growing out of surface-mount resistors. It is generally believed that the mechanical and other properties of the crystal are also pertinent to the subject matter, and that crystal
morphology provides the missing link between growth kinetics and physical properties. The necessary thermodynamic apparatus was provided by
Josiah Willard Gibbs' study of heterogeneous equilibrium. He provided a clear definition of surface energy, by which the concept of surface tension is made applicable to solids as well as liquids. He also appreciated that
an anisotropic surface free energy implied a non-spherical equilibrium shape, which should be thermodynamically defined as
the shape which minimizes the total surface free energy. It may be instructional to note that
whisker growth provides the link between the mechanical phenomenon of high strength in whiskers and the various growth mechanisms which are responsible for their fibrous morphologies. (Prior to the discovery of carbon nanotubes,
single-crystal whiskers had the highest tensile strength of any materials known). Some mechanisms produce defect-free whiskers, while others may have single screw dislocations along the main axis of growth—producing high strength whiskers. The mechanism behind whisker growth is not well understood, but seems to be encouraged by compressive mechanical
stresses including mechanically induced stresses, stresses induced by
diffusion of different elements, and thermally induced stresses. Metal whiskers differ from metallic
dendrites in several respects. Dendrites are
fern-shaped like the branches of a tree, and grow across the surface of the metal. In contrast, whiskers are fibrous and project at a right angle to the surface of growth, or substrate.
Diffusion-control Very commonly when the supersaturation (or degree of
supercooling) is high, and sometimes even when it is not high, growth kinetics may be diffusion-controlled, which means the transport of atoms or molecules to the growing nucleus is limiting the velocity of crystal growth. Assuming the nucleus in such a diffusion-controlled system is a perfect sphere, the growth velocity, corresponding to the change of the radius with time \textstyle \frac{\partial r}{\partial t} , can be determined with Fick's Laws. 1. Fick' s Law: J=-D \nabla c , where \textstyle J is the flux of atoms in the dimension of \textstyle \frac{[quantity]}{[time]\cdot[area]} , \textstyle D is the diffusion coefficient and \textstyle \nabla c is the concentration gradient. 2. Fick' s Law: \frac{\partial c}{\partial t} =D \nabla^2c, where \textstyle \frac{\partial c}{\partial t} is the change of the concentration with time. The first Law can be adjusted to the flux of matter onto a specific surface, in this case the surface of the spherical nucleus: J_{matter} = D 4 \pi \cdot r^2 \frac{\partial c}{\partial r} , where \textstyle J_{matter} now is the flux onto the spherical surface in the dimension of \textstyle \frac{[quantity]}{[time]} and \textstyle 4 \pi \cdot r^2 being the area of the spherical nucleus. \textstyle J_{matter} can also be expressed as the change of number of atoms in the nucleus over time, with the number of atoms in the nucleus being: N(t)=\frac{\frac{4}{3} \pi \cdot r(t)^3}{V_{at}} , where \textstyle \frac{4}{3} \pi r^3 is the volume of the spherical nucleus and \textstyle V_{at} is the atomic volume. Therefore, the change if number of atoms in the nucleus over time will be: \frac{\partial N(t)}{\partial t}=\frac{4 \pi \cdot r(t)^2}{V_at} \frac{\partial r}{\partial t}=J_{matter} Combining both equations for \textstyle J_{matter} the following expression for the growth velocity is obtained: \frac{\partial r}{\partial t}=V_{at} D \frac{\partial c}{\partial r} From second Fick's Law for spheres the equation below can be obtained: \frac{\partial c}{\partial t}=D \frac{\partial }{\partial t} (r^2 \frac{\partial c}{\partial r}) Assuming that the diffusion profile does not change over time but is only shifted with the growing radius it can be said that \textstyle \frac{\partial c}{\partial t}=0, which leads to \textstyle r^2 \frac{\partial c}{\partial r} being constant. This constant can be indicated with the letter A and integrating will result in the following equation: r^2 \frac{\partial c}{\partial r}=A \Rightarrow \frac{A}{r^2} dr=dc \Rightarrow \int_{r}^{r+\delta} \frac{A}{r^2} dr = \int_{c_{0}}^{c_{l}} dc \Rightarrow c_{0}-c_{l}=A[\frac{1}{r}-\frac{1}{r+ \delta}] \Rightarrow A=\frac{c_{0}-c_{l}}{[\frac{1}{r}-\frac{1}{r+ \delta}]} , where \textstyle r is the radius of the nucleus, \textstyle r+ \delta is the distance from the nucleus where the equilibrium concentration \textstyle c_{0} is recovered and \textstyle c_{l} is the concentration right at the surface of the nucleus. Now the expression for \textstyle \frac{\partial c}{\partial r} can be found by: r^2 \frac{\partial c}{\partial r}=A \Rightarrow \frac{\partial c}{\partial r} = \frac{A}{r^2} = \frac{c_{0}-c_{l}}{[\frac{1}{r}-\frac{1}{r+\delta}] r^2 }=(c_{0}-c_{l} ) \cdot (\frac{1}{r}+\frac{1}{\delta}) Therefore, the growth velocity for a diffusion-controlled system can be described as: \frac{\partial r}{\partial t}= V_{at} D(c_{0}-c_{l} ) \cdot (\frac{1}{r}+\frac{1}{\delta}) (
manganese(IV) oxides)
dendrites on a
limestone bedding plane from
Solnhofen, Germany. Scale in mm. Under such diffusion controlled conditions, the polyhedral crystal form will be unstable, it will sprout protrusions at its corners and edges where the degree of supersaturation is at its highest level. The tips of these protrusions will clearly be the points of highest supersaturation. It is generally believed that the protrusion will become longer (and thinner at the tip) until the effect of interfacial free energy in raising the chemical potential slows the tip growth and maintains a constant value for the tip thickness. In the subsequent tip-thickening process, there should be a corresponding instability of shape. Minor bumps or "bulges" should be exaggerated—and develop into rapidly growing side branches. In such an unstable (or metastable) situation, minor degrees of anisotropy should be sufficient to determine directions of significant branching and growth. The most appealing aspect of this argument, of course, is that it yields the primary morphological features of
dendritic growth. ==See also==