Simplified self-consistent theory of colloid dynamics.

Abstract

One of the main elements of the self-consistent generalized Langevin equation (SCGLE) theory of colloid dynamics [Phys. Rev. E 62, 3382 (2000); 72, 031107 (2005)] is the introduction of exact short-time moment conditions in its formulation. The need to previously calculate these exact short-time properties constitutes a practical barrier for its application. In this Brief Report, we report that a simplified version of this theory, in which this short-time information is eliminated, leads to the same results in the intermediate and long-time regimes. Deviations are only observed at short times, and are not qualitatively or quantitatively important. This is illustrated by comparing the two versions of the theory for representative model systems.

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arXiv:0711.1920v1 [cond-mat.mtrl-sci] 13 Nov 2007
Simplified Self-Consistent Theory of Colloid Dynamics
R. Ju´ arez-Maldonado1, M. A. Ch´ avez-Rojo2, P. E.
Ram´ ırez-Gonz´ alez1, L. Yeomans-Reyna3, and M. Medina-Noyola1
1Instituto de F´ ısica “Manuel Sandoval Vallarta”,
Universidad Aut´ onoma de San Luis Potos´ ı,
´Alvaro Obreg´ on 64, 78000 San Luis Potos´ ı, SLP, M´ exico
2Facultad de Ciencias Qu´ ımicas, Universidad Aut´ onoma de Chihuahua,
Venustiano Carranza S/N, 31000 Chihuahua, Chih., M´ exico.
3Departamento de F´ ısica, Universidad de Sonora, Boulevard Luis
Encinas y Rosales, 83000, Hermosillo, Sonora, M´ exico.
(Dated: February 2, 2008)
Abstract
One of the main elements of the self-consistent generalized Langevin equation (SCGLE) theory
of colloid dynamics [Phys. Rev. E 62, 3382 (2000); ibid 72, 031107 (2005)] is the introduction
of exact short-time moment conditions in its formulation. The need to previously calculate these
exact short-time properties constitutes a practical barrier for its application. In this note we report
that a simplified version of this theory, in which this short-time information is eliminated, leads to
the same results in the intermediate and long-time regimes. Deviations are only observed at short
times, and are not qualitatively or quantitatively important. This is illustrated by comparing the
two versions of the theory for representative model systems.
PACS numbers: 64.70.Pf, 61.20.Gy, 47.57.J-
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I. INTRODUCTION
In recent work a new first-principles theory of dynamic arrest has been proposed [1, 2].
This consists essentially of the application of the self-consistent generalized Langevin equa-
tion (SCGLE) theory of colloid dynamics [3, 4, 5, 6] to the description of the singular
behavior characteristic of dynamic arrest phenomena in specific colloidal systems and con-
ditions. The SCGLE theory was originally devised to describe tracer and collective diffusion
properties of colloidal dispersions in the short- and intermediate-times regimes [7, 8]. Its
self-consistent character, however, introduces a non-linear dynamic feedback, leading to the
prediction of dynamic arrest in these systems, similar to that exhibited by the mode cou-
pling theory (MCT) of the ideal glass transition [9]. The resulting theory of dynamic arrest
in colloidal dispersions was applied in recent work to describe the glass transition in three
mono-disperse experimental model colloidal systems with specific (hard-sphere, screened
electrostatic, and depletion) inter-particle effective forces [1, 2]. The results indicate that
the SCGLE theory of dynamic arrest has the same or better level of quantitative predictive
power as conventional MCT, but is built on a completely independent conceptual basis, thus
providing an alternative approach to the description of dynamic arrest phenomena.
There is, however, a possible practical disadvantage of the SCGLE with respect to the
MCT, and it refers to the fact that the MCT only requires the static structure factor of the
system as an external input, whereas the SCGLE theory requires this information plus other
additional static properties involved in the exact short-time conditions that the theory has
built-in [5]. As it happens, however, the long-time asymptotic solutions of the relaxation
equations that constitutes the SCGLE theory are independent of such exact short-time
properties [2]. The questions then arise if a simplified version of the SCGLE theory, in
which this short-time information is eliminated, could be proposed, and to what extent such
a simpler theory will still provide a reliable representation of the dynamics of the colloidal
system not only in the asymptotic long-time regime, but also at earlier stages. In what
follows we demonstrate that there is a simple manner to build this simplified version of the
SCGLE theory, and that it is virtually as accurate as the full version, even in the short- and
intermediate-time regimes. This finding will greatly simplify the application of the SCGLE
theory of dynamic arrest.
Let us summarize the four distinct fundamental elements of the full self-consistent gen-
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eralized Langevin equation theory of colloid dynamics. The first consists of general and
exact memory-function expressions for the intermediate scattering function F(k,t) and its
self component FS(k,t), derived with the generalized Langevin equation (GLE) formalism
[10], which in Laplace space read [3]
F(k,z) =
S(k)
z +k2D0S−1(k)
1+C(k,z)
, (1.1)
FS(k,z) =
1
z +
k2D0
1+CS(k,z)
, (1.2)
where D0is the free-diffusion coefficient, S(k) is the static structure factor of the system,
and C(k,z) and CS(k,z) are the corresponding memory functions.
The second element is an approximate relationship between collective and self-dynamics.
In the original proposal of the SCGLE theory [3], two possibilities, referred to as the ad-
ditive and the multiplicative Vineyard-like approximations, were considered. The first ap-
proximates the difference [C(k,t) − CS(k,t)], and the second the ratio [C(k,t)/CS(k,t)], of
the memory functions, by their exact short-time limits, using the fact that the exact short-
time expressions for these memory functions, denoted by CSEXP(k,t) and CSEXP
S
(k,t), are
known in terms of equilibrium structural properties [5, 11]. The multiplicative approxima-
tion was devised to describe more accurately the very early relaxation of F(k,t) [6], but the
additive approximation was found to provide a more accurate prediction of dynamic arrest
phenomena [2]. In this paper, for “full SCGLE theory” we refer to the theory that involves
the additive Vineyard-like approximation,
C(k,t) = CS(k,t) + [CSEXP(k,t) − CSEXP
S
(k,t)]. (1.3)
The third ingredient consists of the independent approximate determination of FS(k,t)
[or CS(k,t)]. One intuitively expects that these k-dependent self-diffusion properties should
be simply related to the properties that describe the Brownian motion of individual particles,
just like in the Gaussian approximation [7], which expresses FS(k,t) in terms of the mean-
squared displacement (msd) (∆x(t))2as FS(k,t) = exp[−k2(∆x(t))2/2]. We introduce an
analogous approximate connection, but at the level of their respective memory functions.
The memory function of (∆x(t))2is the so-called time-dependent friction function ∆ζ(t).
This function, normalized by the solvent friction ζ0, is the exact long wave-length limit of
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CS(k,t), i.e., limk→0CS(k,t) = ∆ζ∗(t) ≡ ∆ζ(t)/ζ0. Thus, we interpolate CS(k,t) between
its two limits, namely,
CS(k,t) = CSEXP
S
(k,t) +?∆ζ∗(t) − CSEXP
S
(k,t)?λ(k), (1.4)
where
λ(k) ≡ [1 + (k/kc)2]−1
(1.5)
is a phenomenological interpolating function, with kcbeing the position of the first minimum
that follows the main peak of S(k) [5].
The fourth ingredient of our theory is another exact result, also derived within the GLE
approach [10], this time for ∆ζ∗(t). This exact result may, upon a well-defined simplifying
approximation, be converted into the following approximate but general expression [2]
∆ζ∗(t) =
D0
3(2π)3n
?
dk
?k[S(k) − 1]
S(k)
?2
F(k,t)FS(k,t).(1.6)
Eqs. (1.1)–(1.6) constitute the full SCGLE theory of colloid dynamics. Besides the un-
known dynamic properties, it involves the equilibrium properties S(k), CSEXP(k,t) and
CSEXP
S
(k,t), determined by the methods of equilibrium statistical thermodynamics. We
should also point out that Eqs. (1.1) and (1.2) are exact results, and that Eq. (1.6) derives
from another exact result. Hence, it should not be a surprise that the same results are used
by other theories; in fact, the same equations are employed in MCT. The difference lies, of
course, in the the manner we relate and use them. In this sense, the distinctive elements of
the SCGLE theory are the Vineyard-like approximation in Eq. (1.3) and the interpolating
approximation in Eq. (1.4).
The simplified version of the SCGLE theory is now suggested by the form that these
distinctive equations (Eqs. (1.3) and (1.4)) attain for times longer than the relaxation time
of the functions CSEXP(k,t) and CSEXP
S
(k,t). Under those conditions, Eqs. (1.3) and (1.4)
become, respectively,
C(k,t) = CS(k,t). (1.7)
and
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CS(k,t) = [∆ζ∗(t)]λ(k). (1.8)
It is not difficult to see that the original self-consistent set of equations (involving Eqs.
(1.3) and (1.4)) shares the same long-time asymptotic stationary solutions as its simplified
version. Such stationary solutions are given by [2]
lim
t→∞F(k,t) =
λ(k)S(k)
λ(k)S(k) + k2γS(k). (1.9)
and
lim
t→∞FS(k,t) =
λ(k)
λ(k) + k2γ.
(1.10)
where γ is the solution of the following equation
1
γ=
1
6π2n
?∞
0
dkk4
[S(k) − 1]2λ2(k)
[λ(k)S(k) + k2γ][λ(k) + k2γ].(1.11)
The parameter γ is the long-time asymptotic value, of the msd, i.e., γ ≡ limt→∞(∆x(t))2.
In the arrested states, this parameter is finite, representing the localization of the particles,
whereas in the ergodic states it diverges.
It is then natural to ask what the consequences would be of replacing Eqs. (1.3) and
(1.4) of the full SCGLE set of equations by the simpler approximations in Eqs. (1.7) and
(1.8), that no longer contain the functions CSEXP(k,t) and CSEXP
S
(k,t). Our proposal of a
simplified version of the SCGLE theory consists precisely of this replacement, so that the
“simplified SCGLE theory” consists of the exact results in Eqs. (1.1) and (1.2) along with
Eqs. (1.5) and (1.6), complemented by the closure approximations in Eqs. (1.7) and (1.8).
We have made a systematic comparison of the various dynamic properties involved in the
SCGLE theory, including the intermediate scattering function F(k,t), its self component
FS(k,t), and other tracer-diffusion properties such as the time-dependent friction function
∆ζ∗(t), the mean squared displacement or the time-dependent diffusion coefficient D(t) ≡
(∆x(t))2/2t. As expected, the scenario of dynamic arrest exhibited by this simpler theory is
identical to that provided by the full SCGLEscheme. This is probably not surprising since,
as indicated above, both sets of dynamic equations share the same long-time asymptotic
behavior and the same asymptotic stationary solutions. What is surprising, however, is
the degree of accuracy of the simplified theory in the short- and intermediate-time regimes.
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