Crystallization --> Growth

Mechanism of two-dimensional nuclei growth.

by Leonid Sakharov

One of the models of layer by layer growth of crystal is based on the random process of forming two dimensional nucleus that schematically displayed on the following picture:

Fig. 1. Schematic view of two dimensional nucleus on the flat crystal surface. 

A change of free Gibbs energy as a result of formation such disc-like island represents a formula:

ΔG = dπr2×ΔGv + 2πrd×σ                             (1),

where r - radius of nucleus (the presumption of the disc-like shape of nucleus is based on the fact that this form provides minimum addition of surface), d - is height of nucleus and in the same time a size of layer, ΔGv - specific change of free energy per volume, σ - surface energy.

The dependence of change of free energy has maximum on the curve of displaying dependence from size of nucleus as seen on the following picture:

Change of free Gibbs energy depend on radius of two dimensional nucleus for following parameters: cubic like molecule with size d = 0.6 nm, ΔHv = -2×10-28 J/nm3, temperature of liquidus Tl=1500o C, temperature T=1475o C, surface energy σ = 2×10-20 J/nm2.

As seen from chart for size of two dimensional nucleus less than rcr (critical radius of nucleus) a change of free energy with increasing its size is positive that thermodynamically prohibited. Randomly collected molecules into nucleolus with size above critical are able to stable expansion. To find value of critical nucleus the value of deviation of ΔG by radius must be zero:

d ΔG = 2dπΔGv+ 2πσ = 0                                            (2)
d r 

resolving equation (2) toward r an analytical expression can be found:

rcr = -     σ                                             (3)
ΔGv

and change of free energy to form critical nucleus:

ΔGcr = -     πdσ2                                  (4)
ΔGv

Important to point out that ΔGv ( specific change of free energy per volume) has negative sign for temperatures below of temperature of equilibrium that makes critical radius and energy of formation of critical nucleus positive.

For the model of layer by layer growth one can consider critical nuclei as constantly existing fluctuations on surface of growing crystal. Average number of molecules (positions on surface of crystal) to produce one such fluctuation can be calculated by formula:

Ne(   ΔGcr )                                  (5)
kT

where k - Boltzmann's constant  (1.380 6504(24) × 10−23 J K−1 ), T - absolute (Kelvin scale) temperature.

Average radius of surface of the area on crystal filled by N molecules can be approximately defined via effective radius of molecule and presented by formula:

Ri = N0.5× rm                                         (6),

where Ri - average distance of expansion for one island grown from critical nuclei until one layer will be completely filled, rm - effective radius of one molecule.

Velocity rate of expansion of two dimensional island on surface can be taken from the thermal activated growth model where value d - heights of the layer is substitutes of the size of one molecule in plane direction: 2×rm. Taking in mind that filling of on layer equivalents to moving crystal at the distance its height the final formula for growth rate:

Vn = 2×d×γo × e(-ΔGcr/2kT) ×e(-Ea in/kT) × [1 - e(-    ΔH* ΔT ) (7) ,
kTTl

where meaning of parameters are as described for Turnbull formula in  thermal activated growth model. Remarkable is that radius of molecule doesn't manifest itself into final formula. The only difference of growth rate by discussed mechanism is coefficient:

Vn = Cn ×V                                            (9),

where

Cn = 2×e(-ΔGcr/2kT)                               (10).

It is important to point out that by reasoning of the model the value of correction coefficient for two dimensional nuclei growth cannot be large than 1 that contradicts to formula (10) for close to null values of energy of formation of critical nuclei. The model is not valid for radius of critical nuclei that is close to radius of molecule. Actually the consensus opinion is that area of positional applicability of the model is limited to very small supercooling where growth gem like crystals are actually observed if not universally but at least there are such phenomena. There is an easy mathematical trick to construct  normalized coefficient that will be equal to Cn for very small values and goes to 1 for big one:

C'n = (1 - 1/Cn)-1                                             (11).

Interpretation of formulas should be done with great conscious. An observation of flat surface of growing crystal doesn't automatically means actualization of two dimensional nuclei mechanism. Dislocation stimulated growth and intermediate situation between two dimensional mechanism and continues growth could create flat surfaces too. As a rule two dimensional mechanism of crystal growth should represent itself at very small supercooling. When temperature approaches Tl value of specific change of free energy  ΔGv= ΔHv×(Tl - T)/Tl goes to zero and critical radius with energy of creation of critical nucleus to infinity so Cn → 0. Paradoxically at very small temperatures mathematically  Cn → 0 as well and radius of critical nuclei rcr σ/ΔHv so if this value is significantly large than radius of molecule one can expect two switches of shape of growing crystal with cooling down. From gem like crystal just below of temperature of equilibrium dendrites with round surfaces and back to smooth flat polygon like figures at when approaching zero temperature.

One of the commonly accepted criteria for determination of mechanism of growth is value of the power in temperature dependence of rate growth from supercooling. The empirical equation:

V = a*exp(-E/kT)*((To- T)/T)c                      (12),

where a, E, c - coefficients are defined by approximation of experimentally obtained data for crystal growth. The value of c coefficient is supposed to be 1 for continues growth that described by Turnbull formula, for spiral dislocation growth c=2 and for two dimensional nuclei mechanism c commonly agreed to have about 4.5. That is obviously not the case.

Following figure displays temperature dependences of growth rate of Al2O3 calculated by Turnbull with coefficient C'n for series of values of surface energy:

Values used for calculation of growth rate are d×γ= 1×105 cm/s, Tl = 2350 K, ΔH = 2×109  J/m3, Ea = 1.45×109 J/mol and surface energy as shown at figure are taken in range from 0.01  to 0.3 J/m2. On the following table coefficients of empirical equation 12 are shown for approximation of calculated data:

Surface energy, J/m2 a, 105 cm/s E, 105 J/mol   c  
0.01 2.9 1.57 0.84
0.05 3.3 1.58 0.91
0.1 6.8 1.67 1.29
0.2 329 2.13 3.46
0.25 4000 2.46 5.05
0.3 80000 2.86 6.94

Remarkable varying surface energy one can mimic any value of coefficient c. The fact compromise value of power from list of key criteria for determining mechanism of the crystal growth. 

The most pronounces weakness of the model is the premise of strictly layer by layer filling of the surface. Actually only for limited range of parameters all conditions for two dimensional nucleation could be fulfilled. Nevertheless the fact of observation of growth of perfect polygons hints that such mechanism can actually take place in nature.

Submitted at Mar. 4, 2010; 17:31

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