Onsager theory

The distribution of orientations can, in principle, be computed theoretically from a nematic potential that expresses the influence of one rod s orientation on that of its neighbors. 

From the potential Vnem(vt). one can obtain the rod orientation distribution function 1/ (u), and hence the order parameter S, by a self-consistent calculation. At equilibrium, the distribution function V (u) is related to the potential Fnemfu) by Boltzmann s equation  

For low concentrations v—that is, small Uo—the only solution is the trivial one, rfr = const. = 1 /(47t), corresponding to the isotropic state. For a high enough value of Uo, there is in addition to the isotropic solution a stable nontrivial solution corresponding to a nematic state with 5 0. Of these two solutions, the one with the lowest free energy corresponds to the equilibrium state. As Uq increases, the lowest free-energy state changes from the isotropic to the nematic. 

Onsager s potential also applies to oblate particles or disks of high aspect ratio, if the parameter Uq ts redefined as 47rv i, where d is the disk diameter (de Gennes and Frost 1993). Solutions composed of disk-shaped particles or molecules can therefore also undergo a transition to a nematic phase, called a discotic nematic. 

The Onsager theory and its extensions are valid only when the concentration is low enough that pairwise excluded-volume interactions are the dominant ones. Thus, these theories are not likely to apply to solvent-free bulk, or thennntmpir ligiiid crystallinp, phases, for which there are likely to be complex packing interactions and anisotropic energetic interactions, such as those produced by van der Waals forces. 

The diffusion of the ion pair can be described by the Smoluchowski equation. The positive ion is assumed to rest at the origin the probability P(r,t) d r of finding the negative ion at r in a volume d r at time t is given by 

A critical distance, r, is defined which follows from the condition that the Coulomb energy of attraction equals the thermal energy of the medium. 

If the initial separation of the ion pair is much greater than r, then the influence of diffusion is prevailing and the ion pair escapes its mutual attraction. If the initial separation is smaller than r, then the Coulombic attraction eventually leads to recombination. Onsager solved Equation 22 for the steady state, when 3P/3t = 0. Here, we follow the treatment presented by Pai and Enck (1975). The probability, p(r,0,E), for the escape of the ion pair was derived to be 

If we consider many ion pairs which have an isotropic distribution function g(r,0) given as 

For numerical calculations of p(r,0,T), especially at high values of E, a series derived by Mozumder (1974) is very suitable. 

---

These are just a few of the standard examples of explicit applications of the Onsager theory to concrete cases. 

A byproduct of the preceding analysis is that the Onsager theory innnediately detennines the fonn of the fluctuations that should be added to the difhision equation. Suppose that a solute is dissolved in a solvent with concentration c. The difhision equation for this is... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

Incomplete Dissociation into Free Ions. As is well known, there are many substances which behave as a strong electrolyte when dissolved in one solvent, but as a weak electrolyte when dissolved in another solvent. In any solvent the Debye-IIiickel-Onsager theory predicts how the ions of a solute should behave in an applied electric field, if the solute is completely dissociated into free ions. When we wish to survey the electrical conductivity of those solutes which (in certain solvents) behave as weak electrolytes, we have to ask, in each case, the question posed in Sec. 20 in this solution is it true that, at any moment, every ion responds to the applied electric field in the way predicted by the Debye-Hiickel theory, or does a certain fraction of the solute fail to respond to the field in this way In cases where it is true that, at any moment, a certain fraction of the solute fails to contribute to the conductivity, we have to ask the further question is this failure due to the presence of short-range forces of attraction, or can it be due merely to the presence of strong electrostatic forces ... 

Here, Vd = pE is the drift velocity. The recombination and escape probabilities are now given by PR = NR /n+° L0 and Pkc = 1 - Pr. Since Vd = i, but T /r1 these probabilities are independent of mobility. However, the initial separation r0 is expected to depend (increase) with electron mobility, thus making the escape probability indirectly dependent on the mobility. These effects are quite similar to those in the Onsager theory... 

Tachiya M (1988) Breakdown of the Onsager theory of gemmate ion recombination. J Chem Phys 89 6929... 

Wojcik M, Tachiya M (2009) Accuracies of the empirical theories of the escape probability based on Eigen model and Braun model compared with the exact extension of Onsager theory. J Chem Phys 130 104107... 

The world surrounding us is mostly out of equihbrium, equilibrium being just an idealization that requires specific conditions to be met in the laboratory. Even today we do not have a general theory about nonequilibrium macroscopic systems as we have for equilibrium ones. Onsager theory is probably the most successful attempt, albeit its domain of validity is restricted to the linear response regime. In small systems the situation seems to be the opposite. Over the past years, a set of theoretical results that go under the name of fluctuation theorems have been unveiled. These theorems make specific predictions about energy processes in small systems that can be scrutinized in the laboratory. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

The recombination of photogenerated electrons and holes is the bane of all solar cells and a major reason for their less than ideal efficiencies. Excitonic solar cells, in which the electrons and holes exist in separate chemical phases, are subject primarily to interfacial recombination. There is, as yet, no theoretical model to accurately describe interfacial recombination processes, and this is an important area for future research. Wang and Suna [91] have laid a possible foundation for such a model by combining Marcus theory with Onsager theory. 

The most authors consider that the Onsager theory cannot be applied to the photogeneration of charge carriers in polydiacetylenes and transport processes are controlled by deep traps. 

The Onsager theory of geminate recombination was qualitatively consistent for aryl-substituted thiapyrylium salt and dialkylamino-substituted triphenyl-methane dispersed in polycarbonate film [301]. The quantum yield of Hie photogeneration was equal to 0.5 at the electric field strength of 106 V cm-1, mobility of I0 12m2 V-1 s-1. Hole and electron conductivity was established. In a triphenylamine-lexan system doped with a boron diketone acceptor, the... 

Sano (501] has discussed the probability of escape of an anion (e.g, solvated electron) from several stationary cations which are also sinks. This marks an extension of the Onsager theory of ion-pair recombination [321]. If the N cations are located at p1( p2... pw and all quite close to one another, the potential energy of the anion is... 

If we admit that the Onsager theory correctly accounts for the field-facilitated dissociation phenomenon occurring in the present system,... 

Onsager Theory for C(t) for Non-Debye Solvents. Generally solvents have more complex dielectric responses than described by the Debye equation (Eq. (18)). To obtain the time dependence of the reaction field R from Eqs. (12, (15), (16) and (7) an appropriate model for dielectric behavior of a specific liquid should be employed. One of the most common dielectric relaxation is given by the Debye-type form, which is applicable to normal alcohols. 

According to the Onsager theory and to computer simulations of the behavior of hard sphero-cylinders [37], in the absence of additional interactions no LC ordering is predicted for rods with L/D < 4, and therefore DNA double helices with a number of base pairs N < 24 would lack the anisotropy to display mesophase behavior at any concentration (Fig. 10). 

Dielectrics, Onsager theory of, 103 Dienophile, oxygen as a, 248, 252 Diffusion, coefficient, 170... 

---