Revista Mexicana de Física (Suplemento) 32 No. SI (1986) 5189-5208

STRUCTURE ANDDYNAMICS OF CHARGED

SUPERMOLECULARFLUIDSRudo lf Kl ei n

Fakultat für Physik, Universitat Konstanz7750 Konstanz, Germany

RESLMEN

5189

En estas notas se presentan algunos resultados recientes sobresoluciones de macropartículas cargadas. En la primera parte se estudiala estructura estática, y se muestra que la presencia de iones pequeños(contra iones y sal). tiene efectos significativos para soluciones micelares típicas. La segunda parte presenta algunos resultados sobre la dinImica de sistemas coloidales cargados. -

ABSfRACf

In these notes sorne secent results on solutions of charged macroparticles are reviewed. The first part is devoted to the static &ructure,and it is shown that the presence of small ions (counter-ions and salt)has significant effects for typical mycellar solutions. The second partpresents sorne results on the dynarnics of charged colloidal systerns.

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1. INTRODUCTIONComplex fluids cons;sting of charged macroparticles have been

of considerable ;nterest for a number of years1. Roughly, there aretwo ma;n classes of charged macroparticle systems: the prototype ofthe first kind are spherical polystyrene spheres, which are highlycharged (several hunde red elementary charges) and form strongly corre~lated supermolecular fluids for volume fraction below roughly 0.01,above which they form colloidal crystals. The other type are ;on;c m;-celles. These are rather concentrated solutions of sma'l objects whichcarry a smal1 effect;ve charge.

In this paper first, an outline of calculations of the stat;cstructure will be given. The role of the small ;on5 (counter-;ons andsalt) will be shown to have significant effects for typical rn;cellarsolut;ons. The second part rev;ews sorneresults on the dynam;cs ofcharged collo;dal systems, ~s ;t ;s man;fested ;n the dynam;cal struc-ture factor wh;ch can be determined from quas;-elastic l;ght scatter-ing.2. STATIC PROPERTIES OF CHARGED COLLOIDAL SYSTEMS2.1 The Mean-Spher;cal Approxirnation for the Pr;mit;ve Model

The model most frequently used for stat;c and dynam;c propert;esof suspensions of charged particles ;s the so-called one-component rna-crofluid model (OCM)2. Only the suspended macroparticles are treatedexplic;tly ;n this rnodel,whereas the small ;ons (counter-;ons and salt;ons) are taken ;nto account by provid;ng a screen;ng for the Coulomb;nteractions between macro;ons. Th;s leads to the Derjaguin-Landau-Ver-wey- Overbeek potent;al, whose repuls;ve part

u(r) o

di

2Z+ K

(2.1 )

;s then taken together with the hard core (u(r) = ~ , r < d1) as theeffect;ve pair potent;al between macroions, wh;ch are separated by ad;stance r. In eq. (2.1), Z ;s the charge of the rnacro;on of d;ameter

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di • € the dielectr;c constant of the salvent and ~ the Debye screen-;og parameter, g;ven by

(2.2 )

The sum runs ayer al1 spec;es of sma'l ;ons, ni ;5 the number densityand Z; the charge of species ;

The interaction potential (2.1) has beeo taken as the startingpo;nt for the calculation of structural properties of suspensions, suchas the radial distribution funct;on g(r) and the stat;c structure fac-tor S(k) , by us;ng var;ous approximation schemes developed in thetheory of simple liquids3. One uses the Ornstein-Zernike equation

h(r) • e(r) + n fh(lr-r' 1) elr') d'r' 12.3 )

which connects the total correlation function h(r) = g(r)-l w;th thedirect correlat;on funct;on c(r) . Here, n is the number density ofthe macroions. This integral eQuat;on, be;ng essent;ally the def;n;-t;on of c(r) , reQuires an addit;onal relat;onship between the totaland d;rect correlat;on funct;ons, a so-called closure relat;on. Thes;mplest closure ;s prov;ded by the mean-spher;cal approx;mat;on (MSA)

elr) • - a u(r)12.4 )

h(r) • - 1

wh;ch ;5 eQu;valent to the Percus-Yev;ck approx;mat;on ;n the case ofpure hard-sphere ;nteract;ons. For the case of screened Coulomb poten-tials, like eq. 12.1), the MSA solution was 9iven by Waisman4 and by

Blum and H~ye5. It has been put into a conven;ent form by Hay ter andPenfold2. The latter authors were the f;rst to calculate g(r) andS(k) essent;ally analyt;cally for suspens;ons of charged spher;calpart;cles. It was found that the MSA y;elds good results as long as

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the concentration is not too small. For highly charged but dilute sy-stems, such as the polystyrene sphere systems, the MSA leads to nega-t;ve values of g{r) near contact, which are of course unphys;cal.Far such systems Hansen and Hayter6 introduced an ;mprovement, the re-scaled MSA.

As ment;oned earlier, th;s procedure does not treat the smal1;ons explicitly. It ;5, however, possible to generalize ;t to a multi-component system7. This has the advantage of treating the smal1 ionson the same basis as the macro;ons; furthermore, one can ;nvestigatethe effects of the finite s;ze of the smal1 ;ons on the structuralproperties, and, final1y, ;t ;5 possible to treat systems which con-sist of different kinds of macroions (polydispersity) by the same for-malism. Within this rnodel of an m-component system with number den-sities ni • charges Z; and diameters di the ;nteraction potent;als areCoulomb;c

{ r < (di + dj)/2 = dijuij(r) = (2.5 )

Z;Zj r > dij (i,j=l,...,m)<r

Th;s model ;s called the pr;m;t;ve rnodel ;n analogy to electrolytetheory, when the solvent is treated as a cont;nuurn of d;electr;c con-stant E • Instead of one radial d;str;but;on funct;on g(r) and onestatic structure factor S(k) there are now m(m+l)/2 partial functionsgij(r) and Sij(k), respectively. The structural information in gij(r)is the probability density to find a particle of species j at distancer from a particle of species i at the origino Eq. (Z.3) is replaced bya set of coupled equat;ons

m

lk=l (2.6 )

and the MSA cons;sts ;n wr;t;ng

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e;j(r) = - a u;j(r) r > d ..lJ

(2.7 )

h;j (r) = - 1 r , d;j

From the solution of the system (2.6) and (2.7) one obta;ns the part;-al structure factors as

(2.8 )

These Quantities determine the intens;ty of scattered light, x-rays arneutrons by

where

m

I(k). L (n¡ni'" f;(k) fj(k) S;j(k);,j=1

(2.9 )

denotes the scatter;ng amplitude of one particle of spec;es ;. Theforro amplitude b;(k) fer a homogeneous spherical particle ;5

3j,(k d/2)b;(k) = k d./2 (2.11)

1

where jl is the spherical Bessel function of index 1.If there is just one kind of macroions (;=1), whereas a11 other

species (i > 1) are small ;ons, incoming light will on1y be scattered

by the maero;ons so that f,tO ;n eq. (2.10) and f;=O, ; > 1. Althoughthe radiation couples to the multicomponent system only through thespecies of macroions, the partíal structure factor Sll(k) in eq. (2.9)a150 depends on the rema;ning species beca use of the coupling in eqs.(2.6) and (2.8). A systemat;e study of the ehanges of the statie strue-ture ;nduced by changes ;n ;on;c strength, s;ze and/or charge of themacropart;cles, s;ze of the small ;ons, etc., has been performed, us;ng

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the MSA. and results have been obtained7 tor parameter values, where arescaling procedure of the MSA ;5 unnecessary. These correspond te ty-p;cal m;cellar solutions, where nl~ lOlBcm-3, d1z 50 A and Zl = 20 e.Such systems have been studied by light 8 and small angle neutronscattering9 experiments.2.2 Influence of the finite size of 5mall ions

Effects of the finite size of counterions on the micelle-m;cel1estructure factor 311(k) and the corresponding partí al radial distribu-tion function 911(r) have been studied in detail. Because of the;r fi-nite s;ze the counter;ons have a lower screening ability so that there;5 a stronger effective repulsion, resulting in a 15 % ;ncrease in theheight of the roain maximum in SII(k) as compared to point-1ike sma11;ons. With ;ncreas;ng macropart;c1e concentration this difference ;n-creases further. We have a1so compared these results with those of theOCM model by solving the one-component Ornstein-Zernike equation (2.3)

with the pair potential (2.1), using K' = 4 TI n,Z, '/«kBT). This is ofsorne importance for ana1yzing recent static neutron sC3tter;ng data on;on;c m;celles. Bendedouch et al.9 used the OCM mode1 to fit their dataand obta;ned from it va1ues for the aggregat;on number and the degreeof ionizat;on of the micelles. A rather sens;tive measure of the effec-tive charge is the height of the main peak of S,,(k) . Since for thesame va1ue of ZI a higher maximum is obta;ned in treating the smallions as of f;nite s;ze, the OCM overest;mates the effect;ve charge ona micelle. We found that in sorne cases a charge which is about 30 %smaller ca" lead to the same peak height, if the finite size of smallions ;s taken intc account.

The treatment described here gives a1so the distribution of thecounterions around a given macroion in the system of finite concentra-t;on nI . It ;5 found that g12 first drops from a h;gh value at thesurface of the macroion. But ;t does not fa11 monotonica11y as expec-ted from simple Debye-Hückel theory; ;nstead there ;5 a local minimumat the position of the f;rst maximum of gll(r) , ;ndicating a part;a1exc1usion ofcounterions from the region of the Ilf;rst shellll of macro-

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ions. Then, there ;5 a local maximum in 912 between the first and se-cond shells this out-of-phase behavior of 912 and gIl continues.

We have a150 studied the effects of added salt on the structuralproperties and have extended the treatment based on eqs. (2.5) to(2.11) to the case of mixtures of macropart;cles, which are of differ-ent size and carry different charges. The ;nterested reader ;5 re-fered to the original literature7,10,11.

3. DYNAMIC PROPERTIES3.1 Introduction

The investigation of supermolecular fluids by coherent and ;n-coherent scattering of neutrons, by photon correlation spectroscopyand by torced Rayleigh scattering gives informat;on about the time de-penden ce of correlation functions and not just results tor transportcoefficients. Therefore, the ;nformat;on about the dynamics of thesesystems as conta;ned, for instance, ;n the veloc;ty autocorrelationfunct;on of the concentration autocorrelat;on function, is much r;cherthan the one obta;ned from the measurement of only transport coeffi-c;ents. Knowing, for instance, the full time dependence of the mean-square displacement W(t) of a tagged particle gives much more informa-tion about the dynamics of the system than determ;ning the self-diffu-sion coefficient, for which the behav;or of W(t) at large times ;ssuff;cient.

A similar situat;on exists for the spatial structure of the sy-stem and the spat;al behavior of the dynamics. A scattering experimentwh;ch determines the dynam;c structure factor S(k,t) measures the cor-relat;on of the k-th Four;er component of the spatial concentrationfluctuations. In supermolecular systems it is often easy to probe thesystem in a scattering exper;ment on a length scale which is comparableto the typical nearest neighbor distances. Under these circumstancesthe system can no longer be considered as a cont;nuum; instead one ob-tains information about the short range structure and the time develop-ment of this structure.

5196

From these two examp1es it beeomes elear that the detailed beha-vior of the time correlation funct;ons cannot be described by the phe-nomenolog;cal transport equations, such as the diffus;on equation andthe Navier-Stokes equation. To obtain a fu11 understanding of thescattering exper;ments it ;5 necessary to start from sorne 50rt of tran-sport equation on a molecular level, which will reproduce the phenome-no109;ca1 hydrodynamic equat;ons only in the limit of long waves andlong times. For the case of simple liquids such theories, sometimesealled molecular hydrodynamics, have been developed12. They can be usedw;th certa;n modifications to describe the dynamics of interactingBrownian particles. It turns out that there are many s;milarities buta150 character;st;c differences in the description of simple liquidsand supermolecular fluids.

Here, we w;ll first give a quasi-phenomenolog;cal general;zationof the hydrodynam;c transport equations to f;nite frequenc;es and wavevectors and then ;nd;cate how the generalized transport coeff;c;ents,wh;ch determine the full correlation functions, can be obta;ned froma m;croscop;c po;nt of view, starting from the ;nteractions between themacropart;cles13.3.2 General;zed transport equations for collect;ve diffusion

Dynam;c scattering experiments measure the dynamic structure fac-tor

S(k,t) = < e(k,t) e(-k,O) > (3.1

where c(k,t) is the k-th Fourier component of the ;nstantaneous con-centrat;on c(r,t) and the brackets denote an ensemble average over thecanonical equ;libr;um d;str;but;on. The s;mplest approach to determineS(k,t) 1s to start from the cont;nu;ty equat;on

aeC[.t)at = -"i' iC[,t) (3.2

and to relate the part;cle current j(r,t) to the concentration gra-

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dients by Fick's law

(3.3

Here, Oc ;5 the collective or mass diffus;on coefficient, which atth i S 1eve 1 ; s an unknown parameter.

Caleulating S(k,t) from a solution of (3.2) and (3.3) leads toa simple exponential behavior

S(k,t) = S(k) exp(- De k' t). (3.4 )

Experiments on very nearly monodisperse polystyrene lattices14, however,show that outside a rather narrow region around k=O, S(k,t) ;5 no 10n.ger a simple exponential funetion. Therefore, eqs. (3.2) and (3.3) arenot expected to be appropr;ate descriptions of the dynamical behavioroutside the long wavelength limito

Since the cont;nu;ty equation is valid generally. ene has to ge-neralize Fick's law. One has to take ;nto account that in the case ofshort wavelength fluctuations in ancurrent at position I at time tdients at neighboring pos;tions rl

neralized Fick's law to a non-local

interacting system the particlealso depends on concentration gra-and earlier times ti . This ge-

relation:t

( dt' (d3r' D(r-r' ,t-t')b J --- 'V I c(rl.tl)-r - (3.5 )

Here, the generalized diffusion function Q(!,t) replaces the hydro-dynamic transport coefficient Oc in this non-local formulation. Ifthe dynamic structure factor, or rather its Laplace transform, is nowealeulated from (3.2) and (3.5), one obtains

S(k,z) _ f dt e-zt S(k,t) = S(k), z + íí(k,z)k'

(3.6 )

where O(1,z) is the longitudinal component of the Laplace transform

m98

of Q(r,t) . The Laplaee transform of the simple result (3.4) is of thesame form as (3.6), but with ~(k,z) replaeed by De . Sinee in thehydrodynamie limit k ~ O, z ~ O, eq. (3.5) reduces to eq. (3.3), thegeneralized diffus;on funct;on D(k,z) becomes O in this limit:- e

lim O(~,z) = 0(0,0) = Dek,z ~ O

k2/z=constOn the other hand, outside this limit the k and z dependence ofO(k,z) will be responsible for deviations of S(k,t) from a simple ex-ponential form, eq. (3.4). At this stage, however, the detailed formof D(k,z) ;5 still unknown. How this funct;on can be calculated froma transport equation on a molecular level w;ll be ;ndicated latero

The generalization of the transport coeffic;ents to wavevectorand frequency dependent functions can a150 be used for the equation ofmatian of the current fluctuations. Denoting by f(~,t) the Fouriercomponents of the force density, Newton's law ;5

m (3.7

(3.8

The force density consists of twa parts, an equ;libr;um part, wh;ch;5 the grad;ent of the local osmot;c pressure n(r,t), and a fr;ct;onpart, wh;ch arises froID the mot;on of the macropart;cles:

t

~ "(~,t) - f dt' k(~,t-t') .l(~,t')o

The fr;ct;on part has aga;n been wr;tten as a non-local relat;on; itcons;sts of various d;ss;pat;ve processes, such as the fr;ct;on Co ofeach macropart;cle by the solvento the hydrodynamic ;nteract;ons ba-tween d;fferent macropart;cles and the ;nternal fr;ction ar;sing fromthe direct ;nteract;on~ among the macroparticles.

The local osmotic pressure ;n eq. (3.8) can be expressed by con-centration fluctuat;ons by us;ng a k-dependent vers;on of the compress;-

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bility sum rule. The equations (3.2), (3.5) and (3.7) to (3.8) form aclosed set, which leads to a relation between the general;zed diffu-.-sion function 0(1,z) and the generalized (Laplace transformed) fric-tion funct;on Z(~.z)

0(1,z) (3.9 )

Here, rll(~'z) is the longitudinal component of the generalized fr;c-tion tensor. Eq. (3.9) is a generalization to finite k and z of thewel1-known Stokes-Einstein relation.

Usin9 (3.9) in eq. (3.6) gives the dynamic structure factor interms of the longitudinal friction function. For highly charged systemssuch as polystyrene spheres w;thout too much salt, hydrodynamic inter-actions can be neglected. Under this assumption 'Z,,(k,z) cons;sts ofthe one-particle fr;ct;on ~o and the longitudinal viscosity func-tion ~Il(k,z) , which ar;ses from the direct potential interactionsbetween macroparticles,

~ k'e ,,(Lz) = [.0 + e íill(k,z) (3.10)

where e ;5 the mean concentration. A further simplif;cat;on ar;sesfrom the magnitude of the fr;ct;on. For relal;stic systems the Brownian

-srelaxat;on time tB; mico ;s of the order of 10 sec, whereas thequasi-elastic light scattering experiment only resol ves times of the-,order of 10 sec and 10nger. Therefore, S(k,z) becomes

~5(1,z) =z +

S(k)m cT'(k) k'

k' ~Co + e T1uC~,z)

(3.11)

where cT(k) = (kBT/m 5(k») Y,macropart;cle system. Now, the

;5 the isothermal sound velocity of thetask of a molecular transport theory ;5

reduced to a calculat;on of the longitudinal v;scos;ty funct;on.

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One can use the result (3.11) to characterize measured correla-tion funct;ons. A camman method to describe the short-t;me behavior ;5a cumulant expans;on

(3.12)"n(k) nJ--tn!= S(k) expS(k,t) [- L

n=1

The first two cumulants which describe the initial slope of ln(S(k,t)/S(k)) and its first correction are from (3.11)

k T k'B',S(k)

° k'= S(k) , 0eff(k) k' (3.13 )

(3.14)

Eq. (3.13) ;5 a well-known result15 which for 5mal1 wavevectors g;vesthe collective diffusion coefficient as Oc = 0o/S(O) . It shows inparticular that O can be ealculated entirely by SeO) , a stat;c quan-r.tity. The second cumulant (3.14) is determined by the initial value ofthe dynamic v;scosity function ~'I(~.t) • which arises trom potentialinteractions. The short-time part of this funct;on describes the ela-stic behavior of the colloidal solution at high frequenc;es. Therefore,the deviat;on of S{k,t) from apure exponential of t is connectedwith v;scoelastic propert;es of the collo;dal system.

Th;s last point becomes even more pronounced, if one representsthe full time decay of S(k,t) by a mean relaxation time, defined by

- f ill...lll(k) - S(k) dto

~S(k,z=O)S(k) (3.15 )

(3.16 )

Th;s quantity can eas;ly be determ;ned from exper;ment and is on theother hand g;ven from (3.11) by

'o + ",,(k,z=O) k'/c= --------

m cT'(k) k'

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Here,

~.(k,z=O) = 1 dt D.(k,t) - D.(k)o

(3.17)

Is the time Integral of D,,(k,t) , whleh Is just the k-dependent lon-gitudinal v;scos;ty. For k=O we have 11 •• = T'JB + (4/3) IlS • where nBand DS denote the bulk and shear viscoslty, respeetlvely. If ".(k)would vanish, l(k) would be equal to the reciprocal of the first cu-mulant. Therefore, a suitable measure of the non-exponential behaviorof S(k,t) ean be defined by

6(k) =

-,v,(k) - T (k)

v, (k)(3.18)

whieh is glven from (3.13) and (3.16) by

'(k) = ------e 'o + k' D.(k)

(3.19)

Clearly, ,(k ~ O) ~ O , In a9reement wtth the valldlty of eq. (3.4)in this limito

3.3 Molecular hydrodynamics tor interacting macropart;clesThe Quasi-phenomenological approach presented aboye for the case

of the concentration autocorrelation function can be put on a bas;swhich starts froro a molecular level. What ;5 needed ;5 the time depen-dent distribution funct;on of the positions ~1""'~ and the momen-ta P-l' ...• ~ of a system of N ;nteract;ng macroparticles,f(E." ...,E¡;,I" ...,!:¡;,t), f(r,t) . Its equatlon of motlon Is a many-part;cle Fakker-Planck equat;on. which has the form16

.f(r,t) = a f(r t).t '

where the Fokker-Planck operator ;5 g;ven by

(3.20)

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a'[,... (kT-+_1J B al?j

(3.21)1m I?j).

Here, f; ;5 the force on macroparticle by al1 other macroparticles,which ar;ses fram direct interact;ons like the hard core, screened Cou-lomb, van der Waals, etc. The hydrodynamic ;nteractions are describedby tensors (, .. which relate the hydrodynam;c force on particle to_'Jthe motían of a11 other part;cles j . S;nce ii ;5 well establishedthat h: 'odynamic ;nteractions are unimportant in dilute but highlycharged systems they will be neglected in what follows, so that [,.. o_1J(, 6 .. 1 • where 1 is the unit tensor and 'o the one-particle fric-o lJ - -tion coeffic;ent; it ;5 however pointed out that the theory includingthese ;nteract;ons can be developed in the same way, bui the correspon-ding express;ons get more complicated13.

The correlation funct;ons which are of pr;mary interest to de-scribe the dynamical behavior of a l;qu;d are the dynam;c structurefactor (concentration autocorrelation function)

the longitudinal and transverse current correlation functions

C,,(~, t) o < j,,(~) ent j" (-~) , e(t)

CL(~,t) o < J•.r~) ent jL( -~) , e(t)

(3.22)

(3.23)

(3.24)

and the one-part;cle propagator

(3.25)

which is of importance for self-diffusion and related s;ngle-particleproperties. The average s in these definitions are over the phase-space

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equilibrium distribution and B(t) is the unit step function. Thephase-space variables entering (3.22) to (3.25) are the concentrationfluctuations

N -ik'r. -ik'r.c(~) I - -1 I - -1= e - < e >

i=1 ithe current f1 uctuat; ons

B-ik'r.

j(t)- -1= Ei e

iand

(3.26)

(3.27)

-ik.rc,(~) = e - -1 (3.28)

where I1

denotes the coordinate of the tagged particle. The compo-nents of l(~) para"e1 and perpendicular to ~ are denoted by j,,(1)and j .••(~) , respective1y.

The equations of mot;an of the basic phase-space variables canbe obtained by operating with the hermitian adjoint Fokker-Planck opera-toro In this way, phase-space versions of the phenomenological conti-nuity equation (3.2), the equat;on of motían (3.7) and the express;on(3.8) for the force dens;ty are derived. The basic difference comparedto the phenomenological approach ;5 that now a11 quantities are givenin terms of molecular parameters, depencting on the ;nteraction poten-tials or static quantities like S(k).

The time derivatives of c(k) and l(k) behave different'y inA+ A+ Athe limit k ~ O . Whereas n c(~) ~ O for k ~ O, we have n j(~)'O

in this limit due to the presence of friction. This ;5 one of the eha-racter;stic differences of a liquid of Brown;an particles as campa redto a simple liquido Whereas the van;shing of ñ+ c(~) for k ~ O ex-presses the conservation of the total numbe of particles, the finitevalue ter ñ+ j(!) tor k ~ O reflects the fact that the momentum den-

5204

sity of the Brownian particles ;5 no longer conserved. Indeed, mamen-tum and energy are continuously be;ng exchanged between the subsystemof interacting macroparticles and the solvento

Having the equat;ons of matian of the bas;c phase-space variableswhich enter the correlation functions of interest, the latter can becalculated by methods wh;ch are well established ;n the theory of s;mp-le 1;qu;ds12. Taking the time derivative of (3.26) and ;ntroducingLaplace transforms, one finds

i kZ S(k,z) = S(k) - N- ~ _1 A

• < i(k) [z-oJ c(-k) > (3.29)

Since the time der;vative of the concentration fluctuations vanishesfor k ~ O , the c(~) are the slowest variables of the system. Employ-;n9 the projection operator technique of Mor; and Zwanz;g, we there-fore define a projector P • which projects an arbitrary phase-space_ ef~nct;on A(~) on the concentration fluctuations. Us;ng the Mori-Zwanz;g theory leads to express;on (3.6) for S(k,z), where ;n contrastto the phenomenolog;cal approach the funct;on D(k,z);s now g;ven ;nterms of a correlat;on funct;on

l-P e (3.30)

Th;s ;s a longitudinal current correlat;on funct;on, but the d;fferenceof th;s express;on compared to (3.23) ;s the appearance of Qc;n theresolvent operator. In the language of the project;on operator forma-l;sm, eq. (3.6) ;s a memory equat;on for the concentrat;on correlat;onfunct;on and the general;zed d;ffus;on funct;on D(~,z) ;s essent;allythe memory function of S(k,z) .

To proceed further, the Mor;-Zwanz;g method can be appl;ed to(3.30) ;n a similar way as ;t has be en used aboye for S(~,z).To de-rive a memory equat;on for D(~,z) one defines a project;on operatorPj which projects anta the longitudinal current fluctuat;ons. The re-sult is the general;zed Stokes-Einstein relation (3.9), where the new

5205

memory funct;on lS essentially the longitudinal fr;et;an funct;onc,,(k,z) . far which this procedure g;ves an explicit expression in

ter~s of a correlation function13. This express;on for the dynamicalfr;et;an function can be rewritten in the forro of the phenomenologicalequat;on (3.10), where nQW the longitudinal v;scosity funct;on is ex-plicitly given byl3

(3.31)

Here k has been chosen in the z-direction and ózz(~) ;5 the (zz).component of the stress tensor, so that ~1I(Lz) ;5 just a stress-stress correlation function, and the stress ar;ses from the potentialsbetween Brownian part;cles.

The final task of the theory is to ealculate the v;scos;ty func-tion ~1I(!.t). This has been done by a mode-coupling approximation.The projection operator Q in (3.31) projects on the subspace orthogo-nal to our basic phase-space variables and consists, ;n the sp;rit ofthe Mori-2wanzig formalismo of a11 fast variables. The idea behind themode-coupl;ng apprax;matian17 is ta appraximate Q by an aperatarwhich prajects anta the slawest af these fast variables wh;ch ;n aurcase are bilinear praducts af the concentrat;on fluctuat;ons. The re-sult of this procedure ;s13

(3.32)

where the kernel K(~,~') can be expressed in terms of the staticstructure factor S(k). Therefare, the v;scosity function ;s given bythe stat;c structure factor and the dynamic structure factor S(~,t),

and eqs. (3.11) and (3.32) form a self-consistent set of coupled equa-t;ans. The input ;s the knowledge of the static prapert;es g;ven byS(k) which ene can take either frem static experiments or from theoriesas descr;bed in the first part of this papero The advantage of the made-coupling theory ;s the reduction of the dynamics ta the exact statics;

S206

no adjustable parameters appear ;n the dynamical theory.The viseos;ty funetion has been ealeulated by replae;ng S(k,t)

;n (3.32) by the mean-f;eld express;on S(h,t) = S(k) eXP(-Ook2t/S(k))and us;ng for S(k) the Hansen-Hayter6 model for systems of h;ghlycharged polystyrene spheres. A sens;tive test of the theory ;5 the com-parison of the quant;ty ;(k), eq. (3.19), wh;eh ;s the measure of thedev;at;on of the measured correlat;on funct;on from apure exponential.Perform;ng the time integral of (3.32) g;ves, aeeord;ng to (3.17), thelongitudinal k-dependent stat;c viscosity. which determines 6(k).Comparing theory and experiment and keep;ng ;n mind that no adjustableparameters are involved the agreement is faund to be good.

The physical reason tor the non-exponential behavior of the mea-sured correlation function ;n quasi-elastic light scattering ;5 thecoupling of the concentrat;on fluctuations to the stress fluctuat;onsof the highly eorrelated l;qu;d of ;nteraeting eollo;dal partieles,as deser;bed by the result (3.11) for S(k,z). The stress fluetuationsreflect the non-trivial viscoelastic behavior of the system, as descri-bed by the fu11 time and waveveetor dependenee of the v;seosity (3.32).On a short time scale the systems behaves elastic and at long timesthere is viscous flow. This property is reflected in the appearance ofthe initial value ~n(~,t;O)of the viscosity function in the secondcumulant, whereas the mean relaxation time ~(k) depends on the timeintegral of n,,(k,t).

From this short outline of the molecular theory for the concen-tration eorrelation function it should be elear how the quasi-phenome-nological approach of Section 3.2 can be based on a more microscopiclevel. Essentially the same procedure has been used13 to caleulate one-partiele properties by apply;ng the Mor;-Zwanz;g method to the one-par-tiele propagator G(k.t), eq. (3.25). In this way results were ob-tained for the mean-square displacement W(t) • the veloeity autocorre-lation function of a tagged particle and the self-diffusion coeffieientas functions of the eoncentration and the interaction strength, whichcan be changed by adding salto

~07

In conclusion, the theoretical understanding of charged sphericalparticles ;5 well developed. In contrasto much less ;5 known aboutcharged rodlike particles, which are quite important, since many ob-jects of biophysical ;nterest, such as viruses and polyelectrolytes.are often hav;ng such shapes. It is a150 known that in certain micellarsystems apprec;able m;cellar growth of spheres into rods can happen.For these objects even the static properties are not well understood.4. REFERENCES

l. Pecora, R., ed.: Oynamic Li9ht Scattering, Plenum 1985.Oeg;org;o, V. and Cort;. M., eds.: Phys;cs of Amphiphiles:Micelles, Vesicles and Microemulsions, Proc. Intern. Schoolof Physics, Varenna, Course XC, North-Holland 1985.

2. Hayter, J.B. and Penfold, J., Mol. Phys. 42,109(1981).3. Hansen, J.P. and McOonald, I.R.: Theory of Simple Liquids,

Academic Press 1976.4. Waisman, E., Mol. Phys. 25,45(1973).5. H~ye, J.S. and Blum, L., J. Stat. Phys. 1&,399(1977).6. Hansen, J.P. and Hayter, J.B., Mol. Phys. 46, 651 (1982).7. Nagele, G., Klein, R. and Medina-Noyola, M., J. Chem. Phys.

83, ~560 (1985).8. Missel, P.J., Mazer, N.A., Benedek, G.B., Youn9, C.Y. and

Carey, M.C., J. Phys. Chem. 84, 1044 (1980).9. Bendedouch, D., Chen, S.-H. and Koehler, W.C., J. Phys. Chem.

87, 2621 (1983). Bendedouch, O. and Chen, S.-H., ibid. 88,648 (1984).

ID. Klein, R., Nagele, G. and Hess. W., in: Physical Optics ofDynamical Phenomena and Processes in Mcromolecular Systems,Sedlacek, B., ed., p. 335, de Gruyter 1985.

11. Nlgele, G. and Klein, R., unpublished.

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12. Boon, J.P. and Yip, S.: Molecular Hydrodynamics, McGraw-Hill, 1980.

13. Hess, W. and Klein, R., Adv. Phys. 32, 173 (1983).14. Pusey, P.N., J. Phys. A 11, 119 (1978); Dalberg, P.S.,

Boe, A., Strand, K.A. and Sikkeland, T., J. Chem. Phys. 69,5473 (1978); Grüner, F. and Lehmann, W., J. Phys. A lf, L303(1979).

15. Pusey, P.N., J. Phys. A ~, 1433 (1975); Ackerson, B.J.,J. Chem. Phys. 64, 242 (1976).

16. Oeutch, J.M. and Oppenheim, l., J. Chem. Phys. 54, 3547(1971);Murphy, T.J. and Aguirre, J.L., ibid. 57, 2098 (1972).

17. Keyes, T., in: Statistical Mechanics B, Berne, B.J., ed.,p. 259, Plenum 1977.