Total volume fraction

The filled polymer is considered as a collection of repesentative volume elements (RVE) of many spherical or cylindrical composites of various sizes. Each of these contains a filler particle and two concentric spherical shells, a thin one corresponding to the mesophase, and another thicker, representing the matrix respectively. The volume fraction of the filler in each composite is the same, as the total volume fraction of the filler in the filled polymer. 

Fig. 56 Phase diagram of blend of PS-fi-PI with PS. T0dt. o TDMt, Toot- Vertical lines separating microdomain structures are obtained from total volume fraction PS in system. Dashed line results of mean-field calculation for ODT. The OOT line which exists at volume fractions ps 5 ub was obtained during a heating process. From [174]. Copyright 2000 American Chemical Society...

The mean-field theory has a number of shortcomings, including the approximations of a mean concentration around all particles and the establishment of spherically symmetric diffusion fields around every particle, similar to those that would exist around a single particle in a large medium. The larger the particles total volume fraction and the more closely they are crowded, the less realistic these approximations are. No account is taken in the classical model of such volume-fraction effects. Ratke and Voorhees provide a review of this topic and discuss extensions to the classical coarsening theory [8]. 

Fig. 6.37 Interfacial volume fraction profiles calculated for a ternary blend of a PS-PB block copolymer with PS and PB homopolymers in a good solvent (Noolandi and Hong 1982). The diblock has N — 600 and f = The homopolymers have infinite molecular weight. The solid lines are the volume fractions of homopolymer (A = PS) (B = PB), the dashed lines indicate the volume fractions of PS and PB blocks of the diblock. The dots correspond to the total volume fractions of the A and B components and the position is measured in units of a segment length a = 6.95 A.

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

For given values of the control variables Xsw and. Xaw, the maximum in Xgo (or Xgw) was found, using the IMSL subroutine ZXMWD, by solving the implicit eq 3.1 in combination with eq 3.4. As mentioned in section 2, the area per surfactant molecule a, the alcohol-to-surfactant ratio gAi4 si, and the oil-to-surfactant ratio golgsi in the interfacial layer were selected as the three independent variables with respect to which the maximization was carried out. The total volume fraction 4>s of surfactant present in the microemulsion is given by... 

Figure 3. Predicted radii of microemulsion droplets as a function of the volume ratios of alcohol to surfactant in microemulsions. The predictions are for a system consisting of the surfactant SDS, 1-pentanol, 0.3 M NaCl, water, and cyclohexane. The total volume fraction of the surfactant in the microemulsions is 0.01. Filled circles denote O/W droplets, and open circles correspond to W/O droplets.

According to Equation (2F-1), a liquid mixture with a total volume fraction liquid phases with binodal compositions Gibbs energy for the mixture will thus lie on the solid line between solid line is a tangent touching the predicted curve at the binodal compositions. 

After a period of growth, these aggregates occupy a large amount of space due to their large volume to mass ratio. The total volume fraction of all aggregates, is given by... 

If the system follows von Smoluchowski growth kinetics, the total volume fraction, Oj., will have the following time behavior [62] ... 

The total volume fraction, 7 ( ), as a function of time for both reaction limited aggregation (RLA) and diffusion limited aggregation (DLA)... 

Table 4 summarizes the different conditions and compositions that were achieved in a blending study of narrow molecular weight PB with inly = 3 kg/mole incorporated into the KRO-1 particles in PS. As the table indicates in the blending, the total volume fraction of KRO-1 Resin and PB3000 was kept constant to maintain a constant volume fraction of composite particles at 21.7%. In addition, pain was taken to keep the average composite particle size also roughly constant by the use... 

Figure 5.13 Predicted phase diagrams for physical gels made from low-molecular-weight molecules with junctions of unrestricted functionality 4> is the total volume fraction of polymer, and Tr is here the reduced distance from the theta temperature, Tr = — Q/T. The parameter Aq controls the equilibrium constant among aggregates of various sizes. The outer solid lines are binodals, the inner solid lines are spinodals, and the dashed lines are gelation transitions. CP is a critical solution point, CEP is a critical end point, and TCP is a tricriti-cal point. (Reprinted with permission from Tanaka and Stockmayer, Macromolecules 27 3943. Copyright 1994 American Chemical Society.)...

Figure 22. Variation of [ 17ou>s/i w7t toith total volume fraction of the dispersed phases for 9-pm silica sand. (Reproduced with permission from r erence 57. Copyright 1991 Pergamon Press.)...

Figure 8.24 Diagrammatic representation of a solid polymer showing regions of crystallinity and regions which are amorphous the total volume fraction of crystalline regions,

Figure 13. Volume fraction versus salinity Calculated phase boundary (solid line) and experimental data for total volume fraction of drops (dashed line).

We will also consider the apparent phase volume p which is calculated from the mixture theories as the total volume fraction of the microemulsion that is excluded from the transport. Assuming that the transport property of the hydration water is negligible compared to that of the bulk liquid, p would include the hydration water as well as the oil and emulsifier. 

Fig. 25 Transmission electron micrographs (TEM) of a ternary nanocomposite of PS-poly(ethyl propylene) (PEP) diblock copolymer with two types of nanoparticle-Ugand systems AuR]- and SiO2R2-ftmctionalized (R i, R2 are alkyl groups) nanoparticles of total volume fraction 0.02. The former appear along the interface of the lamellar microdomains, whereas the latter reside in the center of PEP microphases. Schematically, the nanoparticle distribution is shown in the inset. Taken from [308]...

Assuming that a stable suspoemulsion (in the colloid sense) could be prepared - for example, by using a polymeric dispersant and emulsifier - the creaming and/or sedimentation behaviour of the suspoemulsion showed different patterns depending on the density difference between the oil droplets and suspension particles, as well as the total volume fraction of the whole systems. 

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