Polarons, Bipolarons and Polaron Pairs

As an extension of the two former cases we consider now the possibility that besides the Ps both PPs and BPs can occur as doubly charged states of the polymer chains. Equilibrium is then described by (2), (3), (17) and, instead of (5) or (16) by 

Nevertheless, one has two problems with this model. At first, the polaron peak is relatively broad whereas in experiments at least in some cases the peak of the spin concentration is indeed narrower than for a one-electron transition. Further, it seems to be hard to find a reason why the BP equilibrium constant should be just roughly one tenth of the PP equilibrium constant. A reasonable alternative will be presented in the next section. 

In the past, the relevance of the thermodynamic derivation of the P-BP equilibrium [47] or the equivalent statistical derivation [48] has not been questioned. However, there is one point which is crucial. This is the assumption of Eq. (5) for the equilibrium, equivalently Eq. (6) for the reaction, and equivalently in the statistical treatment the assumption that both Ps and BPs have the same extension over m monomer units. These assumptions are quite natural for molecules in solution, which can be singly or doubly charged. In the case of charges on the polymer chain connected with a chain distortion, there is no reason that the BP has the same extension as the P, nor that the PP has exactly twice the extension of the P. 

as before we suppose that the P extends over m monomer units. Then a neutral chain segment of the same extension is defined. For the doubly charged species it is now supposed that it extends over Wj monomer units. 

With m m2 2m it can be larger than the BP and smaller than the PP. To distinguish this more general case we denote it as a dipolaron (DP). It must be emphasised, that no information about the type of binding enters this supposition. Counting now the number of segments one finds 

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A revisal of the equilibrium between polarons and doubly charged states of the polymer chains shows that the previous assumptions on the extension of polaron, bipolarons and polaron pairs were not justified. A more general formulation is presented for the equilibrium concentrations and the kinetics. But the differences are almost negligible up to the maximum charge concentrations that can be achieved in accumulation layers. The resulting rate constants for formation and dissociation of (immobile) bipolarons can be estimated using a rate constant for the bipolaron formation determined recently by Salleo and Street, and indicate that these processes can cause the hysteresis on the time scale of the measurements. 

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