Cahn s theory

Following Cahn s theory, more extended versions were proposed by Langer et al. [ O] and Binder et al. [ l] Recently, de Gennes [52], and Pincus [53] applied spinodal decomposition to polymer mixtures. Many of the recent experimental studies on spinodal decomposition of polymer mixtures deal with measuring characteristic scattering maxima with various scattering techniques [5A-60]. 

The first theory for film fluctuations resembles Cahn s theory of spinodal decomposition of unstable bulk systems. A very simple mechanism was adopted for the liquid flow. It was assumed that because of the presence of the soap monolayers, the film surfaces were stagnant (see Section 11) and that the film liquid was pumped back and forth through a slab with thickness h according to Reynolds s law ... 

A t5q)ical SD is shown that follows Cahn s theory, and the size of the heterogeneous structure due to phase separation is estimated to be approximately 1 pm. 

C CP/MAS NMR spectrum, poly(vinyl alcohol) (PVA), 259-61 Cahn s theory, 200 Caldron, 285-7 Carbon bond, 337... 

Cahn s theory was developed for the initial stages of phase separation (remember that the final result of the equilibrium phase separation does not depend on the mechanism and is determined only by the phase diagram of the system). 

We can apply this theory to our case provided phase separation takes no more than 200-300 s (the first linear branch). In our case, during the reaction time when phase separation was observed, the composition of the system changes only slightly and can be considered as constant. This fact gives us the right to apply Cahn s theory for describing the phase separation in our system. However, we believe that Cahn s theory can also be formally applied to the second linear branches in Fig. 9. 

The authors attempted to measure the intensities of light scattered from a NIPA gel during the phase separation induced by a change in temperature [23]. In this study, we examined the applicability of Cahn s linearized theory [24] to the spinodal decomposition of the geL... 

Besides the NIPA gel, we applied Cahn s linearized theory to a NNPA gel and determined the values of Dapp [25], which are presented in Fig. 21. The... 

Dynamical study of the phase transition of the gels in spinodal regimes was described. The evolution of intensity of light scattered from the gels indicated the applicability of Cahn s linearized theory to the phase transition. Our work offers a basis for the determination of diffusion coefficient of gels in their spinodal regimes. 

J.W. Cahn s early contributions to elastic coherency theory were motivated by his work on spinodal decomposition. His subsequent work with F. Larche created a rigorous thermodynamic foundation for coherency theory and stressed solids in general. A single volume, The Selected Works of John W. Cohn [15], contains papers that provide background and advanced reading for many topics in this textbook. This derivation follows from one in a publication included in that collection [16]. 

S.M. Allen and J.W. Cahn. Microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall., 27(6) 1085-1095, 1979. 

The theory of the kinetics of concurrent nucleation and growth reactions has a rich history that includes work by Kolmogorov [1], Johnson and Mehl [2], Avrami [3-5], Jackson [6], and Cahn [7]. Cahn s time-cone method for treating a class of these problems is the most general of these, with the most transparent assumptions, and is presented here. The method of Johnson, Mehl, and Avrami is covered in Section 4 of Christian s text [8]. 

Late stage. The consequence of Cahn s linearized theory is that the growth of dominant concentration fluctuations is ruled by qm t°, i.e. it takes place with no change in the size. Experimental results show that this is fulfilled in the early... 

Fig. 19. From the intersections in this figure, the diffusion coefficients were determined and are presented in Fig. 20. By extrapolating the D pp value to the horizontal axis, the spinodal temperature can be determined. Applying this procedure, we could determine a spinodal temperature of 34.2 °C which was sHghtly higher than the phase transition temperature of 34 °C. Thus, it was verified that the initial sta of the phase separation with the NIPA gel was expressed by Cahn s linearized theory.

Cahn s linear theory for spinodal decomposition is difficult to test in mixtures of simple liquids, because phase separation occurs so rapidly that the range of validity of the linear theory is typically exceeded before data can be acquired. In polymeric mixtures, however, phase separation is much slower, and tests of the theory are easier. For polymers of not-too-... 

The application of the Cahn s linearized theory to polymer mixtures was first developed by de Gennes [40] and Pincus [41], based on the Flory-Huggins free energy expression. Subsequently, Binder [42] incorporated the concept of thermal noise, introduced earlier by Cook [75], into the polymer-based derivation. This theory implies an exponential growth of structure with time ... 

The Kinetics of Spinodal Decomposition. Cahn s kinetic theory of spinodal decomposition (2) was based on the diffuse interface theory of Cahn and Hilliard (13). By considering the local free energy a function of both composition and composition gradients, Cahn arrived at the following modified linearized diffusion equation (Equation 3) to describe the early stages of phase separation within the unstable region. In this equation, 2 is an Onsager-type... 

Owing to the presence of specific interactions, most blends have a phase separation diagram with a lower critical solution temperature, LOST, i.e. phase separation occurs upon heating. Two separation mechanisms are known spinodal decomposition (SD), and nucleation-and-growth (NG). The morphology generated in NG is dispersed, whereas that in SD is co-continuous. Cahn and HiUiard s theory describes well the SD kinetics [1, 2]. 

Spaull, A.J.B., Cahn s perfect wetting theory in determining monolayer capacity, J. Chem. Technol. Biotechnol., 57(1), 87-92(1993). 

EoS models can also be used in the frame of the gradient approximation, such as the Cahn-Hilliard theory [100] of inhomogeneous systems, for the description of surface properties. In the frame of this theory, the Helmholtz s free-energy density r in a one-component inhomogeneous system can be expressed as an expansion of density p and its derivatives ... 

In the one-phase region of the phase diagram, the Cahn-HilHard theory predicts that S q, f) 0 as t 00, which is incorrect since fluctuations are continually appearing and decaying due to thermal motion, so that the structure factor has a finite value at all q. To account for this Cook [40] added a random noise term, 0(r, t), which, in order to ensure the correct equilibrium behaviour, has the following properties. 

---