# The Spatiotemporal Oscillations of Order Parameter  for Isothermal Model of the Surface-Directed Spinodal Decomposition in Bounded Binary Mixtures

Authors: Igor B. Krasnyuk; Roman M. Taranets
Journal: Research Letters in Physics
 (2009)

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## Abstract

The asymptotical behavior of order parameter in confined binary mixture is considered in one-dimensional geometry. The interaction between bulk and surface forces in the mixture is investigated. Its established conditions are when the bulk spinodal decomposition may be ignored and when the main role in the process of formation of the oscillating asymptotic periodic spatiotemporal structures plays the surface-directed spinodal decomposition which is modelled by nonlinear dynamical boundary conditions.


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## Body

## 1. Introduction

The problem of self-organization for order
parameter in binary mixtures, which are confined with two flat walls, is
important in physics of condense matter [1, 2]. At first, for the binary mixtures, this problem was
considered by Binder, Puri, and Frisch. The existence of oscillating
distributions of the order parameter (on a space variable for each fixed moment
time) has been proved [2, 3]. Binder and Frisch considered also the Cahn-Hilliard (CH)
equation for conserved-order parameter with nonlinear dynamical boundary conditions
[1] and described a
phase diagram for the problem. However, it is unknown what is being an
attractor of the considered dynamical binary system, that is, what is being an
asymptotic behavior of the solutions of such imposed boundary-value problem as
a time t→∞.The aim of the present paper is to establish how the
attractor of trajectories for this problem is constructed. The achievements of
the paper are as follows: (i) we show that the
solutions tend to the limit functions ast→∞ with a finite set of discontinuities which are
determined by topological properties of the functions Fk,k=1,2  in the boundary conditions (in
this case, we have the solutions of relaxation
non-Fickian diffusion type); (ii) there are limit solutions with infinite
(countable) and infinite (uncountable) number of discontinuities (in such case,
we have the solutions of preturbulent type and turbulent type; accordingly, the
set of points of discontinuities for the solutions of turbulent type is
familiar to a Cantor-like set); (iii) there exist
the solutions whose limit functions are discontinuities at all points. This
solutions take up a particular place among the solutions and they lead to the
appearance of the so-called “deterministic randomness,” namely, in
order to characterize the long-time behavior of the solutions, it is necessary
to proceed to a probabilistic description [4].In such way, we consider the problem which was introduced in [1], but instead of the
classical (CH) equation, we research the so-called modified Cahn-Hilliard (mCH)
equation which is proposed by Galenko et al. [5, 6]. The modified equation is
the model for diffusion and phase separation which takes into account
hyperbolic relaxation of the solute diffusion flux I.
The flux can be determined by the formula (see [5, 6])
(1)I(⋅,t+τD)=I(⋅,t)+τDIt(⋅,t)+O(τD)=−M(u)(ℱu′(u)−ϵu2uxx)x(⋅,t),
where O(τD)→0 as τD→0, u is the order parameter, τD is the delay time, M is the atomic mobility, ℱu′ is the u-derivative of the free energy ℱ.
We denote that ϵu2=2F0ζb2, where F0 is connection energy [6] which is normalized by kBT. Here, kB is the Boltzmann constant, and ζb is the bulk correlation length. A few
advancements were made for strongly nonequilibrium phase separation. In 1986,
Galenko and Lebedev [6]
generalized the linearized (CH) equation to the case of existence of slowly
relaxing variable. Their calculations show that the instability of the system
is not standard diffusive type but rather it is controlled by the relaxation of
a slow structural variable such as the diffusion flux determined by the formula
(1).In this paper, we consider the case only when the
order parameter varies slowly in the bulk, and we
determine the condition when this supposition is
true. In this case, asymptotically stable equilibrium states of the system may
be stabilized by nonlinear boundary conditions which are described by
surface-directed spinodal decomposition (SDSD) in the binary system. As a
result, the SDSD determines the limit distributions of order parameter which
have the form of traveling waves and its graphes are asymptotic periodic
functions with finite, infinite (countable), or infinite (uncountable) number
of discontinuities (Figure1).Asymptotically
periodic distributions of the order parameter of the relaxation type. The mapΦ:g↦g2+μ. Here, β1 and β3 are stable ordered phases, and β2 is unstable disordered phase. If the amplitude
of initial bubbles is greater than β2, then there are the oscillations. If the amplitude of initial bubbles is less
than β2,
then asymptotic distributions of order parameter
tend to the constant value β1 [4].
(a)(b)(c)


## 2. Formulation of the Problem

We consider binary mixtures with diffusion flux
(1) and, additionally, we suppose that the equation ut+Ix=0 is true. Then, from (1), we obtain the (mCH)
equation:
(2)τDutt+ut=[M(u)(ℱu′(u)−ϵu2uxx)x]x.
In [6], it is shown that (2) takes place for bulk spinodal
decompositions with locally nonequilibrium diffusion. In what follows, we
consider the boundary conditions in the common form [1]:(3)ut|x=xk=Fk(hk,gk,ν,u,ux,uxx,uxxx)|x=xkfor the solutions of (2). Here, x1=0,x2=l and Fk are given functions, hk describes surface attractive potentials (for
magnetic liquid, it may be surface magnetic field [1]), and gk=Js(k)/J(k),
where J(⋅) is modelled the bulk forces and Js(⋅) is modelled surface forces (in detail see
[1, 2]). Next, ν is a constant (or a function) which is
proportional to the bulk correlation length ζb. In [1], for example,
it was considered (3) for Fk=hk+gk(u)+νux−(ν/4)2/3uxx−(ν/4)1/3uxxx.
In the simplest case, if F1=h1+g1u(0,t),
then u(0,t)=u(0,0)exp(g1t)−h1/g1.
Thus, if g1>0,
then the surface order parameter increases in time, consequently, and the wall x=0 is attractive. If g1<0,
the wall x=0 is repulsive. In [2], it was considered (3) for Fk=(τv/τs)(−ux−hk/ν−(1/ν)gk(u)),
where τs and τv are the surface and bulk relaxation times
accordingly. In typical cases τs≪τv [1].Further, the self-organized structures (as in Figure1)
appear if and only if the Pfaffian form F2du1−F1du2=0 is integrable, that is, it can be reduced to
the form dW(u1,u2)=0,
where W is the known function, and uk=u(xk,t).
From the last equality, it follows that
(4)u(l,t)=Φμ(u(0,t)),μ=μ(hk,gk,ν),
where Φμ is some function. So, if the Pfaffian form is
integrable, then (3) may be reduced
to (4). Next, we consider the initial
conditions u(x,0)=u(0)(x),ut(x,0)=u(1)(x). In common case, analytical solutions of this problem are unknown; but the
boundary-value problem may be reduced to some simpler problem for spatial
functions Fk and constant mobility M.
The problem has unique solution such as in Figure 1. We confined ourselves to
the polymer-polymer systems with the components A,B and the power of polymerization NA,NB,
accordingly. The free energy (see [5]) is
(5)ℱ=uANAlnuA+uBNBlnuB+κuAuB,
where uA+uB=1,
and κ is Flory-Huggins parameter. We denote uA=u,uB=1−u,
and we consider, for simplicity, the case NA=NB=N.
Then, we find from (5) by the linear approximation that
(6)ℱu′=(4N−2κ)(u−12)
at a neighborhood of u=1/2.
Then, perturbations u↦u−1/2 of the equilibrium state u=1/2 can be considered as the order parameter. As a
result, due to (6), equation (2) can be rewritten in the form
(7)utt+λut=D1uxx+D2ux2−D3uxxxx−D4uxuxxx,
where t↦t/τv, x↦x/l, λ=τv/τD,
and D1=(M(1/2)τv2/τDl2)(4/N−2κ), D2=(Mu′(1/2)τv2/τDl2)(4/N−2κ), D3=M(1/2)εu2τv2/τDl4, D4=Mu′(1/2)εu2τv2/τDl4. Here, l is the spatial length, τD=Mαj, αJ is the coefficient specifying non-Fickian
diffusion (see [6]).
So, τD is the relaxation time of the diffusion flux
to its steady state. Further, we suppose that D2≪1 and D4≪1,
in particular, it is possible when M(u)~u(1−u).
Hence, we get from (7) that
(8)utt+λut=D1uxx−D3uxxxx.
Next, we consider the boundary conditions
(9)τsτvut(xk,t)=hk+2gku(xk,t)
for the solutions of (8), where hk and gk are some constant (in common case, they may be
given functions). Moreover, we suppose that(10)ux(0,t)=0,ux(l,t)=0.


## 3. Reduction to the Difference Equation with Continuous Time

Let(11)u(x,t)=f(ζ)+g(η),where v=D1, ζ=t+x/v, η=t−x/v,
and f,g are arbitrary functions. Substituting (11)
into (8), we deduce
(12)βv4fζ′′′′+fζ′=0,βv4gη′′′′+gη′=0,
where β=M(1/2)εu2τv/l4.
Integrating (12) along characteristics dx/dt=±v,
we arrive at the differential-difference equation(13)βv4Y′′′(t)+Y(t)=0,
where Y(t)=f(t+l/v)−g(t−l/v).
Here, we used the relations f(t)=g(t) and f′′′(t)=g′′′(t) accurate within a constant. In one's turn,
these relations follow from the first condition of (10). The general solution
of (13) is
(14)Y(t)=C1e−2νt+C2eνtcos(3νt)+C3eνtsin(3νt),
where ν=(1/2)(v4/β)1/3.
It follows from the definition of Y(t) and functional relation (4)
that
(15)f(t+lv)=Φμ(f(t))+Y(t),
where f↦2f.
Henceforth, we consider the case Y(t)≃const,
that is, νt≪1.
Hence, we have two-parametric family of difference equations. The typical
solutions of these equations are asymptotic periodic functions, such as in
Figure 1. For example, let Fk be from (9). Then, from the above procedure,
we find the first integral
(16)c0(h1+2g1u(0,t))g2/g1=h2+2g2u(l,t),where c0 is a constant. Let h1=0,g2/g1=2.
These values correspond to experimental values from [2]. From (16), we
find
(17)u(l,t)=c0g22u2(0,t)−h22g2.For the sake of simplicity, we
choose c0g2/2=1.
Thus, it follows from (17) that(18)f(t+lv)+g(t−lv)=(f(t)+g(t))2+μ^,
where μ^=−h2/2g2.
Next, we deduce from (10), that is, f(t)=g(t)+c1 and f(t+l/v)=g(t−l/v)+c2, that
(19)2f(t+lv)+c2=(2f(t)+c1)2+μ^,
where c1=f(0)−g(0),c2=f(l/v)−g(−l/v),
and f(0),f(l/v),g(0) and g(−l/v) depend on initial functions u(0) and u(1).
Let z(t)=2f(t)+c1.
Then, it follows from (19) that
(20)z(t+lv)=z2(t)+μ,μ=μ^+c1−c2.


## 4. Phase Diagram

The phase diagram for the solutions of (20) is shown on Figure 2.
This diagram is similar to corresponding diagram which was obtained by numeric
simulation in [1, 2], where it was considered
similar boundary problem to (CH) equation [2].Figure 2
Phase diagram
is obtained for the problem with nonlinear boundary conditions, which cause the
surface-directed spinodal decomposition, by the theoretical method. Here (a) is
the region of unique stable phase; (b) is the region of the spinodal
decomposition; (c) is the region of the lamellar phase; (d) is the region of
stochastic distributions of the order parameter.



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*Source: 250203-2009-02-16.xml*