A Stochastic Model of Crystallization in an Emulsion - Laboratoire de ...

phase during the experimentation. We thus consider an emulsion containing the phase-change
material. However, repeating the melting-freezing cycles, destroys the dispersed structure of the
emulsion. To avoid that, a surface-active agent (surfactant) can be used and so the behavior of
the emulsion is closed to the behavior of solid matter. Therefore, the effect of free-convection can
be ignored.
The object of this paper is to talk about the crystallization of an emulsion.
This process is based on two phenomena (cf. Dumas et al [6]) :
1) the undercooling, dened as the difference between the melting temperature TF
and the
crystallization temperature, increases as the sample size of PCM decreases. For example, for water,
3
3
the undercooling is 14 K for a few cm macrosamples and is 38 K for a few m microsamples.
Moreover, any droplet which temperature is less than TF
may crystallize as soon as any perturbation
or shock occurs. Where from the second phenomenon :
2) the main feature of emulsions crystallization in an instable dispersed medium (here an
emulsion with temperature less than TF
) is its stochastic behavior. Samples of PCM which
are apparently identical, will not transform at the same temperature during the cooling process.
This leads to the notion of nucleation rate. The Nucleation Theorie gives us a function J,
the
deterministic part of the crystallization speed, per unit of time. In the sequel, we consider the
nucleation rate as a stochastic perturbation of this speed J.
We assume that J is a positive lipschitz function of the temperature with, J(0) = 0 and J( x)
=
0 , ∀x ∈ [ T , + ∞[
.
∞
F
One mays nd a presentation of the experimental context in the last section.
Let us denote by u the temperature of the emulsion and ϕ( t, x)
the proportion of crystallized
droplets, in the neighborhood of a point x and a time t .
Then, the mathematical modelling proposed by Dumas et al [6] is based on :
the nonlinear heat equation with a heat source term proportional to the speed of crystallization,
i.e.
∂u
∂ϕ
( u)
= ,
∂t
∂t
with some Fouriers type boundary conditions
in
(1)
∂ u
∂
N
( ) ∞
= ( u u ) , on = ∂
where u is a given temperature on the boundary ,
the following stochastic differential equation for the speed of crystallization
∂ϕ
∂t
= (1 ϕ) J( u) + b( u, ϕ)
∂w
, in
∂t
where 1 ϕ is the proportion of uncrystallized droplets, J( u) the nucleation rate, w the standard
real Wiener process and b the random part of the nucleation phenomenon.
A rigorous formulation of our system will be given below.
2 Notations and hypothesis
From now on, we consider the following notations :
is a bounded domain of R with a smooth boundary and an outward unit normal ,
for T > 0 , Q is the cylinder ]0 , T [ and its lateral boundary ]0 , T [ ,
2
(2)
(3)