In order to understand the long known anomalies in the composition dependence
of diffusion and viscosity of binary mixtures, we introduce here two new models
and carry out extensive molecular dynamic simulations. In these models, the two
molecular species (A and B) have the same diameter and mass. In model I the
inter-species interaction is more attractive than that between the pure
components, while the reverse is true for model II. Simulations and also mode
coupling theory calculations reveal that the models can capture a wide variety
of behavior observed in experiments, most interesting among them are the
non-monotonic variation of diffusion and viscosity with the composition and the
re-entrant viscosity dependence of the relaxation time.Comment: 8 pages, 4 figures; submitted to Physical Review Letter

Bagchi, Biman

Mukherjee, Arnab

Srinivas, Goundla

English

arXiv

In order to understand the long known anomalies in the composition dependence of diffusion and viscosity of binary mixtures, we introduce here two new models and carry out extensive molecular dynamics simulations. In these models, the two molecular species (A and B) have the same diameter and mass. In model I the interspecies interaction is more attractive than that between the pure components, while the reverse is true for model II. Simulations and mode coupling theory calculations reveal that the models can capture a wide variety of behavior observed in experiments, including the reentrant viscosity dependence of relaxation time

Publications of the IAS Fellows

Reentrant Behavior of Relaxation Time with Viscosity at Varying Composition
in Binary Mixtures
Arnab Mukherjee, Goundla Srinivas, and Biman Bagchi*
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore-12, India
In order to understand the long known anomalies in the composition dependence of diffusion and
viscosity of binary mixtures, we introduce here two new models and carry out extensive molecular
dynamics simulations. In these models, the two molecular species (A and B) have the same diameter and
mass. In model I the interspecies interaction is more attractive than that between the pure components,
while the reverse is true for model II. Simulations and mode coupling theory calculations reveal that the
models can capture a wide variety of behavior observed in experiments, including the reentrant viscosity
dependence of relaxation time.Transport properties of binary mixtures often show
strong and baffling dependencies on the composition
which have not been understood or even adequately ad-
dressed in a molecular theory. The well-known Raoult’s
law of classical physical chemistry [1] predicts the fol-
lowing simple linear dependence on the composition for a
given property P,
P  x1P1 1 x2P2 , (1)
where xis are the mole fractions and Pis are the values
of the property P of the pure (single component) liquids.
More often than not, significant deviation from Eq. (1) is
observed. Of many anomalies exhibited by binary mix-
tures, the existence of an extremum (sometimes even a
double extrema) in the composition dependence of excess
viscosity [2] and the reentrant type behavior of the relax-
ation time when plotted against viscosity [3] are certainly
the most remarkable. The latter shows, in a dramatic fash-
ion, that viscosity is not a unique determinant of relax-
ation in binary mixtures [3]. Several interesting theoretical
and computer simulation studies on Lennard-Jones binary
mixtures have been carried out recently [4–6], but these
studies have concentrated mainly on the glass transition in
binary mixtures and considered only one particular com-
position and a unique interaction strength. Earlier Heyes
carried out the extensive equilibrium molecular dynam-
ics (MD) simulations of Lennard-Jones binary mixtures to
study the partial properties of transport coefficients in the
inert gas medium [7], by using the Berthelot mixing rule
and found only a weak nonideality. The nonideality in the
case of inert gas mixtures is small, since their mutual in-
teraction strength (eAB) follows the Berthelot mixing rule.
In order to understand the markedly nonlinear compo-
sition dependence of viscosity and diffusion, here we in-
troduce and study two models (referred to as model I and
model II) of binary mixtures in which the solute-solvent
interaction strength is varied by keeping all the other pa-
rameters unchanged. In our models, all the three inter-
actions (solute-solute, solvent-solvent, and solute-solvent)
are described by the Lennard-Jones (12 2 6) potential,Uij  4eij
∑µ
s
rij
∂12
2
µ
s
rij
∂6∏
, (2)
where i and j denote any two different particles. We set
the diameter (s) and mass (m) of both the solute and the
solvent molecules to unity, for simplicity. The solvent-
solute interaction strength lies in the potential well depth
eAB, where A and B represent the solvent and solute par-
ticles, respectively. Throughout this study we keep the in-
teraction strength eAA  1.0 (solvent-solvent), eBB  0.5
(solute-solute). In the two models we use two different
solvent-solute interaction strength values, namely eAB 
2.0 in model I and eAB  0.3 in model II. So, while
model I is a “structure former” (between A and B),
model II is a “structure breaker.” Model I shows glass
transition behavior similar to the Kob-Andersen model.
One advantage of this model is that only the interaction
energy is varied. These models can also capture the
anomalous inverse correlation between excess viscosity
and excess volume observed in many experiments. We
believe that the models introduced here can serve as
starting points to understand the many baffling properties
of binary mixtures.
In addition to the self-diffusion coefficients (Dsi ) and the
viscosity, we also calculated the mutual (or the inter-) dif-
fusion coefficients (DAB) of the binary mixture at various
compositions. The latter is a subject of long-standing in-
terest, although not much is known about the extent of its
nonideality. The ideal value is given by (D0AB)
D0AB  QxBD
s
A 1 xAD
s
B , (3)
where xi and Dsi are the mole fraction and the self-
diffusion coefficient of the ith species in the mixture, re-
spectively, and Q is the well-known thermodynamic factor
of binary mixing [8]. The simulations presented here
shows a marked departure from Eq. (3), due to cross-
velocity correlations. We believe that this is the first time
that such pronounced nonideality in DAB is established.
It is worthwhile to note that Enskog kinetic theory,
which is applicable to hard sphere systems [9], attempts to
explain in a concise way the composition dependence of
the transport properties in binary mixtures. But unfortu-
nately this theory fails completely to explain the observed
strong nonideality in the composition dependence of dif-
fusion in binary mixture.
Extensive MD (microcanonical ensemble, with the usual
periodic boundary condition) simulations have been car-
ried out with a total of 500 particles for two types of mod-
els by varying the solute mole fraction (that is, of B) from
0 to 1. The reduced temperature T kBTe is set equal
to unity in model I and 1.24 in model II, and the reduced
density r  rs3 is 0.85 in both the models. After
many trial runs to verify the existing results on viscosity
[7] of one component liquids, we have selected a time step
Dt  0.002t for model I andDt  0.001t for model II,
where t  s
p
me. We have dealt with six different sol-
ute compositions, namely 0, 0.2, 0.4, 0.6, 0.8, and 1.0. For
each solute composition we have equilibrated the system
up to 1.5 3 105 steps. Simulations carried out for another
2.0 3 105 steps after the equilibration during which all the
relevant quantities have been calculated. For each compo-
sition, we have run three independent simulations and have
taken averages over them. We have checked all the three
partial radial distribution functions to make sure no phase
separation occurs during simulations (for model II). Vis-
cosity values are obtained by integrating the stress time
correlation function which defines the time dependent vis-
cosity by the following relation:
ht  VkBT 21sxz0sxzt , (4)
where the off-diagonal element of the stress tensor sxz for
binary mixture is defined as
sxz 
NX
j1
pxj p
z
jm 1 F
z
j xj . (5)
Here Fzj is the z component of the force acting on the jth
particle, the corresponding x coordinate is xj , and pzj is
the z component of the momentum of jth particle, m being
the mass of the particle. Among total N particles, solvent
particles are labeled from 1 to N1 and solute particles from
N1 1 1 to N .
Diffusion coefficients are calculated both from the mean
square displacement and velocity autocorrelation function
via the Green-Kubo relation. The results of the simulation
are given in Figs. 1–4. We shall discuss the results after
we describe the mode coupling theory employed.
We have carried out mode coupling theory (MCT) cal-
culations of diffusion and viscosity to understand the simu-
lation results, especially the origin of nonmonotonicity.
These calculations have been carried out by using well-
established expressions [10–14]. Note that for binary mix-
tures at normal density and temperature, the short time
dynamics of the relevant time correlation functions are im-
portant and, in fact, can contribute more than 50% of the
total value, just as for one component liquids. Thus, any
solution of MCT equation requires accurate input of theshort time dynamics. For a given transport property P
[P can be viscosity (h) or friction (z )], MCT formalism
[10–14] assumes the following separation into the short
time, binary collision controlled, contribution Pbint and
the contribution from the collective term, which in dense
liquid is dominated by the density term, Prrt. So the
total dynamical quantity Pt can be written as [10],
Pt  Pbint 1 Prrt , (6)
where the binary term is assumed to be Gaussian asPbin 
Pt  0 exp2t2t2P. The resulting expressions of the
characteristic time tP are complicated and given elsewhere
[15]. Pt  0 values are determined mainly by the static
correlation like the radial distribution functions [gijr] or
the direct correlation functions [cijr] which are obtained
from Ornstein-Zernike equations with soft mean spherical
approximation (SMSA) closure [16]. The mode coupling
contribution to viscosity is assumed to be given by the bi-
nary product of the density terms. In case of binary mix-
ture total MCT term hrr can be expressed as the sum of
four hrirj terms [15(a)]. The short time part of time
dependent friction is calculated from short time expansion
of force-force time correlation function [10–14] and the
mode coupling expression follows Bosse et al. [13]. The
frequency dependent diffusion coefficients [Dsi z] [which
is the same as the Laplace transform of velocity autocorre-
lation function Cyiz] are related to the respective fre-
quency dependent frictions [ziz] according to Einstein
relation,
Dsi z  Cyiz 
kBT
miz 1 ziz
. (7)
viscosity ( )
1.5 2 2.5 3 3.5 4
R
el
ax
at
io
n 
tim
e 
( i
)
10
15
20
25
30
FIG. 1. The simulated values showing the reentrant behavior of
the relaxation times ti are plotted against simulated viscosity for
model I. Filled circles represent tA, while open circles represent
tB. The direction of the arrow shows the increasing solute (B)
composition in both the cases. T   1.0, r  0.85.
0.0 0.2 0.4 0.6 0.8 1.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D
iff
us
io
n 
co
ef
fic
ie
nt
solute composition (XB)
FIG. 2. The self-diffusion coefficients obtained from MD
simulation and mode coupling theory are plotted for model I.
Filled and open circles represent the solvent and solute diffusion
coefficients obtained from simulations, respectively. Full and
dashed lines show the MCT results. T   1.0, r  0.85.
Figure 1 shows the remarkable reentrant behavior of
the structural relaxation times (ti) when the viscosity is
changed by varying the composition. The increase in com-
position is indicated by arrows. Here the relaxation time ti
is calculated by using ti  s2Dsi . Thus, the relaxation
time is inversely proportional to the diffusion constant. We
believe that a similar behavior will be observed for rota-
tional relaxation as well. The simulation results here are
averages over three independent long runs; error bars are
typically 60.2 for viscosity and 61.5 for the relaxation
time (that is, 60.002 for diffusion coefficients). Note that
in this figure we have shown only the simulation results,
for clarity — theory shows a similar behavior.
Figure 2 shows the composition dependence of self-
diffusion coefficients obtained from both theory and simu-
lation, for model I. Figure 3 shows the same for model II.
Note the nonmonotonic composition dependence. Diffu-
sion of A and B shows differing behavior, in all the cases.
This demonstrates the sensitivity of the transport proper-
ties to eAB.
Mutual diffusion coefficients DAB (sometimes called in-
terdiffusion coefficients) are also calculated for model I0.0 0.2 0.4 0.6 0.8 1.0
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
D
iff
us
io
n 
co
ef
fic
ie
nt
solute composition (XB)
FIG. 3. The self-diffusion coefficients obtained from MD
simulation and mode coupling theory are plotted for model II.
Filled and open circles represent the solvent and solute diffusion
coefficients obtained from simulations, respectively. Full and
dashed lines show the MCT results. T   1.24, r  0.85.
and model II for three different compositions (0.2, 0.4,
and 0.8) following the scheme of Schoen et al. [8] and
the results are given in Table I. The comparison between
self and mutual diffusion coefficients clearly shows that
cross correlations are NOT negligible in both the models
introduced here. For model I, Enskog theory even fails to
predict the qualitative trend.
Figure 4 depicts the nonideality of viscosity with respect
to composition, for both the models. Though the agree-
ment between theory and simulation is certainly not per-
fect, the trends are similar in both the calculations. Note
that the theoretical calculation does not use any simula-
tion data as input or any adjustable parameter either; thus
the theory and the simulation provide independent tests of
each other which is important for binary mixtures.
We conclude this Letter with the following comments.
(1) It is shown that the simple models can describe the
differing behavior of diffusion and viscosity in binary mix-
tures, when the viscosity is changed by varying the com-
position. This nonmonotonicity of diffusion with viscosity
is most dramatically manifested in the reentrant type be-
havior depicted in Fig. 1. This shows that viscosity is not
a unique determinant of relaxation in binary mixtures.TABLE I. The calculated thermodynamic factor for binary mixing (Q), the ideal mutual diffusion coefficient (D0AB), and the simu-
lated mutual diffusion coefficient (DAB) are listed against the solute composition (XB) for both models I and II. (The definitions of
all these quantities are adopted from Ref. [8].)
Model I Model II
XB Q DAB
D0AB
sim DAB
D0AB
Enskog Q DAB
D0AB
sim DAB
D0AB
Enskog
0.2 10.3702 0.4095 1.1642 0.2318 1.6491 1.0628
0.4 5.3465 0.1708 1.4315 0.0580 3.3832 1.2882
0.8 8.8987 0.3541 1.1334 0.2484 2.3353 1.0711
0.0 0.2 0.4 0.6 0.8 1.0
1.5
2.0
2.5
3.0
3.5
4.0
v
isc
os
ity
 (η
)
solute composition (XB)
FIG. 4. The composition dependence of viscosity obtained
from MD simulations (symbols) and mode coupling theory
(lines) for both the models. Filled (open) circles give simulation
results for model I (model II). The lines give the theories.
(2) Model I and model II seem to reproduce the behav-
ior observed in a large number of systems. We believe
that this is the first time a microscopic model captures the
strong nonideality of diffusion and viscosity. The results
agree with the age old wisdom that structure making inter-
actions between the two constituents (here A and B) lead
to a slower stress relaxation. The opposite has also been
observed for model II which has the structure breaking
interactions.
(3) Both simulations and theoretical calculations reveal
that the main reason for the anomalous composition de-
pendence of viscosity lies in the variation of the mean
square stress fluctuation with the composition of the mix-
ture. Similarly, for friction, it is the Einstein frequency
which shows nonmonotonic behavior. It is thus fair to
say that the anomalies have both structural and dynamical
origins.It is a pleasure to thank Dr. Srikanth Sastry and Dr. Ram
Seshadri for discussions. This work was supported in part
by the Department of Science and Technology, India and
CSIR, New Delhi, India. G. Srinivas thanks CSIR.
*Corresponding author.
Electronic address: bbagchi@sscu.iisc.ernet.in
[1] G. W. Castellan, Physical Chemistry (Addison-Wesley,
Reading, 1971).
[2] L. Qunfang and H. Yu-Chun, Fluid Phase Equilib. 154, 153
(1999), and references therein.
[3] G. R. Fleming, Chemical Application of Ultrafast Spec-
troscopy, The International Series of Monograph on Chem-
istry (Oxford University Press, New York, 1986), p. 144.
[4] W. Kob and H. C. Andersen, Phys. Rev. E 51, 4626 (1995);
Phys. Rev. Lett. 73, 1376 (1994).
[5] R. Mountain and D. Thirumalai, Phys. Rev. A 36, 3300
(1987); 42, 4574 (1990).
[6] S. Sastry, Phys. Rev. Lett. 85, 590 (2000).
[7] D. M. Heyes, J. Chem. Phys. 96, 2217 (1992); Phys. Rev. B
37, 5677 (1988).
[8] M. Schoen and C. Hoheisel, Mol. Phys. 52, 33 (1984).
[9] H. A. Al-Chalabi and E. McLaghlin, Mol. Phys. 19, 703
(1970).
[10] U. Balucani and M. Zoppi, Dynamics of the Liquid State
(Clarendon Press, Oxford, 1994).
[11] J. Bosse, W. Gotze, and M. Lucke, Phys. Rev. A 17, 434
(1978); T. Geszti, J. Phys. C 16, 5805 (1983).
[12] L. Sjogren and A. Sjolandar, J. Phys. C 12, 4369 (1979).
[13] J. Bosse and Y. Kaneko, Phys. Rev. Lett. 74, 4023 (1995).
[14] B. Bagchi and S. Bhattacharyya, Adv. Chem. Phys. 116,
67 (2001).
[15] (a) G. Srinivas, A. Mukherjee, and B. Bagchi, J. Chem.
Phys. 114, 6220 (2001); (b) A. Mukherjee, G. Srinivas,
and B. Bagchi (to be published).
[16] W. G. Madden and S. A. Rice, J. Chem. Phys. 72, 4208
(1980); J. P. Hansen and I. R. McDonald, Theory of Simple
Liquids (Academic Press, New York, 1986).


Reentrant behavior of relaxation time with viscosity at varying composition in binary mixtures

Open Access Repository of IISc Research Publications

Reentrant Behavior of Relaxation Time with Viscosity at Varying Composition in Binary Mixtures

Name not available

Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

Indian Academy of Sciences

Reentrant Behavior of Relaxation Time with Viscosity at Varying Composition
in Binary Mixtures
Arnab Mukherjee, Goundla Srinivas, and Biman Bagchi*
Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore-12, India
In order to understand the long known anomalies in the composition dependence of diffusion and
viscosity of binary mixtures, we introduce here two new models and carry out extensive molecular
dynamics simulations. In these models, the two molecular species (A and B) have the same diameter and
mass. In model I the interspecies interaction is more attractive than that between the pure components,
while the reverse is true for model II. Simulations and mode coupling theory calculations reveal that the
models can capture a wide variety of behavior observed in experiments, including the reentrant viscosity
dependence of relaxation time.Transport properties of binary mixtures often show
strong and baffling dependencies on the composition
which have not been understood or even adequately ad-
dressed in a molecular theory. The well-known Raoult’s
law of classical physical chemistry [1] predicts the fol-
lowing simple linear dependence on the composition for a
given property P,
P  x1P1 1 x2P2 , (1)
where xis are the mole fractions and Pis are the values
of the property P of the pure (single component) liquids.
More often than not, significant deviation from Eq. (1) is
observed. Of many anomalies exhibited by binary mix-
tures, the existence of an extremum (sometimes even a
double extrema) in the composition dependence of excess
viscosity [2] and the reentrant type behavior of the relax-
ation time when plotted against viscosity [3] are certainly
the most remarkable. The latter shows, in a dramatic fash-
ion, that viscosity is not a unique determinant of relax-
ation in binary mixtures [3]. Several interesting theoretical
and computer simulation studies on Lennard-Jones binary
mixtures have been carried out recently [4–6], but these
studies have concentrated mainly on the glass transition in
binary mixtures and considered only one particular com-
position and a unique interaction strength. Earlier Heyes
carried out the extensive equilibrium molecular dynam-
ics (MD) simulations of Lennard-Jones binary mixtures to
study the partial properties of transport coefficients in the
inert gas medium [7], by using the Berthelot mixing rule
and found only a weak nonideality. The nonideality in the
case of inert gas mixtures is small, since their mutual in-
teraction strength (eAB) follows the Berthelot mixing rule.
In order to understand the markedly nonlinear compo-
sition dependence of viscosity and diffusion, here we in-
troduce and study two models (referred to as model I and
model II) of binary mixtures in which the solute-solvent
interaction strength is varied by keeping all the other pa-
rameters unchanged. In our models, all the three inter-
actions (solute-solute, solvent-solvent, and solute-solvent)
are described by the Lennard-Jones (12 2 6) potential,Uij  4eij
∑µ
s
rij
∂12
2
µ
s
rij
∂6∏
, (2)
where i and j denote any two different particles. We set
the diameter (s) and mass (m) of both the solute and the
solvent molecules to unity, for simplicity. The solvent-
solute interaction strength lies in the potential well depth
eAB, where A and B represent the solvent and solute par-
ticles, respectively. Throughout this study we keep the in-
teraction strength eAA  1.0 (solvent-solvent), eBB  0.5
(solute-solute). In the two models we use two different
solvent-solute interaction strength values, namely eAB 
2.0 in model I and eAB  0.3 in model II. So, while
model I is a “structure former” (between A and B),
model II is a “structure breaker.” Model I shows glass
transition behavior similar to the Kob-Andersen model.
One advantage of this model is that only the interaction
energy is varied. These models can also capture the
anomalous inverse correlation between excess viscosity
and excess volume observed in many experiments. We
believe that the models introduced here can serve as
starting points to understand the many baffling properties
of binary mixtures.
In addition to the self-diffusion coefficients (Dsi ) and the
viscosity, we also calculated the mutual (or the inter-) dif-
fusion coefficients (DAB) of the binary mixture at various
compositions. The latter is a subject of long-standing in-
terest, although not much is known about the extent of its
nonideality. The ideal value is given by (D0AB)
D0AB  QxBD
s
A 1 xAD
s
B , (3)
where xi and D
s
i are the mole fraction and the self-
diffusion coefficient of the ith species in the mixture, re-
spectively, and Q is the well-known thermodynamic factor
of binary mixing [8]. The simulations presented here
shows a marked departure from Eq. (3), due to cross-
velocity correlations. We believe that this is the first time
that such pronounced nonideality in DAB is established.
It is worthwhile to note that Enskog kinetic theory,
which is applicable to hard sphere systems [9], attempts to
explain in a concise way the composition dependence of
the transport properties in binary mixtures. But unfortu-
nately this theory fails completely to explain the observed
strong nonideality in the composition dependence of dif-
fusion in binary mixture.
Extensive MD (microcanonical ensemble, with the usual
periodic boundary condition) simulations have been car-
ried out with a total of 500 particles for two types of mod-
els by varying the solute mole fraction (that is, of B) from
0 to 1. The reduced temperature T kBTe is set equal
to unity in model I and 1.24 in model II, and the reduced
density r  rs3 is 0.85 in both the models. After
many trial runs to verify the existing results on viscosity
[7] of one component liquids, we have selected a time step
Dt  0.002t for model I and Dt  0.001t for model II,
where t  s
p
me. We have dealt with six different sol-
ute compositions, namely 0, 0.2, 0.4, 0.6, 0.8, and 1.0. For
each solute composition we have equilibrated the system
up to 1.5 3 105 steps. Simulations carried out for another
2.0 3 105 steps after the equilibration during which all the
relevant quantities have been calculated. For each compo-
sition, we have run three independent simulations and have
taken averages over them. We have checked all the three
partial radial distribution functions to make sure no phase
separation occurs during simulations (for model II). Vis-
cosity values are obtained by integrating the stress time
correlation function which defines the time dependent vis-
cosity by the following relation:
ht  VkBT 21sxz0sxzt , (4)
where the off-diagonal element of the stress tensor sxz for
binary mixture is defined as
sxz 
NX
j1
pxj p
z
j m 1 F
z
j xj . (5)
Here Fzj is the z component of the force acting on the jth
particle, the corresponding x coordinate is xj , and p
z
j is
the z component of the momentum of jth particle, m being
the mass of the particle. Among total N particles, solvent
particles are labeled from 1 to N1 and solute particles from
N1 1 1 to N .
Diffusion coefficients are calculated both from the mean
square displacement and velocity autocorrelation function
via the Green-Kubo relation. The results of the simulation
are given in Figs. 1–4. We shall discuss the results after
we describe the mode coupling theory employed.
We have carried out mode coupling theory (MCT) cal-
culations of diffusion and viscosity to understand the simu-
lation results, especially the origin of nonmonotonicity.
These calculations have been carried out by using well-
established expressions [10–14]. Note that for binary mix-
tures at normal density and temperature, the short time
dynamics of the relevant time correlation functions are im-
portant and, in fact, can contribute more than 50% of the
total value, just as for one component liquids. Thus, any
solution of MCT equation requires accurate input of theshort time dynamics. For a given transport property P
[P can be viscosity (h) or friction (z )], MCT formalism
[10–14] assumes the following separation into the short
time, binary collision controlled, contribution Pbint and
the contribution from the collective term, which in dense
liquid is dominated by the density term, Prrt. So the
total dynamical quantity Pt can be written as [10],
Pt  Pbint 1 Prrt , (6)
where the binary term is assumed to be Gaussian as Pbin 
Pt  0 exp2t2t2P. The resulting expressions of the
characteristic time tP are complicated and given elsewhere
[15]. Pt  0 values are determined mainly by the static
correlation like the radial distribution functions [gijr] or
the direct correlation functions [cijr] which are obtained
from Ornstein-Zernike equations with soft mean spherical
approximation (SMSA) closure [16]. The mode coupling
contribution to viscosity is assumed to be given by the bi-
nary product of the density terms. In case of binary mix-
ture total MCT term hrr can be expressed as the sum of
four hrirj terms [15(a)]. The short time part of time
dependent friction is calculated from short time expansion
of force-force time correlation function [10–14] and the
mode coupling expression follows Bosse et al. [13]. The
frequency dependent diffusion coefficients [Dsi z] [which
is the same as the Laplace transform of velocity autocorre-
lation function Cyiz] are related to the respective fre-
quency dependent frictions [ziz] according to Einstein
relation,
Dsi z  Cyiz 
kBT
miz 1 ziz
. (7)
viscosity ( )
1.5 2 2.5 3 3.5 4
R
el
ax
at
io
n 
ti
m
e 
(
i)
10
15
20
25
30
FIG. 1. The simulated values showing the reentrant behavior of
the relaxation times ti are plotted against simulated viscosity for
model I. Filled circles represent tA, while open circles represent
tB. The direction of the arrow shows the increasing solute (B)
composition in both the cases. T   1.0, r  0.85.
0.0 0.2 0.4 0.6 0.8 1.0
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
D
if
fu
si
on
 c
oe
ff
ic
ie
nt
solute composition (X
B
)
FIG. 2. The self-diffusion coefficients obtained from MD
simulation and mode coupling theory are plotted for model I.
Filled and open circles represent the solvent and solute diffusion
coefficients obtained from simulations, respectively. Full and
dashed lines show the MCT results. T   1.0, r  0.85.
Figure 1 shows the remarkable reentrant behavior of
the structural relaxation times (ti) when the viscosity is
changed by varying the composition. The increase in com-
position is indicated by arrows. Here the relaxation time ti
is calculated by using ti  s2Dsi . Thus, the relaxation
time is inversely proportional to the diffusion constant. We
believe that a similar behavior will be observed for rota-
tional relaxation as well. The simulation results here are
averages over three independent long runs; error bars are
typically 60.2 for viscosity and 61.5 for the relaxation
time (that is, 60.002 for diffusion coefficients). Note that
in this figure we have shown only the simulation results,
for clarity — theory shows a similar behavior.
Figure 2 shows the composition dependence of self-
diffusion coefficients obtained from both theory and simu-
lation, for model I. Figure 3 shows the same for model II.
Note the nonmonotonic composition dependence. Diffu-
sion of A and B shows differing behavior, in all the cases.
This demonstrates the sensitivity of the transport proper-
ties to eAB.
Mutual diffusion coefficients DAB (sometimes called in-
terdiffusion coefficients) are also calculated for model I0.0 0.2 0.4 0.6 0.8 1.0
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
D
if
fu
si
on
 c
oe
ff
ic
ie
nt
solute composition (X
B
)
FIG. 3. The self-diffusion coefficients obtained from MD
simulation and mode coupling theory are plotted for model II.
Filled and open circles represent the solvent and solute diffusion
coefficients obtained from simulations, respectively. Full and
dashed lines show the MCT results. T   1.24, r  0.85.
and model II for three different compositions (0.2, 0.4,
and 0.8) following the scheme of Schoen et al. [8] and
the results are given in Table I. The comparison between
self and mutual diffusion coefficients clearly shows that
cross correlations are NOT negligible in both the models
introduced here. For model I, Enskog theory even fails to
predict the qualitative trend.
Figure 4 depicts the nonideality of viscosity with respect
to composition, for both the models. Though the agree-
ment between theory and simulation is certainly not per-
fect, the trends are similar in both the calculations. Note
that the theoretical calculation does not use any simula-
tion data as input or any adjustable parameter either; thus
the theory and the simulation provide independent tests of
each other which is important for binary mixtures.
We conclude this Letter with the following comments.
(1) It is shown that the simple models can describe the
differing behavior of diffusion and viscosity in binary mix-
tures, when the viscosity is changed by varying the com-
position. This nonmonotonicity of diffusion with viscosity
is most dramatically manifested in the reentrant type be-
havior depicted in Fig. 1. This shows that viscosity is not
a unique determinant of relaxation in binary mixtures.TABLE I. The calculated thermodynamic factor for binary mixing (Q), the ideal mutual diffusion coefficient (D0AB), and the simu-
lated mutual diffusion coefficient (DAB) are listed against the solute composition (XB) for both models I and II. (The definitions of
all these quantities are adopted from Ref. [8].)
Model I Model II
XB Q DAB
D0AB
sim DAB
D0AB
Enskog Q DAB
D0AB
sim DAB
D0AB
Enskog
0.2 10.3702 0.4095 1.1642 0.2318 1.6491 1.0628
0.4 5.3465 0.1708 1.4315 0.0580 3.3832 1.2882
0.8 8.8987 0.3541 1.1334 0.2484 2.3353 1.0711
0.0 0.2 0.4 0.6 0.8 1.0
1.5
2.0
2.5
3.0
3.5
4.0
vi
sc
os
ity
 (
η)
solute composition (X
B
)
FIG. 4. The composition dependence of viscosity obtained
from MD simulations (symbols) and mode coupling theory
(lines) for both the models. Filled (open) circles give simulation
results for model I (model II). The lines give the theories.
(2) Model I and model II seem to reproduce the behav-
ior observed in a large number of systems. We believe
that this is the first time a microscopic model captures the
strong nonideality of diffusion and viscosity. The results
agree with the age old wisdom that structure making inter-
actions between the two constituents (here A and B) lead
to a slower stress relaxation. The opposite has also been
observed for model II which has the structure breaking
interactions.
(3) Both simulations and theoretical calculations reveal
that the main reason for the anomalous composition de-
pendence of viscosity lies in the variation of the mean
square stress fluctuation with the composition of the mix-
ture. Similarly, for friction, it is the Einstein frequency
which shows nonmonotonic behavior. It is thus fair to
say that the anomalies have both structural and dynamical
origins.It is a pleasure to thank Dr. Srikanth Sastry and Dr. Ram
Seshadri for discussions. This work was supported in part
by the Department of Science and Technology, India and
CSIR, New Delhi, India. G. Srinivas thanks CSIR.
*Corresponding author.
Electronic address: bbagchi@sscu.iisc.ernet.in
[1] G. W. Castellan, Physical Chemistry (Addison-Wesley,
Reading, 1971).
[2] L. Qunfang and H. Yu-Chun, Fluid Phase Equilib. 154, 153
(1999), and references therein.
[3] G. R. Fleming, Chemical Application of Ultrafast Spec-
troscopy, The International Series of Monograph on Chem-
istry (Oxford University Press, New York, 1986), p. 144.
[4] W. Kob and H. C. Andersen, Phys. Rev. E 51, 4626 (1995);
Phys. Rev. Lett. 73, 1376 (1994).
[5] R. Mountain and D. Thirumalai, Phys. Rev. A 36, 3300
(1987); 42, 4574 (1990).
[6] S. Sastry, Phys. Rev. Lett. 85, 590 (2000).
[7] D. M. Heyes, J. Chem. Phys. 96, 2217 (1992); Phys. Rev. B
37, 5677 (1988).
[8] M. Schoen and C. Hoheisel, Mol. Phys. 52, 33 (1984).
[9] H. A. Al-Chalabi and E. McLaghlin, Mol. Phys. 19, 703
(1970).
[10] U. Balucani and M. Zoppi, Dynamics of the Liquid State
(Clarendon Press, Oxford, 1994).
[11] J. Bosse, W. Gotze, and M. Lucke, Phys. Rev. A 17, 434
(1978); T. Geszti, J. Phys. C 16, 5805 (1983).
[12] L. Sjogren and A. Sjolandar, J. Phys. C 12, 4369 (1979).
[13] J. Bosse and Y. Kaneko, Phys. Rev. Lett. 74, 4023 (1995).
[14] B. Bagchi and S. Bhattacharyya, Adv. Chem. Phys. 116,
67 (2001).
[15] (a) G. Srinivas, A. Mukherjee, and B. Bagchi, J. Chem.
Phys. 114, 6220 (2001); (b) A. Mukherjee, G. Srinivas,
and B. Bagchi (to be published).
[16] W. G. Madden and S. A. Rice, J. Chem. Phys. 72, 4208
(1980); J. P. Hansen and I. R. McDonald, Theory of Simple
Liquids (Academic Press, New York, 1986).


Arnab Mukherjee

Goundla Srinivas

Biman Bagchi

Crossref

http://repository.ias.ac.in/4214/1/478.pdf

Re-entrant behavior of relaxation time with viscosity at varying composition in binary mixtures

Abstract

Similar works

Full text

Available Versions