The Onsager Formalism

The models that have been most widely used to describe a geminate recombination are based on theories due to Onsager (1934. 1938). The theories are derived from the Smoluchowski (1916) equations, 

In 1938, Onsager described a solution to Eqs. (11) and (12) subject to the boundary conditions 

The Onsager expression that a thermalized electron-hole pair will escape recombination, Q(r,0, ), is 

Substituting Eq. (19) into Eq. (18) and completing the integration (Mozumder, 1974, 1974a) gives 

The applicability of the Onsager theory can be determined from the field dependence of the photogeneration efficiency at low fields. From Eq. (20), the efficiency should have a linear field dependence. From plots of the efficiency versus field, the slope-to-intercept ratio is 

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The probability of the dissociation of the CT state into a pair of separated carriers, and its functional form is determined by the mechanism of final charge separation [Q(F)]. Besides the one-step PF dissociation process (130), the Onsager formalism is often used to describe l(F) [314,315]... 

Figures 1 through 3 show the field and temperature dependencies of r]/7]0 calculated from Eq. (20) for different values of rQ. The dielectric constant was assumed to be 3.0. The Onsager formalism leads to strongly field-dependent photogeneration efficiencies that approach a limiting value at high fields. At...

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. 

Table 1 Thermalization Distances Determined from the Onsager Formalism...

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