The tension field created by a spherical nucleus freezing into its less dense undercooled melt

Abstract When nickel freezes into its less dense melt, that is cooled by more than 175°C below its equilibrium freezing temperature, the solidified material exhibits—as first observed by J. L. Walker—dispersed fine-grain structure (presumably as a result of cavitation induced by huge negative pressures surrounding the growing nuclei), whereas for undercooling less than 175°C the observed structure is coarse grained. The purpose of the present analysis was to provide numerical (theoretical) estimates for the pressures, flow velocities, and time scales involved. This necessitated study of freezing as proceeding from a finite initial embryo. Using a “generalized orthogonalization method” of solution, the freezing process is traced out, taking the pressure dependence of freezing temperature also into account, on the basis of incompressible inviscid fluid dynamics. The solution of the governing differential equation system is represented as a sum Σ0k − 1 Fk of [vector] functions Fk(ξ) (ξ is the dimensionless radial coordinate) whose time dependence (τ is dimensionless time) is determined from orthogonality conditions (boundary layer integral equations), using in the integrand weight functions of type ξkm; k = 0, 1, …, K−1. We refer to approximations I, II, III, … when K = 1, 2, 3, …. (K = 1 corresponds to the conventional boundary layer solution of the type von Karman-PohIhausen-GoodmanVeynik), and to approximations II1, II 1 2 , II0 when K = 2 and = 1, 1 2 , 0. Using for ΣFk(ξ) a sequence of perturbed (inτ) decaying (inξ) exponentials, it was found that the graph of solution II0 is, in its asymptotic behavior (τ → ∞), indistinguishable from the well known rigorous solution of the problem where the nucleus grows from zero radius and pressure dependence of freezing temperature is ignored. However, this asymptotic era is not reached until elapse of about 10−7 s from start of growth, whereas the maximum inrush of fluid on to the growing nucleus (at a speed exceeding 100 m/s) occurs in the first 10−11 s and is accompanied by tensions of several thousand atmospheres. This first portion of the phenomenon (to 10−11 s) may be represented by ascending power series in τ 1 2 in the perturbation factors, the last portion (past 10−7 s) by descending power series in τ 1 2 ; the huge intervening portion must be bridged by numerical integration of the pertinent differential equation system. Besides corroborating the expected pressure distribution, the analysis brought forth an unexpected result. The freezing process, as now described, is, for the case of precisely zero density change, totally different from that for infinitesimal density change. The latter starts from a finite initial radius, with zero velocity, the former with infinite velocity. This discontinuity (with density change) in the solution points to the need for further studies.