Biomedical Engineering Reference
In-Depth Information
In the case of the 2D nucleation of crystals with n * molecules, the homogeneous
nucleation energy
ŒG homo 2D and the radius of critical size ( r c ) 2D are expressed by
( 7.21 , 7.22 ), respectively [ 13 ],
ŒG homo 2D D . step / 2
(7.21)
Œr c 2D D step
(7.22)
where
step is the crystal-liquid interfacial edge free energy.
The widely accepted kinetic model of nucleation (within the cluster approach)
was used first by Farkas [ 3 ] in 1927. It is based on the Szilard scheme of successive
“chain reaction” between monomer molecules and n -sized clusters as shown by
( 7.20 ).
A master equation for the concentration Z n ( t )ofthe n -sized clusters at time t is
written in the form of a continuity equation [ 56 , 58 , 59 ]
d
Z n
d
D J n1 J n ;
(7.23)
t
where J n is the flux through point n on the size axis. Thus in this formulation, J is
the flux through the nucleus size n * , i.e.,
J J n .t/
, so that a time dependence is
introduced in the nucleation rate.
The basic problem in the nucleation kinetics is to solve the master ( 7.20 )in
the unknown cluster size distribution Zn ( t ), as knowledge on Zn ( t ) will allow
the acquirement of the nucleation rate. In this concern, three physically distinct
states of the nucleating system require a special attention: the equilibrium , the
stationary ( or steady ), and the nonstationary states . In the following, we will brief
the equilibrium and the nonstationary states and focus more on the steady state,
which is more relevant in the most nucleation events.
7.3.2.1
Equilibrium State
As nucleation is a dynamic process other than that at equilibrium, this state is
conceptual and never occurs in practices. Introducing this state is (1) to make a
reference for other states and (2) to help in the treatment of the stationary state
mathematically.
Based on the thermodynamic statistical principles, in equilibrium, d Z n /d t
D
0
and J n D
C n is the equilibrium cluster size distribution. ( C n is the
equilibrium concentration of n -sized clusters.) According to the Boltzmann law, one
can easily obtain
0. Then Z n D
X
/ n exp
.C n =Z/ D .C 1 =
. G n =kT/
(7.24)
 
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