Dendrites are the finger-like protrusions generated when a dynamically unstable interface evolves in time. The instability often then occurs again on surface of the initial protrusions, generating branched structures.
Dendrite formation in crystals, artificially grown and in a snowflake. The anisotropy in both these cases is caused by the internal structure of the crystals (from NASA's Isothermal Dendritic Growth Experiment)
Dendrite formation occurs widely, including in the process of metal casting and crystal growth (solid-liquid interface), in the formation of snowflakes (solid-gas interface) and in the fluid-dynamical Saffman-Taylor instability (liquid-liquid interface). It can also occur when discrete solid particles join together through the process of aggregation. While the physical mechanisms for these phenomena are very different, the fundamental instability is often described by a so-called Stefan problem, which in its simplest form reduces to Laplace's equation with a boundary condition (the Stefan condition) applied at the interface.
The Saffman-Taylor instability occurs in a Hele-Shaw cell (a thin incompressible fluid-filled gap between two plates) when a less viscous fluid is injected into a cell containing a more viscous fluid. Assuming that inertia remains negligible (that is, in the low Reynolds number limit), the motion of the viscous fluid is governed by the balance between pressure forces and viscous drag, so for the viscous fluid:
u = - μ ∇p
, a function of the fluid viscosity and cell thickness, is a constant. With the incompressibility condition ∇ · u = 0
, this implies that the pressure in the viscous fluid is harmonic — that is, it satisfies Laplace's equation ∇² p = 0
. If the ratio of viscosities is large, the pressure in the injected fluid is essentially constant, due to there being little viscous drag to induce a pressure gradient. The velocity of the interface in a Hele-Shaw cell is given by the normal fluid velocity there, that is, proportional to the gradient of the pressure normal to the interface.
A simulation of Saffman-Taylor fingering using the boundary integral method (adapted from J. Comp. Phys, 212, pp. 1-5 (2006))
If no surface tension or other effects are present at the interface, the pressure at the interface is continuous and is given by the constant pressure of the injected fluid. When a perturbation occurs on the interface, that the pressure in the viscous fluid satisfies Laplace's equation causes an increase of the normal pressure gradient on the protruding part of the interface. This increases the speed at which the perturbation grows, making the interface unstable.
A linearised analysis shows that this instability grows fastest for infinitesimally small wavelength perturbations to the interface: this is at odds with the observed characteristic finite length-scale seen in Saffman-Taylor fingering. The inhibition of small-scale perturbation growth is due to surface tension on the interface, which causes a discontinuity in pressure of a magnitude proportional to the interface curvature. This has the effect of stabilising small-wavelength perturbations, causing the most unstable wavelength to be a finite one.
An example of Saffman-Taylor fingering. The interface here is affected by surface tension, which leads to a characteristic length scale over which the dendrites occur (adapted from Perspectives in Fluid Dynamics, CUP (2000))
The mechanism of dendritic growth governed by a harmonic field applies to the discrete system known as diffusion-limited aggregation (DLA). Here, free particles are moved under Brownian motion in an environment with a fixed boundary. When a free particle touches either a boundary or a fixed particle, the free particle becomes fixed itself. In this way particles are deposited, forming a growing aggregation of fixed particles.
The results of a simulation of DLA with approximately 10,000 particles. Contours of probability are shaded in blue
Laplace's equation occurs here as the infinitesimal generator of Brownian motion. This implies that the evolution of the probability distribution free particles is determined by the diffusion equation. In the case where diffusion is very much faster than the rate of growth of the aggregation, the problem is essentially static (in the sense that it is dependent on time only through the time-varying boundary conditions), and is governed by the steady-state case of the diffusion equation, Laplace's equation. The velocity at which the boundary moves is proportional to the mean flux of particles in the direction normal to the wall, which is given by the boundary-normal derivative of the particle probability distribution. As for the boundary conditions on the probability distribution, since the fixed objects absorb free particles the probability of finding a free particle at the boundary is zero.
Extending the model
In some situations the problem of dendrite formation cannot be simplified fully to Laplace's equation. This is often the case in solidification problems, where the diffusion rate of temperature (the relevant variable) in a stationary fluid may be of the same order as the boundary velocity. In addition to this, the temperature gradients induced in the fluid by solidification may induce convection currents in the liquid phase, which is then described by an advection-diffusion equation.
The Laplacian model can also be extended in the case of diffusion-limited aggregation. In situations such as electroplating, the free particles are not purely guided by Brownian motion, but are attracted towards the fixed cluster by electrostatic forces. The probability that those particles that collide with the aggregation stick to it can also be small, further modifying the shape of the dendrites.
Simulation of DLA where particles experience an force towards the centre. This increases the probability that particles will travel down 'valleys' between dendrites, filling them in and resulting in a denser aggregation.
Simulation of DLA where a particle colliding with the aggregation sticks to it in 10% of collisions and rebounds otherwise. The 'bushier' aggregation formed is a result of the free particles spending a greater proportion of time near the boundary of the aggregation than in the case of simple DLA.