Spinodal instability

T. Biben, J. P. Hansen. Spinodal instability of suspensions of large spheres in a fluid of small spheres. J Phys Condens Matter 2 F65-72, 1991. 

Fig. 5 illustrates a peculiar kinetic phenomenon which occurs when an initially disordered alloy is first annealed at temperature T corresponding to area b in Fig. 1 and then quenched to the final temperature T into the spinodal instability area d antiphase boundaries "replicate , generating approximately periodic patterns. This phenomenon reflects the presence of critical, fastest growing concentration waves under the spinodal instability (the Calm waves ). Lowering of the temperature to T < T results in lowering of the minority concentration minimum ("c-well ) within APB, while the expelled solute atoms build the c-bank adjacent to the well . Due to the... 

Then the mixture with droplets is quenched into the spinodal instability region to some T < Ta (Concentration c(r) within droplets starts to evolve towards the value C(,(T) > C(,(T ), but the evolution type depends crucially on the value Act = cj(T) — Ch(Ta). At small Act we have a usual diffusion with smooth changes of composition in space and time. But when Act is not mall (for our simulations Act O.2), evolution is realised via peculiar wave-like patterning shown in Figs. 8-10. 

As before, the (nonzero) vector 8p with components <5p,- gives the direction of the spinodal instability the moment densities pt and ptJ that appear in F and M are to be evaluated for the parent distribution p (cr) being studied. Note that Eq. (54) is valid for any p (cr) no specific assumptions about the parent were made in the derivation. We can thus simply drop the (0) superscript For any phase with density distribution p(external control parameters are varied, locates a spinodal instability. 

On the other hand, in a polydisperse melt a spinodal instability would be expected eventually to reach biphasic equilibrum, in which case standard Cahn-Hilliard kinetics would be expected. Phase separation could be due to several effects, including chemical polydispersity, coupling of mass density to the density of polymer ends, and conformation-density-orientation coupling, which has a natural prediliction for inducing phase separation (longer chains are, per unit monomer, more nematic than their short chain cousins ). There are evidently several effects to sort out, and close collaboration between theory and experiment will help to attack the various mechanisms. 

Verma, R., Sharma, A., Banerjee, I., Kargupta, K. Spinodal instability and pattern formation in thin liquid films confined between two plates. J. Colloid Interface Sci. 296(1), 220-232 (2006)... 

H. Tanaka, Phase behaviors of supercooled water Reconciling a critical point of amorphous ices with spinodal instability, J. Chem. Phys. 105(12), 5099-5111 (1996). 

Figure 17 Calculated state diagram for hard globules with diameter a dispersed In nematic rods with axial ratio L/D = too. 0s is the volume fraction of spheres, and 0r that of the rods. L/a = 10. The solid line marks the spinodal instability to the lamellar phase, and the dash-dotted line that to macroscopically demixed phases. The dotted line separates the region where self-assembled chains are of (a) the open type from that where they are of (b) the dense type. (From P. van der Schoot. J. Chem. Phys. 117 3537, 2002. With permission.)... 

Here nMM ( ) the (generally attractive) tail potentials between species M and M. The spinodal instability corresponds to a vanishing of the second compositional derivative of the free energy of mixing yielding the condition... 

For simplicity, we consider the S-ordering tail potential choice and Uab 0. The R-MMSA and linearized R-MPY/HTA expressions for the direct correlations functions and effective chi parameter are identical to the blend cases and hence are deterministic. Thus, at low temperature the reduced repulsive tail potential, jSUabW become arbitrarily large leading to a microphase separation spinodal instability defined as The latter condition is precisely given by ... 

The reduction of thread PRISM with the R-MMSA closure for the idealized fully symmetric block copolymer problem to the well-known incompressible RPA approach " is reassuring. However, in contrast with the blend case, for copolymers that tend to microphase separate on a finite length scale, the existence of critical or spinodal instabilities is expected to be an artifact of the crude statistical mechanical approximations. That is, finite N fluctuation effects are expected to destroy all such spinodal divergences and result in only first-order phase transitions in block copolymers [i.e., Eq. (7.3) is never satisfied]. Indeed, when PRISM theory is numerically implemented for finite thickness chain models using the R-MMSA or R-MPY/HTA closures spinodal divergences do not occur. Thus, one learns that even within the simpler molecular closures, the finite hard-core excluded volume constraint results in a fluctuation effect that destroys the mean-field divergences. 

Since thread PRISM theory at the R-MMSA or R-MPY/HTA closure level predicts a spinodal instability, its description of the disordered phase can be combined with field-theoretic Landau expansion and Brazovskii... 

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