, the specific conductivity is zero, that (ii) in the bulk of the conducting Li2O grains as well as in the interfaces between them, the specific conductivity is σBB, and that (iii) Li2O grains in contact with a B2O3 grain share a highly conducting interface with specific conductivity σA = τσB. The enhancement factor τ is assumed to be of the order of 10 - 10 .

B

2 3

Finally, for calculating the total conductivity of the composite the problem is mapped onto the corresponding diffusion problem by defining appropriate jump rates (propor-tional to the bond conductances) along the bonds. For given values of τ, a, λ and p, the mean square displacement of many random walks is then determined as a function of time t on the largest cluster of each model system. Averaging over all of them, one obtains the diffusion coefficient , which is propor-tional to the dc conductivity (cf. Eq. (9)).

>< )(2 tr

ttrD t 6/)(lim 2 ><= ∞→

Fig. 8. (a) The largest cluster of insulating particles for the brick-layer type model in three dimen-sions at the critical concentration. (b) A single grain and the bonds assigned to conduction in the interior of the grains, along the sides and along the edges of the grain, after [23].

115

The numerical results for the dc conductivity vs. insulator volume content p are shown in Fig. 9 for various grain sizes a and enhancement factors τ. In all model calculations, a fixed interface thickness λ = 1 nm was assumed. The figure shows that this microscopic model, which treats nano- and microcrystalline samples in exactly the same way (the only difference is the size of the grains) is able to reproduce all qualitative features of the ex-perimental results. One feature cannot be reproduced by the model, however, namely the very high insulator concentration where the conductivity drops to zero. We will come back to this point at the end of the next section.

Fig. 9. Numerical results of the normalized dc conductivity σ(p)/σ(0) vs. insulator volume fraction p in the brick-layer type percolation model for different grain sizes a and enhancement factors τ = σA /σB. In all cases the interface thickness λ = 1nm is fixed. Nanocrystalline grains: Δ a = 10 nm, τ = 200; a = 10 nm, τ = 100; ‘ a = 20 nm, τ = 200; Ï a = 20 nm, τ = 100; Microcrystalline grains: a = 10 µm, τ = 200; a = 10 µm, τ = 100; ♦ a = 20 µm, t = 200; a = 20 µm, τ = 100 (after [23]).

B

4.5 Voronoi construction To get a more realistic structure of the composites in the model description (compared to the over-simplified cubic arrangement) a Voronoi approach [24] has been used, see Fig. 10 for a two-dimensional sketch [18]. 2000 seeds which represent the centers of the grains have been

116

Fig. 10. Polycrystalline composite material created by Voronoi construction in two dimensions. Dark grey areas represent the ionic conductor grains and light grey areas represent the insulator grains, after [18]. placed randomly inside a volume of 1503 lattice sites. The borders of the grains are de-fined by the planes perpendicular to the connection line between two neighboured seeds intersecting this line exactly in the middle between both seeds. By this a fully compacted structure of irregular polyhedra is created. The shapes of the individual grains differ sig-nificantly and thus the number of edges of a crystallite does also change which results in a locally varying coordination number. Furthermore the particles are not mono-disperse but show a distribution of grain sizes. The distribution of the local coordination numbers of the Voronoi system is a Gaussian, with an average coordination number close to 15.6 and a standard deviation close to 4.4. The grain volumes follow a log-normal distribution, in agreement with the experimental situation. The structure created by the Voronoi construction seems to represent quite nicely a real polycrystalline material (though pores are not included). The cells in Fig. 10 can now be regarded as insulating with probability p and as ionically conducting with probability (1 - p), irrespective of their size. It is clear that the dc conductivity will show a similar behaviour as in the two models before. The main question that arises is whether the more realistic Voronoi construction is able to describe the conductivity close to p = 1 in a better manner than the previous models. To this end, the percolation probability P(p) for ionically conducting particles to percolate the system was determined. The result is shown in Fig. 11. One can see that the critical concentration pc above which the conducting paths get disrupted is close to 0.86, being even smaller than the value which was obtained for the brick-layer model in the foregoing section.

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A reason for this underestimation of pc might be that Li ion transport is also possible in the interface between insulating nanocrystalline grains, representing an additional Li diffusion passageway of nanometer length. Indeed, the percolation threshold increases to about 0.93 (see Fig. 11) if B2O3/B2O3 interfaces are considered to be permeable for Li ions (thus linking two nondirectly connected Li2O grains) if the length of these interfaces is smaller than the average particle diameter. In the brick-layer model, the assumption of such a ‘nanometer-passageway diffusion’ yields a threshold close to 0.95 [23].

Fig. 11. The percolation probability P(p) of the ionically conducting particles vs. insulator volume fraction p in the three-dimensional Voronoi system shown as white circles. The black squares represent the case where Li ions can pass along BB2O3/B2O3 interfaces being shorter than the aver-age grain diameter, after [18]. We gratefully acknowledge very valuable discussions with Wolfgang Dieterich, Joachim Maier, and H. Eduardo Roman. References [1] Fractals and Disordered Systems, ed. by A. Bunde, S. Havlin (Springer, Berlin, 1996) [2] D. Stauffer, A. Aharony: Introduction to Percolation Theory (Taylor & Francis,

London, 1992) [3] M. Sahimi: Application of Percolation Theory (Taylor & Francis, London, 1994) [4] S. Alexander, R.L. Orbach: J. Phys. Lett. (Paris) 43, L625 (1982) [5] S. Feng, B.I. Halperin, P. Sen: Phys. Rev. B 35, 197 (1987) [6] C.C. Liang: J. Electrochem. Soc. 120, 1289 (1973) [7] For a review see: A.K. Shukla, V. Sharma. In: Solid State Ionics: Materials Applica-

tions, ed. by B.V.R. Chowdari, S. Chandra, S. Singh, P.C. Srivastava (World Scientific, Singapore 1992) p. 91

[8] J. Maier. In: Superionic Solids and Electrolytes, ed. by A.L. Laskar, S. Chandra (Academic Press, New York 1989) p. 137

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[9] A. Bunde, W. Dieterich, H.E. Roman: Phys. Rev. Lett. 55, 5 (1985) [10] H.E. Roman, A. Bunde, W. Dieterich: Phys. Rev. B 34, 3439 (1986) [11] F.W. Poulsen, N.H. Andersen, B. Kinde, J. Schoonman: Solid State Ionics 9/10,

119 (1983) [12] H.E. Roman, M. Yussou: Phys. Rev. B 36, 7285 (1987) [13] H. Gleiter: Progress in Materials Science 33, 223 (1989) [14] R.W. Siegel: Nanophase Materials. In: Encyclopedia of Applied Physics, vol. 11,

ed. by G.L. Trigg, E.H. Immergut, E.S. Vera, W. Greulich (VCH, New York, 1994) pp. 173-200

[15] J. Maier: Prog. Solid State Chem. 23, 171 (1995) [16] P. Heitjans, S. Indris: J. Phys.: Condens. Matter 15, R1257 (2003) [17] S. Indris, P. Heitjans, H.E. Roman, A. Bunde: Phys. Rev. Lett. 84, 2889 (2000) [18] S. Indris, P. Heitjans, M. Ulrich, A. Bunde: Z. Phys. Chem. 219, 89 (2005) [19] A. G. Rojo, H. E. Roman, Phys. Rev. B 37, 3696 (1988) [20] H. E. Roman, J. Phys.: Condens. Matter 2, 3909 (1990) [21] S. Indris, P. Heitjans, Mater. Sci. Forum 343–346, 417 (2000) [22] S. Indris, D. Bork, P. Heitjans, J. Mater. Synth. Process. 8, 245 (2000) [23] M. Ulrich, A. Bunde, S. Indris, P. Heitjans, Phys. Chem. Chem. Phys. 6, 3680

(2004) [24] A. Okabe, B. Boots, K. Sugihara, Spatial Tessellations: Concepts and Applications

of Voronoi Diagrams, Wiley, Chichester, 1992.

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Interdiffusion in Critical Binary Mixtures by Molecular Dynamics Simulation

Kurt Binder,1 Subir K. Das,2 Michael E. Fisher,2 Jürgen Horbach,3 Jan V. Sengers2

1 Institut für Physik, Johannes Gutenberg Universität, Germany 2 Institute for Physical Science and Technology, University of Maryland, USA

3 Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt, Germany

Corresponding author: Kurt Binder Institut für Physik Johannes Gutenberg Universität Mainz Staudinger Weg 7 55099 Mainz, Germany E-Mail: kurt.binder@uni-mainz.de

Abstract A simulation study of the static and dynamic critical behavior of a symmetric binary

Lennard-Jones mixture is briefly reviewed. Using a combination of semi-grand-canonical Monte Carlo (SGMC) and molecular dynamics (MD) methods near the critical temperature of liquid-liquid unmixing, the correlation length and “susceptibility” related to the critical concentration fluctuations are estimated, as well as the self- and interdiffusion coefficients. While the self-diffusion coefficient does not show a detectable critical anomaly, the interdiffusion coefficient is found to vanish when one approaches the critical temperature at fixed critical concentration. It is shown that in the corresponding Onsager coefficient both a divergent singular part and a nonsingular background term need to be taken into account. With appropriate finite-size scaling analysis (the particle numbers studied for the dynamics lie only in the range from N = 400 to 6400), the critical behavior of the interdiffusion coefficient is found to be compatible both with the theoretically predicted behavior and with corresponding experimental evidence.

Keywords: fluid binary mixtures; critical behavior; Monte Carlo method; molecular dynamics simulation; self-diffusion coefficient; interdiffusion coefficient; Onsager coefficient

1. Introduction The interplay of static structure and transport coefficients in fluid and solid binary mixtures has been a topic of longstanding interest [1-6]. Understanding this problem is crucial for ionic conductors [1], disordered metallic alloys [2,5] and the phase separation processes [5] that these systems may undergo, dynamics of glassforming fluids [6], ordering phenomena occurring in monolayers adsorbed on surfaces affected by surface

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, K. Binder 120

diffusion [3,7], etc. A particularly popular concept (e.g., [2]) is the idea to find simple relations between self-diffusion coefficients, which characterize the diffusive motion of “tagged” particles in a mixture, and the interdiffusion coefficient, which describes how concentration gradients spread out. However, even for simple lattice-gas type models this idea is still subject to controversy (e.g. [8,9]). Fluid binary mixtures near the critical point of a miscibility gap in their phase diagram are clearly an example where a simple relation between self-diffusion and interdiffusion coefficients does not exist. According to the conventional van Hove theory of critical slowing down [10,11] the interdiffusion constant DAB of a binary (A,B) mixture would have the same singularity as the inverse of the susceptibility χ describing the concentration fluctuations in the mixture. In terms of the well-known partial structure factors Sαβ (q) [6] of binary systems (α,β = A,B; q is the wave number), χ is defined by (1) ,]1/[)( 22 …++= ξχ qTkqS B

cc

(2) .)()1(2

)()()1()( 22

qSxxqSxqSxqS

ABAA

BBAAAAcc

−−+−≡

Here kB is Boltzmann’s constant, T the absolute temperature, and ξ is the correlation length of the concentration fluctuations, while xA (xB = 1 − xA) are the relative concentrations of A (B) particles in the mixture. The interdiffusion coefficient then can be written as ,/)( χTDAB Λ= (3) where according to the van Hove conventional theory the Onsager coefficient is finite at the critical temperature T

)(TΛc of the mixture. The interdiffusion coefficient vanishes

since χ diverges according to a power law (as well as ξ), for xA = xB = 1/2 being the critical concentration of a symmetric binary mixture, (4) ./)(,, 0 cc TTT −==Γ= −− εεξξεχ νγ

Here Γ, ξ0 are critical amplitudes, and γ,ν the associated critical exponents [12]. Binary mixtures belong to the Ising universality class [12], for which, in d = 3 dimensions [13,14] .629.0,239.1 ≈≈ νγ (5) While the van Hove conventional theory is believed to hold for solid binary mixtures [10,11,15], both mode coupling theory [16-20] and renormalization group theory [10,21,22] imply that it fails for fluid binary mixtures near their critical point. In particular, it was found that contains a singular part )(TΛ )(TΔΛ in addition to a non-

critical background term , )(TbΛ

(6) ,)(),()()( 0λνε −=ΔΛΔΛ+Λ=Λ LTTTT b

L0 being a critical amplitude, and the exponent νλ is related to the exponent η = 2 - γ/ν describing the decay of critical correlations [12] and the exponent xη describing the

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critical divergence of the shear viscosity η*, which exhibits a so-called multiplicative anomaly [10,11,19] ,νν λλ x= with ,1 ηλ η xx −−= (7)

(8) ,068.0,* 0 ≈= −η

ν ηεηη xx

η0 being a critical amplitude prefactor. Since η ≈ 0.03 one predicts νλ = 0.567. In fact, precise experimental data consistent with these results have existed for a long time [23]! However, no experiments exist where both self-diffusion and interdiffusion coefficients are available near the critical point of a fluid binary mixture. In fact, no pronounced critical anomaly is expected theoretically for the self-diffusion constants DA, DB, and this expectation is consistent with simulation studies of lattice-gas models [24]. Nevertheless it is interesting to reconsider the problem of interdiffusion and self-diffusion in critical binary mixtures via computer simulation again in view, in particular, of recent results [25] claiming that the vanishing of DAB is inconsistent with Eqs. (6-8). In the present paper, we hence review a recent study [26,27] where a symmetric binary Lennard-Jones mixture near its critical point has been extensively studied: combining semi-grand-canonical Monte Carlo (SGMC) methods [28-30] with microcanonical molecular dynamics (MD) runs [31] one can study transport phenomena in model systems in which long-wavelength concentration fluctuations are well equilibrated [32]. In this way, reliable information both on static critical properties Eqs. (1), (4) and the transport properties (interdiffusion coefficient Eq.(3), shear viscosity Eq.(8), and the self-diffusion coefficients) can be obtained for the model system studied. Of course, simulations always deal with systems of finite size (particle numbers 400 ≤ N ≤ 12800 were studied [26,27]) and hence a careful assessment of finite-size effects near the critical point is necessary, applying finite-size scaling concepts [33-35].

2. Model and Simulation Techniques We consider a binary fluid of point particles with pair-wise interactions in a box of

volume V = L3 with periodic boundary conditions. The Lennard-Jones potential (9)

is truncated at r])/()/[(4)( 612

ijijijLJ rrr αβαβαβ σσεφ −=ij = rc and modified there to create a potential that is everywhere

continuous and also has a continuous first derivative,

cij rrijLJcijcLJijLJij drdrrrrru

=−−−= |)/)(()()()( φφφ (10)

for rij < rc while u (rij ≥ rc) ≡ 0. The range parameters in Eq. (9) were chosen as σAA = σBB = σAB = σ, while the cutoff was set at rc = 2.5 σ. The particle number N = NA + NB and the volume are chosen to yield a reduced density ρ* = ρσ3 = Nσ3/V = 1. Then neither crystallization nor the liquid-vapor transition is a problem, at the temperatures of interest. Finally, the reduced temperature T* = kBT/ε0 and energy parameters are chosen as [26,27,32] εAA = εBB = 2 εAB = ε0. For equilibration, first a MC run is carried out in the

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canonical ensemble (NA = NB, V, T) starting from particles at random positions. The MC moves tried are random displacements of randomly chosen single particles, applying the standard Metropolis method [31,36]. After about 104 Monte Carlo steps (MCS) per particle, the equilibration is continued by the SGMC method [26-30,32]: after 10 displacement steps per particle N/10 particles are randomly chosen successively, and an identity switch in general is controlled both by the energy change, Δ E, and the chemical potential difference, Δμ, between the particles. However, since we wish to simulate states with an average concentration 2/1== BA xx (with xα = Nα/N) for T >Tc, we chose

Δμ = 0. For T < Tc, this choice yields states either along the A-rich or the B-rich branch of the coexistence curve in the (T,xA)-plane.

Fig.1: Probability distribution P(xA) of the relative concentration xA = NA/N of A-particles for N = 6400, Δμ = 0, at several temperatures (a) below Tc and (b) above Tc. From S. K. Das et al. [27]. In the SGMC method the concentration xA is a fluctuating variable, and hence the

probability distribution P(xA) is straightforwardly sampled. Since P(xA) = P(1−xA) for Δμ = 0, the coexistence curve (for T < Tc) is obtained as

,),1( )2()1(A

coexAA

coexA xxxx =−= (11)

where moments kAx are defined by

kAx = 2 (12)

The concentration susceptibility χ is then obtained from the second moment via

.)(1

2/1∫ AA

kA dxxPx

.,)4/1( 2cAB TTxNTk >−=χ (13)

For the accurate location of the critical temperature, it is useful to record the fourth-order cumulant UL [34]

123

].)2/1(/[)2/1(224 −−= AAL xxU (14)

Fig. 1 shows “raw data” of the simulation for P(xA) for various temperatures T*, while Fig. 2 illustrates the estimation of the critical temperatures from intersections of cumulants for different sizes N [27]. Note that in the finite-size scaling limit (L → ∞, ξ → ∞, L/ξ finite) these cumulants should intersect at Tc in a value, which (for a given universality class and type of system shape and boundary conditions) is universal [34]. The data indeed are consistent with a unique intersection point at

with the predicted value [37] for the Ising universality class.

0005.04230.1* ±=cT 6236.0)( * ≈cL TU

Fig. 2: The fourth-order cumulant UL(T) plotted vs. T* for N = 1600 (circles), 3200 (squares) and 6400

(diamonds). The vertical straight line indicates the resulting estimate for , while the horizontal broken

straight line indicates the theoretical value that should take, for the Ising universality class. From Das et al. [27].

*cT

*L cU (T T )=

The advantage of the SGMC method is not only the relative ease with which P(xA)

and its moments are obtained, allowing then a reliable estimation of Tc, the coexistence curve Eq. (11) and χ Eq. (13); a crucial further advantage is that long wavelength concentration fluctuations are rather well equilibrated, since critical slowing down near Tc is somewhat less of a problem than it would be for other simulation methods. The critical divergence of the largest relaxation time τmax is, of course, always rounded off in a simulation due to finite system size and one expects

(15) where the dynamic exponent z depends on the “dynamic universality class” [10].

,,max cz TTL =∝τ

For a MC simulation in the canonical ensemble the critical unmixing transition falls

into “class D” of the Hohenberg-Halperin classification [10], for which z = 4 – η, while for the SGMC we have “class C” [10] for which z ≈ 2. Clearly the numerical effort for equilibrating the system in the canonical ensemble would be prohibitively large, since the time over which averages are taken should exceed τmax by several orders of magnitude.

124

Also using MD exclusively (both for equilibration and for a study of the critical dynamics) is problematic, since the predicted dynamic exponent is [10] z ≈ 3. Note that MD realizes the microcanonical NVE ensemble (internal energy E being strictly conserved) and then it is difficult to fine-tune the temperature (which would be a fluctuating observable of the simulation [31]). Using the standard recipe of isothermal MD simulations via the coupling of the system to suitable “thermostats” [31] slightly disturbs the (otherwise Newtonian) equations of motions and hence leads to some systematic errors, when dynamic correlations are recorded. Such errors are likely to be significant near criticality.

All these problems are avoided by the present technique where an ensemble of initial

states at the desired temperature is created by SGMC methods, and starting MD runs in the NVE ensemble from these initial states enables well-defined canonical averages of time-displaced correlation functions to be computed [26,27,32]. We choose the masses of the particles identical (mA = mB = m), apply the velocity Verlet algorithm [31] with a time step 48/01.0* =tδ where t*=t/t0 with the MD time scale t0 being set by t0 = (m σ2/ε0)1/2. We used about 106 MD steps for T* = 1.5 and higher, and up to 2.8 x 106 MD steps for T closer to Tc. Within this time scale, no systematic temperature drift occurs, and we think that discretization errors are still well under control.

Self-diffusion constants DA, DB are then extracted simply from the mean-square

displacements of tagged particles [ ] 2)()0()( trrtg iAiAA −= (16)

by applying the Einstein relation (17)

and similarly for . Note that for the computation of Eq. (16), a second set of coordinates (with no periodic boundary conditions) is used, so the displacements are not restricted by the size of the simulation box, and can grow without limit. For T > T

,]6/)([lim)/( 20

* ttgtD At

A∞→

= σ

*BD

c and Δμ = 0 we have 2/1== BA xx , of course, and then the symmetry of the model

requires gA(t) = gB(t) and . To gain statistics, the average in Eq. (16) thus includes an average over all the particles.

**BA DD =

The interdiffusion constant is estimated from Eq. (3) and the Green-Kubo relation

[38]

125

,)()0(*)/()(0

20 tJJdtTNtT AB

xABx∫

∞

=Λ σ (18)

in which the current vector ABJ is defined by

∑ ∑= =

−−=A BN

i

N

iBiAAiA

AB tvxtvxtJ1 1

,, ,)()()1()( (19)

where )(, tvi α denotes the velocity of particle i of type α at time t. The shear viscosity is also obtained from a corresponding relation [26,27,32,38].

Fig 3: Log-log plot of (a) the reduced susceptibility χ* = ε0χ and (b) the reduced correlation length ξ(T)/σ vs. ε. The lines represent fits using the anticipated Ising exponents. All data refer to systems of N = 6400 particles. From Das et al. [27].

Fig. 4: Variation of the reduced self-diffusion coefficient with temperature, using data for N = 6400 at xA = xB = ½. From Das et al. [27].

B

3. Results for Static and Dynamic Critical Properties From the distribution P(xA) the “concentration susceptibility” χ has been extracted

[26,27], paying careful attention to finite-size effects via a finite-size scaling analysis. From the partial structure factors, using Eqs. (1), (2), and the known χ, estimates for ξ have also been extracted. Fig. 3 shows the final results of this analysis. It is seen that both χ and ξ (and also the order parameter [27]) are very well compatible )1()2( coex

AcoexA xx −

126

with the expected Ising critical behavior, although due to finite-size effects only data for ε ≥ 0.02 are available.

Fig. 4 shows results for the self-diffusion coefficient: it decreases weakly and linear

with temperature as the critical point is approached. No detectable sign of a critical anomaly can be seen. Similarly, a study of self-diffusion near the vapor-liquid critical point of a lattice-gas model did not find a critical anomaly either [24] (although this model belongs to “class B” in the Hohenberg-Halperin classification [10]).

The Onsager coefficient for interdiffusion is shown in Fig. 5. While far above

T)(TΛ

c the Onsager coefficient has only a very weak temperature dependence, there occurs a sharp rise when Tc is approached. These data clearly demonstrate that the simple van Hove description (implying a finite nonsingular )( cTΛ ) is invalid: )(TΛ must contain both a critical part and a non-critical background term [which dominates far above Tc], as specified in Eq. (6).

Fig. 5: Onsager coefficient Λ(T) for interdiffusion plotted vs. temperature, for a system of N = 6400 particles. From Das et al. [27]. However, most of our simulation results are in a temperature regime where the effect

of the non-singular background is by no means negligible. As a result, when one analyzes directly the critical behavior of DAB in a naïve way, e.g. by a log-log plot of DAB versus ε, one expects to see a crossover from an exponent γ = 1.239 for ε ≥ 0.4 [where Λ(T) in Fig. 5 is almost constant] to the exponent γ − νλ ≈ 0.672 for ε → 0. For a restricted range of ε, such as 0.02 ≤ ε ≤ 0.1, this crossover may show up as an effective exponent in between these values, and indeed, such an effective exponent (namely ≈ 1.0) is observed [27]. However, in view of Fig. 5 it would be clearly wrong to take such an effective exponent as the asymptotic result, as done in [25].

127

It turns out that for an analysis of the critical behavior of )(TΛ as displayed in Fig. 5

both the non-critical background )(TbΛ in Eq. (6) needs to be estimated, and the finite-

size rounding of the singularity of )(TΔΛ needs to be taken into account. In view of this difficulty Das et al. [26,27] tested the consistency of the simulation results with the following finite-size scaling description

, (20) where Q is a universal amplitude, also predicted from the theories [16-22,39], namely Q = (2.8 ± 0.4) x 10

ξε λν /,)(*)( LyyWQTT ==ΔΛ −

-3 [27], and W(y) is a scaling function with the limiting properties W(y → ∞) = 1 and W(y → 0) ≈ , with WλxyW0 0 an amplitude factor while ννλλ /=x .

To test for the validity of Eq. (20), it is convenient to assume various trial values for

and plot the resulting estimates for the scaled part

versus y. When one achieves optimal data collapse, for

small y a power law proportional to should emerge, while for large y a crossover towards a constant should be visible, from which an estimate for the amplitude Q could be extracted.

)( cbeffb TΛ=Λ

)(*)( yWQTT =ΔΛλνελxy

This strategy is tried in Fig. 6. For convenience of plotting, we choose y/(y0 + y) with

y0 = 7 rather than y as an abscissa variable (since this variable approaches unity when y → ∞), and display four possible assignments of . The filled symbols represent the data at T* = 1.48 for systems sizes L/σ = 7.37, 11.70, 14.74 and 18.57: their reasonably good collapse onto the remaining data (all for L/σ = 18.57, for different T*) and their approach to zero for small y, serve to justify the estimate ; furthermore, the flat part for large y is indeed close to the theoretical value for Q, quoted above, and highlighted by arrows in Fig. 6. From the simulation alone, one would arrive at an estimate Q = (2.7 ± 0.4) × 10

effbΛ

210)8.03.3( −×±=Λeffb

-3, in almost perfect coincidence with the theoretical value!

128

Fig. 6: Finite-size scaling plots of 103 ΔΛ (T) /T*, the critical part of the reduced interdiffusional Onsager coefficient, versus y/(y

λνε0 + y) where y = L / ξ(T) and y0 = 7. The exponent estimate νλ = 0.567 has been adopted.

The solid arrows on the right-hand axis indicate the theoretical estimate for the amplitude Q; see text. From Das et al. [27].

More recently, Das et al. [40] analyzed a theoretical prediction [39] of the background term and found that the value obtained from the above fit is nicely compatible with

this analysis [39,40]. Thus, both the constants Q and deduced from this fit appear to be physically significant.

effbΛ

effbΛ

4. Conclusions and Discussions In this paper a comprehensive simulation study of a model for a fluid binary mixture

with a critical point of fluid-fluid demixing was described. It was shown that methodological advances, such as combination of SGMC and MD, and analysis techniques based on finite-size scaling, allow a rather complete analysis of both static and dynamic critical properties of the model. While the self-diffusion coefficient is nonsingular at the critical point, the behavior of the interdiffusion coefficient is more subtle: the critical vanishing is less strong than expected from the van Hove conventional theory, due to a diverging Onsager coefficient. Far away from Tc, this behavior is masked by a nonsingular background term in the Onsager coefficient. In this way, theoretical predictions, experiments, and simulation results can be reconciled with one another.

This work also shows that MD techniques are now available that can explore the

critical dynamics of simple model systems. We expect that in the future it will become possible to carry out analogous studies for chemically realistic models of simple materials, as well as to extend our approach to critical behavior in complex fluids. But

129

one has to be aware of the fact that finite-size effects turn out to be much more pronounced in simulating dynamic critical behavior than for static critical properties [26,27,40].

Acknowledgements: Two of the authors (M. E. F. and S. K. D.) have received support

from the National Science Foundation under Grant No CHE 03-01101. S. K. D. also acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) via Grant No Bi314/18-2.

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Oxford, 1987. [32] S. K. Das, J. Horbach, K. Binder, J. Chem. Phys. 119 (2003) 1547-1558. [33] M. E. Fisher, in M. S. Green (Ed.) Critical Phenomena, Academic Press, London,

1971, pp. 1-99. [34] K. Binder, Z. Phys. B 43 (1981) 119-140. [35] V. Privman (Ed.) Finite Size Scaling and Numerical Simulation of Statistical

Physics, 2nd ed., Cambridge, 2005. [36] D. P. Landau and K. Binder, A. Guide to Monte Carlo Simulations in Statistical

Physics, 2nd ed. (Cambridge University Press, Cambridge, 2005). [37] N. B. Wilding, J. Phys.: Condens. Matter 9 (1997) 585-612. [38] J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids, Academic, London, 1986. [39] H. C. Burstyn, J. V. Sengers, J. K. Bhattacharjee, R. A. Ferrell, Phys. Rev. A 28

(1983) 1567-1578. [40] S. K. Das, J. V. Sengers, M. E. Fisher (to be published).

131

Diffusion on Diffusing Particles

Yu Imoto*, Takashi Odagaki

Dept. of Physics, Kyushu University, Fukuoka, Japan *Present address: Just System Inc., Tokushima, Japan Corresponding author: Takashi Odagaki Dept. of Physics Kyushu University Fukuoka 812-8581, Japan E-Mail: odag3scp@mbox.nc.kyushu-u.ac.jp

Abstract We investigate random walk of a particle constrained on cells, where cells behave as a

lattice gas on a two dimensional square lattice. By Monte Carlo simulation, we obtain the mean first passage time of the particle as a function of the density and temperature of the lattice gas. We find that the transportation of the particle becomes anomalously slow in a certain range of parameters because of the cross over in dynamics between the low and high density regimes; for low densities the dynamics of cells plays the essential role, and for high densities, the dynamics of the particle plays the dominant role. Key words: Self-diffusion, first passage time, slow dynamics, lattice gas

1. Introduction Diffusion or random walk of particles in random environments has been studied for a

long time because of its versatility applicable to many phenomena [1,2]. It is known in some systems that the random environment fluctuates in time and a particle diffuses in the fluctuating environment. For example, surfactant molecules in a solution of micelles diffuse under the effect of fluctuation of the network structure of the micelles. Surprisingly, the self-diffusion constant shows a minimum as a function of surfactant concentration [3-5]. Another example is the fluid mercury in the super critical region, where electrons move on clusters of mercury which are formed and dissolved incessantly. It is known that the electron mobility becomes anomalously small in a certain range of the temperature and density domain, which is off the liquid-gas critical point [6]. This anomalous behavior of the transport property has not been well understood, although it has been suggested to be related to the competition between two transport regimes; the diffusion of micelles or clusters themselves and the diffusion of particles on the micelles or clusters [5,7].

In this paper, we introduce a generic model that is relevant to the diffusion of a particle on diffusing particles and investigate the transport property of the particle. We consider an assembly of cells distributed on the square lattice at a given density. The cells are assumed to be a lattice gas and the distribution of the cells fluctuates in time. We attach a particle to this system, which moves only on the cells. We assume that the particle makes a random walk within a cell and can jump from one cell to an adjacent cell on its nearest

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, T. Odagaki 132

neighbor sites. As a transport property, we focus on the first passage time of the particle whose Lap lace-Fourier transform is related to the susceptibility for a boundary perturbation experiment [8,9]. The reason that we study the first passage time is that the diffusion constant cannot be obtained accurately by the Monte Carlo simulation in a finite system. In particular, when the system shows anomalous diffusion, the diffusion constant cannot be well defined. We obtain the first passage time as a function of the density and temperature of the cells, which determine the structure of the fluctuating environment. The first passage time shows a maximum, i.e. the transport becomes slow in a certain area in the parameter space.

In the next section, we explain the model system and the method of Monte Carlo simulation. In Sect. 3, we present the results of the simulation where the ratio of the jump rates of the particle and that of the cells is treated as another controlling parameter. Note that a preliminary result, when the dynamics of the particle is much faster than that of the cells, has been reported in a previous paper [10]. We discuss the results in Sect.4 where the analysis of the mean square displacement of the particle is also mentioned briefly. Conclusion is given in Sect. 5. 2. Model and method of simulation 2-1 Model

We prepare a lattice gas on an LL× square lattice and assume each atom is a cell which can accommodate a particle. We assume that a cell itself is an × square lattice [Fig. 1(a)] and the particle, which we call a carrier from now on, makes random walk on these lattice sites. The carrier is assumed to jump from a cell to an adjacent site in the adjacent cell [See Fig 1 (b)]. The temperature dependence of the jump rate can be absorbed into the time scale and is assumed not to depend on the temperature.

(a) (b)

Fig.1 (a) a cell consists of an × square lattice. The carrier makes random walk within the cell if it is isolated. (b) The cells form a lattice gas on an square lattice at a given density. The carrier can hop between two adjacent cells through one of the bonds connecting the cells.

LL×

133

The energy of the lattice gas is given by

∑><

−=ji

jinnJE,

. (1)

Here, the summation is taken over the nearest neighbor pair of sites, is the coupling constant, and when site i is occupied by a cell and

>< ij )0(>J1=in 0=in when site i is

empty. Note that the lattice gas has a fluid-gas critical point given by CT

12

sinh =⎟⎟⎠

⎞⎜⎜⎝

⎛

CBTkJ , (2)

where is the Boltzmann constant. BkWe consider the equilibrium configuration of the lattice gas, that is the cell’s move on

the lattice, randomly under the effect of the interaction energy (1) and the heat bath at a given temperature T . When the density is sufficiently low, the carrier is trapped in one of the cells and the transport of the carrier is determined by the random walk of the cell. On the other hand, when the density is sufficiently high, the cells form a fixed network and the transport property of the carrier is determined by the random walk of the carrier itself on the network. Consequently, we expect a cross over in transport property when the density is increased from zero to unity. 2-2 Monte Carlo simulation and first passage time

The equilibrium distribution of cells is produced by the standard Metropolis algorithm. Namely, a randomly chosen cell is tried to move to a site in its neighboring sites. When it is occupied the attempt is stopped. When the site is vacant, the cell occupies the new site with probability when ]/exp[ TkE BΔ− 0>ΔE where EΔ is the difference in energy before and after the move: When , the cell is always moved to the new site. When this step is taken times where is the number of cells in the system, then the Monte Carlo time is advanced by one. We impose the periodic boundary condition in the horizontal direction and assume the reflecting boundary at the top and bottom edges.

0≤ΔEN N

During one Monte Carlo step, the carrier performs random walk on the cells. We introduce a parameter m which represents the speed of the carrier. In one Monte Carlo step, the carrier makes m steps of random walk which are randomly distributed within

trials of the dynamics of cells. NWe investigate the first passage time as the transport property of the carrier. In order to

define the first passage time, we attach the carrier to one of the cells on the top boundary at . The first passage time is defined by the time that it takes to reach the opposite boundary and escape from the system for the first time. As the boundary conditions for the dynamics of the carrier to meet the measurement, we set the periodic boundary condition in the horizontal direction, the reflecting boundary at the upper edge and the absorbing boundary just out side of the bottom edge.

0=t

The first passage time shows some distribution and the first passage time distribution plays the role of response function in the boundary perturbation experiment [8,9]. Here we focus on the mean first passage time );,( mTF ρ as a function of the density 2/ LN=ρ

134

and the temperature T for different values of m . When the mean first passage time is long, the diffusion is supposed to be slow. 3. Results

We set and for our simulation and obtained the mean first passage time by averaging over 10,000~20,000 samples. Although the first passage time has a wide distribution, the mean can be determined with sufficient accuracy.

32=L 5=

3-1 Infinite temperature limit Figure 2 shows the density dependence of );,(F mTρ in the high temperature limit,

where the cells make simple random walk. It is seen that )1;,( ∞ρF and )1000;,( ∞ρF are monotonically increasing and decreasing functions of the density, respectively, and

)25;,( ∞ρF is approximately constant.

Fig. 2 The density dependence of the mean first passage time at ∞=T for 1 and 25 ,1000=m .

3-2 Finite temperatures

In order to produce equilibrium distribution of the cells, we first prepared the system by annealing it at for 50000 Monte Carlo steps, and then reduced the temperature by . At each temperature, the system was again annealed for 50000 Monte Carlo steps before measuring the first passage time. Figure 3 shows the density dependence of

3/ =CTT1.0)/( =Δ CTT

);,( mTF ρ for various temperatures for =m 1000, 25 and 1. For (Fig. 3(a)), 1000=m )1000;,( TF ρ is a monotonically decreasing function of ρ for

high temperatures and shows a maximum (slow dynamics) for lower temperatures. We can define the cross over temperature by the condition XT

0)1000;,(lim0

=∂

∂→ ρ

ρρ

XTF . (3)

We found , which coincides with the value obtained for 5.2/ ≈CX TT ∞=m [10]. It is interesting to note that the maximum observed at occurs below the critical density

1/ ≥CTT5.0=ρ . As the temperature is reduced below , a new broad peak emerges CT

135

(a)

(b)

(c) Fig. 3 Density and temperature dependence of the mean first passage time );,( mTF ρ .

(a) fast carrier m , (b) intermediate carrier1000= 25=m and (c) slow carrier . 1=m

136

around 5.0=ρ . This broad peak is due to the phase separation of the lattice gas where a giant cluster (fluid phase) coexists with the gas phase and the giant cluster tends to be formed as a horizontal band because of the boundary conditions imposed. Once the carrier is trapped in the giant cluster, it takes a long time to escape from it and the mean first passage time becomes longer. However, in case of higher densities, the cluster can form a percolating channel connecting the top and bottom edges, though it fluctuates in time, and the carrier can travel along the channel with its own dynamics and the mean first passage time becomes shorter and shorter as the density approaches 1=ρ .

For (Fig. 3(b)), 25=m )25;,( TF ρ has a broad peak for all temperatures we investigated. The peak position is shifted toward 5.0=ρ as the temperature is reduced due to the same reason for explained above. 1000=m

For (Fig. 3(c)), the trend of the density dependence of 1=m )1;,( TF ρ is opposite to that of )1000;,( TF ρ , since the dynamics of the carrier is slower than that of the cells. Therefore the mean first passage time in the limit of 0=ρ is shorter than that at 1=ρ . 4. Discussion

We first note that at 1=ρ , the carrier makes a simple random walk on square lattice with jump rate . For an

LL ×4/m MM × regular square lattice, the mean first passage

time of a simple random walker with jump rate MF ω is rigorously given by [11]

∑−

= ⎟⎠⎞

⎜⎝⎛

++

⎟⎠⎞

⎜⎝⎛

++

−+

=1

0 3

2

21212sin

21212cos

)1()12(2

1 M

M

M

MM

Fμ

μ

πμ

πμ

ω . (4)

For our system, thus the mean first passage time is given by which explains the value observed at

m/515201=ρ .

In the limit of 0=ρ , the first passage time for 1000=m and 25 is basically determined by the time during that a cell arrives at the bottom for the first time on LL× square lattice, which is 1984, since the dynamics in the cell is so fast that the first passage of the carrier is essentially identical to the first arrival of the cell at the lower boundary. For , however, the situation is rather complicated and the exact expression has not been obtained. Our result shows the mean first passage time is about 50% longer than that of the cell.

1=m

The enhancement of );,( mTF ρ below the critical temperature is due to the strong spatial fluctuation of the structure: At higher densities, the carrier moves on the percolating network of cells, and as the density is reduced, the connectivity of the network becomes less effective and the mean first passage time becomes longer. In the low-density side, the carrier is transported by a cell on which it resides, and as the density is increased, the dynamics of the cell becomes slow due to cluster formation and the first passage time becomes longer

When the dynamics of the carrier is fast ( 1000=m ), we observed the cross over temperature above which cluster formation does not affect the first passage time

137

significantly. The cross over occurs when the cluster formation reduces the mobility of the cell. When the two-cell cluster breaks easily with probability larger than 50%, then

)1000;,( TF ρ is a decreasing function of the density, otherwise it will increase as the density is increased. Therefore, the cross over temperature can be determined by the condition 5.0)/exp( ≈− TkJ B , and thus is given by XT ≈CX TT / . 2ln/)2(sinh2 1− 5.2≈

In passing, we would like to refer to the diffusion constant. Using a larger system and , we investigated the mean square displacement for various values of

parameters. We found that the carrier shows anomalous diffusion in the area where the mean first passage time is long, and thus the diffusion constant cannot be well defined and may not be the relevant quantity to represent the dynamics of the carrier in the entire parameter space.

64=L 10=

5. Conclusion

We have shown that the cells providing the space for the random walk of the carrier have two competing effects on the dynamics, and the transport of the carrier shows an extremum as a function of density in a certain range of the temperature. This density dependence is in line with experiments for micelles and fluid Hg [3-7]. These competing effects for diminish when the temperature is higher than . 1000=m XT

It should be emphasized that the first passage time can be related to a boundary perturbation experiment [9] and the first passage time distribution can be measured by experiments such as the intensity modulated photocurrent spectroscopy [12].

Acknowledgement This work was supported in part by the Grant-in-Aid for Scientific Research from the

Ministry of Education, Culture, Sports, Science and Technology.

References [1] R. Burridge, S.Childress, G. Papanicolaou (eds), Macroscopic Properties of

Disordered Media, Springer-Verlag, Berlin, 1982. [2] J. Kärger, F. Grinberg, P. Heitjans, Diffusion Fundamentals, Leipziger Universitäts

verlag, Leipzig, 2005. [3] T. Kato, T. Terao, M. Tsukada, T. Seimiya, J. Phys. Chem. 97 (1993) 3910-3917. [4] T. Kato, T. Terao, T. Seimiya, LANGMUIR 10 (1994) 4468-4474. [5] S. Mandal, S. Tarafdar, A. J. Bhattacharyya, Solid State Commun. 113 (2000) 611-

613. [6] M. Yao, Z. Phys. Chem. Bd. 184 (1994) S73-84. [7] F. Hensel, W. W. Warren, Fluid Metals, Princeton University Press 1999. [8] M. Kawasaki, T. Odagaki, K. Kehr, Phys. Rev. E61 (2000) 5839-5842. [9] T. Okubo, T. Odagaki, Phys. Rev. E73 (2006) 026128-1-6. [10] T. Odagaki, H. Kawai, S. Toyofuku, Physica A266 (1999) 49-54. [11] H. Kawai, Master thesis, Kyushu University (1998). [12] P. E. de Jongh and D. Vanmaekelbergh, Phys. Rev. Lett. 77 (1996) 3427-3430.

138

Genetic and Cultural Diffusion

L.Luca Cavalli-Sforza (Genetics Dept., Stanford University)

A population growing demographically will soon reach saturation in the environment in which it lives, and then if it can expand to other territory in the neighborhood or more distant it will often spread to it, and continue increasing in numbers. This trend may create a repeated colonization process, and this is how Homo sapiens sapiens expanded to the whole world in a process that went on during the last 2000-5000 generations. In this process our species grew numerically by a factor of a million-fold, but kept losing progressively genetic diversity with distance from the place of origin according to a regular “serial founder effect” which is very clear genetically. Founders of new colonies are usually few, and therefore the colonies may lose genetic diversity with respect to the mother colony, because of random genetic drift. If the expansion continues at a sufficient rate there will be a continuous loss of genetic diversity toward the periphery. During it there will be inevitably a process of intra-specific genetic differentiation because of drift, accompanied by adaptation to new environments occupied during the expansion.

Modern humans seem to fit this model of genetic diffusion remarkably well, with a complication almost unique to them: the development of a strong culture which has generated the accumulation of knowledge, much of which is useful, is transmitted across generations and evolves over the centuries, at a rate which is increasing dramatically and has become almost explosive in the last few decades. It is the outcome of two characteristics that have developed in the course of human evolution and made our species unique. One of them is a remarkable inventiveness that has generated many tools directed to solve everyday problems felt to be of practical importance, and usually answering common needs or wishes. Its products have led to distinguishing our genus by calling it Homo, given by archeologists to our ancestors of almost three millions years ago. This development must be related to the quadrupling of size of our brain in the last 5 or six million years, and was helped by the freeing of our hands made possible by bipedalism. The other advance is the development of language, the last phase of which is probably much more recent, and allows to communicate reasonably well (even if not always unambiguously). Neither advance is limited to humans and is found also in many other animals, but never at the same degree as in our species. Migration is responsible for genetic diffusion and in part for cultural diffusion, but cultural diffusion has become faster at a rapidly increasing rate, mostly through the invention of new techniques of communication, and is now almost independent of space and time.

The two evolutions, genetic and cultural, interact with each other. It is also often difficult to decide if differences among individuals are genetic or cultural, but it is clear that behavioral differences among groups (distinct from physical ones that are mostly due to exposure to different climates, like skin color and many other somatic differences) are largely, if not exclusively, due to culture. Culture is also responsible for the origin and transmission of innumerable forms of prejudice. Racism, the persuasion that behavioral differences among groups are genetic, is one, and must be considered a social disease.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, L. Cavalli-Sforza 139

Cellular Automata Modeling of Diffusion under Confinement

Pierfranco Demontis, Federico G. Pazzona, Giuseppe B. Suffritti

Università degli Studi di Sassari, Dipartimento di Chimica, via Vienna 2, 07100 Sassari, Italy, E-Mail: fpazzona@uniss.it

1. Introduction Both thermodynamic and transport properties of molecular species are strongly influ-

enced by the effect of confinement exerted by microporous materials such as zeolites. The nature of particle-framework interactions, along with geometric effects (size, shape, and connectivity of the pores), provides the energy landscape for the transport process and plays a major role in determining the aptitude of the diffusing species to migrate from pore to pore. [1] Geometrical restrictions can cause a sharp separation on the time scales involved in the diffusion process: intracage motion (short times) and intercage migration (long times). [2]

Zeolites provide a three-dimensional framework (connected channels and cages with finite capacity) which, when reduced to its essential constituents, can be represented as a set of struc-tured lattice points (cells) evolving in time according to well defined local rules: these are the basic ingredients of Cellular Automata (CA) models.

With their parallel, space-time discrete nature, CA algo-rithms represent a very convenient environment in which physical systems can be modelled in a reductionistic approach, in order to cover large scales of space and time. [3] Fig. 1: a CA unit cell

(cubic symmetry). We constructed a CA satisfying detailed balance to model intercage diffusion and equilibrium properties of particles ad-sorbed in a ZK4 zeolite. [4, 5, 6]

2. The Model The structure of our CA is constituted by L3 cubically

arranged cells (L is the number of cells per side of the cube) at constant temperature. N adsorbed particles can diffuse from cell to cell. A cell is pictured in Fig. 1, while in Fig. 2 a sketch of their connection is repre-sented. A single cell has a total number of K adsorption sites, which can be grouped into 6 exit sites with poten-tial adsorption energy εex, and K–6 inner sites with en-ergy εin. Each site can accommodate only one particle, and each cell can exchange particles with its 6 first-neighboring cells. Jumps may occur only between adja-cent exit sites of adjacent cells, therefore the topology of the exit sites (dark grey cubes in Fig. 1) turns out to be

Fig. 2: some connected cells: a small 3D portion of the system.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, F. Pazzona 140

automatically defined. Instead, a particle cannot migrate to another cell if it occupies an inner site, therefore we choose to neglect their spatial arrangement (this is why we repre-sented them with the pale gray sphere in Fig. 1).

At each time step, the evolution of the system is given by a randomization procedure in which each cell (treated as a closed system) independently of each other can change the configuration of the guest particles, and a propagation procedure in which independ-ent pairs of adjacent exit sites can synchronously exchange particles. The output of each operation is determined stochastically according to probabilities defined by Boltzmann’s statistics. Each operation is carefully designed in order to satisfy detailed balance.

In our first calculations the particles interact with each other only by mutual exclu-sion. We found that a differentiation between εex and εin is enough to produce various types of diffusivity profiles, while the adsorption isotherm is a dual-Langmuir isotherm. Even in this simple case, many features of the model (e.g., the separation between the mean life times of differently occupied cells and the relaxation time of local density fluc-tuations) turn out to behave in a way similar to that observed through Molecular Dynam-ics simulations.

The introduction of mutual interactions between particles is represented by a depend-ence of the energy parameters εex and εin on the local density of each cell. This fact arises naturally when a systematic coarse-graining is performed to transfer the essential features of a cell equipped with adsorption sites structured in space and pair interparticle poten-tial into a less structured cell such the one pictured in Fig. 1. The introduction of this dependence allows to use effective energy potentials as flexible parameters by means of which the model can exhibit a wide range of behaviors, and therefore opens a way to generate coarse-grained models of diffusion in zeolites.

3. Conclusion We constructed a Cellular Automaton to capture the essential features of confinement

by means of a probabilistic scheme satisfying detailed balance. The model works with few flexible parameters, which depend on local observables and rule the adsorption iso-therm, the diffusivity profile, the separation of time scales.

References [1] J. Klafter and J. M. Drake, Molecular Dynamics in Restricted Geometries (Wiley,

New York, 1989) [2] P. Demontis, L. Fenu, and G. B. Suffritti, J. Phys. Chem. B 109 (2005) 18081 [3] B. Chopard and M. Droz, Cellular Automata Modeling of Physical Systems, (Cam-

bridge University Press, Cambridge, England, 1998) [4] P. Demontis, F. G. Pazzona, and G. B. Suffritti, J. Phys. Chem. B 110 (2006) 13554 [5] P. Demontis, F. G. Pazzona, and G. B. Suffritti, J. Chem. Phys. 126 (2007) 194709 [6] P. Demontis, F. G. Pazzona, and G. B. Suffritti, J. Chem. Phys. 126 (2007) 194710

141

Driven Polymer Translocation through a Nanopore: a Manifestation of Anomalous Diffusion

Johan Dubbeldam, Andrey Milchev, Vakhtang Rostiashvili, Thomas Vilgis

Max-Planck-Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany, E-Mail: rostiash@.mpip-mainz.mpg.de

1. Introduction Recently, single molecule experiment probing single-stranded DNA or RNA

translocation through a membrane nanopore attracted widespread attention [1]. Nevertheless the physical nature of the translocation process is still not well understood.

In this report we suggest a unique physical picture based on the mapping of the 3d – problem on a 1d translocation coordinate s (or a translocated number of segments). We study the translocation dynamics of a polymer chain threaded through a nanopore with and without an external force. By means of diverse methods (scaling arguments, fractional calculus and Monte Carlo) we show that the relevant dynamic variable, the translocation number of segments s (t), displays an anomalous diffusion behaviour even in the presence of an external force. The anomalous dynamics of the translocation process is governed in both (i.e. with and without a force) cases by the same universal exponent α = 2/(2ν + 2 - 1γ ), where ν is the Flory exponent and 1γ - the surface exponent. The process is described by a fractional Fokker – Planck equation which is solved exactly in the interval 0 < s < N ( where N is a chain length) with appropriate boundary and initial conditions. We have obtained a closed analytic solution for the probability function W (s, t) in terms of the polymer chain length N and the applied drag force f. This solution also enables to derive the expression for the probability distribution of translocation times Q (t) as well as the variation with time of statistical moments :

<s (t)> and <s (t)> - <s (t)> which provide full description of the diffusion process. It is found that the translocation time in the absence of an external force goes as

whereas in the driven anomalous diffusion case .

Also the corresponding time dependent statistical moments, <s (t)> and <s (t)> , reveal unambiguously the anomalous nature of the translocation dynamics and permit direct measurement of

2 2

122 γντ −+∝ N 112 γντ −+∝ N 1−fαt∝

2 ∝ α2tα in experiments. These findings are tested

and found to be in perfect agreement with extensive Monte Carlo study. Fig. 1: How a fold squeezes through a nanopore. The driving force f is caused by a chemical potential gradient μ = 1μΔ - 2μ .

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, V. Rostiashvili 142

2. Dynamics in terms of a translocation coordinate Figure 1 shows how a polymer fold of the length s overcomes an entropic barrier

11()( γ−=Δ sE ) T ln caused by a narrow pore. s

This consideration enables to calculate the exponent of the anomalous diffusion

122 γνα −+= . For d = 3, ν = 0.588 and 1 =γ = 0.680 we obtain α 0.801. Then for

the free-force translocation time we have . Remarkably, in d = 2,

ατ /2∝ 5.2≈=

N Nν 0.75 , 1γ = 0.945 and one finds α = 0.783, i.e.α is almost dimensionally

independent! This explains why the measured exponents in 2d [4] and in 3d [5] are so close. The influence of external force f could be treated within the formalism of fractional Fokker-Planck equation (FFPE) [6]. In our case the dimensionless force is caused by a chemical potential gradient 1μμ =Δ - 2μ (see Fig. 1). The analytical solution of FFPE for the translocation times distribution function Q (t) has been compared with results of Monte Carlo study and Fig.2 shows that the agreement between both methods is very good.

3. Conclusion

Fig. 2: First passage time distribution function at N = 128 and different forces as calculated from MC – data (symbols) and the theoretical predictions (solid lines). The average translocation time which can be calculated from such curves (where the chain length ranges in the interval 32 ≤ ≤ ) follows the law256N 1fτ −∝ 3 2/N .

The polymer translocation is considered as squeezing of subsequent chain fragments (folds) through a narrow pore. This consideration gives rise to an universal scaling exponent for anomalous diffusion 122/(2 γνα −+= ) so that the translocation time in

the absence of drag scales as . The presence of external force f modifies this relationship to . This principal result is unambiguously confirmed by calculation of the mean first passage times (the average translocation times) from the derived analytic expression for translocation time distribution function Q (t) as well as by comparison to the Monte Carlo simulation.

ατ /2N∝1−∝ fτ 112 γν −+N

References [1] A. Meller, J. Phys. Condensed Matter. 15 (2005) R581. [2] J. Dubbeldam, A. Milchev, V. Rostiashvili, T. Vilgis, arXiv:cond-mat/0701664. [3] J. Dubbeldam, A. Milchev, V. Rostiashvili, T. Vilgis, arXiv:cond-mat/0702463. [4] J. Chuang, Y. Kantor, M. Kardar, Phys. Rev. E 65 (2004) 021806. [5] D. Panija, G. Barkema, R.C. Ball, arXiv:cond-mat/0610671. [6] R. Metzler, J. Klafter, Physics Rep. 339 (2000) 1.

143

Molecular Dynamics Study of Carbon Diffusion in Cementite

Alexander V. Evteev, Elena V. Levchenko, Irina V. Belova and Graeme E. Murch

Diffusion in Solids Group, Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, NSW 2308 Callaghan, Australia,

E-Mail: Elena.Levchenko@newcastle.edu.au

1. Introduction Although cementite (Fe3C) is a most important phase in steels, at the same time, there is very

little known in regard to the fundamental properties of Fe3C. This lack of information is largely a result of this compound being metastable with respect to its decomposition products: C-saturated ferrite or austenite (depending on temperature) and graphite. It is very difficult to obtain ‘pure’ Fe3C in the sizes and amounts necessary for many fundamental studies of its properties including investigation of its thermodynamics and diffusion behaviour. For example, conventional radiotracer diffusion experiments are essentially impossible and the measured chemical diffusivities show considerable error and lack information on their compositional dependence.

In the last few years, the role of diffusion in Fe3C in the technologically important metal-dusting process has attracted considerable interest [1] but the above mentioned problems mean that revealing the mechanism of diffusion in Fe3C from experiments will be especially difficult. The simulation method of molecular dynamics has now reached a level of maturity and reliability that it can very profitably be used to understand for the first time the mechanism of C diffusion in Fe3C. This is the subject of this paper.

2. The model The Fe3C structure was simulated as a calculation box with periodic boundary conditions

consisting of 10×10×10 simple orthorhombic unit cells with lattice parameters a=4.523 Å, b=5.089 Å and c=6.743 Å and 4 Fe atoms of type 1 (0.833, 0.040, 0.250; 0.167, 0.960, 0.750; 0.667, 0.540, 0.250; 0.333, 0.460, 0.750), 8 Fe atoms of type 2 (0.333, 0.175, 0.065; 0.667, 0.825, 0.935; 0.167, 0.675, 0.435; 0.833, 0.325, 0.565; 0.667, 0.825, 0.565; 0.333, 0.175, 0.435; 0.833, 0.325, 0.935; 0.167, 0.675, 0.065) and 4 C atoms (0.430, 0.870, 0.250; 0.570, 0.130, 0.750; 0.070, 0.370, 0.250; 0.930, 0.630, 0.750) per unit cell (the atomic positions are in units of the lattice parameters). Fe (1) and Fe (2) are two stoichiometrically different iron sites. Thus, this model contains 12000 Fe atoms and 4000 C atoms. The Fe-Fe interaction was described using the Johnson empirical pair–potential [2]. To describe the Fe-C pair interaction we used the potential proposed by Johnson, Dienes and Damask [3]. The use of the Johnson–Dienes–Damask potential together with the Johnson potential permits an adequate description of the behaviour of C interstitials in α-Fe and in martensite [3]. The structure and relative energy of Fe3C [3] have also been investigated by means of these potentials. The situation with the weak C–C potential is more complicated. There is no agreed and detailed data about the preferred type of interaction for C–C pairs in Fe-C alloys. That is why as a first approximation we do not consider the direct interaction between C–C atoms. However, we prevent the situation when two C atoms occupy the same positions, and, in the present study for describing the C-C interaction, we chose a purely repulsive Born–Mayer potential with a cut–off radius of 1.5 Å which is much

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, E. V. Levchenko 144

smaller than the distance between the first nearest neighbour carbon positions. To initiate the diffusion process, atoms were given initial velocities according to the Maxwell distribution at a given temperature, and an isothermal annealing procedure was performed in a temperature range of 1173–1373 K with a step of 50 K. The MD simulations consist of a numerical integration of the equations of atomic motion using a time step Δt = 1.5×10-15 s according to the Verlet algorithm. Periodically, the system was transferred to a state at T = 0 K where atoms occupied equilibrium positions in a local potential minima. This was done by making use of the static relaxation method. After this, the C movements that occurred were analyzed.

3. Results and discussion It was found that C diffusion in Fe3C is realized by means of interstitial sites, which form a

base-centered orthorhombic sublattice in cementite with sublattice parameters aI=a, bI=b and cI=0.5c and 2 sublattice points 0.0, 0.0, 0.0 and 0.5, 0.5, 0.0 in units of aI, bI and cI or, in other words, 4 interstitial positions per Fe3C unit cell 0.0, 0.0, 0.0; 0.5, 0.5, 0.0; 0.0, 0.0, 0.5 and 0.5, 0.5, 0.5 in units of a, b and c. It should be noted that the distance between the nearest neighbours interstitial sites in [110] and ]011[ directions (3.404 Å) is very close to the distance between the nearest neighbour interstitial sites in the [001] direction (3.372 Å). The interstitial sites are less energetically favourable for C atoms than their original positions. However, with an increase of temperature, because of the growth of entropy, some fraction of the C atoms occupies the interstitial sites and the corresponding same fraction of originally C positions becomes vacant. In effect, this is equivalent to the formation of Frenkel defects. In particular, at the highest temperature 1373 K in our computer simulation the ‘equilibrium’ fraction of C atoms on interstitial sites is ~ 0.22. We defined the three principal carbon diffusion coefficients Dx, Dy and Dz along the three orthogonal crystallographic axes x, y and z at each temperature at ‘equilibrium’ by making use of the Einstein equation and the mean square displacements of the C atoms. Then the Arrhenius parameters of C diffusion in Fe3C along the three orthogonal crystallographic axes were determined. It was established that the fastest C diffusion is along the [001] z-direction, followed by the [100] x- and [010] y-direction. For example, at 1373 K we have that Dx/Dz≈0.73 and Dy/Dz≈0.45.

4. Conclusions We have performed molecular dynamics simulations to investigate C diffusion in

cementite. The assumption that C atoms can interact with each other only indirectly (via neighbouring Fe atoms) has been used. We have elucidated the interstitial mechanism of C diffusion in Fe3C as well as determining the ‘equilibrium’ fraction of carbon atoms on interstitial sites and parameters of C diffusion in Fe3C along the three orthogonal crystallographic axes.

Acknowledgements: The support of the Australian Research Council is acknowledged.

References [1] D.J. Young, Mat. Sc. Forum, 522-523 (2006) 15. [2] R.A. Johnson, Phys. Rev., 134 (1964) A1329. [3] R.A. Johnson, G.J. Dienes and A.C. Damask, Acta Metall., 12 (1964) 1215.

145

Carbon Diffusion in Austenite: Computer Simulation and Theoretical Analysis

Alexander V. Evteev, Elena V. Levchenko, Irina V. Belova and Graeme E. Murch

Diffusion in Solids Group, Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, NSW 2308 Callaghan, Australia,

E-Mail: Alexander.Evteev@newcastle.edu.au

1. Introduction Carbon interstitial diffusion in austenite (γ-Fe) has been studied experimentally and

analytically for many years. Two experimental studies have been made of carbon diffusion in austenite at low carbon contents. One is a measurement of the carbon chemical or intrinsic diffusion coefficient at 1075 K, 1124 K and 1273 K, and as a function of carbon content [1]; the other is a measurement of the tracer carbon diffusion coefficient at the single temperature 1273 K as a function of carbon content [2]. These data demonstrate that both the carbon tracer and chemical diffusion coefficients increase with carbon content. The data have been analyzed on many occasions with two different diffusion models.

The first is the well-known interacting lattice gas model in which nearest neighbour pair interactions are supposed between the carbon atoms and an inter–site transition rate based on those interactions is formulated; see, for example, [3]. Repulsive interactions between the carbon atoms are required in order to make quantitative contact with the diffusion data. A difficulty with the lattice gas model is one of uniquely describing 'rotational jumps' (a second atom making a rotational jump from one nearest neighbour site to another of a given atom) in the f.c.c. lattice when simple inter–site transition probabilities are used.

In the second model [4] (which is appropriate only at very low carbon content) it is supposed that the interstitials can diffuse only as isolated atoms or as bound nearest neighbour pairs. Because this model is based on a specification of fundamental jump frequencies it is a more general starting point for a diffusion kinetics analysis than the first, which started with effective interactions from which jump frequencies can be derived. In this second model, four atom – vacant site exchange frequencies are explicitly specified. These are: an atom – vacant site exchange frequency for jumps of isolated interstitial atoms, a rotational atom – vacant site frequency of one atom about the other when the pair of interstitial atoms are first nearest neighbours, a dissociative exchange frequency of a first nearest neighbour pair and the frequency of the reverse jump (associative) to form a pair. All other interstitial jumps are assumed in this model to occur with the same frequency as for jumps of isolated interstitial atoms. The increase in diffusion with carbon content could be ascribed to a higher mobility of pairs than isolated carbon atoms. The most important assumption of the 4–frequency model for interstitial diffusion is that the frequencies of dissociation and association of the first nearest neighbour pairs do not depend on whether dissociation occurs to the second, third or fourth nearest neighbour sites or whether association takes place from any of these sites.

2. Model In the present work, a kinetic 10–frequency model for interstitial diffusion by octahedral

voids in f.c.c. lattice, which considers the specific role of the transition probabilities during association and dissociation of the first nearest neighbour interstitial pairs through the

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, A. V. Evteev 146

second nearest neighbour sites, was developed. Next, we performed molecular dynamics simulations to investigate carbon interstitial diffusion in austenite at low carbon contents. The assumption that carbon atoms can interact with each other only indirectly (via neighbouring iron atoms) has been used. We have determined the Arrhenius parameters of the interstitial carbon jump frequencies as identified by the 10–frequency model and have also implemented Monte Carlo calculations of correlation factors.

3. Results and discussion Comparison of the molecular dynamics and Monte-Carlo results with the

experimental data at 1273 K in the context of the 10–frequency model is then made. It is shown that direct repulsion between the carbon atoms at the first nearest neighbour pairs in dilute austenite is undoubtedly present. Jump frequency ratios at 1273 K for the carbon first nearest neighbour pairs have been adjusted according to the experimental data.

It was established that direct repulsive interaction between the first nearest neighbour carbon atoms and indirect repulsive interaction via neighbouring iron atoms between the second nearest neighbour carbon atoms lead to significantly lower values (by ∼ 42 % and by ∼ 24 %) of both concentrations of the first and second nearest neighbour carbon pairs respectively near 1273 K in comparison with a random distribution. At the same time, the concentrations of the third and the fourth nearest neighbour carbon pairs are very close to a random distribution. At 1273 K the initial increase of both the tracer and the chemical diffusion coefficients with increasing carbon content can be explained as increased rates of dissociation of the carbon first and second nearest neighbour pairs to the third nearest neighbour sites.

4. Conclusions A kinetic 10–frequency model for interstitial diffusion via octahedral interstices in the

f.c.c. lattice was developed. In this model, the specific role of the transition probabilities during association and dissociation of the first nearest neighbour interstitial pairs through the second nearest neighbour sites is considered. Molecular dynamics was used to investigate carbon interstitial diffusion in austenite at low carbon contents with the assumption that carbon atoms can interact with each other only indirectly (via neighbouring iron atoms). The Arrhenius parameters of interstitial carbon jump frequencies were determined and compared with experimental data at 1273 K in the context of the 10–frequency model. It was shown that a small direct repulsion between carbon atoms at first nearest neighbours should be included. It was found that the initial increase (with increasing carbon content) in both the tracer and the chemical diffusion coefficients is shown to be a result of increased rates of dissociation of carbon from first and second nearest neighbour pairs to third nearest neighbour sites.

Acknowledgements: The support of the Australian Research Council is acknowledged.

References [1] R.P. Smith, Acta Metall., 1 (1953) 578. [2] D.C. Parris and R.B. McLellan, Acta Metall., 24 (1976) 523. [3] G.E. Murch and R.J. Thorn, J. Phys. Chem. Solids, 40 (1979) 389. [4] a) R.A. McKee, Phys. Rev., B21 (1980) 4269. b) R.A. McKee, Phys. Rev., B22 (1980) 2649.

147

Analytical and Kinetic Monte–Carlo Study Shrinkage by Vacancy Diffusion of Hollow Nanospheres and Nanotubes

Alexander V. Evteev, Elena V. Levchenko, Irina V. Belova and Graeme E. Murch

Diffusion in Solids Group, Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, NSW 2308 Callaghan, Australia,

E-Mail: Elena.Levchenko@newcastle.edu.au

1. Introduction Recently, Sun et al. [1] have developed a simple and generic approach to the synthesis of

hollow noble metal nanostructures. The key step of this process is the redox reaction between a silver template and the solution of the appropriate salt precursor. A quite different method for synthesizing hollow (binary) nanostructures makes use of the Kirkendall effect of diffusion [2]. Such hollow nanostructures have a wide range of technological applications such as catalysis, vehicles for drug delivery, containment of environmentally sensitive species.

However, it has been noted [3,4] that hollow nanospheres should in fact be unstable in principle and, with time, they will tend to shrink into a solid nanosphere. The mechanism of shrinking can be considered as resulting from the vacancy flux from the inner surface to the external surface.

2. Results and discussion In this work, shrinking via the vacancy mechanism of a pure element hollow

nanosphere and nanotube is described analytically. Using Gibbs-Thomson boundary conditions in quasi steady-state at the linear approximation, we determine the collapse time as a function of geometric size both for hollow nanoshpere (Eq. 1) and nanotube (Eq. 2):

( )( )( )

( )( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

−−−

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−

−

+−= 33

0

030

003

00

34

330

34

30

300

11

112

111

1

11ln)(δεδεδ

εεε

δεδεδε

ε

εεδτ , (1)

( ) ( )( )

( ) ( ) ( ) ( )

( )[ ] ( ) ( ) ( )[ ] ( )0020202

01010020

220

lnln1

arcsinarcsin11

11ln45

21

21

εεϕδεϕδδεϕ

δεϕεϕεδεε

δεδτ

−+−+

+−+−+⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

−+=

(2)

with

( )( ) ( )xx

xxx+−

++=

112

3121

2

2

1ϕ , ( )( ) ( )xx

xxxxx+−

++−+=

112

2232223

2

432

2ϕ , (3)

where 3fV

eqV6 rtDc βτ = ( and are the equilibrium concentration and the diffusion

coefficient of the vacancies, respectively;

eqVc VD

kTΩ= γβ 2 and kTΩ= γβ for hollow nanospheres and nanotubes, respectively; γ is the surface energy, Ω is the atomic volume and is the radius of a collapsed compact nanosphere or nanorode),

fr

0εεδ = and ei rr=ε is the ratio of the inner and the external radii of the hollow nanosphere or nanotube at time (ir er t 0εε = at ). 0=t

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, E. V. Levchenko 148

Kinetic Monte Carlo (KMC) simulation by the Metropolis algorithm confirms the predictions of the analytical model (as an example see Figs. 1 and 2).

104 105 1060.0

0.2

0.4

0.6

0.8

1.0

δ

MCSA Fig. 1. KMC simulation of the shrinking kinetics of a f.c.c. pure element nanotube with 5.00 =ε ,

ar 38f = (a is the lattice parameter) and the pair interaction energy 2.1−=kTφ .

0.0 0.5 1.0 1.50.0

0.1

0.2

0.3

τMCSA×10-6

Fig. 2. Test of Eq. 2 on the basis of the results of KMC simulation shrinking kinetics of a f.c.c. pure element nanotube with 5.00 =ε ,

ar 38f = and 2.1−=kTφ (see Fig. 1).

It was shown on the basis of this simulation that the averaged surface energy per unit area (γ) of the external surface of such nanoobjects during shrinking is at least no lower than γ of the 110 crystallographic surface. It is well-known from both experimental data and molecular dynamics (MD) simulations using the embedded-atom method that the 110 surface inclines to reconstruction with increasing atomic density. Therefore, we can anticipate that the reconstruction processes will occur in areas of high energy facets on the external surface of such nanoobjects both in real experiments and in MD simulations by the embedded-atom method.

3. Conclusion The shrinkage via the vacancy mechanism of a pure element hollow nanosphere and

nanotube has been described. Using Gibbs-Thomson boundary conditions an exact solution has been obtained of the kinetic equation in quasi steady-state at the linear approximation. The collapse time as a function of the geometrical sizes of hollow nanospheres and nanotubes has been determined. KMC simulation of the shrinkage of these nanoobjects was performed: it completely confirmed the predictions of the analytical model. However, it has been shown on the basis of this simulation that under real conditions reconstruction of the external surface can occur. This reconstruction could not be taken into account either in the theoretical analysis or KMC simulation.

Acknowledgements: The support of the Australian Research Council is acknowledged.

References [1] Y. Sun, B. Mayers and Y. Xia, Adv. Mater., 15 (2003) 641. [2] Y. Yin, R.M. Rioux, C.K. Erdonmez et al, Science, 304 (2004) 711. [3] K.N. Tu and U. Gösele, Appl. Phys. Lett., 86 (2005) 093111. [4] A.M. Gusak, T.V. Zaporozhets, K.N. Tu and U. Gösele, Phil. Mag., 85 (2005) 4445.

149

Formation of a Surface–Sandwich Structure in Pd-Ni Nanoparticles by Interdiffusion: Atomistic Modelling

Alexander V. Evteev, Elena V. Levchenko, Irina V. Belova and Graeme E. Murch

Diffusion in Solids Group, Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, NSW 2308 Callaghan, Australia,

E-Mail: Alexander.Evteev@newcastle.edu.au

1. Introduction Bimetallic Pd-Ni nanoparticles are currently attracting a great deal of interest due to

their physical and chemical properties, which are determined by their size, shape, structure and composition. In particular, it is well-known that Pd is a metal with important existing and potential applications as a catalyst in heterogeneous catalysis. The use of Pd as a single active metal component in catalysis has received considerable attention on the basis of its remarkable activity for oxidation reactions and the availability of cleaner fuels, but at the same time it has negative economical aspects due to its high cost [1]. An interesting approach in these respects consists of alloying Pd with lower cost and higher surface energy metals, since it would be economically attractive to design bimetallic catalyst nanoparticles in which the precious and catalytic Pd atoms segregate to the surface. Promising results in this sense have been obtained by using Ni [1-3].

Understanding and controlling of the atomic structure of alloy nanoparticles are important in both fundamental science and technological applications. Atomistic simulation techniques such as molecular dynamics (MD) have become a powerful tool in the field of nanotechnology by providing physical insight in understanding phenomena on an atomic scale and predicting many of the properties of nanomaterials.

In the present study, long–time scale molecular dynamics simulation in combination with the embedded atom method is used to investigate the effect of surface segregation phenomena on the atomic structure of Pd alloy nanoparticles (of diameter of ∼ 4.5 nm) containing ∼ 30 at. % Ni.

2. The model A spherical core–shell f.c.c. structure was chosen as the initial state for bimetallic Pd–Ni

nanoparticle wherein a core of Ni atoms is surrounded by shell of Pd atoms. Then the static relaxation procedure was applied to accommodate the core and shell atoms especially at interface, since Pd atoms (the equilibrium bulk lattice constant is 3.89 Å) are noticeably larger than Ni atoms (the equilibrium bulk lattice constant is 3.52 Å) with difference in atomic sizes of ∼ 10%. Following this, atoms were given initial velocities according to the Maxwell distribution at the temperature of 1000 K and the MD procedure of isothermal annealing was performed. The MD procedure consisted of a numerical integration of the equations of atomic motion according to the Verlet algorithm with a time step Δt=1.5 fs. Periodically without an effect to the continuity of the annealing procedure the system was transferred to a state at T=0 K where atoms occupied equilibrium positions in local potential minima by making use of the static relaxation method. After this, the atomic movements and structure transformations occurring in the model were analyzed.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, A. V. Evteev 150

3. Results and Discussion It is found that such nanoparticles

form a surface–sandwich structure by interdiffusion. In this structure, the Ni atoms, which mostly accumulate in a layer just below the surface, at the same time are located in the centres of interpenetrating icosahedra and generate a subsurface shell as a Kagome net. Meanwhile, the Pd atoms occupy the vertices of the icosahedra and cover this Ni layer from inside and outside as well as being located in the core of the nanoparticle forming a Pd–rich solid solution with the rest of the Ni atoms. A basic structure of such nanoparticles nucleating during interdiffusion in the system is a Ni spherical subsurface shell (layer) having a structure of the Kagome net with ‘sequence faults’ (see Fig. 1). Each Ni atom of the spherical subsurface layer almost always has 4 Ni and 8 Pd nearest neighbours. It should be noted that planar layers of Kagome nets are the basic structure of the well–known bulk Frank–Kasper phases of certain intermetallic alloys.

Fig. 1. Snapshot of the Ni subsurface shell having a structure of the Kagome net with ‘sequence faults’ after ∼0.3 μs of annealing at T = 1000 K. Every Ni atom of the subsurface shell is located in the centres of interpenetrating icosahedra and almost always has 4 Ni and 8 Pd nearest neighbours. For better clarity only a half of the shell and only Ni atoms at centres of icosahedra are shown on the perspective projection with size and grey-scale graduation.

4. Conclusion Long–time scale molecular dynamics simulation in combination with the embedded atom

method has been used to investigate the effect of surface segregation on the atomic structure of Pd alloy nanoparticles (of diameter of ∼ 4.5 nm) containing ∼ 30 at. % Ni. We have found that such nanoparticles form a surface–sandwich structure where the Ni atoms, which mostly accumulate in a layer just below the surface, at the same time are located in the centres of interpenetrating icosahedra and generate a subsurface shell as a Kagome net. Meanwhile, the Pd atoms occupy the vertices of the icosahedra and cover this Ni layer from inside and outside as well as being located in the core of the nanoparticle forming a Pd–rich solid solution with the rest of the Ni atoms.

Acknowledgements: The support of the Australian Research Council is acknowledged.

References [1] A.B. Hungría, J.J. Calvino, J.A. Anderson, A. Martínez-Arias, Appl. Catal., B62 (2006) 359. [2] P. Miegge, J.L. Rousset, B. Tardy, J. Massardier, J.C. Bertolini, J. Catal., 149 (1994), 404. [3] L. Porte, M. Phaner-Goutorbe, J.M. Guigner, J.C. Bertolini, Surf. Sci., 424 (1999) 262.

151

Molecular Dynamics Study of Diffusion in Palladium Hollow Nanospheres and Nanotubes

Alexander V. Evteev, Elena V. Levchenko, Irina V. Belova and Graeme E. Murch

Diffusion in Solids Group, Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, NSW 2308 Callaghan, Australia,

E-Mail: Elena.Levchenko@newcastle.edu.au

1. Introduction The synthesis of nano–scale materials is a rapidly developing field of materials

science. Recently, Sun et al. [1] have demonstrated the preparation of a number of hollow nanostructures of Au, Pd and Pt by a replacement reaction with Ag starting with solid Ag templates. Such structures have very considerable promise in a wide range of technological applications such as catalysis, precise drug delivery and many others.

However, it has been noted [2,3] that hollow nano-objects should be unstable in principle and, with time, they will tend to shrink. According to [2,3] the mechanism of shrinking can be considered as resulting from the vacancy flux from the inner surface to the external surface. The driving force for this flux is the difference between the vacancy concentrations on the inner and external surfaces. We have described the shrinkage via the vacancy mechanism of a pure element hollow nanosphere and nanotube (see our abstract titled ‘Analytical and kinetic Monte–Carlo study shrinkage by vacancy diffusion of hollow nanospheres and nanotubes’). Using Gibbs-Thomson boundary conditions an exact solution was obtained of the kinetic equation. The collapse time as a function of the geometrical sizes of hollow nanospheres and nanotubes was determined. Kinetic Monte-Carlo (KMC) simulation of the shrinkage of these nano-objects was performed: it confirmed the predictions of the analytical model. However, it has been shown on the basis of this simulation that under real conditions reconstruction of the external surface can occur. This reconstruction could not be taken into account either in the theoretical analysis or KMC simulation. In the present paper, we study the diffusion in a pure Pd hollow nanosphere and nanotube by performing a molecular dynamics (MD) simulation using the embedded-atom method (EAM).

2. The model The initial hollow nanosphere and nanotube were cut from a perfect bulk f.c.c. Pd

lattice and consisting of 76657 and 51068 atoms in the nanoshells with inner and external radii ri0≈28 Å and re0≈65 Å, respectively. The periodic boundary conditions along the axis of the nanotube, coinciding with the [001] direction (z–direction), were imposed. Then, the MD procedure of isothermal annealing was performed.

3. Results and discussion It was established that the investigated Pd MD models of hollow nano-objects melt at 1500 K.

Therefore, our MD study of diffusion in Pd hollow nano-objects was carried out just below the melting temperature at 1450 K. Before the diffusion calculations, a vacancy concentration was permitted to approach equilibrium (this took the first 2.4 ns of our experiment). Then, for different distance intervals in nanoshells, we have calculated the mean square displacement

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, E. Levchenko 152

(MSD) only for those Pd atoms, which during the time of experiment were in the same distance interval (as an example, see the results for hollow nanosphere in Fig. 1). It should be noted that outside the intervals indicated by Fig.1 the MSD is much larger. Therefore, the average diffusion length was confined to less than intervals of 3 Å. Thus, outside these distance intervals it was impossible to determine correctly the diffusion coefficient by the vacancy mechanism. The diffusion coefficient of Pd atoms in different distance intervals in the nanoshell was calculated from the Einstein equation. As can be seen in Fig. 2, the diffusion coefficient in the Pd hollow nanosphere increases with proximity to the surfaces. It is well known that the diffusion coefficient is proportional to the vacancy concentration; therefore, these results demonstrate that the vacancy concentration is larger near the inner and external surfaces compared with the middle part of the nanoshell. Similar results were obtained for nanotubes, with the only difference that an anisotropy of diffusion in the radial direction (D⊥) and along the cylinder axis (D||) is observed D⊥/D||~0.7. Therefore, the MD results obtained provide quite a strong argument that a pure element hollow nanosphere or nanotube will not shrink readily via the vacancy mechanism.

4. Conclusion MD simulation in combination with the EAM

has been used to investigate the diffusion by the vacancy mechanism in a Pd hollow nanosphere and nanotube. We found that the diffusion coefficient in a Pd hollow nanosphere and nanotube is larger near the inner and external surfaces compared with the middle part of a nanoshell. This is quite a strong reason to argue that a pure element hollow nanosphere or nanotube will not shrink readily by the vacancy mechanism.

0 1 2 3 4 5 6 7 8 9 100

152.5 Å < r < 55.5 Å

Time, ns

0

149.5 Å < r < 52.5 Å

0

146.5 Å < r < 49.5 Å

0

143.5 Å < r < 46.5 Å

Mea

n Sq

uare

d D

ispl

acem

ent,

Å2

0

140.5 Å < r < 43.5 Å

34.5 Å < r < 37.5 Å

0

137.5 Å < r < 40.5 Å

0

1

2

Fig. 1. MSD of Pd atoms by the vacancy mechanism in the hollow nanosphere.

30 33 36 39 42 45 48 51 54 57 60 631.0

1.5

2.0

2.5

3.0

3.5

D(r

)×10

13, m

2 s-1

r, Å

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

1.5

2.0

2.5

(r-ri0)/(re0-ri0)

D(r

)/Dm

in

Fig. 2. Diffusion coefficient versus radius.

Acknowledgements: The support of the Australian Research Council is acknowledged.

References [1] Y. Sun, B. Mayers and Y. Xia, Adv. Mater., 15 (2003) 641. [2] K.N. Tu and U. Gösele, Appl. Phys. Lett., 86 (2005) 093111. [3] A.M. Gusak, T.V. Zaporozhets, K.N. Tu and U. Gösele, Phil. Mag., 85 (2005) 4445.

153

Effects of Superspreaders in Spread of Epidemic

Ryo Fujie, Takashi Odagaki

Kyushu University, Department of physics, Faculty of science, Fukuoka 812-8581, Japan, E-Mail: r.fujie@cmt.phys.kyushu-u.ac.jp

1. Introduction Spread of epidemic can be regarded as diffusion or random walk of disease on the

fixed discrete elements. Severe Acute Respiratory Syndrome (SARS) which spread around the world during 2003 is no exception. It is now believed that the sharp increase and decrease in the number of SARS patients was caused by the existence of superspreaders. According to WHO, the patients are defined as superspreaders if they infect more than 10 people. However, the cause why such superspreaders appeared is not yet clear. Possible origins of appearance of them might be (i) the superspreaders have constitutional or hereditary strong infectiousness or (ii) the superspreaders have many social connections. In this presentation, we investigate the effects of such superspreaders on diffusion process, and show that the basic reproductive number plays a critical role.

2. Model and Results In our study, we introduce two models as superspreaders in relation to two possible

origins of appearance of them [1]. We investigate the percolation probability, the propagation speed, the epidemic curve and the distribution of secondary cases on the basis of a random walk model.

We consider N individuals randomly distributed on an L×L continuous space who take one of three possible state; susceptible (S), infected (I) and recovered (R). Superspreaders are mixed in normal individuals group, and they are characterized through the infection probability w(r). We investigate two models for the superspreaders.

At first, we introduce the strong infectiousness model corresponding to the possible cause (i) whose infection probability w(r) is assumed as

(i)

⎪⎩

⎪⎨

⎧⎟⎟⎠

⎞⎜⎜⎝

⎛−=

<

≤≤

rr

rrrrwrw

0

00

0

0

1)( 0α

. (ii)

We set 2=α for the normal infection probability and 0=α for superspreaders (Fig. 1-i). As an alternative model corresponding to the possible cause (ii), we introduce the hub model whose infection probability w(r) is assumed as

Fig. 1: Distance dependence of infection probability w(r); (i) the strong infectiousness model, (ii) the hub model.

⎪⎩

⎪⎨

⎧⎟⎟⎠

⎞⎜⎜⎝

⎛−=

<

≤≤

rr

rr

h

hhrrwrw

0

1)( 02

0 .

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, R. Fujie 154

We set for the normal infection probability and 0rrh =

06 rrh = for superspreaders (Fig. 1-ii).

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

ρcπr

0

λ

(simulation)(R0=Rc)

Hub model (simulation)(R0=Rc)

Strong infectiousness model

2

From Monte Carlo simulation, we obtained the following results. (1) As the fraction of superspreaders is increased, the critical density of percolation transition shifts to the lower density. If all individuals are superspreaders for the strong infectiousness model, this epidemic threshold problem reduces to the question of overlap random disk percolation. We related the critical density of this case to the basic reproductive number R0 (the mean of the number of newly infected individuals resulting from a single infected), and obtained the relation between the critical density and the basic reproductive number for the system consisting of nomals and superspreaders and for the hub model (Fig. 2).

Fig. 2: Dependence of critical density on the fraction of superspreaders. The circles and the squares show the simulation results. The critical lines show R0=RcAbove the critical curves, the disease percolates, and below the curves, the disease does not percolate.

(2) Both the size of infection and the velocity for the hub model are lager than those for the strong infectiousness model. (3) The distributions of the number of links of the infection route network with superspreaders on both models show the feature of the distribution of secondary cases of SARS (large peak at zero and long tail) (Fig. 3).

Fig. 3: Distribution of the number of links on the infection route network, both models for the superspreaders.

3. Conclusion In this poster, we have studied the effects of

superspreaders in spread of epidemic. We introduced two models of superspreaders assuming two kinds of distance dependence of infection probability. From Monte Carlo simulation, we obtained the percolation probability as functions of the density for the different fraction of superspreaders. The percolation transition appears at the critical density which decreases as the fraction of superspreaders is increased. We showed that the critical density coincides with the density at the critical basic reproductive number R0=Rc. This result suggests that percolation transition can be understood by basic reproductive number R0 for any mixing ratio in binary mixed percolation system.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20the number of links

Strong infectiousness model

Hub model

Reference [1] R. Fujie, T. Odagaki, Physica A 374 (2005) 843-852.

155

Residence Times of Reflected Brownian Motion

Denis S. Grebenkov

Laboratoire de Physique de la Matière Condensée, CNRS - Ecole Polytechnique F-91128 Palaiseau, France, E-Mail: denis.grebenkov@polytechnique.edu

1. Introduction In the course of diffusive motion, each species explores different regions of the bulk.

Since reactive zones are often heterogeneously distributed in the bulk (e.g., in chemical reactor or biological cell), the net outcome and the whole functioning of the system strongly depend on how long the diffusing species remains in these zones. These so-called residence or occupation times are relevant for various diffusion influenced reactions e.g., energy transfer or fluorescence quenching [1]. In most cases, the motion of diffusing species is restricted by a geometrical confinement, resulting in drastic modifications of the transport. The resulting process is known as reflected Brownian motion [2]. Here, we propose a general solution to the problem of finding the probability distribution of its residence times [3].

2. Eigenmode expansion For a given function B(r) in a bounded domain Ω with a smooth boundary ∂Ω, we

consider the random variable ϕ = ∫0t ds B(Xs)

Xs being a random trajectory of the reflected Brownian motion in Ω, started with a given initial density ρ(r). Intuitively, the function B(r) can be considered as a distribution of “markers” to distinguish different points or regions of the confining domain. When the diffusing species passes through these regions, the random variable ϕ accumulates the corresponding “marks”. In other words, different parts of the trajectory are weighted according to the function B(r), “encoding” thus the whole stochastic process.

The classical Kac's result [4] allows one to relate the statistics of ϕ to a solution w(r, t) of the diffusion equation

[∂t – ∆ + hB(r)] w(r, t) = 0 (in the bulk) with the Neumann boundary condition ∂nw(r, t) = 0 at the boundary ∂Ω, the initial condition w(r, t=0) = ρ(r), and a positive constant h. The Laplace transform of the probability distribution of ϕ is then

E exp(–hϕ) = ∫Ω dr w(r, t) ρa(r) where the expectation includes the average of the functional exp(–hϕ) over all random trajectories Xs0≤s≤t of the reflected Brownian motion between the starting point r0 at time 0 and the arrival vicinity of point r at time t, as well as the average over all r0 and r with given initial and arrival densities ρ(r0) and ρa(r), respectively.

Expansion of the solution w(r, t) over the complete orthonormal basis of the Laplace operator eigenfunctions um(r) reduces the problem to a set of ordinary differential equations for the unknown coefficients cm(t). Thinking of cm(t) as components of an

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, D. Grebenkov 156

infinite-dimensional vector, one finds the solution of these equations that yields a compact matrix form of a scalar product for the expectation:

E exp(–hϕ) = (U exp[–(hB + Λ)t] U’) (1) where the infinite-dimensional matrices B and Λ and vectors U and U’ are

BBm,m’ = ∫Ω dr um*(r) B(r) um’(r) Λm,m’ = δm,m’ λm

Um = V1/2 ∫Ω dr um*(r) ρ(r) U’m = V-1/2 ∫Ω dr um(r) ρa(r) λm being the Laplace operator eigenvalues, and V the volume of the domain. Note that a formal substitution of h = –iq into Eq. (1) gives the characteristic function of ϕ. Its inverse Fourier transform yields the probability distribution of ϕ, while the series expansion of exp(iqϕ) generates all its moments.

If A is a subset of the confining domain Ω and B(r) is taken to be the indicator of A (IA(r) = 1 for r in A, and 0 otherwise), the random variable ϕ is the residence time showing how long the diffusing species resides onto this subset. The indicator function IA(r) can be thought of as a “counter” which is turned on whenever the diffusing species resides in A. One can estimate for instance the “trapping” time that particles spend in deep and almost enclosed pores (like fjords) of a catalyst.

The closed matrix form (1) is the central result. It provides a complete probabilistic description of the random variable ϕ. In addition, this matrix representation allows one to study the moments of ϕ in detail (not presented, see [3]). Most importantly, the present approach is efficient for numerical calculation of the residence times. The increase of eigenvalues λm enables one to truncate both matrices Λ and B to moderate sizes, allowing rapid and very accurate computation of the matrix exponentials in Eq. (1), in comparison to conventional methods like Monte Carlo simulations (see Ref. [2] for the use of similar matrix formalism in nuclear magnetic resonance).

3. Conclusion We proposed a general solution to the problem of finding the probability distribution

of residence times of a Brownian particle confined by reflecting boundaries. The Fourier and Laplace transforms of this distribution were derived in a compact matrix form involving the Laplace operator eigenbasis. When the eigenbasis (or its part) is known, the numerical computation of the residence time is straightforward and very accurate. The present approach can also be applied to investigate other functionals of reflected Brownian motion describing, in particular, restricted diffusion in an external field or potential (e.g., nuclei diffusing in an inhomogeneous magnetic field [2]). The developed concepts can be extended to more complicated stochastic processes governed by a general second-order elliptic differential operator having a complete eigenbasis.

References [1] G. H. Weiss, Aspects and Applications of the Random Walk (North-Holland,

Amsterdam, 1994). [2] D. S. Grebenkov, Rev. Mod. Phys. 79 (2007) in print. [3] D. S. Grebenkov, Phys. Rev. E, submitted. [4] M. Kac, Trans. Am. Math. Soc. 65 (1949) 1-13.

157

Surface Resistance to Heat and Mass Transfer in a Silicalite Membrane. A Non-Equilibrium Molecular Dynamics Study.

Isabella Inzoli a, Jean Marc Simon a,b, Signe Kjelstrup a.

a Department of Chemistry, Norwegian University of Science and Technology, Trondheim, Norway ( Email: isabella.inzoli@nt.ntnu.no)

b Institut Carnot de Bourgogne, UMR-5209 CNRS-Université de Bourgogne, 9 av. A. Savary, 21000 Dijon, France

1. Introduction Zeolites are microporous materials which are ideally suited for different industrial applications, e.g. as separators or catalysts. Their applicability is determined by the dynamical and the adsorptive properties of the molecules inside their micropores. Adsorption kinetics in zeolite has been explained as a two-step process, adsorption on the external crystal surface and subsequent inter-crystalline diffusion [1]. Recently much attention has been given to the resistance to transport of molecules at the external surface. It has been attributed to the crossing of potential barriers at the entry of the pores [1] or/and to the probability that a gaseous molecule sticks to the external surface [2]. Moreover, in the early 80’s, Ruthven suggested the presence of additional heat resistances at the external surfaces that slow down the kinetics of the exothermic adsorption processes [3]. Here, using non-equilibrium molecular dynamics simulations (NEMD), we give evidences for the presence of mass and heat resistances at the external surface of a silicalite-1 membrane when n-butane is adsorbed [4].

2. Model and simulation details The initial system was composed of an infinite membrane of silicalite-1 in contact with a gas of 250 molecules of n-butane. The external surfaces of the crystal were flat and normal to the straight channels (y directions). The silicalite was composed of 18 unit cells ([2 3 3]) and the butane molecules were modeled using a united atom model. Both were flexible and interacted with Lennard-Jones potentials. At time t=0 contact was made between the two phases and, by integrating the equation of motion, the gas molecules moved and part of them was adsorbed in the zeolite pores until equilibrium was reached. A velocity rescaling thermostat was applied on the gas phase at the end of the simulation box in order to mimic the real uptake-experiments. The heat released by the adsorption process first accumulated in the crystal and subsequently flowed to the gas phase where it was removed by the thermostat. More details can be found elsewhere [4].

3. Results and discussion The adsorption was followed until the equilibrium state and was characterized by two stages as shown in Fig.1. The first stage is governed by a large chemical driving force, which leads to a rapid uptake of butane, and the large enthalpy of adsorption (ΔH = -61.6 kJ/mol butane) explains the accompanying temperature rise in the crystal.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, I. Inzoli 158

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18 20 22 24 26Time / ns

Load

ing

/ mol

ecul

es p

er u

nit c

ell .

290

310

330

350

370

Tem

pera

ture

/ K

LoadingTemperature

Fig. 1 Loading and average temperature of the membrane as function of time

0 10 20 30 40 50 60 70 80 90 100Distance/ Å

380

360

340

320

300

280

Tem

pera

ture

/ K

1.5 - 2 ns3.5 - 4 ns6.5 - 7 ns

13.5-14 ns

0 10 20 30 40 50 60 70 80 90 100Distance/ Å

380

360

340

320

300

280

Tem

pera

ture

/ K

1.5 - 2 ns3.5 - 4 ns6.5 - 7 ns

13.5-14 ns

Fig. 2 Temperature profile as a function of the distance from the center of the zeolite during different periods of the second stage of the adsorption.

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14 16 18 20 22 24 26Time / ns

Load

ing

/ mol

ecul

es p

er u

nit c

ell .

290

310

330

350

370

Tem

pera

ture

/ K

LoadingTemperature

Fig. 1 Loading and average temperature of the membrane as function of time

0 10 20 30 40 50 60 70 80 90 100Distance/ Å

380

360

340

320

300

280

Tem

pera

ture

/ K

1.5 - 2 ns3.5 - 4 ns6.5 - 7 ns

13.5-14 ns

0 10 20 30 40 50 60 70 80 90 100Distance/ Å

380

360

340

320

300

280

Tem

pera

ture

/ K

1.5 - 2 ns3.5 - 4 ns6.5 - 7 ns

13.5-14 ns

Fig. 2 Temperature profile as a function of the distance from the center of the zeolite during different periods of the second stage of the adsorption.

In the second stage the chemical driving force is very low and the thermal driving force created across the external surface relaxes slowly to zero as shown in Fig. 2. A thermal conductivity of 3.4 x 10-4 W/K was found for the surface. As the temperature decreases the chemical potential of butane goes down, leading to a corresponding slow influx of butane. In order to study the effect of the structure of the external surface on the kinetics, additional simulations were performed considering a non-flat surface, which exhibited half zig-zag channels. The kinetics was characterized by a 25 % faster first step (smaller mass resistance), by no change in the relaxation time of the thermal driving force (second step), and by the same equilibrium state. The mass resistance is related to the probability of adsorption/ desorption of gas in zeolite. The better access to the pore openings for the second surface may thus explain the faster kinetics in this case. Heat conduction takes place by collisions, but in the absence of a net mass flux, so a constant thermal conductivity is understandable. The end state is a thermodynamic state that does not depend on the structure.

3. Conclusion In agreement with previous analysis based on experimental results [3], we found that the adsorption kinetics of n-butane on silicalite is characterized by two non-isothermal steps. The adsorption process appears to be limited by the relaxation time of the thermal force due to the presence of a large thermal resistance at the external surface. This may help explain why non-equilibrium techniques can give much smaller diffusion coefficients than equilibrium techniques [5]. We have also found that the nature of the external surface does not affect the kinetics of the heat exchange but only the mass transfer.

References [1] R.M. Barrer, J. Chem. Soc. Faraday Trans. 86, 1123 (1990). [2] A. Schuring, S. Vasenkov, S. Fritzsche, J. Phys. Chem. B, 109, 861 (1997). [3] D. M. Ruthven, L. K. Lee, and H. Yucel, AICHE J. 26, 16 (1980). [4] I. Inzoli, J. M. Simon, S. Kjelstrup, and D. Bedeaux, J. Col. Int Sci., accepted (2007). [5] J. Karger, Adsorption 9, 29 (2003).

159

Irreversible A + B → 0 Reaction – Diffusion Process with Initially Separated Reactants: Exponential Temporal Asymptotics

Slava Kisilevich1, Misha Sinder2, Joshua Pelleg2, and Vladimir Sokolovsky1

1Ben-Gurion University of the Negev, Physics Department, P.O.Box 653, Beer Sheva, 84105, Israel

2Ben-Gurion University of the Negev, Department of Materials Engineering, P.O.Box 653, Beer Sheva, 84105, Israel, E-mail: micha@bgu.ac.il

We study theoretically and numerically the irreversible A + B → 0 reaction-diffusion process of initially separated reactants occupying the regions of lengths LA, LB comparable with the diffusion length (LA, LB ~√ Dt, here D is the diffusion coefficient of the reactants). It is shown that the process can be divided into two stages in time. For t << L2/D the front characteristics are described by the well-known power law dependencies on time, whereas for t > L2/D these are well approximated by exponential laws. To confirm the obtained theoretical results, a numerical simulation of the reaction has been carried out. The simulation is based on the Monte-Carlo methods. The results of the numerical simulation are presented in Figs. 1 and 2. The numerical simulation properly confirms the obtained asymptotical temporal dependencies of the reaction front characteristics.

0 20 40 60 80 1000

500

1000

1500

2000

2500

(a)

BA

Cordinate, x

Rea

ctan

t Con

cent

ratio

n

1000 2000 3000 4000 5000 6000 7000 8000 9000

100

1000

10000

10 100 1000

100

1000

10000

(b)R (t) ~ e- 0.000493 t

Time

Tota

l Rea

ctio

n Ra

te, R

(t )

Tota

l Rea

ctio

n R

ate,

R(t

)

Time

R (t) ~ t - 0.506

0 2000 4000 6000 8000 100001

10

100

1000

10000

(c)

R f (t) ~ e- 0.000636 t

Time

Max

imum

Rea

ctio

n Ra

te, R

f (t)

0 2000 4000 6000 8000

1

(d)w(t) ~ e+ 0.000166 t

Time

Reac

tion

Zone

Wid

th, w

(t)

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, M. Sinder 160

Fig. 1. Numerical simulation of the reaction-diffusion process for the case of equal diffusion coefficients, D = DA = DB = 0.5 (the probabilities of unit jumps to the left and to the right are 0.5): the distribution of the particle density at t = 8000 (a); the temporal dependencies of the total (b) and maximum (c) reaction rates and of the reaction zone width (d). Insert in (b) presents the temporal dependence of the total reaction rate at t << L2/D. Reaction probability is 0.0001, the initial particle concentrations are 100,000. The numerically obtained constants in the exponent indexes agree with the theoretically predicted constant, which for the chosen parameters D and L is 0.000493.

0 10 20 30 401

10

100

1000

10000

100000

(a)

BA

Rea

ctan

t Con

cent

ratio

ns

Coordinate, x0 500 1000 1500 2000

100

1000

10 100 1000

100

1000

(b)R(t) ~ e - 0.00147 t

Tota

l Rea

ctio

n R

ate,

R(t

)

Time

R(t) ~ t - 0.55

Time

Tota

l Rea

ctio

n Ra

te, R

(t )

0 500 1000 1500 200010

100

1000

(c)

Rf (t) ~ e - 0.00150 t

Time

Max

imum

Rea

ctio

n R

ate,

Rf (t

)

0 500 1000 1500 20000.0

0.2

0.4

0.6

0.8

1.0

1.2

(d)Reac

tion

Zone

Wid

th, w

(t)

Time

Fig. 2. Numerical simulation of the reaction-diffusion process for the case of immovable B particles. The initial particle concentrations are: NB0 = 100,000, NA0 =50,000, and DB = 0, DA = 0.5. The distribution of the particle density at t = 2000 - (a); the temporal dependencies of the total (b) and maximum (c) reaction rates and of the reaction zone width (d). Insert in (b) presents the temporal dependence of the total reaction rate at t << L2/D. Reaction probability is 0.00001. In the asymptotic time regime LA is about 29 (see (a)). The numerically obtained constants in the exponent indexes agree with the theoretically predicted constant, which for the chosen parameters D and LA is 0.00147.

161

Kinetic Monte Carlo Study of Binary Diffusion in MFI-Type Zeolite

Nicolas Laloué(1,2), Catherine Laroche(1), Hervé Jobic(2), Alain Méthivier(1)

(1)Institut Français du Pétrole, BP 3, 69390 Vernaison, France (2)Institut de Recherche sur la Catalyse, 2 avenue A. Einstein, 69626 Villeurbanne, France

E-Mail: catherine.laroche@ifp.fr

1. Introduction Separation induced by kinetic effects has been investigated, using zeolites as a shape

selective adsorbent [1]. The control of diffusion process, requiring a precise knowledge of self-, transport and corrected diffusivities (Dself, Dt and DC, respectively) of molecules inside the nanoporous material, is essential for the design of industrial applications. In the case of strongly-binding or tight-fitting guest-zeolite systems, residence times in adsorption sites are much longer than travel times between sites so that diffusion becomes a slow activated process. Diffusion can then be viewed as successive random jumps from an adsorption site to another by crossing free energy barriers. Combining Kinetic Monte Carlo (KMC) algorithm and lattice model is the most adapted technique to study diffusion at finite loadings of poorly connected systems [2]. By its flexibility, it allows to consider different type of sites and to account for their local environment.

In this work, we report a study of the loading dependence of single-component diffusion of linear hexane (nC6) and of binary diffusion of nC6 and 2,2-dimethylbutane (22DMB) mixture in MFI-type zeolite at 300K, using a lattice model approach combined with KMC.

2. Kinetic Monte Carlo Simulation In this study, the silicalite framework is represented as a three-dimensional network of

intersecting straight (str) and zigzag (zz) channels. Molecules are randomly positioned on the lattice and can move from one site to the neighbouring site via hops, with a probability determined by transition rates estimated by ad hoc atomically detailed simulations [3,4]. Local molecular interactions are taken into account by modifying the magnitude of the free energy barrier according to the methodology developed by Reed and Ehrlich [5] and adapted for KMC applications with the introduction of a parameter f [6]. A standard KMC algorithm with variable time step is implemented to propagate the system. Dself is determined using the Mean Square Displacement (MSD) of the N individual particles. DC, also called Maxwell-Stefan diffusivity, is related to the displacement of the centre of mass of the N adsorbed particles. MSD calculations are performed using the order-n algorithm [7,8], mapping the variable time scale on a periodic time scale as done in Molecular Dynamics simulations. The source code developed here has been validated by quantitatively reproducing the results of the loading dependence of Dself and DC of CH4 in MFI on cubic and MFI lattices [6].

The loading dependence of single-component diffusion of nC6 in silicalite at 300K, has been investigated using a KMC model based on original anisotropic transition rates. In order to account for the molecule distribution within the zeolite framework, two

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, C. Laroche 162

different approaches have been developed to this model enclosing adsorption features via the introduction of the Langmuir constants of the DSL model. Moreover, guest-guest interactions are accounted for. The results of these KMC simulations for one of the approach developed are in good agreement with recent experimental QENS measurements [9], for both Dself and DC. The KMC model used for the study of binary diffusion at 300 K of nC6 and 22DMB mixture in silicalite is based on information coming from the single component investigation and accounts for the different saturation capacities of both species. It also allows for all type of molecular interactions to be considered by introducing fij parameters. As this model only accounts for nearest neighbours influence, the interactions between molecules of 22DMB are neglected. Results have shown a larger impact of interaction between linear and double-branched molecules than interactions between two linear molecules. Besides, the presence of 22DMB molecules in intersection sites has a significant impact on nC6 self- and corrected diffusivities compared to its behaviour as a single component. Near Θ22DMB,sat, the nC6 diffusivities sharply decrease towards 22DMB diffusivity values due to the reduction of vacant intersection sites.

22DMB

nC6

fnC6/22DMB

Fig. 1: 22DMB loading dependence of self-diffusion of nC6 and 22DMB.

3. Conclusion Kinetic Monte Carlo combined with a lattice model approach is used to simulate single-component and binary diffusion in an MFI-type zeolite. A good agreement is obtained with experimental behaviours of Dself and DC for nC6 diffusion. The binary diffusion study shows the impact of inter-species interaction, as well as site blocking effect. Deceleration/acceleration effect on molecules has also been confirmed.

References [1] Cavalcante C.L.J., Ruthven D.M., Ind. Eng. Chem. Res. 34 (1995) 185-191. [2] Keil F.J., Krishna R., Coppens M.O., Rev. Chem. Eng. 16 (2000) 71-197 [3] June R.L., Bell A.T., Theodorou D.N., J. Phys. Chem. 95 (1991) 8866-8878 [4] Dubbeldam D., Beerdsen E., Vlugt T.J.H., Smit B., J. Chem. Phys. 122 (2005)

224712 [5] Reed A.D., Ehrlich G., Surf. Sci. 102 (1981) 588-601 [6] Krishna R., Paschek D., Baur R., Microporous and Mesoporous Mater. 76 (2004)

233-246 [7] Frenkel D., Smit B., "Understanding Molecular Simulations : From Algorithms to

Applications", 2nd Ed, Academic Press, San Diego, USA (2002) [8] Auerbach S.M., Henson N.J., Cheetham A.K., Metiu I.H., J. Phys. Chem. 99 (1995)

10600-10608. [9] Jobic H., Laloué N., Laroche C., van Baten J.M., Krishna R., J. Phys. Chem. B 110

(2006) 2195-2201.

163

Modelling of Diffusion Saturation of (α+β) Titanium Alloy by Oxygen in Rarefied Gaseous Medium

Ya. Matychak, O.Yaskiv, V. Fedirko, I. Pohreljuk, O.Tkachuk

Physico-Mechanical Institute of National Academy of Sciences of Ukraine, 5, Naukova St., L`viv 79601, Ukraine, E-Mail: matychak@ipm.lviv.ua

Titanium (α+β) alloys are considered as promising structural materials for modern air-craft and space industries. Thermodiffusion saturation by oxygen under low partial pressure is one of the effective methods of surface strengthening of titanium alloys [1]. The problems of optimization of phase-chemical composition of alloys, gaseous medium pressure as well as scientific-founded choice of the temperature - time parameters for the process are not completely solved.

The aim of the present research is to create a model for diffusion saturation of (α+β) titanium alloy by oxygen under rarefied atmosphere with taking into consideration singularities of interaction on the interface and to estimate the effect of temperature and time of process.

The interaction between (α+β) titanium alloy and rarefied oxygen is schematically shown by the following processes with corresponding parameters (Figure 1): A (left): supply of molecules to titanium surface with, further, their adsorption, dissociation and chemisorptions (h is the coefficient of mass transfer, [cm/s]); B (centre): oxygen segregation on defects in contact layer (with mass capacity ω, [cm]) due to the chemical interaction with metal (reaction rate – k, [cm/s]; C (right): diffusion dissolution of interstitial elements in metal (D is the diffusion coefficient, [cm2/s]) [2].

Figure 1: Scheme of mass fluxes in the vicinity of an interface (α+β)Ti / O2.

It should be noted, that the processes mentioned in item A can be presented as two-stage reaction which includes diffusion stage described by rate constant hD and stage of irreversible reaction (chemisorption) with rate constant hR. Then, corresponding to summation of kinetic resistors 111 −−− +=

RDhhh , non-equilibrium processes mentioned

above in points A and B should be described by an equation of the mass balance on the interface using adequate definition of boundary condition [2].

Phase-structural aspect mostly determines the diffusion dissolution of oxygen in metal core (point C). The two-phase structure is a feature of (α+β) alloy. That means that there

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, O. Yaskiv 164

are the regions with the different oxygen dissolution potential ability as well as with the different diffusivity of impurity. For instance, oxygen solubility in α-phase (area 1) is higher than in β-phase (area 2). However, oxygen diffusion coefficient in β-phase is one order higher than in α-phase [1]. Therefore, even in the case of small volumetric quantity of β-phase, these areas could be considered as, in a sense, the paths for enhanced diffusion which correct the kinetics of strengthened diffusion zone formation.

For the formulation of the corresponding diffusion problem a phenomenological approach was applied. This approach is based on a model of continuous spectrum in a medium consisting of two diffusion paths. The quasi-chemical reaction of transfer by oxygen [ ] [ ]αβ OO ⇔ with corresponding coefficients (k1 and k2), which characterize the rate of such transfer, takes place between these channels [3]:

,/,/ 22112222211111 CkCkCCDCkCkCCD +−∂∂=Δ−+∂∂=Δ ττ (1) Here, Ci and Di (i =1, 2) are the concentration and the oxygen diffusion coefficient in the corresponding α and β areas. When we introduce the concentration C=C1+C2, the system of equations (1) can be reduced to a “non Fickian” differential equation of high order (fourth order following a spatial value) [3]. Assuming local dynamic equilibrium, such an equation attains a Fickian form, in which the effective diffusion coefficient of oxygen (Deff) is presented by the relation:

τ∂∂=Δ /CCDeff , where )/()( 211221 kkkDkDDeff ++= (2)

As the object of investigation the half-space (0≤x<∞), with the initial (τ=0) oxygen concentration C=C0 and the following boundary conditions, was chosen [2]:

difJRJJddC −−=)/( τω for 0+=x , 0CC = for +∞=x . (3)

It should be noticed, that the non-stationary boundary condition, Eq. 3, reflects the mass balance on interface (Fig.1). The difference between transport flux of oxygen

( ) and diffusion flux jJ 0−=x diff into the metal determines the kinetics of oxygen segregation on the interface caused by chemical reaction. The segregations take place in the contact layer on defects prototyped as “traps” for the diffusant [2]. The solution of diffusion problem Eq. 2, Eq. 3 was obtained in an analytic form.

Based on the obtained solution, the regularities and peculiarities of kinetics of diffusion saturation of (α+β) titanium alloys by oxygen have been illustrated. They correlate well with experimental results. It was shown that the depth of strengthened diffusion zone increases with increment of volumetric quantity of β-phase. The effect of temperature and time on the kinetics of the process is estimated. The peculiarities of oxygen redistribution on the interface in dependence on temperature of isothermal exposure are explored. It is established that, for the same duration of the process, an increase of temperature does not always tend to increase the impurity concentration on the interface.

References [1]. Fedirko V. M. et al., Materials Science, 41, 2 (2005) 208-216. [2] Prytula A., Pohreljuk I., Vedirko V., Matychak Ya., Defects and diffusion forum,

237-340 (2005) 1312-1318. [3] Aifantis E.C. Acta Met., 27 (1979) 683-691.

165

Diffusion of Water Molecules in Narrow Carbon Nanotubes and Nanorings

Biswaroop Mukherjee, Prabal K. Maiti, Chandan Dasgupta, A. K. Sood

Department of Physics, Indian Institute of Science, Bangalore 560 012, India. cdgupta@physics.iisc.ernet.in

Introduction: It is known that fluids confined in nanometric spaces behave very differently from their bulk counterpart [1]. Lately, the process of transport of confined water through narrow channels has attracted a lot of attention, since knowledge of these processes is important in understanding transport through ion channels [2]. Numerical studies by Lee et al. [3] and Mao et al. [4] of the diffusion of oxygen and methane molecules, respectively, through narrow carbon nanotubes suggest the occurrence of single file diffusion. However in all these simulations, the time over which the mean squared displacement (MSD) is measured is rather short (100-500 ps). Additionally, due to finite size effects (the consequences of which are elucidated in the present work), it is difficult to draw a firm conclusion about the true nature of diffusion of these molecules inside the nanotube. The goal of our study is to develop a better understanding of the nature of diffusion of water molecules in narrow carbon nanotubes. Results: We have used [5] extensive atomistic molecular dynamics (MD) simulations to study the structure and dynamics of water molecules in narrow carbon nanotubes immersed in a bath of water [6], and in isolated carbon nanorings. The diameters of the tubes and rings are chosen to be such that only a single file of water molecules is allowed inside. The water molecules inside the nanotube show solid-like positional ordering at room temperature, which we quantify by calculating the pair correlation function. This behavior is a consequence of the formation of strong hydrogen bonds between neighboring water molecules inside the nanotube. Our studies show that even for the longest observation times, the mode of diffusion of the water molecules inside the nanotube is Fickian (normal) and not sub-diffusive. The MSD initially increases linearly with time and then saturates at long times (see Fig.1). This is a finite-size effect, arising from the fact that the MSD is measured only for the molecules inside the finite-sized open-ended nanotube. We propose a one-dimensional random walk model for the diffusion of the water molecules inside the nanotube. Two versions of the random walk model are considered, one in which the time is treated as a continuous variable, and another, in which time is discrete, since MD observations are made at discrete times. We find good agreement between the MSD calculated from both versions of the random walk model with that obtained from MD simulations, with the discrete version faring slightly better (Fig. 1). This confirms that water molecules undergo normal-mode diffusion inside the nanotube. We attribute this behavior to strong positional correlations that cause all the water molecules inside the nanotube to move collectively as a single object. We also measure the survival probability of the water molecules inside the nanotube from MD simulations and calculate it from the random walk models. There is good agreement between the simulation results with those obtained analytically from the random walk models.

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© 2007, C. Dasgupta 166

These results suggest that single file diffusion in this type of systems can be observed only when the molecules inside the channel form several clusters. In order to achieve this, we have performed simulations of water molecules inside narrow carbon rings that are partially filled with water. At intermediate fillings (between 10 to 50 %), we find that the water molecules inside the ring form two oppositely polarized clusters, each containing about twenty molecules. These clusters behave as single “particles” because of the strong hydrogen bonding discussed above and they repel each other due to electrostatic interactions. In this situation and at even lower fillings where the confined molecules form a gas-like phase, we find evidence for the occurrence of single file diffusion with the MSD scaling as the square root of time (Fig. 2). Normal diffusion is found when the molecules inside the ring form a single cluster.

Fig.1. The MSD of water molecules inside a nanotube, from MD and random walk models.

Fig.2. The MSD for bipolar water clusters inside a carbon nanoring.

Conclusions: Our results show that in order to observe single file diffusion (such as that observed, for example, in experiments [7] on confined colloidal particles) of water molecules in narrow carbon nanotubes and nanorings, the system parameters must be such that the confined molecules form several clusters. Normal (Fickian) diffusion is found when the confined water molecules form a single, tightly bound cluster due to the formation of hydrogen bonds. References: [1] R. Zangi, J. Phys. Condens. Matter 16 (2004) S5371. [2] E. Tajkhorshid, P. Nollert, M. O. Jensen, L. J. W. Miercke, J. O'Connell, R. M. Stroud, K. Schulten, Science 296 (2002) 525. [3] K. H. Lee, S. B. Sinnott, Nano Letters 5 (2005) 793. [4] Z. Mao, S. B. Sinnott, J. Phys. Chem. B 104 (2000) 4618. [5] B. Mukherjee, P. K. Maiti, C. Dasgupta, A. K. Sood, J. Chem. Phys. 126 (2007) 124704. [6] G. Hummer, J. C. Rasaiah, J. P. Noworyta, Nature 414 (2001) 188. [7] C. Lutz, M. Kollmann, C. Bechinger, Phys. Rev. Lett. 93 (2004) 026001.

167

Modeling of Surface Diffusion for Stepped Surfaces: Transfer Matrix Approach

),ba Alexander V. Myshlyavtsev, Marta D. Myshlyavtseva )a

)a Omsk State Technical University, Omsk, Russia,e-mail: myshl@omgtu.ru

)b Institute of Hydrocarbons Processing SB RAS, Omsk, Russia

1. Introduction Surface diffusion is of considerable intrinsic interest and also important for

understanding the mechanism of surface reactions [1,2]. It is well known that diffusion of chemisorbed particles occurs, as a rule, via jumps between nearest- neighbor sites and can be rather accurately described in the frameworks of the transitient state theory and lattice-gas model. For simplicity we assume that the tops of diffusion potential barriers are the same for all lattice sites. In this case, the chemical diffusion coefficient can be expressed as

001)/exp()0()( P

RTRTDD

θμμθ∂∂

= , (1)

where D(0) is the chemical diffusion coefficient at the low coverage, μ is the chemical potential of adparticles, R the universal gas constant, T the absolute temperature in K, θ the surface coverage, the probability to find an empty couple of the nearest neighbor

site [2]. To calculate the chemical diffusion coefficient the cluster approximation [1] or Monte Carlo simulation [2] are usually employed. It was shown that this problem can be also solved by the transfer matrix technique [3]. In this paper we employ the latter approach for study of the simplest model of stepped surface.

00P

2. Model of surface and results

The simplest model of stepped surface was considered in [4]. This model is shown in Fig. 1. The

width of terraces is noted as L. We assume that adsorption sites belonging to each first row are represented by a potential well with the depth Δ. We take into account the nearest – neighbor lateral interactions as well. The Hamiltonian for the developed model can be written as

Fig. 1: The model of surface.

∑∑∑ Δ−−=>< k

ki

inn

ji nnnnH με , (2)

where ε is the lateral interaction energy, the occupation number for ith lattice site. Here, <nn> means summation over all pairs of the nearest lattice sites. In the second term

in

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© 2007, A. Myshlyavtsev 168

one takes the summation over all lattice sites. In the third term one takes the summation over the sites belonging to the first rows (the rows are oriented along the X axis).

As the model under consideration is anisotropic the probability is also anisotropic. In this case we should use the following equations

00P

∑∑∑∑ Δ−−+=><>< k

ki

inn

jiynn

jix nnnnnnHyx

μεε , (3)

θε

211,00 −∂Ω∂

+=x

x RTP , (4)

where Ω is the grand thermodynamic potential. It has been calculated by the transfer matrix method. Thus the equations (1), (4) allow to obtain the coverage dependence of the chemical diffusion coefficient for the simplest model of stepped surface. The obtained results are shown in Fig. 2. The upper curves from each pair correspond to the X axis. The main feature is the independence of the anisotropy of the chemical diffusion coefficient at large coverage from the ratio Δ/ε. Notice, that for the special case L = 2 this anisotropy has strong dependence on the ratio Δ/ε.

Fig. 2: Chemical diffusion coefficient

3. Conclusion 1. The transfer matrix method is an efficient tool in studying of the chemical diffusion

for heterogeneous lattice models with 2D-translational groups. 2. It was shown in the framework of the model under consideration that the chemical

diffusion coefficient is essentially anisotropic at large surface coverage. 3. It was shown that for L = 2 the chemical diffusion coefficient anisotropy has strong

dependency on the ratio Δ/ε and the one is independent from this ratio for L ≥ 3.

References [1]. V.P. Zhdanov, Elementary Physicochemical Processes on Solid Surfaces, Plenum, New York, 1991. [2]. K. Binder, D Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction, Springer, Berlin, 2002. [3]. A.V. Myshlyavtsev, A.A. Stepanov, C. Uebing, V.P. Zhdanov, Phys. Rev. B. 52 (1995) 5977-5984. [4]. A.V. Myshlyavtsev, M.D. Myshlyavtseva, Appl. Surf. Sci. 253 (2007) 5591-5597.

169

Kinetic of Void Growth in fcc and bcc Metals.

Andrei V. Nazarov1,2, A.A.Mikheev3, I.V.Valikova1, Aung Moe1, A.G.Zaluzhnyi1,2

1Moscow Engineering Physics Institute (State University), Kashirskoe shosse, 32, 115409, Moscow, RUSSIA, 1avn46@mail.ru

2Institute of Theoretical and Experimental Physics, Moscow, RUSSIA

3Moscow State University of Design and Technology, RUSSIA

1. Introduction

We examine how elastic stress, arising from voids, influences the diffusion vacancy fluxes and growth of voids in cubic metals. Usually, the equation of diffusion in the presence of stress field has the following form [1]:

⎟⎠⎞

⎜⎝⎛ ∇

+∇−=kT

UccDJ , (1)

where U is an interaction potential of the diffusing atoms with the defects generating stress fields. Some authors consider point defects as the centers of dilatation [2]. In this approximation:

σπβ SprU 303

4= (2)

where r0 is the radius of the matrix atom, r0(1 + β) is the effective radius for impurity atom or a defect and Spσ is the trace of stress tensor. Note that in a case of elastic stress, arising from void, ConstSp =σ , and a second term in Eq.(1) equals zero [2]. Consequently, the elastic stress, arising from voids, does not influence the diffusion vacancy fluxes and growth of voids. This queer result is a sequent of the Eq.(1). In particular therefore some authors considered a possibility of generalization of this equation [3-5]. The aim of our work is to examine the elastic stress influence on diffusion flux of vacancies using the approach, developed by us earlier [4] and to obtain a kinetic equation for the growth rate of voids in cubic metals

2. Main moments of theory of diffusion under stress and simulation of void growth This approach takes into consideration, that the strains can alter the surrounding atom

configuration near the jumping one and consequently the local magnitude of the activation barrier. Knowing this change, we can calculate the atomic jump rate and obtain an equation for the vacancy flow [4]. In this case, the vacancy flux depends on the matrix of the diffusion coefficients. Each of these coefficients depends on the strain tensor components in a nonlinear way. In corresponding nonlinear equations, the functional

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© 2007, A. Nazarov 170

dependence on strain is determined by coefficients, which are the main characteristics of the strain influence on diffusion (SID coefficients). These coefficients are very sensitive to atomic structure in the nearest vicinity of defect and still more to atomic structure of the saddle-point configuration. We have built an advanced model to evaluate them. SID coefficient simulation is the first step of this work.

Then we used the results of theory of elasticity concerned with the displacement field around a sphere and obtained the equation for vacancy flux, where the influence of elastic stress near the void on this flux was taken into account.

Next step is the calculation of the dependence of the diffusion coefficient on the coordinates near the different size voids. To analyze these complicated dependences we used visualization.

The diffusion equation for vacancies in which the influence of elastic stress near the void on flux was taken into account is linearized and resolved. The obtained kinetic equation for the growth rate of voids contains the additional terms conditioned by strains, arising from voids. These terms change the kinetics of void growth (See Fig.1). Using this kinetic equation, we simulate void growth in fcc and bcc metals for different temperatures and vacancy supersaturation.

-0,008

-0,006

-0,004

-0,002

0

0,002

0,004

0 100 200 300 400 500 600 700 800

dRp/d

t LS

Our results

Fig. 1. Growth rate of voids versus their radius (A) in Cu. (T=250 C).

3. Conclusion We have developed the model of growth rate of voids taking into consideration the elastic stress, arising from voids. It is shown that this elastic stress can fundamentally alter the kinetics of void growth in fcc and bcc metals

References

[1] P.G. Shewmon, Diffusion in Solids, McGraw-Hill Company, 1963. [2] B.Ya. Lybov, Kinetic Theory of Phase Transformations, Metallurgiya, Moscow, 1969. [3] J. Philibert, Metal Phys. and Adv. Technologies, .21, (1999) N1, 3-9. [4] A.V. Nazarov, A.A .Mikheev, Def. and Dif. Forum, 143-147 (1997) 177-185. [5] A.V. Nazarov and A.A. Mikheev, Physica Scripta, T108, (2004) 90-95.

171

The Effect of the Dislocation Elasticity on the Thermal Motion of Attached Particle

Sergei Prokofjev1,2, Victor Zhilin1, Erik Johnson2,3, Ulrich Dahmen4

1 Institute of Sold State Physics RAS, 142432 Chernogolovka, Russia 2 Nano Science Center, NBI, University of Copenhagen, Copenhagen, Denmark 3 Materials Research Dept., RISØ National Laboratory, Roskilde, Denmark 4 National Center for Electron Microscopy, LBNL, Berkeley, CA, USA

E-Mail: prokof@issp.ac.ru

1. Introduction Results of our recent in-situ TEM studies of the thermal motion of liquid Pb nanoparticles attached to fixed dislocations in Al matrix indicate that their motion is affected by a dislocations elasticity [1,2]. Analysis of this effect is presented here. It justifies the method used to determine diffusion coefficients of particles attached to dislocations [1-3].

2. Elastic action of dislocation on the thermal motion of attached particle In-situ TEM studies show that nanoparticles of liquid Pb trapped by fixed dislocations

in Al matrix oscillate in the vicinity of the dislocation lines [1,2], see Fig. 1. It can be explained assuming that the dislocations act on the attached particles as an elastic string, Fig. 2.

10 nm

Fig. 1: Trajectory of 15 nm trapped particle recorded at 722 K for 50 seconds. The trajectory is elongated in the direction of dislocation line.

F

0

ρ

Z

P

L-L z

ρ

Fig. 2: Action of fixed dislocation segment on attached particle. Point P is its center.

Indeed, transverse displacements ρ of the particle caused by thermal fluctuations produce an elastic restoring force (the arrow F in Fig. 2) due to the linear tension of the dislocation. A projection of this force on the dislocation line (z-axis) causes a repulsion of the particle from closer end of the dislocation. That explains the observed oscillatory motion of the attached liquid Pb particles [1,2]. According to Fig. 2, the displacement of the particle to a point (ρ, z) causes an increase in the dislocation energy

( )L)zL()zL(UU o 22222 −+−+++= ρρΔ , where 2L is the length of the dislocation, Uo is the energy of unit length of the dislocation. ΔU = ΔU(ρ, z) defines the elastic energy field determining the elastic restoring force F = -grad(ΔU). Assuming an axial symmetry,

zU

z)z,(fU)z,(fff

zzUU

zz ∂∂

=∂

∂=−−=

∂∂

−∂

∂−=

Δρρ

Δρ

ρΔρρ

Δρρ

1 and 1 where,or , zρFzρF

are the force constants of transverse and longitudinal constituents of the oscillatory motion, respectively. In the assumption (ρ/L)2 << 1

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© 2007, S. Prokofjev 172

(1b) 1

2 and (1a)

12

222

2

2 )(LLU

f)(L

Uf o

zo

λρ

λρ −≅

−≅

Here λ = z/L. As (ρ/L)2 << 1, then, fρ >> fz , and a frequency of transverse oscillations is much higher than that of longitudinal oscillations. Therefore, the longitudinal motion can be averaged over the transverse oscillations, i.e. ρ2 in Eq 1b can be replaced by its average value <ρ2> = 2kT/fρ (fρ doesn’t depend on ρ, see Eq. 1a, then, the transverse oscillations are harmonic, and mean potential energy of the particle is equal to its mean kinetic energy kT). Thus, Eq. 1b transforms to

)(LkTfz 22 1

2λ−

≅ . (2)

Equation 1a and Eq. 2 show that fρ is determined by the dislocation elastic energy UoL, and fz is governed by the energy of thermal motion kT, then, fρ >> fz as the energy of the dislocation elasticity is much larger than the thermal energy.

As attached particles are located mostly in the middle part of dislocations, i.e. λ2 << 1 is good enough fulfilled, then, fz º 2kT/L2 = const, Eq. 2. 1D motion of a Brownian particle in the harmonic potential was considered by Smoluchowski, who had obtained the dependence of mean squared displacement <Δz2> of the particle from its initial position on elapsed time Δt:

)]/tDexp([z zpz222 12 σΔσΔ −−>=< , (3)

where σz2 = kT/fz , and Dp is the diffusion coefficient of the particle [4,5]. This equation

allows to determine Dp from the 1D trajectory z(t) of the particle. Eq. 3 was used to study the mobility of liquid Pb particles in Al as a function of their size and temperature [1-3].

3. Conclusion The effect of dislocation elasticity on the thermal motion of a particle, attached to it,

is considered analytically. The results of the analysis agree good enough with our experimental observations. It follows from the analysis that the Smoluchowski equation for 1D motion of Brownian particle in the harmonic potential can be used to determine the diffusion coefficients of particles attached to dislocations.

The work was supported by the Russian Foundation for Basic Research (Project 05-03-33141), the Danish Natural Science Research Council, and the Director, Office of Basic Energy Sciences, Materials Science Division, US Department of Energy, under contract DE-AC3-76SF00098.

References [1] S. Prokofjev, V. Zhilin, E. Johnson, M. Levinsen, U. Dahmen, Def. Diff. Forum

237-240 (2005) 1072-1077. [2] E. Johnson, S. Prokofjev, V. Zhilin, U. Dahmen, Z. Metallk. 96 (2005) 1171-1180. [3] S. Prokofjev, V. Zhilin, E. Johnson, U. Dahmen, Def. Diff. Forum 264 (2007) 55-61 [4] M. Smoluchowski, Bull. Int. de l’Acad. de Cracovie, Serie A (1913) 418-434. [5] M. Smoluchowski, Sitzungsber. Kais. Akad. Wissensch. Wien (IIa) 123 (1914)

2381-2405.

173

Diffusional Atomic-Ordering Kinetics of Close-Packed Solid Solutions: Models for L12 and D019 Phases

Taras Radchenko1, Valentyn Tatarenko1, Hélèna Zapolsky2

1 Department of Solid State Theory, Institute for Metal Physics, N.A.S. of Ukraine, 36 Vernadsky Blvd., 03680 Kyiv-142, Ukraine, E-Mail: Taras.Radchenko@gmail.com

2 Institut des Materiaux, UFR Sciences et Techniques, Université de Rouen, Avenue de l’Université—B.P. 12, 76801 Saint-Etienne du Rouvray, France

1. Introduction Ordering in solid solutions is a topic of growing technological and scientific interest

within the field of material science. This is because of the advantageous high temperature and corrosion properties of alloys which are linked to the effect of long-range ordering [1]. Long-range order kinetics is one of the diffusion processes occurring within the atomic ranges. X-ray diffraction technique is the most convenient experimental instrument to provide us detailed information on this process. There are also different theoretical methods for this one. One of the models is based on the Onsager-type microscopic-diffusion equation [1].

2. Kinetics model To investigate the kinetics of the diffusional ordering process in close-packed, i.e.

face-centered cubic or hexagonal close-packed, lattices, we take a model based on the Onsager-type microscopic-diffusion equation [1]. For the exchange substitutional diffusion mechanism in A1−cBBc solid solution, the single-site occupation-probability function, P(R,t), is proportional to the thermodynamic driving force, δF/δP(R′,t):

( , ) ( )( , )

dP t FLdt P t′

δ′∝ − −′δ∑

R

R R RR

.

Here, t is a time, L(R – R′) is the kinetic coefficient representing the exchange probability of an elementary diffusional jump between a pair of atoms from site R of to R′, and vice versa, during the time unit, F is a configurational free energy.

Using the above-mentioned equation, the self-consistent field (mean-field) approximation along with the static concentration waves’ approach, an equation for the time dependence of long-range order parameter, η, in the L12 and D019 types structures is as follows:

( 3 4)(1 4)( ) ln* (1 3 4)( 4)

B B

B B

c cd Ldt T c c

⎛ ⎞+ η − + ηη η= − +⎜ ⎟− − η − η⎝ ⎠

k ,

where T* is a reduced temperature, is the Fourier transform of the Onsager-type kinetic coefficients, and k is a superlattice wave vector generating the L1

( )L k2 or D019 types

superstructures, cB is an atomic fraction of B atoms.

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© 2007, T. Radchenko 174

Fig. 1: The reduced-time dependences of L12- and D019-type LRO parameters for different atomic fractions of alloying component at the reduced temperatures T* = 0.12 (left) and T* = 0.20 (right).

Fig. 2: Equilibrium long-range order parameter vs. atomic fraction of alloying component for L12 and D019 types phases at different reduced temperatures.

3. Results and conclusions The last equation permitted obtaining the time dependence of LRO for a wide

temperature–concentration range (Fig. 1). The curves (showing the order-parameter relaxation kinetics) are also obtained for both temperature-independent interaction parameters and temperature-dependent ones. The kinetics results confirm the statistical-thermodynamics ones (Fig. 2). Firstly, equilibrium LRO parameters, ηeq, coincide within the frameworks of both models. Secondly, for the non-stoichiometric alloys (where an atomic fraction of alloying component B is more than cB

st = 0.25), the equilibrium LRO

parameter can be higher than it is for stoichiometric ones at high temperatures (Figs. 1 and 2). The experimental phase diagrams confirm the predicted (ordered or disordered) states for close-packed f.c.c. Ni–Fe and h.c.p. Ti–Al alloys.

References [1] A.G. Khachaturyan, Theory of Structural Transformations in Solids, John Wiley &

Sons, New York, 1983.

175

Diffusion of n-Pentane in Zeolite ZK5

Oraphan Saengsawanga, Andreas Schüringa,b, Ton Dammersc, David Newsomec, Marc-Olivier Coppensc,d, Siegfried Fritzschea

aUniversität Leipzig, Institut für Theoretische Physik, Postfach 100920, 04009 Leipzig, Germany, E-mail: oraphan@pcserv.exphysik.uni-leipzig.de

bUniversität Leipzig, Institut für Experimentelle Physik I, Linnéstr. 5, D-04103 Leipzig, Germany cDelft University of Technology, Delft Chem Tech, Physical Chemistry and Molecular

Thermodynamics, Julianalaan 136, 2628 BL Delft, The Netherlands dRensselaer Polytechnic Institute, Howard P. Isermann Department of Chemical and Biological

Engineering, Ricketts Building, 110 8th Street, Troy, NY12180, U.S.A.

1. Introduction Zeolite ZK5 has an interesting framework topology (Fig. 1) which contains two diffe-

rent types of cages, gamma (γ) and alpha (α) cages, connected via eight-membered oxygen rings with a free diameter of 3.9 Å. Recently, Magusin et al. [1] have investigated n-pentane in zeolite ZK5 using 1D- and 2D-exchange 13C NMR techniques. Diffusion

takes place by rare jumps between neigh-bouring cages. For an interpore distance of ~10 Å these hopping rates correspond to intra-zeolite self-diffusion coefficients between 10-18-10-15 m2s-1 [1]. Though the methyl and methylene groups fit through the eight-membered oxygen ring, the diffusion of n-alkanes with n>2 is known to be much slower than for methane and ethane [2]. This results from the C-C-C bond angle. The slow exchange of the guest molecules between the neighbouring cages through the eight-membered oxygen ring is

in the center of this work. We study the system using molecular dynamics (MD) simulations and transition-path sampling (TPS). For the first method, the determination of diffusion coefficients is in the considered case only feasible at high temperature. The Arrhenius law is used to extrapolate the diffusion coefficient to lower temperatures. TPS allows the calculation of transition rates also at lower temperatures.

Fig. 1: KFI topology type of framework structure viewed along [001] and schematic illustration of half of the cubic pore structure. The α and γ cages are indicated by large and small circle, respectively [1].

2. Model The united-atoms approximation is used in which the CH3 and CH2 groups are treated

as spherical force centers. They interact with the corresponding force centers in other pentane molecules and with the zeolite lattice atoms by Lennard-Jones pair potentials [3]. The loading is set to one molecule per cage according to the experimental conditions in Ref. [4]. The lattice is assumed to be rigid whereas the pentane molecule is flexible. C-C bonds are modeled by harmonic potentials. The self-diffusion coefficient has been calculated from the mean square displacement using the momentum method proposed in

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© 2007, O. Saengsawang 176

ref. [5]. The determination of hopping rates between the cages by use of the TPS method is subject of ongoing work. MD simulations are carried out at high temperature, T. Using the Arrhenius equation, D = D0 exp (-Ea /RT), where D0 is a pre-exponential factor [6], the activation energy, Ea, can be computed and diffusion coefficients at lower temperatures can be extrapolated.

3. Results and Discussion In the previous work, the rotational motion of n-pentane in the γ-cages of zeolite ZK5 has been studied. In addition, the effect of confinement on molecular orientation has been investigated by the distribution of intramolecular vectors of the pentane molecule. Fig. 2

shows some configurations of the molecule trying to escape from the γ-cage to a neighbouring α-cage at high temperature. An energetic barrier occurs in the transition state between two cages which is formed by the eight-membered oxygen ring. The experimental value of the activation energy, Ea=28 ± 5 kJ mol-1 was obtained from 1D- and 2D-exchange 13C NMR measure-ments in ref. [1]. From MD simula-tions, the activation energy can be explicitly obtained by fitting the

temperature dependence of the diffusion coefficient at high temperature to the Arrhenius equation. In preliminary results, the fitted activation energy value of 57 ± 5 kJ mol-1 overestimates the experimental value ranging 28 ± 5 kJ mol-1. The reason could be the constraints of the lattice during simulations or the interaction parameters used. The extra-polated self-diffusion coefficient at 300 K is on the order of magnitude of 10-18 m2 s-1, which approximately agrees within the range of values (10-18- 10-15 m2 s-1) concluded from the experiment [1].

Fig. 2: Orientations of the molecule in the cagerepresented by projection of the intramolecular vectorsC15 (end-to-end) at low (250 K) and high (450 K) temperature, respectively. Each dot represents oneorientation registered from the trajectories.

4. Conclusions Extremely slow diffusion in zeolites is explored by MD simulations at high temperature and transition-path sampling. In first results for n-pentane in zeolite ZK5, good agreement is found between the values extrapolated using the Arrhenius law and the experimentally determined values from Ref [1].

References [1] P. C. M. M. Magusin, D. Schuring, E. M. van Oers, J. W. de Haan, R. A. Santen, Magn. Reson. Chem.

37 (1999) 108. [2] W. Heink, J. Kärger, H. Pfeifer, P. Salverda, K. P. Datema, A. Nowak, J. Chem. Soc. Faraday Trans. 88

(1992) 515. [3] A. Loisruangsin et al. to be published. [4] W. J. M. van Well, J. Jänchen, J. W. de Haan, R. A. van Santen, J. Phys. Chem. B 103 (1999) 1841. [5] S. Fritzsche, R. Haberlandt, J. Kärger, H. Pfeifer, M. Wolfsberg, K. Heinzinger, Chem. Phys. Lett. 198

(1992) 283. [6] C. Bussai, S. Fritzsche, R. Haberlandt, S. Hannongbua, J. Phys. Chem. 108 (2004) 13347.

177

The Probability that a Molecule Enters a Porous Crystal Andreas Schüring

Institut für Theoretische Physik and Institut für Experimentelle Physik I, Universität Leipzig, Vor dem Hospitaltore 1, D-04103 Leipzig,

E-Mail: schuring@uni-leipzig.de

1. Introduction What is the probability that a molecule on the surface of a porous zeolite crystal will

be able to enter the intracrystalline space? This question is of importance for practically all applications of zeolites and is often connected with the term “surface barrier”. The flux of particles from the gas phase arriving at the surface is reduced by this entering probability. Separation processes make use of the fact that too big molecules can not pass the narrow pores while a smaller species easily reaches the region of strong adsorption inside the crystal. When zeolites act as catalysts, a fast access of the reactants to the pores is crucial for the catalytic performance. It is desired to estimate the entering probability since it is an important parameter for industrial processes involving zeolites. This contribution presents an analytical formula to calculate the entering probability through the intracrystalline self-diffusion coefficient in the case of an ideal crystal surface [1].

2. Theory

The entering probability (sometimes named sticking probability) is defined as the ratio between the number Nin of entering and the number Nenc of molecules encoutering the surface of the porous crystal, hence

enc

inenter N

NP = . (1)

The number of molecules passing a barrier is proportional to the number of molecules available. The proportionality coefficients are transition rates, which can be calculated using transition-state theory (TST). As such, Eq. (1) may be transferred into a ratio of transition rates, respectively, as a ratio between configuration integrals in the transition states. The intracrystalline self-diffusion coefficient is given in terms of transition rates as

Figure 1: The poten-tial energy landscape in the transition region gas-adsorbent.

(2) )/exp( Bdes

22D TkEkLLkD Δ==

Between the rate of desorption, kdes, and the rate of intracrystalline diffusional jumps, kD, a Boltzmann factor with kB being Boltzmann’s constant and T the temperature, has to be considered, which takes into account for a change in the (free) energy in the pore region (see Fig. 1). The equilibrium constant, K, which is the ratio between the concentrations in the zeolite and the gas phase may be written also as the ratio between the configuration integrals in these regions.

B

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© 2007, A. Schüring 178

Using these relations, we may transform Eq. (1) into a relation which contains as variables the intracrystalline self-diffusion coefficient, D, the equilibrium constant, K, and the energy difference ΔE, so that we obtain

1

B

Benter )/exp(

2/1

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ−+=

TkEKDmTkL

Pπ

. (3)

3. Results from MD simulations

Fig. 2: The entering probability as a function of the self-diffusion coefficient.

PenterAll quantities occuring in Eq. (3) can be obtained independently from the MD simulations. Penter is calculated directly from the molecular trajectories using Eq. (1). ΔE is accessible in MD simulations from the potential energy landscape. In the considered case, ethane as diffusing molecule and zeolite Si-LTA as a host, ΔE was practically zero and had, thus, no influence on the result. However, in other cases a large influence may result due to the appearance of ΔE in the Boltzmann factor. Fig. 2 shows a comparison between the results obtained by Eq. (1) and Eq. (3) as a function of D. D and Penter span over several orders of magnitude. In all cases, there is excellent agreement between both results. 4. Conclusions

The relation between the quantities given by Eq. (3) improves the understanding of surface barriers and sets it into relation to the resistance given by the intracrystalline diffusion. It helps to explain the reported low values of the entering probability of benzene on silicalite (1.7·10-6) [2] and the high values reported from MD simulations in Ref. [3]. In the latter case, the self-diffusion coefficient of the considered molecule (n-butane) was much larger. Additionally, real crystals have been considered in the experiment where additional effects due to blocked pores occur. The discrepancies between the results of Refs. [2] and [3] may thus be explained by 1. the difference in the diffusion coefficients, 2. the surface energy effect, and 3. the effects of pore blockage occuring for real crystals.

References [1] A. Schüring, J. Phys. Chem. C 2007, in press. [2] A. Jentys, H. Tanaka, J. A. Lercher, J. Phys. Chem. B 109 (2005), 2254. [3] J.-M. Simon, J.-P. Bellat, S. Vasenkov, J. Kärger, J. Phys. Chem. B 109 (2005) 13523.

179

Transport in the Transition Region Gas/Adsorbent Studied by Molecular Dynamics Simulations

A. Schüring1,2*, J. Gulín-González2,3, S. Fritzsche1, J. Kärger2, and S. Vasenkov4

1. Institut für Theoretische Physik, Universität Leipzig, Vor dem Hospitaltore 1, D-04103, Leipzig, Germany, E-Mail: schuring@uni-leipzig.de.

2. Institut für Experimentelle Physik I, Universität Leipzig, Linnéstr. 5, D-04103, Leipzig, Germany.

3. Department of Mathematics, University of Informatics Science (UCI), Boyeros, La Habana, Cuba.

4. Chemical Engineering Department, University of Florida, P.O. Box 116005, Gainesville, FL 32611-6005

1. Introduction Molecular transport through the external surface of an adsorbent differs from the

diffusion in the bulk of the adsorbent material. The effect of the surface transport on the overall diffusive transport in nanoporous materials increases with the surface-to-volume ratio of the adsorbent. For applications, e.g., catalysis, it is desirable to have an estimate showing whether surface effects may be neglected or must be considered, and, to which extent they influence the overall transport. We study such boundary effects using molecular dynamics simulations of tracer exchange in zeolite membranes [1,2]. Zeolites are crystalline nanoporous materials with well-defined pore sizes. Beside normal diffusion (ND), one-dimensional zeolite channels exhibit anomalous single-file diffusion (SFD) of the molecules. Both cases are considered in this contribution. In analytical considerations, the parameters needed to quantify the magnitude of boundary effects have been related to further system parameters. These relations have been verified by the simulation results [2-4].

2. Tracer-exchange simulations As model system we consider a zeolite membrane to study both, diffusion in the

crystal and through the crystal surface. The membrane is in sorption equilibrium with the surrounding gas phase and the exchange of (labelled) particles initially in the membrane by (unlabelled) particles from the gas phase is observed. Simulations of single-file diffusion have been carried out with a spherical model of neo-pentane in an analytical potential function describing the potential energy landscape of neo-pentane in the zeolite-like material AlPO4-5. Simulations of normal diffusion were performed in the same potential landscape but with much smaller size of the diffusing particles allowing them to pass one another in the channels.

3. Results From the tracer-exchange simulation one obtains concentration profiles of the

exchanged particles. Typical examples are shown in Fig. 1. These profiles are fitted to the solutions of Fick’s second law using appropriate boundary conditions. Beside the diffusion coefficient, additional parameters are needed to describe these concentration profiles correctly. In the case of SFD (Fig. 1 top), the inner part of the profile may be described by the analytical solution, but the range where this solution is valid is reduced

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© 2007, A. Schüring 180

to a length L* < L, where L is the thickness of the membrane. At the boundaries, equilibrium is reached faster compared to the inner part. This indicates a higher mobility of the molecules close to the boundaries. The length of this region is given by the parameter

l=(L-L*) λ/2 (1)

l l

where λ is the distance between adsorption sites in the channel.

On the other hand, for normal diffusion, it can be seen in Fig. 1 (bottom) that sorption equilibrium at the boundaries is reached slowly. This behaviour can be described with the boundary condition [5]

J(t)=α(C0-C(t)), (2)

where J(t) is the flux density through the zeolite surface at time t, C0 the concentration needed to maintain equilibrium with the surrounding gas phase, C(t) the concentration at time t, and α is a constant of proportionality.

From analytical considerations we obtained [3]

l2=2L λ2(1-θ)2θ, (3)

Figure. 1: Tracer-ex-change concentration profiles. Top: SFD in comparison to ND with large value of α, bottom ND, low value of α.

where θ is the fractional site occupancy, and [2]

α=(D/λ) exp(-βΔE), (4)

where D is the self-diffusion coefficient and ΔE is an energy difference occuring at the crystal boundaries, which can be obtained from force field calculations. 4. Conclusions Surface effects significantly influence the shape of the concentration profiles and, thus, the overall transport. The relative importance decreases with increasing system size. However, in applications like catalysis, where small crystals are employed, a significant effect is probable. With help of the relations Eqs. (3) and (4) presented here, the magnitude of these effects can be calculated.

References [1] A. Schüring, S. Vasenkov, and S. Fritzsche, J. Phys. Chem. B 109 (2005) 16711. [2] J. Gulín-González , A. Schüring, S. Fritzsche, J. Kärger, and S. Vasenkov, Chem. Phys. Lett. 430 (2006) 60. [3] S. Vasenkov, A. Schüring, and S. Fritzsche, Langmuir 22 (2006) 5728. [4] A. Schüring, J. Phys. Chem. C 2007, in press. [5] J. Crank, The Mathematics of Diffusion, Claredon Press, Oxford, 1956.

181

The Influence of Interstitial Impurity Atom – Vacancy Complex on Diffusivity of Interstitial Atom in α-Iron

Liudmila V. Selezneva, Andrei V. Nazarov

Moscow Engineering Physics Institute (State University), Department of Material Science, Kashirskoe shosse 31, 115409 Moscow, RUSSIA, E-Mail: selezneva_lv@bk.ru

1. Introduction This paper is devoted to simulation of the effect of interaction between impurity and

point defect on interstitial atom diffusion. In initial part the activation barrier set for different atomic configurations of impurity

atom (i.e. carbon) and vacancy in α-iron have been calculated using a molecular static method (MS). In second part the diffusion coefficients depending on temperature have been calculated for carbon and vacancy in perfect lattice and in defect lattice using kinetic Monte-Carlo method (MC). Because of simulation results the anomalous diffusion of carbon in α-iron is analysed.

2. Model and simulation procedure We consider a body-centered cubic lattice, in which there are two different types of

point defects such as a vacancy and interstitial impurity atom. The simulation consists of two steps: - the calculation of the activation barriers for carbon atom and a vacancy in the

different configurations of carbon–vacancy complexes, - the simulation of carbon atom and vacancy migration and the calculation of carbon

atom and vacancy diffusion coefficients depending on the temperature and the density of vacancies. Molecular static method and calculation of the potential barriers. The aim of the MS-method is to determine the minimum energy configuration. In the present calculation about 1220 atoms surrounding the defect were treated as individual particles, each with three degrees of freedom. These atoms are represented as a spherical crystallite, which is located in the centre of atoms frozen at their perfect lattice positions. Atom interactions are described by potential functions for Me-Me and Me-C. In this work, an empirical interatomic potential for the description of C interstitial impurity in metal is employed. Metal-metal interatomic potential of the EAM type is also used in simulation of metal interactions. In the vicinity of a vacancy atomistic structure is distorted, and so equilibrium positions for metal atoms and carbon atom are determined by relaxation.

For the determination of the minimum energy configuration every atom of the central part is consecutively displaced as long as the energy of this atom reaches a minimum value. Then, also the total energy of the system reaches a minimum. This procedure is called relaxation. Then we move one of the atoms from its equilibrium position to a vacant nearest-neighbor equilibrium position step by step. At the same time we relax the

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© 2007, L. Selezneva 182

system and calculate the moving atom energy and total energy on every step. Consequently the activation barrier for an atom jump is obtained.

Monte-Carlo method and determination of the diffusion coefficients. Using the values of the activation barrier which are obtained with the help of the MS-method the jump rates are calculated as:

Γi = ν*exp[ - (Q0 +ΔQi ) / (kB*T)], (2) B

where Γi is the frequency of an atom jump in i-direction; ν is a frequency factor; Q0 is the activation energy for atom migration in a perfect lattice; ΔQi is the change of the activation energy for atom migration in a lattice with point defect (in the i-direction); kB is Boltzmann’s constant; T is the temperature. On basis of these data the atom migration are simulated by using the MC-method.

The mean distance is calculated in every experiment after a certain number of trials K. Then the root-mean-square displacement < RK

2> is calculated for different temperatures and averaged over a certain number of experiments (where RK is the distance between the initial position of the migrating atom and its final position in trial K). The diffusion coefficient is obtained by Einstein’s formula:

D = < RK 2> / (6*t) (3)

and t = K*Δτ , (4)

where t is the migration time which is much more than the mean time between jumps of the atoms (Δτ). The relation of the diffusion coefficients is calculated as:

Ddef / Dp.l. = < RK 2>def / < RK

2>p.l. , (5) where Dp.l. and < RK

2>p.l. are the carbon diffusion coefficient and the root-mean-square displacement of the carbon in the perfect lattice; correspondingly, Ddef and < RK

2>def are the carbon diffusion coefficient and the root-mean-square displacement of the carbon in the lattice with point defects, respectively.

3. Results Thus we calculated a set of activation barriers for the jumps of atoms in different

directions and different configurations of the carbon-vacancy complex by the MS-method.

The diffusion coefficients for vacancy and carbon migration are simulated in the perfect lattice and in a lattice with point defects (i.e. vacancies) by the MC-method using the set of activation barriers. We computed the diffusion coefficients of carbon in dependence on the temperature and the vacancy density.

The vacancy and the carbon atom migrate as a bonded complex in a definite temperature range. In this case the mechanism of a dynamic couple is realised and an acceleration of carbon diffusion is observed. This allows to interpret the anomalous diffusion of carbon in α-iron at high temperatures.

4. Conclusion We have developed a model taking into consideration that point defects can alter the

surrounding atom configuration and the local magnitude of the activation barrier for interstitial atomic jumps. The simulations reveal an acceleration of carbon diffusion.

183

Size Dependence of Solute Diffusivity and Stokes-Einstein Relationship: Effect of van der Waals Interaction

Manju Sharma and S. Yashonath

Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012 India. E-Mail: yashonath@sscu.iisc.ernet.in

1. Introduction Diffusion of solutes in solution was investigated by Einstein1. With the help of kinetic

theory, he derived an expression for the self diffusivity, D=RT/NA6πηru. The motion of the solute can be described by the random walk the solute performs due to collisions with the solvent molecules described by the Einstein’s Theory of Brownian Motion and also reported by Sutherland2 and more recently, Chandrasekhar3. Innumerable groups4 have investigated the range of validity of the above expression. Dependence of D on viscosity, mass of the solute and interaction (hard sphere, square well, soft sphere, Lennard-Jones, etc) have been examined. This relationship has even been employed frequently to determine Avogadro number as well as radius of the solute. Willeke and several others5-8 have reported higher diffusivities than what is predicted by the Stokes-Einstein expression given above. Here we report molecular dynamics study of solute diffusion in solution, interacting via Lennard-Jones potential. The solute radius is varied to obtain the dependence of D on solute radius, ru.

2. Methods Molecular dynamics simulations have been carried out in the microcanonical

ensemble using periodic boundary conditions. Both the solute and solvent interact via the Lennard-Jones interaction with: σvv =4.1Å; εvv=0.25 kJ/mol; σuv=1.0-4.7Å; εuv=1.5 kJ/mol; σuu=0.3-4.0Å; εuu=0.99 kJ/mol employed in this work. Solute-solvent interaction parameter σuv= σuu+ 0.7 and both σuv and σuu are varied. Note that σuu = 2ruu.

All simulations have been carried out with 500 solvent and 50 solute particles at ρ* = 0.933 and T* = 1.663. Time step of 15fs was used for all solutes sizes with the exception of 0.3Å for which 10fs was employed. Energy conservation was better than 1 x 10-5. Equilibration was for 2ns and production for 6ns. Properties were stored for analysis usually every 1ps with the exception of velocity autocorrelation functions and intermediate scattering function for which 100fs was the interval. 3. Results and Discussion Figure 1 plots the variation of diffusivity with the ratio of the solvent Lennard-Jones diameter to the solute Lennard-Jones diameter, σvv/σuu, that is κ-1, where κ = σuu/σvv. The variation of D with κ-1 should be a straight line if the Stokes-Einstein(SE) relationship is valid9. This is the case for large ru. However, the deviation from SE is evident when κ-1

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© 2007, M. Sharma 184

0 2 4 6 8 10 12 14σ

vv/σ

uu

0

0.5

1

1.5

2

Dx1

08 , m2 s-1

MD (with dispersion)

fit (σvv/σuu)2

fit (σvv/σuu)

MD (without dispersion)

6 8 10 12 14

0.65

0.7

0.75 fit (σvv/σuu)2

1 1.2 1.4 1.6 1.8 2

σvv

/σuu

0

0.05

0.1fit (σvv/σuu

)

x 0.25

is larger than 3. For 3 < κ-1 < 6 Ds are higher than predicted by SE. For κ-1 > 6, D ∼ 1/ ru

2 dependence. This surprising deviation from SE relation explains the enhanced diffusivity seen by several groups. Willeke et al as well as others5-8 found in their study of Lennard-Jones binary mixture that the SE relation is not valid for κ < 1 as compared to deviations only for κ < 0.33 found here. There are several others who have found enhanced self diffusivity. For example, Noworyta et al. found higher D for small solutes in water than predicted by SE8. Bhattacharya and Bagchi found higher D for small solutes7. Ould-Kaddour and Barrat10 found higher D for 0.066 < κ < 0.66. We see here an enhanced D for 0.17 < κ < 0.48.

Figure 1 Plot of self diffusivity D as a function of the σvv/σuu with and without dispersion term. Our results here indicate that deviation is a

consequence of the existence of diffusivity maximum seen for solutes diffusing in porous solids or simple liquids and known as the Levitation Effect (LE). Diffusivity from simulations carried out without the dispersion term (that is with the purely soft repulsive term 1/r12 and no 1/r6 term) between the solute and the solvent show no such deviation for up to κ-1 ≈ 7. The enhanced diffusion seen for intermediate sizes of the solute is therefore clearly shown to have its origin in the dispersion interaction.

3. Conclusion Simulations reported suggest that the Stokes-Einstein relation between D breaks down

for small solute sizes. It also suggests that this breakdown has its origin in the Levitation Effect or, equivalently, the presence of the dispersion interaction is responsible for the enhanced self diffusivity.

References [1] A. Einstein, Ann. Phys. 19 (1906) 371; Ann. Phys. 19 (1906) 289. [2] W. Sutherland, Philos. Mag. 9 (1905), 781. [3] S. Chandrasekhar, Rev. Mod. Phys. 15 (1943) 1. [4] R. Walser, A. E. Mark, W. F. Van Gunsteren, Chem. Phys. Lett. 303 (1999) 583;

D. Kivelson, S. K. Jensen, M-K Ahn, J. Chem. Phys. 58 (1973) 428; B. Alder, W. E. Alley, J. Stat. Phys. 19 (1978) 341; B. Bernert, D. J. Kivelson, J. Phys. Chem. 83 (1979) 1401; R. Lamanna, M. Delmelle, S. Cannistraro, Phys. Rev. E 49 (1994) 5878; R. Yamamoto, A. Onuki, Phys. Rev. Lett. 81 (1998) 4915.

[5] M. Willeke, Mol. Phys. 101 (2003) 1123. [6] A. J. Masters, T. Keyes, Phys. Rev. A 27 (1983) 2603. [7] S. Bhattacharyya, B. Bagchi, J. Chem. Phys. 106 (1997) 1757. [8] J. P. Noworyta, S. Koneshan, J. Rasaiah, J. Am. Chem. Soc. 122 (2000) 11194. [9] M. Sharma and S. Yashonath, J. Phys. Chem. B 110 (2006) 17207. [10] F. Ould-Kaddour, J.-L. Barrat, Phys. Rev. A 45 (1992) 2308.

185

Adsorption Kinetics of Mixtures of n-Hexane and 2-Methylpentane on Silicalite by Nonequilibrium Molecular Dynamics.

Jean-Marc Simon, Jean-Pierre Bellat

Institut Carnot de Bourgogne, UMR-5209 CNRS-Université de Bourgogne, 9 av. A. Savary, 21000 Dijon, France, jmsimon@u-bourgogne.fr

1. Introduction In the context of the separation processes of mixtures the use of nanoporous systems

has been revealed as a good complement to classical distillation procedure when the molecules have very similar boiling points as it is the case for many isomers. This has been done successfully for instance with xylene isomers. In gasoline, branched alkanes have higher octane numbers compared to normal ones and they are for this reason more desirable. The aim of this work is to describe, at a molecular level, the mechanism implied in the kinetics of adsorption of mixtures of two valuable compounds, n-hexane (HEX) and 2-methylpentane (2MP), on an infinite membrane of silicalite.

2. Model and simulation details Transient nonequilibrium molecular dynamics (NEMD) simulation was used here to

mimic gravimetric uptake experiments [1] of equimolar mixtures of hexane on silicalite.

Fig 1. Uptake curves of the 3 mixtures. In green first case (20), in red second case (40), in black third case (80). The lines with X on them are the 2MP (lower curves) while the simple lines refer to HEX.

The initial system consisted of an infinite membrane of

d an united atom mo

silicalite with an external surface perpendicular to the straight channel (y direction) in contact with a gas of HEX and 2MP mixture. With time, by integrating the equation of motion, the adsorbate entered the pore until equilibium is reached. The thickness of the zeolite was three unit cells in the y direction (about 60 Å), two and three unit cells in x and z directions, respectively (about 40 Å). Three equimolar mixtures were studied containing 20, 40 and

80 molecules of n-hexane, we will refer to as first, second and third mixture in the text. The silicalite and the alkanes were simulated using an all atom andel, respectively. Both the silicalite and the hexane were flexible and they interacted

via Lennard-Jones potentials. Simulation details can be found elsewhere [1].

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© 2007, J.-M. Simon 186

3. Results and discussion

Figure 1 show the comparison of the uptake of the three mixtures, due to computational limitations the third did not end but the trends are clear enough to be discussed. As expected from previous simulations on pure n-hexane and pure 2MP, the n-hexane was adsorbed faster than the 2MP, its self diffusion coefficient was 2 oders of magnitude higher. HEX had barely the same kinetics in the first and second mixtures but it was much lowered in the denser system. As the amount of 2MP increased its adsorption kinetics decreased, being 6 times slower from 20 to 40. Compared to the pure compound, the presence of HEX strongly reduced the diffusion of 2MP in the pore.

An analysis of the molecular position can give insight to explain these differences. HEX in the pores was located in the straight and zig-zag channels and in their intersections, while 2MP was only located in the intersections. 2MP is much more cumbersome than n-hexane. It has to cross a high-energy barrier to go from an intersection site to another one, which is not the case for HEX. In the first and second case the number of 2MP was not sufficient enough to occupy all the intersections, HEX molecules had then enough free intersections to diffuse nearly as if they were alone. This was not the case for 2MP. Here, as a prerequisite for a jump, it is required that the other intersection is free, that no HEX molecule is located in the channel and that its conformation is favorable. The probability to move is then highly dependent on the density of HEX.

At high loading, case 3, 2MP filled nearly all the intersections and both the kinetics of n-hexane and 2MP was reduced. However, the flexibility of HEX was large enough to overtake 2MP, the kinetics of HEX was then faster than 2MP.

3. Conclusion Using NEMD, we have shown that two isomers with very different shapes (n-hexane

and 2-methylpentane) exhibit a very different behaviour when they are adsorbed together on silicalite. The uptake curve for HEX is similar to the pure compounds except for high loading of 2MP. At the contrary the adsorption kinetics of 2MP decreases when the density of HEX increases. This difference is related to the diffusion “style” of the two molecules, 2MP diffuses in the pores by jumps while HEX exhibits a liquid-like diffusion.

Reference [1] J.M Simon, A. Decrette, J.P. Bellat, J. M. Salazar, Mol. Sim. 30 (2004) 621.

187

Dynamical Behaviour of H2 Molecules on Graphite Surface. A Molecular Dynamics Study.

Jean-Marc Simon1, Ole-Erich Haas2, Signe Kjelstrup2, Astrid Lund Ramstad2

1. Institut Carnot de Bourgogne, UMR 5209 CNRS-Université de Bourgogne, France. jmsimon@u-bourgogne.fr

2. Institutt for kjemi, NTNU, 7491 Trondheim, Norway

1. Introduction The background of this work is the prediction made by Meland et al. [1] that access to

the three-phase contact line in porous gas electrodes is via the interfaces that surround the contact line. Electrodes of this type are central in the polymer electrolyte fuel cell. In this work we investigated the dynamical behaviour of hydrogen molecules on the graphite carbon surface by molecular dynamics (MD) simulations.

2. Model and simulation details The system was composed of two infinite planes of graphite distant by about 160 Å in

contact with different amounts of hydrogen molecules. The surface was in the plane of the hexagonal structure. A rectangular basic simulation cell of about 44 Å long and 34 Å large in that plane was extended to infinity by the use of periodic boundary conditions. Both the graphite and the hydrogen were simulated using an atomic model and they were allowed to interact through inter and intra-molecular potentials and to move by integrating the equation of motion.

Systems with 50, 100, 150, 200 and 300 molecules of hydrogen were simulated at different temperatures 70K, 100K, 130K and 300K. By an analysis of the molecular trajectories at the surface, we could reach different dynamical properties like the self-diffusion coefficients. We also found information on the thermodynamics properties of adsorption.

Figure 1: Arrhenius plot comparing our work (MD and QENS) with Narehood and Bienfait et al. [2-3]

3. Results and discussions Our simulation conditions were such that the density on the surface was lower than

that of a monolayer, it varied from 1.5 × 10-5 to 1.8 × 10-7 mol/m2. The self-diffusion coefficients are presented in an Arrhenius plot in figure 1. They are compared with

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© 2007, J.-M. Simon 188

results from Quasi Elastic Neutron Scattering (QENS) obtained by us and from the literature. Despite our simple model, the results show a good agreement both on the slope and on the absolute values.

The residence time tr for a molecule on the surface slightly decreased at each temperature with increasing surface density. The mean value for each temperature is also plotted on figure 2 in an Arrhenius plot, it decreases as the temperature increases. From that curve we can deduce an activation energy of 3.0 kJ/mol in good agreement with the adsorption energy, but higher than the activation energy of the surface diffusion 2.2kJ/mol. At 300K, the residence time is about 5.7 ps, with an average self-diffusion coefficient of about 2 10-6 m2.s-1, the average distance travelled by a molecule on the surface before desorption is then around 50 Å.

Figure 2. Arrhenius plot of the residence time of hydrogen

4 Conclusion and perspectives We have simulated hydrogen molecules adsorbed on graphite at different

temperatures. Their dynamical properties were investigated showing that at 300K, despite the fact that the number of adsorbed molecules is low, the travelled distance on the surface is quite substantial, 50 Å. This result supports the proposal of Meland et al.[1] that hydrogen gas reaches the catalytic site (Pt) via the interfaces of the porous structure leading up to the site.

References [1] A. K. Meland, D. Bedeaux and S. Kjelstrup, J. Phys.Chem. B 109 (2005) 21380 [2] D.G. Narehood, J.V. Pearce, P.C. Eklund, P.E. Sokol, R.E. Lechner, J. Pieper, J.R.D.

Copley, J.C. Cook, Physical Review B, 67 (2003) 205409. [3] M. Bienfait, P. Zeppenfeld, R.C. Ramos, J.M. Gay, O.E. Vilches, G. Coddens,

Physical Review B, 60 (1999) 11773

189

Method of Fractional Derivatives in Time-Dependent Diffusion

Sergey D. Traytak

Institute of Applied Mechanics RAS, 32-a Lenin Av., GSP-1, Moscow 119991, Russia, E-Mail: sergtray@mail.ru

Tatyana V. Traytak

National Institute Higher School of Management, 23-6 Denisovskiy lane., Moscow 105005, Russia, E-Mail: 89035326173@mail.ru

1. Introduction Many theoretical problems of diffusion lead to the solution of the initial boundary-value problem

( ) ( ) ( )22 1 0, , , 0t x xu x t u x t u x t uα α α∂ − ∂ − ∂ + = in Ω (1)

00

tu

== , ( )0x

u tϕ== , 0

xu

→∞→ , (2)

where and for the sake of simplicity we assume that ( ) (0, 0,Ω = ∞ × ∞) ( )tϕ is a continuous

function and ( ) ( ),k x t Cα ∞∈ Ω 0,1,2k ( = ) with ( )2 0, 0tα ≠ . It is well known that the exact analytical solution to the problem (1), (2) is often difficult if not impossible to obtain. However, particularly for diffusion-limited reactions, the value of the local flux on the boundary proportional to the function ( )

0x xj t u∞ =

= −∂ is of main

interest. It turned out that one can find the function of interest ( )j t∞ without knowing the solution of the posed problem using the method of fractional derivatives (MFD) suggested by Babenko

( ) ( ) ( )1

2

0

n

n tn

j t f t tϕ−∞

∞=

= ∂∑ . (3)

Here ( ) ( ) ( )20, 0,n nf t a t tα= , where ( ),na x t are functions to be determined from the

recurrent relations obtained with the help of the corresponding operator equation; 1

2n

t

−

∂ are operators of fractional differentiation with respect to time and defined by the following formula

( ) ( ) ( ) ( )0

11

t

tdg t t g d (dt

μμ τ τ τμ

−∂ = −Γ − ∫ , ),1μ ∈ −∞ ,

where ( )zΓ is the gamma function.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, S. Traytak 190

2. Method validation and some applications However, so far the range of validity of expansion (3) was unknown. We established the convergence radius of the functional series (3): 2R q−= , where ( ) ( )lim ,k

kkq a x t tϕ

∞ ∞→∞= .

Moreover, a useful uniform upper bound for the solution ( ),u x t in an important particular

case when ( )2 , 1x tα ≡ , ( )1 ,x tα 0≡ was proved. Using the MFD we also found the exact

connection between the function ( )j t∞ and the function ( )0h x h x

j t u=

= −∂ corresponding to

the initial boundary-value problem (1), (2) with and more general Robin’s boundary condition at

hu u→0x = , i. e.

( ) ( )00,x h hx

u h t u tϕ=

−∂ = −⎡ ⎤⎣ ⎦ , where is a positive constant: h

( ) ( ) ( ) ( )0

1 t

hd

hj t j t j t j dh dt

τ τ τ∞ ∞= − −∫ . (4)

We also applied the MFD to the very important particular case of diffusion-controlled

reactions between ions. In this case ( )0 , 0x tα ≡ , ( )1 , 1x tα ≡ , ( )22, 1x tx x

βα ⎛≡ −⎜⎝ ⎠

⎞⎟ , where

2cr Rβ μ= , is the Onsager length [1], cr ( )0 0μ < > for attraction (repulsion). For the attractive potentials the MFD gives

( ) ( ) ( )21 1; 1 exp erfcj t t t Ot

β β β ββπ∞

⎛ ⎞⎛ ⎞= + − − + ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

as β →∞ ,(5)

where Dt t t= , 2Dt R D= is the characteristic relaxation time for pure diffusion, is the

reaction radius and is the sum of the diffusion coefficients for ions. We proved that, contrary to what we have assumed before, the characteristic relaxation time for the attractive Coulomb potential is less than

RD

Dt and has the form

( ) ( ) 2 2exp 2 1 4rt Dtβ β= −⎡ ⎤⎣ ⎦ β . (6) Finally, with the aid of the MFD, we calculated the correction for the effect of the hydrodynamic interaction on the diffusion-controlled rate coefficient.

3. Conclusion The rigorous proof for validity of the method of fractional derivatives is presented. Using this method we found the relationship between the diffusion flux in the case of the Dirichlet and Robin boundary conditions. Application of the method of fractional derivatives to diffusion-controlled reactions between ions allows us to derive an analytical formula for the total diffusive flux, the corresponding realaxation time and the correction for the effect of the hydrodynamic interaction.

References [1] S.A. Rice, Diffusion-limited Reactions, Elsevier, Amsterdam, 1985.

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