Transient analysis of surface supersaturation after crystal face submersion using the analytical and transmission-line matrix (TLM) methods

<title>Transient surface supersaturation after crystal submersion: II</title>

“…In our previous papers 1, 2 we studied, using the surface diffusion theory 3 where n =-n(xt) is the local concentration of growth units at the surface; n. is the equilibrium concentration of growth units at the surface; a is the relative supersaturation just above the surface and very far from a step, so it is the same as the bulk supersaturation; X is the mean diffusion distance of the growth unit adsorbed on the surface and r denotes the relaxation time for leaving the surface adsorption layer. The relaxation time r is related to the activation free energy AG for desorption by Eyrings' formalism 3: h (AG ' where h is Planck's constant, kB is the Boltzmann constant and T denotes temperature.…”

Accelerated convergence in numerical simulations of surface supersaturation for crystal growth in solution under steady-state conditions

“…At this point we will invoke the use of electrical analogues. Rak et al [4] have shown that the BCF problem can be represented by a one-dimensional network comprising a symmetrical series connection of resistor ðRÞ and transmission line (impedance Z) arranged on either side of a node to which are connected in shunt a current source ðI m Þ and a conductance ðGÞ: In electrical terms R and Z represent a dispersive delay line, the analogue of diffusion. The other end of the current source is connected to a notional ground well away from any perturbations of the surface and represents adsorption from the solution bulk.…”

“…Rak et al [4] have discussed these for the BCF problem: ds s =dx ¼ 0 at the centre of the step and s s ¼ 0 at the extremes. The SOR scheme for the numerical solution expresses the value of s s at the new iteration, ðk þ 1Þ in terms of the neighbouring values at iteration ðkÞ: It also contains a fractional contribution from the previous time ðk À 1Þ; but this is dependent on the 'error', the distance from the convergent values.…”

“…The surface supersaturation on a stepped crystal surface can be described by means of a differential equation using the surface diffusion or 'BCF' theory of Burton Cabrera and Frank [1][2][3]. The transient super-saturation at the surface immediately after submersion of a crystal in a supersaturated solution has been analysed using the transmission line matrix (TLM) numerical method [4]. The results obtained by both methods have been in excellent agreement.…”

“…* Given that we have a TLM solution of the steady-state BCF equation [4] is it possible to vary parameters (as was done with the Laplace equation) to accelerate convergence? * How does this compare with the optimum conditions for the equivalent SOR solution using finite differences?…”