Onsager critical

Excluded volume interactions will further decrease the density This is a strong argument suggesting that overlapping coils (semi-dilute solutions are necessary to obtain liquid crystalline order in solutions of worm-like macromolecules. 

We assume, for this model, that only excluded volume type interactions are important. We then approximate the rigid rod Onsager interaction— by the lowest angular dependent term in its multipole expansion which is of quadrupolar symmetry  

In order to simplify the algebra, let us specialize to the case of sharp helix-coil transitions, i.e., E 0 as is common for the synthetic polypeptides. (The general situation will be discussed in a later publication.) For this limit, a given chain is either a flexible coil or a rigid helix, i.e., V = 0 or N (Fig. 

The free energy is separable into two contributions (1) the intra-molecular interactions associated with the helix-coil transition (2) the intermolecular excluded volume interactions which are responsible for any nematic order. The difference in free energy per chain between a helical and randomly coiled molecule arising from intramolecular interactions is -N n s. The intermolecular excluded volume interactions in the mean field Meier-Saupe like approximation [Eqs. (III.2) - (III.6) with V = N] gives a contribution to the free energy difference, (AF) which is sketched in 

For s 1, a phase transition may only occur if the gain in free energy of ordering exceeds the entropic cost in loss of flexibility of the individual chains. In the (s,c) plane, the phase boundary will then be given by 

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Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

However, as discussed below critical concentrations for cellulose, in a variety of solvents, and based on optical observations under crossed polars are much lower than predicted using eauation 1 and kw = 2 q. Como et al. (4 point out one has to consider the possibili that the lattice model does not accuratelv predict the values of V2 and that V2 values using the Onsager (28) and Isihara (30) theories are about half that predicted by equation 1. 

As for the theory of this phenomenon, it was first observed by Onsager [27a] that, since in the limit a — 0 an LCD a is expected to yield a singularity of the type —surface potential, the statistical-mechanical phase integral for counterions should diverge for a greater than some critical value, characteristic of a given valency. Indeed consider a counterion (for definiteness anion) of valency z. The appropriate phase integral is of the form... 

While there are conflicting views about the performances and benefits of all these versions, there are some criteria that may serve for a critical assessment. These include (a) Onsager s well-known lower bound for the mean electrostatic energy per ion [253] in its reformulation by Totsuji [254] and (b) Gillan s upper bound for the free energy [255]. Moreover, the condition for thermal stability requires the configurational isochoric heat capacity to be positive. 

A semi-quantitative picture of positronium formation in a spur in a dense gas was developed by Mogensen (1982) and Jacobsen (1984, 1986). If the separation of the positron from an electron is r, and there is assumed to be only one electron in the spur (a so-called single-pair spur), then the probability of positronium formation in the spur, in the absence of other competing processes, can be written as [1 — exp(—rc/r)] here rc is the critical, or Onsager, radius (Onsager, 1938), given for a medium of dielectric constant e by... 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

For = 3.0, the ratio is 3.5 x 10-5 cni/V at 296 K. Although based on the assumption that g(r,0) is spherically symmetric, the ratio is independent of the function selected to represent the distribution of thermalized pair separations and contains no adjustable parameters. It thus provides a very critical test of the theory. Batt et al. (1968,1969) were the first to demonstrate the applicability of the Onsager formalism by use of the low-field slope-to-intercept ratio. The primary quantum yield and the thermalization distance can be determined by comparing experimental and theoretical values of the field dependence of the photogeneration efficiency at high fields, or by the temperature dependence of the zero-field quantum efficiency. The latter technique is based on the assumption that the primary quantum yield is independent of temperature. In most cases, thermalization distances and primary quantum yields have been determined from the field dependencies of photogeneration efficiencies at high fields. 

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