The evolution of the system is thus described by the system:

\begin{array}{cc}{\dot{\psi}}_{0}^{j}& =-{A}_{j}{\psi}_{0}^{j},\hfill \\ {\dot{\psi}}_{n}^{j}& =-{A}_{j}{\psi}_{n}^{j}+{f}_{j}\sum _{k}{A}_{k}{\psi}_{n-1}^{k},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}

(3.1)

with initial conditions:

{\psi}_{0}^{j}(0)={f}_{j},\phantom{\rule{2em}{0ex}}{\psi}_{n}^{j}(0)=0,\phantom{\rule{1em}{0ex}}n\ge 1.

(3.2)

### 3.1 Numerical solution

It is straightforward to solve this system of ordinary differential equations numerically. In Figure 3, we show the time evolution curves into the first three layers of the solid for the case where there are two species 1 and 2 being etched by a single acid. Figure 4 shows a typical solution for the fraction of exposed sites as a function of depth into the crystal at large times. We see that the ‘interface’ (where {\psi}_{n}^{j}, the fraction of exposed sites at depth *n* into the crystal of type *j*, is positive) is diffuse (that is, it spreads out as it moves down into the crystal), and propagates downwards at an essentially constant rate. Note also that the discrete solution appears to be well approximated by a continuously varying site occupation density for each species.

### 3.2 Analysis of the discrete model

To solve the equations (3.1), we define the Laplace transform of {\psi}_{n}^{j} as

{\Psi}_{n}^{j}={\int}_{0}^{\infty}{\psi}_{n}^{j}{e}^{-\lambda t}\phantom{\rule{0.2em}{0ex}}dt,

(3.3)

so that the equations (3.1) become

\begin{array}{cc}{\Psi}_{0}^{j}& =\frac{{f}_{j}}{\lambda +{A}_{j}}\phantom{\rule{1em}{0ex}}\text{and}\hfill \\ {\Psi}_{n}^{j}& =\frac{{f}_{j}}{\lambda +{A}_{j}}\sum _{k}{A}_{k}{\Psi}_{n-1}^{k},\phantom{\rule{1em}{0ex}}n\ge 1,\hfill \end{array}

(3.4)

respectively. To solve this, define the function

g(\lambda )=\sum _{k}\frac{{A}_{k}{f}_{k}}{\lambda +{A}_{k}};

(3.5)

by induction, we then find that

{\Psi}_{n}^{j}=\frac{{f}_{j}}{\lambda +{A}_{j}}g{(\lambda )}^{n}.

(3.6)

Solutions {\psi}_{n}^{j} are found by taking the inverse Laplace transform,

{\psi}_{n}^{j}=\frac{1}{2\pi \mathrm{i}}{\int}_{\Gamma}\frac{{f}_{j}}{\lambda +{A}_{j}}g{(\lambda )}^{n}{e}^{\lambda t}\phantom{\rule{0.2em}{0ex}}d\lambda ,

(3.7)

where the contour \Gamma =[\gamma -i\infty ,\gamma +i\infty ] lies to the right of the poles of the integrand (for example, take \gamma =0). From (3.5) and (3.7), these are

The integral (3.7) can be solved explicitly by calculating the residues at the poles -{A}_{k}. First note that we can write, for any value of *s*,

(\lambda +{A}_{s})g(\lambda )={f}_{s}{A}_{s}+(\lambda +{A}_{s}){g}_{s}(\lambda ),

(3.9)

where

{g}_{s}(\lambda )=\sum _{k\ne s}\frac{{f}_{k}{A}_{k}}{\lambda +{A}_{k}},

(3.10)

from which it follows that, for any *s*,

g{(\lambda )}^{n}=\sum _{r=0}^{n}\frac{{}^{n}C_{r}{({f}_{s}{A}_{s})}^{r}{g}_{s}{(\lambda )}^{n-r}}{{(\lambda +{A}_{s})}^{r}}.

(3.11)

Since {g}_{s} is regular at -{A}_{s}, we can use (3.11) to determine the coefficients of the Laurent series expansion of (3.7). We can therefore calculate the residues {r}_{jk} of the integrand of (3.7) at -{A}_{k} as

\begin{array}{cc}{r}_{jj}& ={f}_{j}{G}_{j}^{n}{e}^{-{A}_{j}t},\hfill \\ {r}_{jk}& =\frac{n{f}_{j}{f}_{k}{A}_{k}}{{A}_{j}-{A}_{k}}{G}_{k}^{n-1}{e}^{-{A}_{k}t},\phantom{\rule{1em}{0ex}}k\ne j,\hfill \end{array}

(3.12)

where

{G}_{j}={g}_{j}(-{A}_{j})=\sum _{l\ne j}\frac{{f}_{l}{A}_{l}}{{A}_{l}-{A}_{j}}.

(3.13)

We then have the explicit formula

{\psi}_{n}^{j}=\sum _{k}{r}_{jk}.

(3.14)

Suppose that the species *j* are ordered in terms of increasing reaction rate so that {A}_{1}<{A}_{2}<\cdots; it then follows that at large *t*,

{\psi}_{n}^{j}\sim {r}_{j1}\sim \{\begin{array}{ll}{f}_{1}{G}_{1}^{n}{e}^{-{A}_{1}t},& j=1,\\ \frac{n{f}_{j}{f}_{1}{A}_{1}}{{A}_{j}-{A}_{1}}{G}_{1}^{n-1}{e}^{-{A}_{1}t},& j>1,\end{array}

(3.15)

for each fixed *n*. Since (3.13) implies that {G}_{1}>0, it follows that there are two cases to consider. If {G}_{1}<1 then solutions decay in both *n* and *t*, but this tells us nothing about the mean etching rate. When {G}_{1}>1, (3.15) indicates that {\psi}_{n}^{j} increases with *n*; however, this asymptotic result must become inappropriate when n\sim t, since conservation of sites implies

\sum _{j}\sum _{n=0}^{\infty}{\psi}_{n}^{j}\equiv 1,

(3.16)

as in (2.13). Thus {\psi}_{n}^{j} is bounded and in fact decreases towards zero at large *n*. We wish to focus attention on the penetration depth, or wavefront location, of the etchant so we will denote its location by {n}_{w}(t).

#### 3.2.1 Solutions for large *n* and *t*

To examine the large time behaviour when the penetration depth {n}_{w}\sim t, we use asymptotic methods to determine the behaviour of {\psi}_{n}^{j} as t\to \infty directly from the integral in (3.7). From Figure 4, we see that the site densities {\psi}_{n}^{j} appear to spread at a constant rate and diffuse as they propagate. This suggests making the ansatz

{n}_{w}={v}_{p}t+\xi \sqrt{t},

(3.17)

where {v}_{p}>0, and we will consider the asymptotic form of (3.7) for large *t* with {v}_{p} and *ξ* fixed. In particular, we will choose {v}_{p} to be the speed of the wavefront. We define

\rho (\lambda )=\lambda +{v}_{p}lng(\lambda ),

(3.18)

so that (3.7) takes the form

{\psi}_{n}^{j}=\frac{1}{2\pi \mathrm{i}}{\int}_{\Gamma}\frac{{f}_{j}}{\lambda +{A}_{j}}exp[t\rho +\sqrt{t}\xi lng]\phantom{\rule{0.2em}{0ex}}d\lambda .

(3.19)

Suppose that there are *J* species. The integrand of (3.19) has poles at -{A}_{1},-{A}_{2},\dots ,-{A}_{J}, at which *g* is infinite; between these values, *g* is monotonically decreasing, and therefore *g* has J-1 real zeroes at {\lambda}_{1},\dots ,{\lambda}_{J-1}, where -{A}_{j}>{\lambda}_{j}>-{A}_{j+1}. It follows from this that

Re\rho =\lambda +{v}_{p}ln|g|

(3.20)

has 2J-1 logarithmic branch points at -{A}_{j} and {\lambda}_{k}; also since *g* is convex and decreasing for \lambda >-{A}_{1}, it follows that *ρ* is real and convex for \lambda >-{A}_{1}, with a unique minimum at \lambda ={\lambda}^{\ast}, say. The form of Re*ρ* as a function of *λ* is shown in Figure 5.

To evaluate the integral (3.19) asymptotically for large *t*, we aim to deform the contour \Gamma =[\gamma -i\infty ,\gamma +i\infty ] to one passing through a saddle point of \rho (\lambda ), where {\rho}^{\prime}(\lambda )=0, since the size of the exponent in the integrand is dominated by *ρ*. We can then use the method of steepest descents. The obvious such saddle point is at {\lambda}^{\ast}, and this is indeed the correct choice. To understand why, we need to describe the steepest ascent and descent paths in the complex *λ* plane.

These are given by the curves Im\rho = constant, which trace out trajectories in the complex *λ* ‘phase plane’. (If we think of *ρ* as a complex velocity potential, then the curves Im\rho = constant are the streamlines.)

With arrows denoting direction of increasing Re*ρ* on these curves, the points -{A}_{j} are like sinks (Re\rho \to -\infty), and the points {\lambda}_{k} are like sources (Re\rho \to \infty). On the real axis, the source lines from (-\infty ,{\lambda}_{J-1},\dots ,{\lambda}_{1},{\lambda}^{\ast}) are directed to the sinks at -{A}_{j}, as shown in Figure 6. Nearby trajectories (of constant Im*ρ*) must do the same, passing to the sinks on either side; hence there must be a dividing trajectory which must go to infinity.

However, \rho \sim \lambda at infinity, and it follows from this that firstly, the steepest descent trajectories from {\lambda}^{\ast} asymptote to ∞ horizontally in the left hand part of the plane, and consequently so must also the lines of constant Im*ρ* which reach infinity from the sinks at -{A}_{j}. Most of the streamlines from the sources terminate on the adjoining sinks, but the dividing streamline, being sandwiched between adjoining trajectories which originate at Re\lambda =-\infty, must also originate there. But this is only possible if Re*ρ* decreases, and this requires that each dividing streamline from the J-1 sources passes through a saddle point, as shown in Figure 6. Since *ρ* has real-valued coefficients, the phase plane is symmetric about the real axis, so that there are another J-1 saddles in the lower half plane.

Together with {\lambda}^{\ast}, we have accounted for 2J-1 saddles. This constitutes all of the saddles since direct calculation of {\rho}^{\prime} gives

{\rho}^{\prime}(\lambda )=1+{v}_{p}\frac{{g}^{\prime}(\lambda )}{g(\lambda )}=0,

(3.21)

whence the saddles are the 2J-1 roots of the polynomial

\sum _{k}{A}_{k}{f}_{k}(\lambda +{A}_{k}-{v}_{p})\prod _{l\ne k}{(\lambda +{A}_{l})}^{2}=0.

(3.22)

The appropriate choice of {v}_{p} follows from the conservation law in (3.16). Clearly the maximum value of {\psi}_{n}^{j} can neither grow nor decay exponentially, and thus we choose {v}_{p} such that

\rho ({\lambda}^{\ast})=0.

(3.23)

By inspection, we see that this is satisfied, together with (3.21), providing

{\lambda}^{\ast}=0,\phantom{\rule{1em}{0ex}}{v}_{p}={\left(\sum _{k}\frac{{f}_{k}}{{A}_{k}}\right)}^{-1},

(3.24)

and thus in the vicinity of the saddle point,

\rho (\lambda )t+\xi g(\lambda )\sqrt{t}\approx \frac{1}{2}{\rho}_{0}^{\u2033}{\lambda}^{2}t-\frac{\xi}{v}\lambda \sqrt{t},

(3.25)

where {\rho}_{0}^{\u2033}={\rho}^{\u2033}(0). Putting \lambda =is gives the local approximation to the steepest descent trajectory, and integrating the resultant approximation to the integral leads to the asymptotic solution

{\psi}_{n}^{j}\sim \frac{{f}_{j}}{{A}_{j}\sqrt{2\pi t{\rho}_{0}^{\u2033}}}exp(-\frac{{[{n}_{w}-{v}_{p}t]}^{2}}{2t{\rho}_{0}^{\u2033}{v}_{p}^{2}}).

(3.26)

### 3.3 Summary of solutions

In summary, we obtained numerical solutions of the basic model in the previous section. Using Laplace transforms we then found that the exact solution of (3.1) and (3.2) for {\psi}_{n}^{j}, the fraction of vacant sites of type *j* at depth *n* into the crystal is given by (3.14), incorporating (3.12) and (3.13). For large times, this can be simplified to (3.15) which is inappropriate when n\sim t. For large *t* and n\sim t, we then found that the asymptotic limit giving the key result for the penetration {n}_{w} of the wavefront into the crystal, at time *t*, as

{n}_{w}\sim {v}_{p}t,\phantom{\rule{1em}{0ex}}{v}_{p}={\left(\sum _{k}\frac{{f}_{k}}{{A}_{k}}\right)}^{-1},

(3.27)

where {f}_{k}, {A}_{k} are the initial fractions and reaction rates of the species *k* in the solid. The corresponding asymptotic result for {\psi}_{n}^{j} is given by (3.26).

In the context of Figure 4, the asymptotic solution (3.24) in conjunction with (3.17) predicts that the interface, neglecting its diffusion, moves at a speed {v}_{p}={(\frac{{f}_{1}}{{A}_{1}}+\frac{{f}_{2}}{{A}_{2}})}^{-1}=1.54, that is, {n}_{w}\sim 1.54t. It is apparent that the individual etchant rates sum like electrical resistors in parallel. The fact that the numerical solutions in Figure 4 are so smooth, and that they are so close to the asymptotic solution, suggests strongly that a continuum model should be appropriate. We now examine this possibility.