Cahn plot

Fig. 14.a Composition-depth cf)(z) profiles near the surface (at z=0) of a binary mixture at bulk concentration bi at lower (T—>TW ) and upper (T—>TW+) limit of the first order wetting transition point b,c Cahn constructions with trajectories -2kV< ) plotted for profiles cf)(z) with decreasing (solid lines) and increasing (dashed lines) slopes. Surface boundary condition (Eq. 26) is met at points (marked by ) where surface energy derivative (-dfs/d< ))s (idotted line) intersects trajectories -2kV< > at concentrations reached at the surface. Cahn plot b corresponding to the first order transition depicted in a Cahn construction c typical for a critical wetting trajectories -2kV< > with larger extrema correspond to temperatures T[Pg.37]

A mean field theory has recently been developed to describe polymer blend confined in a thin film (Sect. 3.2.1). This theory includes both surface fields exerted by two external interfaces bounding thin film. A clear picture of this situation is obtained within a Cahn plot, topologically equivalent to the profile s phase portrait d( >/dz vs < >. It predicts two equilibrium morphologies for blends with separated coexisting phases a bilayer structure for antisymmetric surfaces (each attracting different blend component, Fig. 32) and two-dimensional domains for symmetric surfaces (Fig. 31), both observed [94,114,115,117] experimentally. Four finite size effects are predicted by the theory and observed in pioneer experiments [92,121,130,172,220] (see Sect. 3.2.2) focused on (i) surface segregation (ii) the shape of an intrinsic bilayer profile (iii) coexistence conditions (iv) interfacial width. The size effects (i)-(iii) are closely related, while (i) and (ii) are expected to occur for film thickness D smaller than 6-10 times the value of the intrinsic (mean field) interfacial width w. This cross-over D/w ratio is an approximate evaluation, as the exact value depends strongly on the... 

Fig. 6a. Structure factor S(q, t) plotted vs time for a nearly symmetrical critical mixture (< >c = 0.486) of perdeuterated and protonated 1,4 polybutadiene degrees of polymerization Nh = 3180, Nd = 3550, polydispersity indices (Nw/Nn)h = 1.03 and ((Nw/Nn)d = 1.07 quenched from T0 75 °C to T 49 °C (Tc = 61.5 1.5 °C) for several representative scattering wave numbers q. Since the scattering intensity is plotted on a logarithmic scale, straight lines imply an exponential growth and their slope hence yields 2t 1 - Eq. (76). Arrows show the time tm where nonlinear effects start to limit the growth, b Cahn plot R(q) /q2 vs q2, for the quenching experiment of a), cf. Eq. (85). Deviations from linearity here are attributed to the neglect of thermal noise - i.e. only the first term on the right hand side of Eq. (76) is kept. From Bates and Wiltzius p6]...

Cahn plot 206-209, 259 Cahn-type theory 258, 259 Carbocation 1... 

The growth rates for the dynamic SCF theory with local Onsager coefficient are displayed in Fig. 16a and compared to the results of the Monte Carlo simiflations. The maximal growth rate occurs at too large wavevectors and the growth rate for qlinear behavior (cf. Eq. 132) in contrast to the simulations. 

Dilatometric methods. This can be a sensitive method and relies on the different phases taking part in the phase transformation having different coefficients of thermal expansion. The expansion/contraction of a sample is then measured by a dilatometer. Cahn et al. (1987) used dilatometry to examine the order-disorder transformation in a number of alloys in the Ni-Al-Fe system. Figure 4.9 shows an expansion vs temperature plot for a (Ni79.9Al2o.i)o.s7Feo.i3 alloy where a transition from an ordered LI2 compound (7 ) to a two-phase mixture of 7 and a Ni-rich f c.c. Al phase (7) occurs. The method was then used to determine the 7 /(7 + 7O phase boundary as a function of Fe content, at a constant Ni/Al ratio, and the results are shown in Fig. 4.10. The technique has been used on numerous other occasions,... 

Figure 4.9. Expansion vs temperature plot for a (Ni79.9Al2o.i)o.87Feo,ij alloy showing y / y + 7-phase boundary at 1159°C from Cahn el al. (1987).

The Cahn constructions used are presented in Fig. 14. The allowed trajectories -2kV( )(( )) are marked by +2(KAf)1/2 and -2(icAf)1/2 curves plotted for bulk composition equal to a binodal one (]) =( ). They are equal to zero at coexistence compositions ( ) and and have extrema at a concentration close to the critical value ( )c. Both binodal values are shifting towards ( )c as temperature is increased. Simultaneously, the width of the interface between ( ) and (( increases (see Figs. 2 and 7) leading to smaller humps in +2(kM)1/2 and -2(kM)1/2. The temperature independent surface energy derivative (—dfs/d<())s, corresponding to Eq. (28), is represented in Fig. 14 by a straight dotted line. 

Fig. 20.a Results of the Cahn construction performed for the segregation data [16] of Fig. 19. Composition derivatives of bare surface free energy (-dfs/d( ))s calculated for different temperatures (symbols A, , O, and for T=99,142,165, and 184 °C, respectively) are fitted well by dashed lines, generated by the function (pf+g /ll+Y s). The hatched area marks the surface energy difference -Afs. b Surface energy derivatives (—dfs/d( >)s (dashed lines) and trajectories -2kV< ) (solid lines) plotted for T=99 °C and 184 °C. For T= 184 °C the surface boundary condition (Eq. 26) is met at point at ( >s>( >2, indicating complete wetting regime. If (—dfs/d([ )s was independent of temperature (and equal to that found at 184 °C) then the boundary condition (O) at 99 °C would correspond to partial wetting (c >s<( >2). In practice, however, (—dfs/d([ )s varies with temperature and the real boundary condition at 99 °C ( ) indicates complete wetting again...

Fig. 28. The surface -%s vs bulk % parameter re-plotted from Fig. 27 for four microstructur-ally identical blends Xj/x2 (x1>x2) 66/52 (O), 86/75 (A), 75/66 (A) and 52/38 ( ). For each pair a point with higher (-%s) and % values corresponds to the mixture with deuterated more branched component (dxj/hx. The Cahn constructions performed for available segregation isotherm data suggest two types of behavior i) a relatively extended (even to ca. 100 °C) critical point wetting region (TW TC) is observed for blends with (-%s, %) loci on the plot surrounded by circles ii) wetting point located close to Tc (TW

Fig. 32a-c. A thin blend film bounded by antisymmetric surfaces exerting opposing fields [93] a Cahn construction with trajectories -2kV( ) (solid and dashed lines) plotted for Ap= 0 and (here) for c —The bare surface free energy derivatives (—dfsL/dcf))s and (+dfsR/d( >)s due to left (L) and right (R) surface are marked by dotted lines. Boundary conditions (Eq. 51) are met at points 1 and 2 b the profile 1-2 as determined by Cahn construction (a) for a rather thick (due to the limit (ft,-Hfi) hlm c the corresponding bilayer equilibrium morphology with the interface between phases and < >2 running parallel to both surfaces... 

One way of testing such numerical implementations is to recover known results. In the present setting, this means recovering the results of Johnson and Cahn for the transition from circular to elliptical shapes. In fig. 10.17 we show the results for the critical size (the inverse of this size is actually plotted) at which the bifurcation from circular to elliptical shapes takes place as a function of the inhomogeneity factor 8 = where jx and jx are the shear moduli for the precipitate and the... 

Demixing, Fig. 2 (a) Characteristic lengths (a) and (b) nucleation barrier AF plotted versus concentration cb. Full curves show the predictions of the Cahn-Hillard mean-field theory of nucleation and spinodal decomposition for the critical wavelength and the correlation length of concentration fluctuations in a metastable... 

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