3.2 Validity of the continuum approximation in multilayers - Continuum models deteriorate on the nanoscale

Cook et al. showed that the continuum and discrete approximations give the same results only if the wavelength of the modulation $\Lambda$ is at least six times longer than the interatomic distance, $d$ , in the direction of the diffusion $(\Lambda > 6d)$ . Cahn and Yamauchi and Hilliard found the similar range of validity of the continuum approach for intermixing of multilayers. These conclusions, however, are obtained in linear approximation, i.e. assuming that the diffusion coefficient is independent of concentration. The treatment of the effects of this type of nonlinearity is very complicated even if one neglects the stress effects. Tsakalakos and Menon and de Fontaine tried to treat this problem analytically, or by solving the continuum equations numerically, considering a concentration dependence no stronger than a quadratic one in the diffusion coefficient, although even in ideal solutions it can be stronger and is better described by an exponential dependence.

We showed that in case of an exponential dependence of the diffusion coefficient (2.5) the discrete and continuum approximations give the only if modulation length of a multilayer is larger than $3$ to $50$  nm ( $\Lambda > 3 - 50$  nm), depending on $m'$ , i.e. on the strength of the composition dependence of the diffusion coefficientm or other words, how large the diffusion asymmetry is. For details see Ref. [2] and [1].

Figure 3.1: Values for the critical modulation length, $\Lambda_c$ , above which the continuum model is valid, as a function of $m'$ .