Crystallization --> Growth
Dislocation induced crystal growth.
The model bypasses the necessity of random formation of two dimensional nuclei on the flat completely filled surface of crystal by presuming an constant presence of dislocations. Dislocations are local deviations from perfect crystal structure on the atomic level. Sources of dislocations could be isomorphic substitution of one atom by other with close chemical properties, holes inside crystal due not complete filling one layer that will be covered by other. Some positions could be left not filled and trapped inside. Taking aside for time the source of dislocation just their presence is postulated as axiom.
A dislocation on the surface could be considered as effective cutting the top of activation barrier for formation of critical nucleus on the flat surface. Analytical solution of such situation for general case of distribution dislocations with different acceleration effects to for formation of overcritical nuclei on the surface of growing crystal is if not imposable but obviously useless until can not be compared directly with experiment. That after invention of force microscopy is maybe will be performed earlier than later. But for now one safely can just add correction coefficient to the theoretical value of energy of formation of critical nuclei to make reasonable estimation.
A special case is the so named spiral (screw) dislocation growth. If in crystal structure there is partial shift in direction perpendicular to the surface like shown in the picture:
The ledge on surface can internally reproduce itself creating preferable area for surface growth. Further expansion can take a form of spiral like is shown on the photo:
or like pulsar sending wave by wave this covered surface of crystal one layer of molecules by other.
The rate of growth of surface limits by the rate of the top of mountain formed on the most energetically preferable defect. The average curvature of terrace for most fast growing dislocation has to provide most fast vertical growth. The fastest curvature on the tip of fastest spiral cannot be less than radius of critical nuclei as soon such formation is tended to dissolve not expand. The rate of vertical growth of the tip can be described by formula:
Vtip = (Sstep/Stip) * Vstep (1),
where Vstep - the rate of growth of the step of terrace near top that is equal to rate for continuous growth defined by Turnbull formula with correction of change of free energy to the curvature of the tip, Sstep is an area situated just in contact with the step and Stip whole area around. If Rtip is average curvature of the tip there could be following equation are valid:
Sstep = 2πdRtip (2),
Stip = πRtip2 (3),
where d is effective size of the molecule in parallel to surface directions. Formula (1) can be transformed to:
Vtip = 2(d/Rtip) *Vstep (4).
The rate of expansion for the step Vstep should has following expression:
where meaning of parameters are given in thermal activation model
For small supercooling formula 5 can be simplified taking into account that1-exp(-x) -> x when x approaches to zero:
Vstep =Vflat*(1-Rcr/Rtip) (6),
where Vflat growth rate for flat step when curvature of surface is approaching to infinity. Here is an point of possible confusion. Infinite curvature here is presumes for infinite rough surface. That looks like internal contradiction statement until you image analogy - differently grade of bending of sand paper. Flat sand paper is here analogous of surface with infinite curvature.
Vtip = 2(d/Rtip)*(1-Rcr/Rtip)* Vflat (7).
Equation 7 permits analytical finding expression for Rtip that produces maximum growth rate:
Rspiral = 2*Rcr (8),
Vspiral = 0.5*(d/Rcr)*Vflat (9).
Obviously equation 9 has limitations for large supercooling when radius of critical nuclei is smaller or just comparable to size of molecule. For such circumstances a continues growth rate should be expected.
It is worst to note that even not simplified version of equation (7) permits analytical solution with caviar that final formulas are too massive and not on the front of importance to present here.
Taking into account that Vflat is linear function of supercooling the same as invert value of radius of critical nuclei hence the growth rate by spiral mechanism should have square function of supercooling.
Submitted at Mar. 4, 2010; 17:32
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