Osmotic equation of state

Information on the equation of state comes from two experimental sources light-scattering at low ( 1%) volume fractions via equation (20), and direct compressibility studies at high ( 10%) volume fractions. For testing statistical theories of concentrated dispersions, we therefore look towards compressibility studies on well defined systems. The available compression data on lattices show only qualitative agreement with the Monte Carlo results they are insufficiently precise to test the pairwise-additivity assumption employed in the computer simulations. 

Evans and Napper have proposed a simple expression for the equation of state of a concentrated, hexagonally close-packed dispersion. The osmotic pressure is given by 

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This relation can be inverted to yield fip(n) as function of c.p(nr) (eh Appendix A 4.1). Substituting the result into 77[/xp], we find the osmotic pressure as function of the concentration, which is the standard form of the osmotic equation of state. Also all the other thermodynamic quantities can be calculated from n[fip]. The excess free energy due to the solute, for instance, takes the form... 

These approximations can then be used in the osmotic equation of state to obtain the compressibility factor. Monte Carlo simulations using the above-discussed Monte Carlo techniques have been performed to assess the approximations inherent in the generalized Flory theory of hard-core chain systems. This theory does quite well in predicting the equations of state of hard-core chains at fluid densities. The question then arises, why does it do so well since the theory typically only incorporates information from a dimer fluid as a reference state ... 

Equation (12) is an exact expression for the pr ure in terms of the insertion probabilities. It is a form of the osmotic equation of state. The underlying physical picture is one of building up the system of N chains by adding chains one by one in the volume V. To make Eq. (12) practically useful, we need a way to calculate the insertion probabilities p (i,V,T). 

To relate macroscopic observables to forces acting between particles is the objective. The relevant measurable properties are (a) the phase diagram, (b) scattering of photons, neutrons, and X-rays, (c) the osmotic equation of state, and (d) rheological behaviour. This Report covers (a), (b), and (c) with emphasis on the transition between ordered and disordered states. Discussion of (b) is limited to light-scattering. Current problems in relation to (d) are set out (up to mid-1980). 

Distribution and Correlation Functions.—We consider a single spherical particle with position r and velocity v at time t in a concentrated dispersion of mean number density p. The distribution function measures the probability of finding a particle (the same or another particle) with position r" and velocity v" at time t". The osmotic equation of state is related to a time-averaged distribution function that depends on r alone, whereas the dynamic behaviour depends on time-dependent functions. A basic premise of statistical mechanics is that a time-average is equivalent to an ensemble average at fixed time the ensemble average is denoted by angular brackets (...). 

A completely different nonstandard technique to obtain a first overview of the equation of state was recently proposed by Addison et al. [269], whereby a gravitation-like potential is applied to the system, and the equilibrium density profile and the concentration profile of the center of mass of the polymers is computed to obtain the osmotic equation of state, fii this sedimentation equilibrium method one hence considers a system in the canonical MVT ensemble using a box of linear dimensions L x L x H, with periodic boundary conditions in x and y directions only, while hard walls are used at z = 0 and at z = H. An external potential is applied everywhere in the system ... 

From the partition function the free energy Fjy follows and hence all thermodynamic quantities of interest can be estimated (entropy, chemical potential, osmotic pressure...). Ottinger applied this technique to test the osmotic equation of state for dilute and semidilute polymer solutions for N <60. Extension of this technique to off-lattice systems has also been made. ... 

In polymer science, weight-based concentrations are almost always more useful than molar units, and accordingly the osmotic equation of state is customarily written in the form... 

The validity of the virial series expansion of the osmotic equation of state becomes questionable for moderately concentrated (or semidilute ) solutions, in which the overall number density N c/wo... 

A film at low densities and pressures obeys the equations of state described in Section III-7. The available area per molecule is laige compared to the cross-sectional area. The film pressure can be described as the difference in osmotic pressure acting over a depth, r, between the interface containing the film and the pure solvent interface [188-190]. 

As pointed out earlier, the contributions of the hard cores to the thennodynamic properties of the solution at high concentrations are not negligible. Using the CS equation of state, the osmotic coefficient of an uncharged hard sphere solute (in a continuum solvent) is given by... 

A. Milchev, K. Binder. Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions Monte Carlo simulations of a bead-spring model. Macromol Theory Simul 5 915-929, 1994. 

The relation between the osmotic pressure II and the polymer concentration, referred to as the equation of state for the solution, is often used for a critical comparison between theory and experiment (or simulation). Kubo and Ogino... 

Write the equation of state for the osmotic pressure of a solution that behaves ideally. 

Flory [3] formalized the equation of state for equilibrium swelling of gels. It consists of four terms the term of rubber-like elasticity, the term of mixing entropy, the term of polymer solvent interaction and the term of osmotic pressure due to free counter ions. Therefore, the gel volume is strongly influenced by temperature, the kind of solvent, free ion concentrations and the degree of dissociation of groups on polymer chains. 

In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients y ko of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3-4 such as that based on the Carnahan—Starling equation of state (see Appendix B for the derivation) ... 

Taking the difference between the Gibbs and Helmholtz free energies, we obtain the equation of state for the osmotic pressure of the small-ion gas as follows ... 

For calculation of various foam parameters (compressibility, equation of state, vapour pressure above foam, etc.) osmotic pressure, as proposed by Princen [51,84,107] proves to be a suitable characteristic. 

As we mentioned earlier in this section, it was van t Hoff who saw an analogy between the properties of dilute solutions and the gas laws. Just as an equation of state can be written for an ideal gas, so can an equation of state be written for an ideal solution in terms of its osmotic pressure, as shown in Table 12-2. It is then a straightforward matter of remembering the definition of a mole (the weight of something divided by its molecu-... 

For a polymer we must write an ideal equation of state for the osmotic pressure in terms of a sum over all the moles of chains of different length, x (Equation 12-7). 

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