Binodal surface

The domain structure and crystalline texture of AB and ABA type block copolymers of ethylene oxide (EO) and isoprene (Is) are studied, and the effects of the casting solvents and the fractional compositions of each block segment are determined. The domain structures of EO-Is copolymers are essentially identical to those of EO-Is-EO copolymers, but they strongly depend on the fractional compositions and the casting solvents. The role of casting solvent in the different domain formation mechanisms is interpreted in terms of an interrelation of two binodal surfaces that represent the critical concentration for crystallization of the EO segment and the critical concentration for micelle formation of the incompatible EO and Is segments. 

Figure 3.10. State (jia-gram of the ternary system O + Pi + F-2 where O is I..MVVL, Pi and P arc polymer hoinologucs with different chain Icnglh.s. DDCAK is the section of the binodal surface by the plane T = ron.Hl KCL is the section of the spinodal surface is a qua.sibinary. section CoC C is the critical point curve (Koningsveld, 1968, 1970b) [Reprinted from Advances in ( olloid and Interface Science 2 (1968) 1.51-215. Copyright 1968 with kind permission of Elsevier Science NL,. Sara Biirgerhart-straal 25, 1055 KV Amsterdam, The. Netherlands]...

I he curve of intersection of the plane TOX with the binodal surface is a quasibinary section (see Figure 3.10), also called a cloud-point curve (CPC) according to a way of its experimental determination when a solution (a system with an UCST) is being cooled and then temperature of its turning cloudy is fixed by means of a colorimeter or nephelometer, cus well as visually. 

I he cusp lines are entirely under the thermodynamically stable binodal surface (sec the points 1) and B in Figure 3.95) except their end points C and D (see Figure 3.97) where tlniy appear on the binodal surface. I hus, the cusp lines are boundary curves between the mctastablc and unstable binodal surfaces. 

At last, the three-phase line, appearing when two binodal surfaces intersect, reflects all the stable three-phase equilibria in the system the horizontal line x, in Figure 3.97 intersects the three-phase one just three times, i.c. at the points corresponding to three conjugate phases. 

Of the three characteristic curves (Figure 3.97), only the three-phase line is thermodynamically stable over its entire length. Recall that the critical line becomes metastable where it is dropped under the stable binodal surface on the segments GB and HA, and b[Pg.490]

The conditions of equilibrium in the two-phase state lead to relationships, which require special computational methods to obtain a (t 4- 1 )-dimensional binodal surface. The intersections of the plane T vs ip ip is the concentration of a polymoleculau polymer) with the hinospinodal surfaces give the boundary of the phase separation region (also called a cloud-point curve (CPC)) and the. spino[Pg.502]

Figure 1.14. Examples of liquid-liquid phase diagrams for four-component mixtures (a) with one binary two-phase liquid constituent (binodal surface is...

Over a century ago, Gibbs W developed the mathematical criteria of criticality. He defined the critical phase as the terminal state on the binodal surface and reasoned that it has one less degree of freedom than the binodal surface (i.e., f = n-1). He then developed two equations as the criteria of criticality these two equations, when imposed upon the Fundamental Equation, reduce the degrees of freedom from n+1 to n-1. 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

Under high(6)the binodal is transformed to the curve for the gel-forming system. It can testify about the change of the mechanism of the phase separation of solutions. At the high shear rate the macromolecules can form the adsorbt.ion-entanglement layers on the surface of the rotor and stator [21 that can lead to the formation of the gel-like particles and, finally to the phase separation. 

In Figure 2F-1 the composition where d2( G)/d 22 s equal to zero, or at the inflection point on the Gibbs energy surface, is defined as the spinodal composition. This corresponds to the boundary between an unstable solution and a metastable solution. If the necessary amount of free energy is supplied to the metastable system, the solution will phase separate into two phases with binodal compositions unstable system will always phase separate into the two phases. The temperature where the two points of inflection on the energy surface merge into a single point is defined as the critical solution temperature. 

Soybean oil was supplied by Procter and Gamble Inc., Toronto, Ontario, Canada. Methyl ester was made by the one-phase method employing a methanol/oil molar ratio of 27 1, tetrahydrofu as cosolvent and 1.0 wt.% sodium hydroxide based on the oil. The methyl ester accounted for 99,7% of the total ester present. The Alcan AA 200 activated alumina used was supplied by Alcan Chemicals, Brockville. Ontario, Canada. The catalyst s BET surface area is 270-290 /g, and its pore volume is 40 cmVg. The pore distribution is binodal, with 66% of pore volume in pores Jess than 30 radius and the rest in larger pores. Its bulk density is 0.75 g/cm... 

The experimental approach examines bilayers with a limited precision in depth z (8=a few nanometers) and in volume fraction < > (a few percent). It assumes that at least the central part of the analyzed profile ( >(z) describes only the internal interface between coexisting phases ([q and 2. This is not necessarily true when surface segregation regions, adjacent to both external interfaces, cannot be neglected as it is for very thin films. Related finite size effects are discussed in detail in Sect. 3.2 theoretical models and computer simulations expect that size effects modify the intrinsic profile internal interface. Therefore size effects may lead in principle to systematic errors [6] of binodal values determined for films which are very thin or are profiled at T—>TC (where the ratio D/w is also small due to the diverging w). 

Another insight into the wetting phenomena, alternative to that yielded by the contact angle 0, has been given by the profile (]>(z) (composition < > vs depth z) [8,53,61,153,158]. The surface of a two component liquid mixture which favors one of the components will be enriched in that component, A say. When the region far from the surface (bulk region) is occupied by the B-rich phase the surface concentration < >s (( >ls or < >2S in Fig. 14) is higher than this binodal value... 

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. 19.a Local d66 volume fraction as a function of depth z, determined for the d66/h52 blend monolayers with different initial compositions following 2 h of annealing at 99 °C [16]. Horizontal solid and dashed lines indicate the respective binodal values and ( )2 and their estimated uncertainty. The hatched area marks the d66 surface excess z. The inset marks on the phase diagram bulk compositions ([) for which z was determined. Solid curve in the inset denotes binodal determined previously [91] and described by %= (0.327/T+3.48X10 4)(l+0.222( )). b Segregation isotherm data [16] plotted as normalized surface excess z /[w(( )2-< )i)] vs normalized bulk volume fraction (1) (1),. Solid lines are generated by Eq. 44 to fit the data. Dashed horizontal lines are normalized surface excess values for the bulk phase enriched at the surface to the compositions ( )s such that (([>2—s)/(ct)2— ())1)=10%, 5%, and 1%, respectively... 

Performed Monte Carlo simulations [178,179] confirm the conclusion, that surface segregation of the minority component plays an important role in the suppression of the phase diagram. The segregation occurs on a scale of the correlation length for any film with varied thickness D, but only for small D (>4 [178]) it leads to visible divergence of the average compositions <(]> > and <(t>2> from their bulk (binodal) values ((h and ( >2. 

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