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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.
Contents lists available at ScienceDirect
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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
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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.
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