Charge transport in polyaniline doped with 3-nitro-1,2,4-triazol-5(4H)-one

Polyaniline (PANI) base was protonated in aqueous solutions of an organic acid, 3-nitro-1,2,4-triazol-5(4H)-one (NTO). The temperature dependence of DC conductivity of PANI-NTO seems to correspond to the theory of variable range hopping (VRH) in three dimensions. The frequency dependence of AC conductivity also reflects the hopping nature of mobile charges. The activation energy for the polymers with protonation degree above 0.12 remains constant with increasing dopant concentration and DC conductivity. The value of this constant may correspond to the energy needed for the ionization of dopant counterion. The fit of the electric relaxation function to the stretched exponential function ϕ(t) = exp[−(t/τ)^β] gives the stretch parameter β about 0.35, which shows that the distribution of relaxation times is broad and indicates a high inhomogeneity in the distribution of a dopant.     

On the interpolation constants for variable Lebesgue spaces

For $\theta \in (0,1)$ and variable exponents $p_0(·),q_0(·)$ and $p_1(·),q_1(·)$ with values in [1, ∞], let the variable exponents $p_{\theta }(·),q_{\theta }(·)$ be defined by $$1/p_{\theta }(·):=(1-\theta )/p_0(·)+\theta /p_1(·),1/q_{\theta }(·):=(1-\theta )/q_0(·)+\theta /q_1(·).$$ The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space $L^{p_j(·)}$ to the variable Lebesgue space $L^{q_j(·)}$ for $j=0,1$, then $${\parallel T\parallel }_{L^{p_{\theta }(·)}\to L^{q_{\theta }(·)}}\le {C\parallel T\parallel }_{L^{p_0(·)}\to L^{q_0(·)}}^{1-\theta }{\parallel T\parallel }_{L^{p_1(·)}\to L^{q_1(·)}}^{\theta },$$ where C is an interpolation constant independent of T. We consider two different modulars ${\varrho }^{\max }(·)$ and ${\varrho }^{\mathrm{sum}}(·)$ generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants C_max and C_sum, which imply that $C_{\max }\le 2$ and $C_{\mathrm{sum}}\le 4$, as well as, lead to sufficient conditions for $C_{\max }=1$ and $C_{\mathrm{sum}}=1$. We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that $p_j(·)=q_j(·)$, $j=0,1$ are Lipschitz continuous and bounded away from one and infinity (in this case, ${\varrho }^{\max }(·)={\varrho }^{\mathrm{sum}}(·)$).