Mixed gels - Physical Gels from Biological and Synthetic Polymers - page 290

ternary systems the phase diagram is three-dimensional, with stability limits de ned as

surfaces.
Starting at a high temperature Ti i in the one-phase region, and cooling down to Ti f in the
two-phase region, an initially clear solution becomes turbid when the temperature is
lowered below the binodal. In Figure 10.1b we can see that the first derivative with

respect to concentration is zero at the two free-energy minima, corresponding to the two
miscible phases, and the second derivative is zero at the points of in

exion. (The above
treatment considers the case of upper critical solution temperatures or UCSTs.)

10.3 Phase dynamics: nucleation and growth versus

spinodal decomposition

10.3.1 Nucleation and growth

For this case, consider the dynamics of the phase transformations induced by an
instantaneous quench from the initial temperature Ti i to a

final temperature Tf f . If a system
is quenched from an equilibrium state in the one-phase region to a state below the
coexistence line, phase separation tends to occur, and in the final equilibrium state both

phases coexist, with equilibrium concentrations c 1 and c 2 .
However, if the system is quenched to the

final temperature Tf f in the metastable region
between the spinodal and the binodal, phase separation starts with the nucleation of small
droplets whose structure and properties vary from that of the parent phase material, but
approach those of the more stable phase. A nite fluctuation is required to render the

solution unstable. This fluctuation results in the formation of a nucleus, and the work of

forming such a nucleus is a measure of the metastability of the phase.
For a nucleus to form through thermal composition

fluctuations, the nucleation barrier,
typically of the order of several k B T, has to be overcome. Therefore, there is a critical
droplet size r* which is in unstable equilibrium with the exterior phase. With any
in nitesimal increase in size, such a droplet (or critical nucleus) can continue to grow

without further external intervention, because the free energy decreases. In the metastable
region below the binodal, the solution is supersaturated and nucleation is homogeneous.
As shown in Figure 10.2 , the nucleus rich in component A is surrounded by a concen-
tration gradient of Awhich provides the driving force for solute diffusion and gives rise to
its growth. The growth rate is controlled by the rate at which molecules A diffuse towards
the interface or by the rate at which they cross the interface. Once the molecules are
incorporated into the growing interface, the solution is depleted of these molecules and
diffusion is slower.
The work required to create a nucleus is strongly in

uenced by surfaces such as
inclusions (seed particles) which may catalyze the nucleation of the new phase. For
example, solidi cation in supercooled solutions allows the solid to start growing from an

existing surface. The critical radius of the nucleus is unchanged, because it depends
mainly on the temperature, but the free energy for the nucleus to overcome is decreased
by so-called heterogeneous nucleation.