Incompressible systems

The relaxing Gaussian network of Green and Tobolsky (4) is the earliest version of this model. Lodge (12) and Yamamoto (J5) independently derived constitutive equations for similar systems, based on a stress-free state for each newly created strand and a distribution of junction lifetimes which is independent of flow history. For Gaussian strands in an incompressible system ... 

The pressure P0 represents the arbitrary additive contribution to the normal components of stress in an incompressible system, 8i is the Kronecker delta, C[ j 1(t t) is the inverse of the Cauchy-Green strain tensor for the configuration of material at t with respect to the configuration at the current time t [a description of the motion (221)], and M(t) is the junction age distribution or memory function of the fluid. 

Preliminary Thermodynamic Relations. Here it will be assumed that we are dealing with incompressible systems this is a very good assumption for aqueous solutions since the isothermal compressibility of water is so small. At constant temperature the equilibrium condition for any mixed association (see Equations 1 and 2 for example) is... 

The theoretical description of the cake formation in the constant pressure filtration is based on Darcy s Permeation Law, which describes the single-phase laminar flow of an incompressible fluid through a porous incompressible system due to an applied pressure difference. It can be written as follows (2, 3) ... 

Since the spatially averaged model for incompressible systems is used occasionally in chemical engineering practice, we will briefly outline the Re3molds like spatial decomposition and averaging procedure for completeness. In this case the 0 represents the spatial deviation of the point variable tjjk from the intrinsic volume average value ipk)vk7 is defined by ... 

In brief, if the solution is incompressible, the parameters concerning the solvent can be eliminated very simply. However, no completely incompressible system exists, and the mathematical transformations presented above may look rather crude. More refinements1 could be introduced and the elimination could be done in a more rigorous way but the results would not be very different. Moreover, it is always possible to take the pressure variations into account empirically by allowing variations of the quasi-potentials that appear in the theoretical expressions (see Chapter 10). 

F Helmholtz free energy of an incompressible system or when the compressibility of the solvent is ignored (F = E— ff lS)... 

For incompressible systems having strong interactions, e.g., acid-base type, the directional-specific model of segmental interactions may be used [Walker and Vause, 1982 ten Brinke and Karasz, 1984]. The following expression was derived ... 

We now consider the problem of minimization of the free energy of a system with a fixed total volume and neglect compressions of the molecules. In the two-phase region of the phase diagram, the system consists of two coexisting phases (i = 1,2) with a volume fraction 0/ of B particles in each of the phases. In addition to the volume fractions, we have to specify the number of sites (i.e., volume in an incompressible system) occupied by each macroscopic phase and we denote as Ni the number of sites in each phase. Since the total number of sites, N, in the incompressible system is fixed, N and N2 are constrained ... 

When Eqs. (13.2) and (13.3) are inserted into Eq. (10.7) and external forces are neglected, we get the following diffusion equation for T for incompressible systems at constant temperature ... 

On the other hand, results from recent experiments where the displacement of markers aeross a polymer-polymer interface has been observed [75,81] have been interpreted to favour Eq. (15). This conclusion has been based largely on arguments concerning the compressibility of the system. Equation (14) implies an incompressible system, whereas Eq. (15) implies a compressible one. 

One of the most powerful methods to assess such phenomena theoretically is the self-consistent field (SCF) theory. Originally introduced by Edwards [8] and later Helfand et al. [9], it has evolved into a versatile tool to describe the structure and thermodynamics of spatially inhomogeneous, dense polymer mixtures [ 10-13]. The SCF theory models a dense multi-component polymer mixture by an incompressible system of Gaussian chains with short-ranged binary interactions and solves the statistical mechanics within the mean-field approximation. 

The only forces driving U towards the saddle point are now those related to the single chain fluctuations (cf. Sect. 4.2), and U will be widely distributed on the imaginary axis. Thus it is not clear whether the Monte Carlo method will work for incompressible systems. 

For an incompressible system, Sy(Q) - 0. The validity of equation (4) has been demonstrated for polyisoprene, polycarbonate, and polystyrene/ The experiments on polystyrene showed that Sy is zero in this system. The radius of gyration of polystyrene derived from 5g(0, was identical to the values obtained by earlier workers using low tag concentrations (see ref. 59). 

For nonpolar systems, the long-range interactions are normally modelled by contact interactions. Finite range corrections are also included sometimes, but they have negligible effect on the phase diagrams [35,41] and we do not include them here. The result, as will become clear below, is that effective interactions in incompressible systems can be modeled by simple Flory parameters, Xkk -The situation is more complicated in compressible systems. 

The same as classic thermodynamics, polymer thermodynamics is function of pressure, temperature and composite. But in many cases, pressure effects on polymer thermodynamics was neglected, because polymer thermodynamics were often studied under atmosphere. The classic theory of polymer thermodynamics is Flory-Huggins hard lattice theory. In this theory, the hard lattice is incompressible. A rigorously incompressible system should be unaffected by pressure. However, since experimental results show that the critical temperature for polymer demixing system is strongly affected by pressure, it is clear that polymer containing systems show significant departures from this ideal limit. We wish... 

There have been sophisticated calculations of x these systems. In the Flory-Huggins model x is independent of molecular weight and composition. In reality, this parameter has been shown, by scattering experiments and by theoretical calculations to be a function of Af, T and 0. 3,40,49 Theoretical attempts, which are beyond the scope of the mean field predictions of the Flory-Huggins approach which applies stri ct to incompressible systems, have been made to address these questions. 3,4o 49 The theories of Bates and Muthukumar 3 and of Schweizer and Curro °> both have predictions which may be written in the following form... 

While the evaluation of pi can be simphfied using Euler relations for certain classes of monomer structures, these Euler relations [50] do not apply, for example, to the three polyolefin chains depicted in Fig. lb because they have short side chains. (Of the monomer structures in Fig. la, the Euler relations for pi are valid only for PEE and PHI.) Therefore, the geometrical index pi = /Mi = nPVsi is evaluated by directly enumerating all sets of three sequential bonds (nP ) that traverse a monomer of species i. [46] siunmarizes the details of these calculations and tabulates values of p,- for several monomer structures, so we pass now to a consideration of how thermodynamic properties depend on n and p, in the high molecular weight, incompressible system limit of the LCT. 

For an incompressible system, we can simplify this expression further, but we will not do this here. Once we have computed Ap, we can estimate the interfacial free-energy ys by using our numerical information about the nucleation barrier AG, using... 

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