Nonlocal phase separation problems for multicomponent

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Nonlocal phase separation problems for multicomponent systems

Collaborator: J.A. Griepentrog, H. Gajewski

Description: To describe phase separation processes we consider a closed multicomponent system with interacting particles of type k $\in$ {0, 1,..., n} occupying a spatial domain. In our model we assume that the particles jump around on a given microscopically scaled lattice following a stochastic exchange process (see [2]). On each lattice site sits exactly one particle (exclusion principle). Two particles of type k and l change their sites x and y with probability p_kl(x, y) due to diffusion and nonlocal interaction. The hydrodynamical limit leads to a system of conservation laws for k $\in$ {0, 1,..., n},

u_k' + $\displaystyle \nabla$ ^.j_k = 0 in (0, T) x $\Omega$ , $\displaystyle \nu$ ^.j_k = 0 on (0, T) x $\partial$ $\Omega$ , u_k(0) = g_k in $\Omega$ , (1)

for (scaled) particle densities u₀, u₁,..., u_n , their initial values g₀, g₁,..., g_n, and current densities j₀, j₁,..., j_n . Here, (0, T) denotes a time interval and $\nu$ is the outer unit normal on the boundary $\partial$ $\Omega$ of the bounded m-dimensional Lipschitz domain $\Omega$ . Due to the exclusion principle of the stochastic process we can assume $\sum_{{k=0}}^{n}$ u_k = 1, $\sum_{{k=0}}^{n}$ g_k = 1, and $\sum_{{k=0}}^{n}$ j_k = 0, that means, only n of the n + 1 equations in (1) are independent of each other. Hence, we can drop out one equation, say the equation for the zero component, and describe the state of the system by n-component vectors u = (u₁,..., u_n) having in mind the notation u₀ = 1 - $\sum_{{k=1}}^{n}$ u_k .

To establish thermodynamical relations between current densities, particle densities, and their conjugated variables, we minimize the free energy functional under the constraint of particle number conservation. In contrast to the classical Cahn-Hilliard theory we consider diffuse interface models and free energy functionals with nonlocal expressions. As a straightforward generalization of the nonlocal phase separation model for binary systems (see [1]) we define a free energy functional F = F₁ + F₂ by

F₁(u) = $\displaystyle \int_{\Omega}^{}$ f (u(x)) dx , F₂(u) = $\displaystyle {\frac{{1}}{{2}}}$ $\displaystyle \sum_{{k=0}}^{n}$ $\displaystyle \int_{\Omega}^{}$ (Ku)_k(x) u_k(x) dx ,

(2)

f (u) = $\displaystyle \sum_{{k=0}}^{n}$ u_klog(u_k) ,(Ku)_k(x) = $\displaystyle \sum_{{l=0}}^{n}$ $\displaystyle \int_{\Omega}^{}$ $\displaystyle \kappa_{{kl}}^{}$ (x, y) u_l(y) dy .

(3)

The convex function f and the symmetric (n + 1) x (n + 1)-matrix kernel $\kappa$ define the chemical part F₁ and the nonlocal interaction part F₂ of the functional F, respectively. Minimizing F under the constraint of particle number conservation, we identify the conjugated variables of the densities as grand chemical potential differences

$\displaystyle \lambda_{k}^{}$ = $\displaystyle {\frac{{\partial F}}{{\partial u_k}}}$ = $\displaystyle \mu_{k}^{}$ + w_k , k $\displaystyle \in$ {1,..., n} ,

where $\mu_{k}^{}$ and w_k are chemical and interaction potential differences, respectively,

$\displaystyle \mu_{k}^{}$ = $\displaystyle {\frac{{\partial F_1}}{{\partial u_k}}}$ = log(u_k) - log(u₀) , w_k = $\displaystyle {\frac{{\partial F_2}}{{\partial u_k}}}$ = (Ku)_k - (Ku)₀ , k $\displaystyle \in$ {1,..., n} .

(4)

The hydrodynamical limit process (see [2]) yields current densities

j_k = - $\displaystyle \sum_{{l=1}}^{n}$ a_kl(u) $\displaystyle \nabla$ $\displaystyle \lambda_{l}^{}$ , k $\displaystyle \in$ {1,..., n} ,

where the mobility has the form a(u) = d (u)(D²f (u))^-1, and d (u) denotes the diffusivity. Hence, we can interpret the above nonlocal phase separation model as a system of drift-diffusion equations with semilinear diffusion and nonlinear nonlocal drift terms, if we rewrite the currents as

j_k = - $\displaystyle \sum_{{l=1}}^{n}$ d_kl(u) $\displaystyle \nabla$ u_l - $\displaystyle \sum_{{l=1}}^{n}$ a_kl(u) $\displaystyle \nabla$ w_l , k $\displaystyle \in$ {1,..., n} .

In [4] we consider the case d_kl = $\delta_{{kl}}^{}$ . Then, an elementary computation of the inverse Hessian matrix (D²f (u))^-1 yields the following expressions for the mobility coefficients

a_kl(u) = $\displaystyle \delta_{{kl}}^{}$ u_k - u_lu_k , k, l $\displaystyle \in$ {1,..., n} .

For the functional analytic formulation of our problem we use standard spaces

H = [L²( $\displaystyle \Omega$ )]ⁿ , V = [H¹( $\displaystyle \Omega$ )]ⁿ , L = [L( $\displaystyle \Omega$ )]ⁿ ,

respectively, and their generalizations suitable for evolution systems,

$\displaystyle \mathcal {H}$ = L²((0, T);H) , $\displaystyle \mathcal {V}$ = L²((0, T);V) , $\displaystyle \mathcal {L}$ = L((0, T);L) , $\displaystyle \mathcal {W}$ = {u $\displaystyle \in$ $\displaystyle \mathcal {V}$ : u' $\displaystyle \in$ $\displaystyle \mathcal {V}$ ^*} .

Having in mind u₀ = 1 - $\sum_{{k=1}}^{n}$ u_k , we define simplices S $\subset$ L and $\mathcal {S}$ $\subset$ $\mathcal {L}$ by

S = {g $\displaystyle \in$ L : 0 $\displaystyle \le$ g₀, g₁,..., g_n $\displaystyle \le$ 1} , $\displaystyle \mathcal {S}$ = {u $\displaystyle \in$ $\displaystyle \mathcal {L}$ : 0 $\displaystyle \le$ u₀, u₁,..., u_n $\displaystyle \le$ 1} .

Furthermore, we introduce the drift-diffusion operator $\mathcal {A}$ : $\left[\vphantom{ \mathcal{V} \cap \mathcal{L}^\infty }\right.$ $\mathcal {V}$ $\cap$ $\mathcal {L}$ $\left.\vphantom{ \mathcal{V} \cap \mathcal{L}^\infty }\right]$ x $\mathcal {V}$ $\longrightarrow$ $\mathcal {V}$ ^* by

$\displaystyle \langle$ $\displaystyle \mathcal {A}$ (u, w), $\displaystyle \varphi$ $\displaystyle \rangle$ = $\displaystyle \sum_{{k=1}}^{n}$ $\displaystyle \int_{0}^{T}$ $\displaystyle \int_{\Omega}^{}$ $\displaystyle \nabla$ u_k^. $\displaystyle \nabla$ $\displaystyle \varphi_{k}^{}$ dx ds + $\displaystyle \sum_{{k=1}}^{n}$ $\displaystyle \sum_{{l=1}}^{n}$ $\displaystyle \int_{0}^{T}$ $\displaystyle \int_{\Omega}^{}$ a_kl(u) $\displaystyle \nabla$ w_l^. $\displaystyle \nabla$ $\displaystyle \varphi_{k}^{}$ dx ds ,

for (u, w) $\in$ $\left[\vphantom{ \mathcal{V} \cap \mathcal{L}^\infty }\right.$ $\mathcal {V}$ $\cap$ $\mathcal {L}$ $\left.\vphantom{ \mathcal{V} \cap \mathcal{L}^\infty }\right]$ x $\mathcal {V}$ , $\varphi$ $\in$ $\mathcal {V}$ . More general than in the above description of the model, we assume that the interaction between particles can be described by means of a (possibly nonlinear and nonlocal) Lipschitz continuous interaction operator $\mathcal {P}$ : $\mathcal {H}$ $\longrightarrow$ $\mathcal {V}$ .

Applying fixed-point arguments and comparison principles in [4], we show that for every initial value g $\in$ S there exists a solution (u, w) $\in$ $\left[\vphantom{ \mathcal{W} \cap \mathcal{S} }\right.$ $\mathcal {W}$ $\cap$ $\mathcal {S}$ $\left.\vphantom{ \mathcal{W} \cap \mathcal{S} }\right]$ x $\mathcal {V}$ of the evolution system

u' + $\displaystyle \mathcal {A}$ (u, w) = 0 , w = $\displaystyle \mathcal {P}$ u , u(0) = g .

(5)

Moreover, under some natural regularity assumption on the interaction operator $\mathcal {P}$ : $\mathcal {H}$ $\longrightarrow$ $\mathcal {V}$ in [4] we also get the unique solvability of our problem. In fact, we additionally assume that $\mathcal {P}$ has the Volterra property and that the restriction of $\mathcal {P}$ to $\mathcal {L}$ is a Lipschitz continuous operator from $\mathcal {L}$ into a certain Sobolev-Morrey space $\mathcal {X}$ $\subset$ $\mathcal {V}$ for some parameter $\sigma$ > m . Then, our regularity theory for initial boundary value problems with nonsmooth data in Sobolev-Morrey and Hölder spaces (see [3]) enables us to prove the unique solvability of problem (5).

To illustrate our results, we consider an example of a ternary system, where the interaction operator $\mathcal {P}$ : $\mathcal {H}$ $\longrightarrow$ $\mathcal {V}$ is defined by ( $\mathcal {P}$ u)_k = (Ku)_k - (Ku)₀ , k $\in$ {1,..., n} (see (3), (4), (5)). The corresponding matrix kernel $\kappa$ is chosen such that particles of the same type attract and particles of different type repel each other with the same range and strength of interaction. Figures 1 and 2 show simulation results of phase separation processes in a unit square. Notice that both initial configurations contain equal numbers of black, white, and red particles, respectively. Moreover, the final states are local minimizers of the free energy functional F under the constraint of particle number conservation.

**Fig. 1:** Phase separation process for an initial value which is constant in the vertical direction. The stripe pattern is preserved during the whole evolution.
$\ProjektEPSbildNocap{0.96\textwidth}{straight.eps}$

**Fig. 2:** Phase separation process for a symmetric initial value. There occur metastable states. Finally, the phases are separated by a straight line and circular arcs.
$\ProjektEPSbildNocap{0.96\textwidth}{upper.eps}$

References:

H. GAJEWSKI, K. ZACHARIAS, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), pp. 11-31.
G. GIACOMIN, J.L. LEBOWITZ, R. MARRA, Macroscopic evolution of particle systems with short- and long-range interactions, Nonlinearity, 13 (2000), pp. 2143-2162.
J.A. GRIEPENTROG, Zur Regularität linearer elliptischer und parabolischer Randwertprobleme mit nichtglatten Daten, Logos-Verlag, Berlin, 2000, 219 pages.
, On the unique solvability of a phase separation problem for multicomponent systems, WIAS Preprint no. 898, 2004, to appear in: Banach Cent. Publ.