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The mathematical description of the morphology instability in supercooled liquid
is well-known as Mullings-Sekerka Instability [97, 98, 132–134]. As in many instability
problems, not every perturbation becomes unstable. The typical wavelength
of Mullings-Sekerka instability is λ ms = 2π(2α L /v n l 0 ). Note that λ ms is the geometric
mean of the microscopic capillary length l 0 and the macroscopic diffusion
length 2α L /v n . It is roughly the right scale to characterize dendritic structure. A
planar solidification front moving at speed v n is linearly unstable against sinusoidal
deformations whose wavelengths are larger than λ ms .
The growth rate of the dendritic structure is of interest referring to the combination
with drop impact hydrodynamics, because the ice dendrite would influence
the impact process if its growth rate is faster than, or comparable with the impact
process. It is known that the product of the growth rate, v tip , of the dendrite and its
tip radius R tip are determined uniquely by the supercooling St SC L as the Ivantsov
relation [62]:
St SC L = P ec e P ec
E 1 (P ec ), (2.52)
where P ec = R tip v tip /2α L is the Peclet number, and E 1 is the exponential integral:
E 1 (y) =
∫ ∞
e −y′
y ′
y ′ d y ′ . (2.53)
The tips of the dendrite often look very paraboloidal under small supercooling,
qualitatively indicating that the Ivantsov relation is satisfied under these conditions.
In order to get the tip velocity, the tip radius needs a size estimation. Langer and
Müller-Krumbhaar postulated that a dendrite with a tip radius R tip appreciably
larger than λ ms must be unstable against sharpening or splitting. In this sense, the
dynamic process of dendrite formation might naturally come to rest at a state of
marginal stability [72, 73, 96], for which the dimensionless group of parameters,
P LM K = 2α Ll 0
v tip r 2 = λ ms
, (2.54)
2πr
tip tip
is a constant, independent of supercooling St SC L . Setting R tip equals to λ ms , P LM K
becomes 0.025. This insightful speculation has delivered encouragingly consistent
predictions with a wide range of experimental observations as Figure 2.22 shows.
Comparing this graph to Figure 2.19, it is easy to find out that the real growth rate
40 2. Background