5Rheological Model J Mol Liq 2014

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Rheological model for micelles in solution from molecular dynamics

E.O. Castrejn-Gonzlez a,, V.E. Mrquez Baos a, J.F. Javier Alvarado a, V. Rico-Ramrez a,J. Castillo-Tejas b, H. Jimnez-Islas c

a Departamento de Ingeniera Qumica, Instituto Tecnolgico de Celaya, Celaya Gto. 38010, Mexicob Facultad de Ciencias Bsicas, Ingeniera y Tecnologa, Universidad Autnoma de Tlaxcala, Apizaco Tlax., 90300, Mexicoc Departamento de Ingeniera Bioqumica, Instituto Tecnolgico de Celaya, Celaya Gto. 38010, Mexico

a b s t r a c ta r t i c l e i n f o

Article history:

Received 24 February 2014

Received in revised form 29 June 2014

Accepted 14 July 2014

Available online 23 July 2014

Keywords:

Rheology

Micelles

Structure factor

Molecular dynamics

Shear ow

Non-equilibrium molecular dynamics was performed to determine the rheological behavior of micelles insolution. Different concentrations of surfactants molecules immersed into two types of solvents were studied.

Three regions in the ow curve were obtained: i) Arst Newtonian plateau, ii) a second shear-thinning region

and iii) a third shear-thickening region. Data from ow curve were well tted to a proposed model based on

the CarreauYasuda equation. Molecular snapshots and static structure factor were determined to complement

the structure information. Results are in qualitative agreement with experimental data.

2014 Elsevier B.V. All rights reserved.

1. Introduction

An interesting topic for researchers[ 1]has recently emerged.A complex uid is characterized by its microstructure and it is generally

measured through scattering techniques such as small angle light,

neutron, or X-ray scattering[2]. Colloids, liquid crystals, polymers and

micelles are some examples of complexuids[2,3]. In complexuids,

spontaneous formation of molecularaggregation canoccur when a strong

asymmetry between the solute and solvent molecules is presented [4,5];

such as in the case of surfactants.

Surfactant molecules are amphiphilic, as they posses both hydro-

philic heads and hydrophobic tails. When the amphiphilic molecule

concentration in aqueous solution is larger than a critical micelle

concentration (CMC) micellar aggregates are formed.

The shape of the aggregates depends on several factors such as:

the molecular interactions, the physical-chemistry conditions of the

solution, the temperature and, if it were the case, the ow conditions.

One of such self-assembled aggregates is known in the literature as

wormlike micelles[6].

Micellar solutions have been used as rheology modiers in paints,

detergents, pharmaceutical products, lubricants and emulsions where

properties of theuids are strictly controlled[7]. Also, micellar solutions

have been widely used in drying agrochemical products, ink-jet print-

ing, reduction of turbulence resistance and enhanced oil recovery[8,9].

The rheological behavior of surfactants is a consequence of the

microstructure of micelles and scission-recombination kinetics.

Although several experimental studies have been developed[8,10

12],a full understanding of this mechanism remains elusive[13].

Due to the wide industrial and commercial application of the

micellar solutions, a detailed comprehension of the behavior at

different ow regions is required[14].

Molecular simulations have been an important tool to understand

the ow behavior of complex liquids. Particle based simulations of

wormlike micelles may be performed on many different lengths and

time scales, from the atomistic to the mesoscopic. To realistically simu-

late the rheology, one would ideally use an atomistic model (all-atoms)

or a model in which each amphiphilic molecule was individually repre-

sentedby a properly coarse-grained version of it [15]. Thissimplication

can considerably reduce the compute time for a very long time scales.

In recent years, several surfactant models have been proposed

[1624] to reproduce the self-assembled aggregates; however,

rheological description of these systems is not widely understood.

Krguer and Makhlou [25] proposed theFENE-C (nitely extensible

nonlinear elastic, C forcut) potentialto simulate the scission and recom-

bination process. With the same potential mentioned above, Padding

and Boek[26]studied the inuence of the shear ow on ring formation

in a micellar solution; however, they did not observe shear-thickening,

which is actually encountered in experimental data[2729].

Castillo-Tejas et al.[30]studied wormlike micelles in systems

constituted by united atom model immersed in a LennardJones

solvent, where they found a shear-thickening behavior at high

shear rates; however, the study lacks structural information.

Journal of Molecular Liquids 198 (2014) 8493

Corresponding author. Tel.: +52 461 6117575.

E-mail address:[email protected](E.O. Castrejn-Gonzlez).

http://dx.doi.org/10.1016/j.molliq.2014.07.016

0167-7322/ 2014 Elsevier B.V. All rights reserved.

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The aim of this study is to increase the information about the

structure of micellar solutions under shear ow and to propose a new

rheological model to t the complex behavior of these systems.

2. Methodology

2.1. Model and simulation

A united atom model was used to describe the surfactant molecule;the tail was considered as four hydrophobic sites with a size of ,

whereas the head was an hydrophilic site with size 41/3. Surfactant

molecules were immersed in two types of solvents, trying to reproduce

the qualitatively behavior of both water and oil. Each solvent particle

was considered as a site of size.

The truncated and shifted LennardJones (LJ) potential given by

Eq.(1) was used to model pair interactions between particles with

same afnity which include: oiloil, waterwater, tailtail, headhead,

waterhead, and oiltail interactions.

ULJij 4

rij

!12

rij

!6#

; rijrc 0; rijNrc

"( 1

the cut-off ratio for the solvent-solvent interaction was

xed torc= 2.5. To reproduce the chemical bond between sites forming the

molecule, a harmonic potential is used:

Ub rij

K rijij

2

2

whereKis an energy-level parameter, and ijis calculated using the

LorentzBerthelot combining rules given byij= 0.5(ii+jj).

Repulsive potential (R) is given by:

URij

1:05ijrij

!9

: 3

The above potentials are implemented in LAMMPS (Large-scale

Atomic/Molecular Massively Parallel Simulator)[31]; parameters areincluded inTable 1.Fig. 1summarizes the particle interactions.

NEMD simulations in the NVT ensemble were performed using the

SLLOD equations of motion for a homogeneous shear ow[32], which is

equivalent to the p-SLLOD equations for planar Couetteow[33]. The

NosHoover thermostat[34,35]was applied to keep the temperature

constant. The equations of motion are:

qiapiamia

qia v 4

pia Fiapia v pia 5

pQ

; Q3NkBT 2

6

pX

i

Xa

p2iamia

3NkBT; 7

where subscripts i and a are used to distinguish molecules from

particles, respectively, and the dot notation means time derivative.

The symbolsqia,pia,Fia, andmiarepresent the position, momentum

and force vectors and the mass of particleain moleculei, respectively.

N is the total number of particles, T is the temperature, and kB is

the Boltzmann constant. Q, , pand represent the inertial mass,coordinates, momenta and dimensionless time, respectively, of the

NosHoover thermostat. The velocity gradient tensorvin shear

ow is given by

v0

0

000

000

24

35 8

where is the applied shear rate.

The wholeset of variablesare in reduced units;the relevant parameters

are: = 3, T = kBT/, U = U/, P = P3/, t = t(/2m)1/2,

m=2 1=2

, and =2/(m)1/2, whereis the local density,

Uis the energy, Pis the pressure tensor, is the viscosity and tis the

time. For simplicity, hereafter the asterisk notation will be omitted.

2.2. Structure factor

The structure factor S(k) is dened as the autocorrelation

function:[36]

Sk 1

N kkh i 9

where

kXNj1

exp ikrj

10

is the Fourier transform of the microscopic (total) density and rjdenotes the position of particlej , with 1 j N. This equation,

along with the Euler's identity exp( ia) = cos(a) isin(a) allows

computing static structure factors from molecular simulation. The

minimumkvector for a box of lengthL for the Nparticle system

is (2/L)[37]and therefore, the components of the kvector are

restricted, due to periodic boundary conditions, to multiples of

(2/L). Ifkhas only componentsx and y, then:

k kx^xky^y nx2

L

^xny

2

L

^y 11

wherenxand nyare integer numbers.

The structure factor can be measured directly through radiation

scattering experiments; in particular, from Static Light Scattering(SLS). This experimental technique has been a useful tool to elucidate

the molecular structure of materials. It has been used to determine

density uctuations of a system due to external light perturbation

with a wavelength of 2/k. In particular, the SLS has been widely used

to characterize micelles in solution[11,38,39].The radius of gyration

Rgand the second virial coefcientB2are some of properties that can

be obtained from SLS experiments.

In a dilute solution formed by npchains (np N N1), withNsites per

molecule, the static structure factor, also called scattering function, for

a single molecule can be obtained from Eq.(12)[40]:

S1 k 1

NXN

j;k1

exp ik rirj h iD E 12

Table 1

Dimensionless variables and parameters used in the simulations.

Variable Symbol Value

Particle mass m 1.0

Particle tail and solvents diameter 1.0

Particle head diameter 41/3

Energy 1.0

Density 0.84

Temperature T 1.35

K constant K 30

Maximum bond length R0 1.5

85E.O. Castrejn-Gonzlez et al. / Journal of Molecular Liquids 198 (2014) 8493


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Head

Hydrophilic

Tail

Hydrophobic

type 1

type 2

L-J

R

4 1/3

R L-JL-J

L-J

L-J

R

L-JR

water like

oil like

Fig. 1.Particle interaction scheme. Red circles correspond to hydrophilic heads and blue circles correspond to h ydrophobic tail beads in the surfactant molecules. Two types of solvent

are represented by light (1) and dark (2) circles being water like (W) and oil like (O), respectively. Interactions between tails and heads, oil like and water like, oil like and heads, and

water likeand tailare represented by a repulsive potential (R) (see Eq. (1)). Interactions between related particles are represented by truncated and shifted the LennardJones potential

(Eq.(2)) whereas bonded chains are represented by a harmonic potential (Eq.(3)). (For interpretation of the references to color in this gure legend, the reader is referred to the web

version of this article.)

Fig. 2. Molecular snapshots of = 0.23at slow shearratefora) 0,b) 10, c)50 andd) 70% ofO solvent.Biggest(red)spheres correspond toheads;bluespheres representthetails, grayand

brown spheres areWand O solventparticles,respectively.Micellar aggregatesare welldened.(For interpretation of thereferencesto color in thisgurelegend, the reader is referred to

the web version of this article.)

86 E.O. Castrejn-Gonzlez et al. / Journal of Molecular Liquids 198 (2014) 8493


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( )1Xk

( )S k(

)

S

k

a)

( )S k

b)

( )S k

(

)

S

k

c) d)

( )S k

(

)

1

Yk

( )1Xk

(

)

1

Yk

Fig. 3.Static structure factor in XY plane projection for = 0.23 at zero shear rate for a) 0, b) 10, c) 50 and d) 70% of O solvent.

a b

c d

Fig. 4. Shear viscosity vs.shear rate forthe 0.01(),0.02 (),0.03 (),and 0.04 () solutions at a) 0%,b) 10%, c) 50%and d) 70%of O solvent,respectively. Proposedrheological modelts

well the simulated data in the whole shear rates region.



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wherekis the scattering vector which is the difference between the

scattered beam wave vectorksand the incident beam wave vectorko.

For nite concentrations, the interference between sites of neighboring

chains must be taken into account. In this case, the structure factor is

given by

Sk S1 k npN

XNi;j1

exp ik r1ik2j

h iD E: 13

The second term adds up the correlations between nearby

molecules 1 and 2. At low concentrations, however, the statistical

average for different molecules is mostly zero, andS(k) becomes

identical toS1(k).

2.3. Rheological model proposed

Complex rheological behavior is presented in various systems,

such as Newtonian, shear thinning and shear thickening simultaneous

effects which are presented in micellar solutions [30,41], colloids

and suspensions[4244]and some polymer solutions[45,46]; some

complex behaviors such as second shear thinning region are presented

in cellulose and ionic liquid mixtures[47,48].The lack of rheological

Table 2

Parameters of the modied CarreauYasuda model that ts simulation data.

Oil % 0 min K a b n1 n2

0% 1 0.8770 0.5027 0.0470 0.1983 0.0055 2.2324 3.6471 0.2776 2.0052

2 0.9580 0.8900 0.0789 0.2308 0.0091 2.2921 5.6154 0.2187 1.6810

3 1.0300 1.1500 0.0789 0.2212 0.0065 2.4116 3.5801 0.2225 1.7099

4 1.3200 2.2800 0.2000 0.2489 0.0085 2.8041 4.6986 0.2485 1.7273

10% 1 0.9020 0.7100 0.0711 0.2211 0.0065 2.2248 3.9423 0.2843 1.9293

2 1.0070 1.0700 0.1000 0.2306 0.0071 2.0060 3.8298 0.2892 1.9310

3 1.0600 1.5500 0.1382 0.2250 0.0080 2.2821 4.1279 0.2833 1.92974 1.3500 2.9900 0.2700 0.2550 0.0105 2.5890 5.6273 0.3006 1.9427

50% 1 0.9650 0.7400 0.0613 0.2312 0.0075 1.6877 5.1130 0.2909 1.9489

2 1.0020 1.3000 0.1000 0.2196 0.0095 2.0030 5.2340 0.3073 1.3346

3 1.0600 1.2000 0.1000 0.2129 0.0079 2.2001 3.9485 0.3120 1.3585

4 1.4200 3.7320 0.2680 0.2483 0.0130 1.9555 5.8000 0.3383 1.1314

70% 1 0.9400 0.7100 0.0700 0.2247 0.0068 1.8675 3.2665 0.2700 1.9601

2 1.0000 1.0700 0.0700 0.2039 0.0075 2.3427 4.1853 0.2915 1.9829

3 1.1100 1.6350 0.1430 0.2125 0.0107 2.5399 5.7135 0.3300 2.6572

4 1.4000 3.6500 0.2630 0.2328 0.0100 2.2977 5.7849 0.3497 10.7768

a b

d c

Fig. 5.Snapshots for= 0.23 and 0% ofOsolvent particles for a)= 1, b)

= 4, c)

= 20 and d)

= 100.



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models in the literature for representing, simultaneously, more than

two changes in the

ow curve with only one equation, has been the mo-tivation to propose a new rheological model. In fact, our model is

intended tot well experimental data even in the cases of four or ve

ow curve changes.

Considerations used to describe the shear behavior of the micellar

solutions that include shear-thickening and shear-thinning behavior

are based on a CarreauYasuda[49,50]model. It is well known that

the CarreauYasuda equation has proved to be an excellent model to

t thewhole shear rate interval forpolymer solutionswith both regions,

Newtonian and Shear thinning[51].The equation is given by:

0

1 2h ip 14

where 0 is the zero shear rate viscosity, is a characteristic timeconstant,is the shear rate and p is a numerical exponent. However,

it is not possible to reproduce the whole shear behavior in micellar

solutions. We proposed the following model, where it is possible to

reproduceve regions in the ow curve, such as rst Newtonian pla-

teau, two shear thinning or shear thickening regions, intermediate

Newtonian region and a last Newtonian plateau at high shear rates:

0in1 1

an1 in1 2

bm1 15

where 0 is the viscosity at rst Newtonian plateau, in is the

intermediate Newtonian viscosity,is the viscosity at last Newtonian

plateau;1and2are the relaxation times. If b 1=1the rst Newto-

nian region is depicted, else if

N

1=2 , the last Newtonian plateau

is presented. Parametersaandbare used to describe the transitions

between constant viscosity and the shear thinning regions with apower law behavior. Finally n and mparameters are the exponents

analogous to power law expressions. If Newtonian plateaus 0and

are unknown, then these could be used as adjustable parameters.

2.4. Material functions

The pressure tensorPis given by a sum of site-site contributions:

Pkl 1

V

Xi

pkiplimi

1

2

Xi

XjNi

rkij Flij

24

35 16

wherePklis the pressure tensor component acting alongl direction

through a normal plane to thekaxis,pkiandpliare thekandlcompo-

nents of the momentum of particle i and mi its associated mass.Further-

more, Vis the system volume, rkij is the k-component of the scalar

distance rij, and Flij is the l-component of the force resulting from the in-

teraction between particlesiandj. Therst term of Eq.(16)represents

the kinetic contribution, and the second arises from pair interactions.

For the system studied here, the main contribution to pressure tensor

stems from the pair potentials.

The shear viscosityis dened by:

Pxy

D E

17

wherePxyis the average of thexycomponent of Pressure tensor over

the time in the production step.

ba

dc

(

)

1

yk

(

)

1

yk

( )1xk ( )1xk

(

)

,0

x

S

k(

)

,0

X

S

k

(

)

,0

y

S

k(

)

,0

X

S

k

( )0,kyS ( )0,kyS

( )0,kYS ( )0,kyS

Flow direction

Fig. 6.Static structure factor for = 0.23 and 0% ofOsolvent particles for a) = 1, b) = 4, c) = 20 and d) = 100.



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3. Results and discussion

Four surfactant concentrations were studied: 0.05, 0.10, 0.15 and0.20; all of them were immersed into two types of solvent,W-like and

O-like; four proportions ofO and Wsolvents were studied 0, 10, 50

and 70% ofOsolvent. The concentrations were calculated by Eq.(18):

NchLch

NchLchNS18

whereNch is the number of surfactant molecules with a lengthLch;NSis

constituted by two types of solvent,NS= SW+ SO, whereSWis the

number of water-like solvent sites and SO is the number of oil-like

solvent sites.

All of the systems were composed of 12, 000 sites at reduced

temperature and density of 1.35 and 0.66, respectively. To generatedifferent concentrations, we varied the number of surfactant molecules

Nchfrom 120 to 580.

Since a simple model potential was used, it was possible to develop

large simulations achieving to verify the micellar assemblies.

3.1. Equilibrium molecular dynamics

Molecular structures at equilibrium conditions ( 0) are shown

inFig. 2. InFig. 2-a) wormlike micelles are well dened, heads point

out to the aggregate because of the afnity with the solvent. As O

particles increase, the worm shape of aggregates varies from wormlike

(0% ofO) to lamellar shape (70% ofO particles). InFig. 2-b) 10% ofO

solvent is included andit is encapsulatedinto micelles;Fig.2-c) includes

50% of O solvent, which is also encapsulated into the cylindrical

micelles; andFig. 2-d) contains 70% ofO solvent; the shape of the

aggregate is a lamella that encapsulates theWparticles.

The corresponding structure factors for the equilibrium microstruc-tures were calculated; results are depicted in Fig. 3. The contribution

of the solvent to the structure factor has been omitted, and only the

coordinates of the surfactant molecules are considered for its

evaluation. In Fig. 3-a) (0% ofO solvent) the structure factor presents

two broad peaks oriented at an angle of 45. The axis of symmetry is

shown as a white line, on which the structure factor is projected as a

one-dimension prole. The formation of these peaks is the result of the

solution's anisotropy caused by the micellar aggregates. In Fig. 3-b), the

scattering pattern of 10% ofOsolvent system is shown; there are two

symmetric regions of peaks with a maximum value ofS(kx,ky) = 6.76

in the y-direction. Along the x-direction, there are also two incipient

peaks located in the center of the pattern with a maximum value

ofS(kx,ky) = 2.53. Further, in this contour plot, two white lines are

included in bothx andy directions, on which the one dimension proleof the structure factor is shown. The shapes of these peaks are in

qualitative agreement with SANS (Small Angle Neutron Scattering)

experiments[52].

In Fig. 3-c) and -d), thestructure factors of 50% and 70% ofO solvent

are depicted. A white line wasplotted on the scattering pattern in order

to visualize the orientation of the highest peaks.Also, these peaks are an

indicative of the well-dened structures; namely, cylinder and lamella,

such as those shown inFig. 2-c) and -d), respectively.

3.2. Non-equilibrium molecular dynamics

Fig. 4shows the rheological behavior of the four systems with a

proportion of (a) 0%, (b) 10%, (c) 50% and (d) 70% ofO solvent. It is

possible to nd three regions in each ow curve i) rst Newtonian

a b

c d

Fig. 7. Snapshots for = 0.23and 10% ofO solventparticlesfor a) = 1 (Newtonian region),b) = 4 (Shear-thinning region),c) = 10(At theminimumshear viscosity)and d) = 60

(Shear-thickening region).



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region, ii) shear thinning region and iii) shear thickening region, and

some of the ow curves tend to present a second Newtonian region.Proposed rheological model (see Eq. (15)) ts well to all the data

regions.Table 2contains thetting parameters of the proposed model

for all of the systems analyzed here.

Despite the evident differences between the equilibrium structures

shown inFig. 2, there are no signicant differences in the rheograms

depicted inFig. 4. The zero shear rate viscosities,0, are almost the

same, as those listed inTable 2. This behavior is a consequence of two

main considerations: the rst is that the densities of all of the systems

studied are the same. Notice that the mass of each particle is equal to

1.0, with no distinction between the solvent-like particles. The second

consideration is related to the applied force eld; the molecular model

is simpleusingonly repulsivepotential and theLennardJones potential

to represent the particle interactions. The aforementioned molecular

model considers a weak intermolecular interaction, so it does notcause a signicant change in the viscosity values as related to the

molecular structure. On the other hand, the effect of surfactant con-

centration generates a considerable change in the viscosities values.

For the concentration: =0.23and0%ofO-solvent, the effectof the

shear rate on molecular structures is shown inFig. 5. Snapshots of the

system are depicted for the key shear rates. Fig. 5-a) shows the

well-dened micelles. As the shear rate increases, the micelles align in

the ow direction until lower viscosity is reached. At this point, the

micelles are subject to a disaggregation process where the surfactant

molecules are being dispersed across the volume of simulation

[see Fig. 5-c)]. This break-dispersion mechanism generates the

shear-thickening phenomenon, as reported by Edwards et al. [45].

Complementary structural informationis presented in Fig. 6 only for

the highest concentration,= 0.23, and 0% ofOsolvent, at the same

shear rates as inFig. 5. At low shear rate ( 1:0) the structure factor

has peaks in the center of the pattern, which corresponds to welldened wormlike micelle; as the shear rate is increasing, the peaks

move away from the center and new peaks emerge in the gradient

direction(y-axis); which is an indicative that thestructure of aggregates

has been deformed. For the lowest apparent viscosity [seeFig. 6-c)]

there are many peaks aligned in the y-axis and small peaks begin to

emerge in the x-direction; these small peaks become higher inFig. 6-d),

which suggests that molecular dispersion is major.

As illustrations, snapshots for the highest concentration of the

surfactant with different proportions of O solvent are depicted in

Figs. 7, 8 and 9. When a 10% ofOparticles is added, these particles are

encapsulated into thin cylindrical micelles [seeFig. 2-b)]. At low shear

rates, the cylinders are aligned into the ow direction keeping the

solvent encapsulated. InFig. 7we present the snapshots of the system

at the shear rates of 1:0; 4:0; 10:0 and 60:0 ; corresponding toNewtonian plateau, shear-thinning region, minimum viscosity, and

shear-thickening region, respectively. In the thinning region the cylin-

ders are aligned into the ow direction, and surfactant molecules

begin to separate from the aggregates. The shear-thinning region is

apparently associated to the rupture of the wormlike micelles and also

to the subsequent banded distribution of the surfactant molecules

along to the ow direction[30,53].

The system with 50% ofOsolvent begins with a cylindrical struc-

ture when equilibrium conditions are presented [see Fig. 2-c)].

In this gure, theO solvent is encapsulated into the cylinder with

the hydrophilic heads pointing out to the micelle. The shear thinning

region appears at the (low) shear rate of gamma = 1.0 where the

micelle is well dened and the cylinder is oriented into the ow

direction, as shown inFig. 8-a). Only in the

rst snapshot the O

a b

c d

Fig. 8.Snapshots for= 0.23 and 50% ofOsolvent particles for a) = 1 (At the starting of the thinning region), b) = 4 (Shear-thinning region), c) = 20 (At the minimum shear

viscosity) and d) = 100 (Shear-thickening region).



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solvent particles are displayed. However, for the sake of clarity these

particles are eliminated fromFig. 8b) to d). In the shear thinning

region, at 4:0 thecylinder is still well dened, but the orientation

in theow direction is more evident [seeFig. 8-a)]. Some surfactant

molecules are even separated from micelles and aligned in the

ow direction; such alignment causes the thinning effect. At the

minimum apparent viscosity, 20, the cylinder is broken and the

surfactant molecules are totally aligned into the ow direction;

however, these surfactant molecules stay at the same position of

the simulation box. At the highest shear rate of 100, the surfactant

molecules disperse across the entire simulation box [seeFig. 8-d)],

causing the shear-thickening behavior[45].

InFig. 9the snapshots for 70% Osolvent are depicted. In this case,

only the rst snapshot displays the Wsolvent particles. At the lowshear rate of 1:0, the ow regime is Newtonian and a lamella is

well dened. In this case, theWsolvent is encapsulated into the micelle

and the head of the surfactant molecules point inward the micelle

[seeFig. 9-a)]. When a shear rate of 4:0 is imposed, the lamella is

deformed and tends to form a cylinder with the head of surfactant

pointing inwards [seeFig. 9b)]. At this ow regime, shear thinning is

observed. At the minimum apparent viscosity, 20, the cylinder is

deformed and some of the surfactant molecules are disaggregated

and aligned in the ow direction [seeFig. 9-c)]. At the high shear rates

of 100, the surfactant molecules are completely disaggregated and

dispersed in the entire simulation box [seeFig. 9-d)]; causing again

the shear-thickening phenomenon.

All the molecular structure snapshots were developed in the OVITO

(The Open Visualization Tool) software[54].

4. Conclusions

NEMD simulations were performedto determinethe rheological be-

havior of micelles in solution. The ow curve obtained presents three

regions i) a Newtonian plateau, ii) a shear-thinning region, and iii) a

shear-thickening region; we propose a rheological model which repro-

duces accurately all the data obtained by molecular dynamics. Despite

the simplicity of the force eld used to simulate surfactants in solution,

the micelle formation is in accordance to the results reported by others

authors. Structurefactoris determined to complement thestructural in-

formation about micelle aggregates; peaks in the scattering pattern

emerge along the gradient direction where shear-thinning is present;

the formation of peaks in the ow direction is observed in the shear-

thickening region. All of these phenomena are a consequence ofow;specically, the shear-thickening effect is a consequence of the micelle

breaking and the subsequent dispersion of the surfactant molecules.

Acknowledgments

The nancial support of DGEST (No. 4529.12-P), CONACYT

(Nos. 129962 and 104672) and UATx (No. CACyPI-UATx-2014),

is gratefully acknowledged.

References

[1] A.P.Deshpande,J.M.Krishnan, S. Kumar,Rheology of ComplexFluids,Springer,2010.[2] R.G. Larson, The Structure and Rheology of Complex Fluids (Topics in Chemical

Engineering), Oxford University Press Inc., 1999.

[3] J.N. Israelachvili, Intermolecular and Surface Forces, University of California, 1991.

a b

c d

Fig. 9.Snapshots for= 0.23 and 70% ofOsolvent particles for a) = 1, clear particles correspond toWsolvent (included for clarity purposes), b) = 4 (Thinning region), c) = 20

(At the minimum shear viscosity) and d) = 100 (Shear-thickening region).

http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0005http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0010http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0010http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0015http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0015http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0010http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0010http://refhub.elsevier.com/S0167-7322(14)00329-8/rf0005


10/10

[4] S. Dietrich, Phase Transitions and Critical Phenomena, Academic Press, 1998.[5] R.J. Hunter, Modern Colloid Science, Oxford University Press, 1993.[6] J.F. Berret, in: P. Terech, R. Weiss (Eds.), Molecular GelsElsevier, 2005.[7] V.J. Anderson, J.R.A. Pearson, E.S. Boek, Rheol. Rev. 2006 (2006) 217253.[8] J.L. Zakin, N.W. Bewerdorff, Rev. Chem. Eng. 14 (1998) 253320.[9] S. Ke, J. Lee, T. Pope, P. Sullivan, E. Nelson, A. Hernandez, T. Olsen, M. Parlar, B.

Powers, A. Roy, A. Wilson, A. Twynam, Oileld Rev. 16 (2004) 1016.[10] I. Coulliet, T. Hughes, G. Maitland, F. Candau, S.J. Candau, Langmuir 20 (2004)

95419550.[11] M.W. Liberatore, F. Nettesheim,P.A. Vasquez, M.E.Helgeson, N.J.Wagner, E.W.Kaler,

L.P. Cook, L. Porcar, Y.T. Hu, J. Rheol. 53 (2009) 441458.

[12] R. Zana, E.W. Kaler, Giant Micelles: Properties and Applications, CRC Press, 2007.[13] C. Zhang, J. Wei, Chem. Eng. Sci. 102 (2013) 544550.[14] J.P. Rothstein, Rheol. Rev. (2008) 146.[15] J.T. Padding, E.S. Boek, W.J. Briels, J. Chem. Phys. 129 (2008) 111 (074903).[16] M. Sammalkorpi, M. Karttunen, M. Haataja, J. Phys. Chem. B 111 (2007)

1172211733.[17] S.J. Marrink, D.P. Tieleman, A.E. Mark, J. Phys. Chem. B 104 (2000) 1216512173.[18] M. Jorge, J. Mol. Struct. 946 (2010) 8893.[19] S. Jalili, M. Akhavan, Colloids Surf. A Physicochem. Eng. Asp. 352 (2009) 99102.[20] P.K. Maiti, Y. Lansac, M.A. Glaser, N.A. Clark, Langmuir 18 (2002) 19081918.[21] D.S. Yakovlev, E.S. Boek, Langmuir 23 (2007) 65886597.[22] B. Fodi, R. Hentschke, Langmuir 16 (2000) 16261633.[23] B.J. Palmer, J. Liu, Langmuir 12 (1996) 746753.[24] J. Gao, W. Ge, G. Hu, J. Li, Langmuir 21 (2005) 52235229.[25] M. Krger, R. Makhlou, Phys. Rev. E. 53 (1996) 25312536.[26] J.T. Padding, E.S. Boek, Phys. Rev. E. 70 (2004) 031502.[27] Y.T. Hu, P. Boltenhagen, D.J. Pine, J. Rheol. 42 (1998) 11851208.[28] Y.T. Hu, Philippe Boltenhagen, D.J. Pine, J. Rheol. 42 (1998) 12091226.[29] V. Lutz-Bueno, J. Kohlbrecher, P. Fischer, Rheol. Acta 52 (2013) 297312.

[30] J. Castillo-Tejas, J.F.J. Alvarado, S. Carro, F. Prez-Villaseor, F. Bautista, O. Manero, J.Non-Newtonian Fluid Mech. 166 (2011) 194207.

[31] S.J. Plimpton, J. Comput. Phys. 117 (1995) 119.[32] D.J. Evans, G.P. Morris, Statistical Mechanics of Non-equilibrium Liquids, Academic,

London, 1990.[33] J.M. Kim, B.J. Edwards, D.J. Keffer, B. Khomami, J. Rheol. 54 (2010) 283310.[34] S. Nos, Mol. Phys. 52 (1984) 255268.[35] W.G. Hoover, Phys. Rev. A 31 (1985) 16951967.[36] J .P. Hansen, I . McDonald, Theory of Simple Liquids, third ed. Elsevier,

Amsterdam, 2006.[37] M. Schoen, R. Vogelsang, C. Hoheisel, Mol. Phys. 57 (1986) 445471.[38] M.Y. Lin, H.J.M.Hanley,S.K.Sinha, G.C. Straty,D.G.Peiffer,M.W.Kim,Phys.Rev.E. 53

(1996) R4302R4305.

[39] I.A. Kadoma, J.W. van Egmond, Phys. Rev. Lett. 76 (1996) 4432

4435.[40] I. Teraoka, Polymer Solutions. An Introduction to Physical Properties, Wiley,New York, 2002.

[41] T. Yamamoto, K. Sawa, K. Mori, J. Rheol. 53 (2009) 13471362.[42] E. Brown, N.A. Forman, C.S. Orellana, H. Zhang, B.W. Maynor, D.E. Betts, J.M.

DeSimone, H.M. Jaeger, Nat. Mater. 9 (2010) 220224.[43] L. Bergstrm, Colloids Surf. A Physicochem. Eng. Asp. 133 (1998) 151155.[44] R.G. Egres, N.J. Wagner, J. Rheol. 49 (2005) 719746.[45] B.J. Edwards, D.J. Keffer, C.W. Reneau, J. Appl. Polym. Sci. 85 (2002) 17141735.[46] K.C. Tam, R.D. Jenkins, M.A. Winnik, D.R. Bassett, Macromolecules 31 (1998)

41494159.[47] H. Song, J. Zhang, Y. Niu, Z. Wang, J. Phys. Chem. B 114 (2010) 60066013.[48] Q.L. Kuang, J.C. Zhao, Y.H. Niu, J. Zhang, Z.G. Wang, J. Phys. Chem. B 112 (2008)

1023410240.[49] P.J. Carreau, Trans. Soc. Rheol. 16 (1972) 99127.[50] K. Yasuda, R.C. Armstrong, R.E. Cohen, Rheol. Acta 20 (1981) 163178.[51] O. Castrejn-Gonzlez, J. Castillo-Tejas, O. Manero, J.F.J. Alvarado, J. Chem. Phys. 138

(2013) 111 (184901).[52] P.S. Goyal, V.K. Aswal, Curr. Sci. Bangalore 80 (2001) 972979.

[53] J.R. Melrose, D.M. Heyes, J. Chem. Phys. 98 (1993) 58735886.[54] A. Stukowski, Model. Simul. Mater. Sci. Eng. 18 (2010) 015012.

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