Features and applications Cahn–Hilliard equation

evolution of random initial data under cahn–hilliard equation



γ
=
0.5


{\displaystyle \gamma =0.5}

,



c
=
0


{\displaystyle c=0}

, demonstrating phase separation.



there transition layer between segregated domains, profile given function



c
(
x
)
=
tanh
⁡

(


x

2
γ



)

,


{\displaystyle c(x)=\tanh \left({\frac {x}{\sqrt {2\gamma }}}\right),}

, hence typical width





γ




{\displaystyle {\sqrt {\gamma }}}

because function equilibrium solution of cahn–hilliard equation.
of interest fact segregated domains grow in time power law. is, if



l
(
t
)


{\displaystyle l(t)}

typical domain size,



l
(
t
)
∝

t

1

/

3




{\displaystyle l(t)\propto t^{1/3}}

. lifshitz–slyozov law, , has been proved rigorously cahn–hilliard equation , observed in numerical simulations , real experiments on binary fluids.
the cahn–hilliard equation has form of conservation law,






∂
c


∂
t



=
∇
⋅


j


(
x
)
,


{\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot {\mathbf {j}}(x),}







j


(
x
)
=
d
∇
μ


{\displaystyle {\mathbf {j}}(x)=d\nabla \mu }

. phase separation process conserves total concentration



c
=
∫

d

n


x
c

(
x
,
t
)



{\displaystyle c=\int d^{n}xc\left(x,t\right)}

,






d
c


d
t



=
0


{\displaystyle {\frac {dc}{dt}}=0}

.
when 1 phase more abundant, cahn–hilliard equation can show phenomenon known ostwald ripening, minority phase forms spherical droplets, , smaller droplets absorbed through diffusion larger ones.

the cahn–hilliard equations finds applications in diverse fields: in complex fluids , soft matter (interfacial fluid flow, polymer science , in industrial applications). solution of cahn–hilliard equation binary mixture demonstrated coincide solution of stefan problem , model of thomas , windle. of interest researchers @ present coupling of phase separation of cahn–hilliard equation navier–stokes equations of fluid flow.