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.

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 π‘‘β†’βˆž.

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 as π‘‘β†’βˆž with a finite set of discontinuities which are determined by topological properties of the functions πΉπ‘˜,π‘˜=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 β€œdeterministic 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]) 𝐼⋅,𝑑+πœπ·ξ€Έ=𝐼(β‹…,𝑑)+πœπ·πΌπ‘‘ξ€·πœ(β‹…,𝑑)+𝑂𝐷ℱ=βˆ’π‘€(𝑒)ξ…žπ‘’(𝑒)βˆ’πœ–2𝑒𝑒π‘₯π‘₯ξ€Έπ‘₯(β‹…,𝑑),(1) where 𝑂(𝜏𝐷)β†’0 as πœπ·β†’0, 𝑒 is the order parameter, 𝜏𝐷 is the delay time, 𝑀 is the atomic mobility, β„±ξ…žπ‘’ is the 𝑒-derivative of the free energy β„±. We denote that πœ–2𝑒=2𝐹0𝜁2𝑏, where 𝐹0 is connection energy [6] which is normalized by π‘˜π΅π‘‡. Here, π‘˜π΅ is the Boltzmann constant, and πœπ‘ 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 (Figure 1).

2. Formulation of the Problem

We consider binary mixtures with diffusion flux (1) and, additionally, we suppose that the equation 𝑒𝑑+𝐼π‘₯=0 is true. Then, from (1), we obtain the (mCH) equation: πœπ·π‘’π‘‘π‘‘+𝑒𝑑=ℱ𝑀(𝑒)ξ…žπ‘’(𝑒)βˆ’πœ–2𝑒𝑒π‘₯π‘₯ξ€Έπ‘₯ξ€»π‘₯.(2) 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]:𝑒𝑑||π‘₯=π‘₯π‘˜=πΉπ‘˜ξ€·β„Žπ‘˜,π‘”π‘˜,𝜈,𝑒,𝑒π‘₯,𝑒π‘₯π‘₯,𝑒π‘₯π‘₯π‘₯ξ€Έ||π‘₯=π‘₯π‘˜(3)for the solutions of (2). Here, π‘₯1=0,π‘₯2=𝑙 and πΉπ‘˜ are given functions, β„Žπ‘˜ describes surface attractive potentials (for magnetic liquid, it may be surface magnetic field [1]), and π‘”π‘˜=𝐽𝑠(π‘˜)/𝐽(π‘˜), where 𝐽(β‹…) is modelled the bulk forces and 𝐽𝑠(β‹…) 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 πœπ‘. In [1], for example, it was considered (3) for πΉπ‘˜=β„Žπ‘˜+π‘”π‘˜(𝑒)+πœˆπ‘’π‘₯βˆ’(𝜈/4)2/3𝑒π‘₯π‘₯βˆ’(𝜈/4)1/3𝑒π‘₯π‘₯π‘₯. In the simplest case, if 𝐹1=β„Ž1+𝑔1𝑒(0,𝑑), then 𝑒(0,𝑑)=𝑒(0,0)exp(𝑔1𝑑)βˆ’β„Ž1/𝑔1. Thus, if 𝑔1>0, then the surface order parameter increases in time, consequently, and the wall π‘₯=0 is attractive. If 𝑔1<0, the wall π‘₯=0 is repulsive. In [2], it was considered (3) for πΉπ‘˜=(πœπ‘£/πœπ‘ )(βˆ’π‘’π‘₯βˆ’β„Žπ‘˜/πœˆβˆ’(1/𝜈)π‘”π‘˜(𝑒)), where πœπ‘  and πœπ‘£ are the surface and bulk relaxation times accordingly. In typical cases πœπ‘ β‰ͺπœπ‘£ [1].

Further, the self-organized structures (as in Figure 1) appear if and only if the Pfaffian form 𝐹2𝑑𝑒1βˆ’πΉ1𝑑𝑒2=0 is integrable, that is, it can be reduced to the form π‘‘π‘Š(𝑒1,𝑒2)=0, where π‘Š is the known function, and π‘’π‘˜=𝑒(π‘₯π‘˜,𝑑). From the last equality, it follows that 𝑒(𝑙,𝑑)=Ξ¦πœ‡ξ€·ξ€Έξ€·β„Žπ‘’(0,𝑑),πœ‡=πœ‡π‘˜,π‘”π‘˜ξ€Έ,𝜈,(4) where Ξ¦πœ‡ is some function. So, if the Pfaffian form is integrable, then (3) may be reduced to (4). Next, we consider the initial conditions 𝑒(π‘₯,0)=𝑒(0)(π‘₯),𝑒𝑑(π‘₯,0)=𝑒(1)(π‘₯). 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 πΉπ‘˜ 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,𝐡 and the power of polymerization 𝑁𝐴,𝑁𝐡, accordingly. The free energy (see [5]) is 𝑒ℱ=𝐴𝑁𝐴ln𝑒𝐴+𝑒𝐡𝑁𝐡ln𝑒𝐡+πœ…π‘’π΄π‘’π΅,(5) where 𝑒𝐴+𝑒𝐡=1, and πœ… is Flory-Huggins parameter. We denote 𝑒𝐴=𝑒,𝑒𝐡=1βˆ’π‘’, and we consider, for simplicity, the case 𝑁𝐴=𝑁𝐡=𝑁. Then, we find from (5) by the linear approximation that β„±ξ…žπ‘’=ξ‚€4𝑁1βˆ’2πœ…ξ‚ξ‚€π‘’βˆ’2(6) at a neighborhood of 𝑒=1/2. Then, perturbations π‘’β†¦π‘’βˆ’1/2 of the equilibrium state 𝑒=1/2 can be considered as the order parameter. As a result, due to (6), equation (2) can be rewritten in the form 𝑒𝑑𝑑+πœ†π‘’π‘‘=𝐷1𝑒π‘₯π‘₯+𝐷2𝑒2π‘₯βˆ’π·3𝑒π‘₯π‘₯π‘₯π‘₯βˆ’π·4𝑒π‘₯𝑒π‘₯π‘₯π‘₯,(7) where 𝑑↦𝑑/πœπ‘£, π‘₯↦π‘₯/𝑙, πœ†=πœπ‘£/𝜏𝐷, and 𝐷1=(𝑀(1/2)𝜏2𝑣/πœπ·π‘™2)(4/π‘βˆ’2πœ…), 𝐷2=(π‘€ξ…žπ‘’(1/2)𝜏2𝑣/πœπ·π‘™2)(4/π‘βˆ’2πœ…), 𝐷3=𝑀(1/2)πœ€2π‘’πœ2𝑣/πœπ·π‘™4, 𝐷4=π‘€ξ…žπ‘’(1/2)πœ€2π‘’πœ2𝑣/πœπ·π‘™4. 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 𝐷2β‰ͺ1 and 𝐷4β‰ͺ1, in particular, it is possible when 𝑀(𝑒)βˆΌπ‘’(1βˆ’π‘’). Hence, we get from (7) that 𝑒𝑑𝑑+πœ†π‘’π‘‘=𝐷1𝑒π‘₯π‘₯βˆ’π·3𝑒π‘₯π‘₯π‘₯π‘₯.(8) Next, we consider the boundary conditions πœπ‘ πœπ‘£π‘’π‘‘ξ€·π‘₯π‘˜ξ€Έ,𝑑=β„Žπ‘˜+2π‘”π‘˜π‘’ξ€·π‘₯π‘˜ξ€Έ,𝑑(9) for the solutions of (8), where β„Žπ‘˜ and π‘”π‘˜ are some constant (in common case, they may be given functions). Moreover, we suppose that𝑒π‘₯(0,𝑑)=0,𝑒π‘₯(𝑙,𝑑)=0.(10)

3. Reduction to the Difference Equation with Continuous Time

Let 𝑒(π‘₯,𝑑)=𝑓(𝜁)+𝑔(πœ‚),(11)where βˆšπ‘£=𝐷1, 𝜁=𝑑+π‘₯/𝑣, πœ‚=π‘‘βˆ’π‘₯/𝑣, and 𝑓,𝑔 are arbitrary functions. Substituting (11) into (8), we deduce 𝛽𝑣4π‘“πœξ…žξ…žξ…žξ…ž+π‘“ξ…žπœπ›½=0,𝑣4π‘”πœ‚ξ…žξ…žξ…žξ…ž+π‘”ξ…žπœ‚=0,(12) where 𝛽=𝑀(1/2)πœ€2π‘’πœπ‘£/𝑙4. Integrating (12) along characteristics 𝑑π‘₯/𝑑𝑑=±𝑣, we arrive at the differential-difference equation𝛽𝑣4π‘Œξ…žξ…žξ…ž(𝑑)+π‘Œ(𝑑)=0,(13) where π‘Œ(𝑑)=𝑓(𝑑+𝑙/𝑣)βˆ’π‘”(π‘‘βˆ’π‘™/𝑣). Here, we used the relations 𝑓(𝑑)=𝑔(𝑑) and π‘“ξ…žξ…žξ…ž(𝑑)=π‘”ξ…žξ…žξ…ž(𝑑) accurate within a constant. In one's turn, these relations follow from the first condition of (10). The general solution of (13) is π‘Œ(𝑑)=𝐢1π‘’βˆ’2πœˆπ‘‘+𝐢2π‘’πœˆπ‘‘ξ€·βˆšcosξ€Έ3πœˆπ‘‘+𝐢3π‘’πœˆπ‘‘ξ€·βˆšsinξ€Έ3πœˆπ‘‘,(14) where 𝜈=(1/2)(𝑣4/𝛽)1/3. It follows from the definition of π‘Œ(𝑑) and functional relation (4) that 𝑓𝑙𝑑+𝑣=Ξ¦πœ‡ξ€·ξ€Έπ‘“(𝑑)+π‘Œ(𝑑),(15) where 𝑓↦2𝑓. Henceforth, we consider the case π‘Œ(𝑑)≃const, that is, πœˆπ‘‘β‰ͺ1. 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 πΉπ‘˜ be from (9). Then, from the above procedure, we find the first integral 𝑐0ξ€·β„Ž1+2𝑔1𝑒(0,𝑑)𝑔2/𝑔1=β„Ž2+2𝑔2𝑒(𝑙,𝑑),(16)where 𝑐0 is a constant. Let β„Ž1=0,𝑔2/𝑔1=2. These values correspond to experimental values from [2]. From (16), we find 𝑐𝑒(𝑙,𝑑)=0𝑔22𝑒2β„Ž(0,𝑑)βˆ’22𝑔2.(17)For the sake of simplicity, we choose 𝑐0𝑔2/2=1. Thus, it follows from (17) that𝑓𝑙𝑑+𝑣𝑙+π‘”π‘‘βˆ’π‘£ξ‚=𝑓(𝑑)+𝑔(𝑑)2+ξπœ‡,(18) where ξπœ‡=βˆ’β„Ž2/2𝑔2. Next, we deduce from (10), that is, 𝑓(𝑑)=𝑔(𝑑)+𝑐1 and 𝑓(𝑑+𝑙/𝑣)=𝑔(π‘‘βˆ’π‘™/𝑣)+𝑐2, that 𝑙2𝑓𝑑+𝑣+𝑐2=ξ€·2𝑓(𝑑)+𝑐1ξ€Έ2+ξπœ‡,(19) where 𝑐1=𝑓(0)βˆ’π‘”(0),𝑐2=𝑓(𝑙/𝑣)βˆ’π‘”(βˆ’π‘™/𝑣), and 𝑓(0),𝑓(𝑙/𝑣),𝑔(0) and 𝑔(βˆ’π‘™/𝑣) depend on initial functions 𝑒(0) and 𝑒(1). Let 𝑧(𝑑)=2𝑓(𝑑)+𝑐1. Then, it follows from (19) that 𝑧𝑙𝑑+𝑣=𝑧2(𝑑)+πœ‡,πœ‡=ξπœ‡+𝑐1βˆ’π‘2.(20)

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].