Incompressible volume, segmental

The structure of the interface formed by coexisting phases is well described by the Cahn-Hilliard approach [53] (developed in a slightly different context by Landau and Lifshitz [54]) extended to incompressible binary polymer mixtures by several authors [4,49,55,56]. The central point of this approach is the free energy functional definition that describes two semi-infinite polymer phases <]), and 2 separated by a planar interface (at depth z=0) and the composition ( )(z) across this interface. The relevant functional Fb for the free energy of mixing per site volume Q (taken as equal to the average segmental volume V of both blend components) and the area A of the interface is expressed by... 

To solve the SCF equations, we make use of the discretisation scheme of Scheutjens and Fleer [69], It is understood that here we cannot give full details on the SCF machinery. For this we refer to the literature [67,70-72]. However, pertinent issues and approximations will be mentioned in passing. The radial coordinate system is implemented using spherical lattice layers r = 1,..., tm, where layers r = 1,..., 5 are reserved for the solid particle. The number of sites per layer is a quadratic function of the layer number, L r) o= i. The mean-field approximation is applied within each layer, which means that we only collect the fraction of lattice sites occupied by segments. These dimensionless concentrations are referred to as volume fraction (p r). We assume that the system is fully incompressible, which means that in each layer the volume fraction of solvent = 1 — (r) — volume fractions are the segment potentials u r). The segment potentials can be computed from the volume fractions as briefly mentioned below. 

It is most convenient to treat the equation of state in dimensionless form. This may be done by writing the compressibility factor z = pV/NkT) as a function of the packing fraction (rj = Nvo/V, where Vo is the volume of a sphere). Consider first a hard sphere fluid in D = l. A one-dimensional sphere is a line segment, so this is nothing but a fluid of hard rods interacting along a line. Clearly the fluid should approach ideality (z —> 1) at very low density (rj 0), and should become incompressible z —> oo) at closest packing ( — 1). In fact, the equation of state is known exactly [4], and is just... 

The incompressible chi-parameter defined in Eq. (6.16) has also been extensively studied. Many of the numerical results for site volume and/or statistical segment length asymmetric athermal Gaussian drain blends are adequately reproduced at a qualitative level by the analytic thread model discussed in Sect. 2. For an athermal stiffness blend of very long Gaussian threads the k = 0 direct correlation functions are [23,62] ... 

The WSL approach for the description of the order-disorder transition, ie, the transition between the microphase-separated block copolymer and the disordered melt, where the two blocks mix with each other, has been developed (74,88,89) using the random phase approximation. This transition is ofl en called the microphase separation transition (MST), and Toot is the temperature at which the order-disorder transition occurs. In this picture the system is described by a so-called order parameter, which is related to the space-dependent volume fraction or segment density of one of the components, say, component A. Again, the system is considered to be incompressible. The order parameter is then given by the deviation of the local segment density from the mean composition value. 

Here the integral extends over the interface region, a and v are the statistical segment length and volume, respectively, and b(x) = 1 - [Pg.294]

Figure 1.5 schematically shows the pressure of a fluid as a function of molar volume for several fixed temperatures, with one curve for each fixed temperature. These constant-temperature curves are called isotherms. For temperatures above the critical temperature there is only one fluid phase, and the isotherms are smooth curves. The liquid branch is nearly vertical since the liquid is almost incompressible while the gas branch of the curve is similar to the curve for an ideal gas. For subcritical temperatures, the isotherm consists of two smooth curves (branches) and a horizontal line segment, which is called a tie line. A tie line connects the two points representing the molar volumes of the coexisting liquid and gas phases. As subcritical temperatures closer and closer to the critical temperature are chosen the tie lines become shorter and shorter... 

The basic idea of these theories is to look at the distribution of conformations of a chain molecule attached to the surface. The conformational probability distribution function is written in terms of the non-local interaction field induced by the other chain molecules. This field is anisotropic, i.e., it depends on the direction perpendicular to the surface, because the presence of the surface and the inhomogeneous variation of the density of polymer segments and solvent molecules as a function of the distance from the surface. The non-local mean-field is determined by packing constraints that take into account the fact that the volume (at all distances from the surface) must be filled by polymer segments or solvent molecules. These self-consistent criteria represent the incompressibility assumption at all distances from the surface. 

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