In a supersaturated solid solution, precipitation may lead to the formation of grains or particles of a new

phase. It is found that, after an initial stage of this process that involves nucleation, there is a second stage

in which the particles grow to much larger sizes. In the second stage the degree of supersaturation is slight,

and the rate of particle growth is controlled by diffusion. This coarsening of particles, called Ostwald

ripening, has important effects on the physicals properties of various alloys. An extensive analysis of the

kinetics of the second stage of this process was given by Lifshitz and Slyozov $(1961).$ The objective here is

to derive one of their simpler results.

Consider a dilute solid solution consisting of components A and B$,$ with B the abundant species. The

particles being formed by precipitation are composed of pure B$.$ The densities of the particle and the

solution are the same. Because B is the abundant species, we do not need to consider its transport. We

only need to account for transport of species A$.$ Assume that at any given time the precipitate consists of

widely dispersed spherical particles of radius R$($t$).$ Accordingly, it is sufficient to model one representative

particle, for which CA $=$ CA$($r$,$ t$)$ in the surrounding solution. Far from the particle the concentration of A

is constant at CA$\backslash$infty A key aspect of this problem is that the concentration of A at the particle surface is

influenced by the interfacial energy and therefore depends on particle size. The relationship is:

CA$[$R$($t$),$ t$]=$ CA$\backslash$infty $+$

$\backslash$Gamma

R$($t$)$

where the constant $\backslash$Gamma is proportional to the interfacial energy. It is the interfacial energy term that provides

the driving force for diffusion.

$($a$)$ Draw the physical system described in the problem. Since we are not modeling transport of B$,$ do not

show species B in the drawing. What is CA inside the particle? What is CA on the particle surface at

r $=$ R$($t$)?$ What is CA far away from the particle at r $=\backslash$infty $?$ Indicate these on the drawing.

$($b$)$ Assume diffusion of A outside the particle is pseudosteady. Starting with the pseudosteady ODE and

boundary conditions, derive the pseudosteady concentration profile CA$($r$,$ t$)$ outside the particle. Note, the

ODE can be solved by direct integration.

$($c$)$ An expression for the growth rate dR$/$dt of the particle can be derived from the interfacial conservation equation at the particle surface. Start from the general form of the interfacial conservation equation

$($considering only the r scalar components of all vectors$)$

$[$f $+$ CA$($v $$ vI $)]$out $[$f $+$ CA$($v $$ vI $)]$in $=$ Rs $(1)$

Here, f is the diffusive flux, v is the background flow velocity, CAv is the convective flux, vI is the interface

velocity, Rs is the surface generation rate of new A species, and the $$out$$ and $$in$$ subscripts indicate on

which side of the particle surface the quantities are evaluated on$.$ Note that some of the terms are zero and

that the diffusive flux f is specified by a constitutive equation $($Fick$$s law$).$ Use your pseudosteady state

profile from part $($b$)$ to obtain the expression for dR$/$dt$.($Hint: Where would dR$/$dt enter this interfacial

balance?$)$

$($d$)$ Without solving for R$($t$),$ what happens to R as t $->\backslash$infty $?$ Considering this limit of dR$/$dt$,$ what is the

ODE for R$($t$)$ at very large times t $1?$

$1$

$($e$)$ By solving the ODE, show that, eventually, R$($t$)$ t

$1/3$

$.$ While solving, you may assume that R$($t $=0)=$

R$0,$ but R$0$ cannot appear in the proportionality R$($t$)$ t

$1/3$

$.$

$($f$)$ Define the process time tp as the characteristic time it takes the particle to grow an amount roughly equal

to its size, i$.$e$.$R$/$R $1.$ Starting with your equation for dR$/$dt from part $($d$),$ use order$-$of$-$magnitude

reasoning to get tp$.$

$($g$)$ We can define the diffusion time td as the characteristic time it takes for A to diffuse an amount roughly

equal to the particle size, i$.$e$.$ td $=$ R$2/$D$.$ Using your estimate for the process time and this definition of

diffusion time, determine the conditions on R for the pseudosteady assumption to be valid.