Dynamic charge-density correlation function in weakly charged polyampholyte globules

We study solutions of statistically neutral polyampholyte chains containing a large fraction of neutral monomers. It is known that such solutions phase separate at very low concentrations, even if the quality of the solvent with respect to the neutral monomers is good. The precipitate is semidilute if the chains are weakly charged. This paper considers theta solvents and good solvents, and we calculate the dynamic charge density correlation function g(k,t) in the precipitate, using the quadratic approximation to the Martin-Siggia-Rose generating functional. It is convenient to express the results in terms of dimensionless space and time variables. Let xi be the blob size, and let tau be the characteristic time scale at the blob level. Define the dimensionless wave vector q = xik, and the dimensionless time s = t/tau. In the regime q<1, corresponding to length scales larger than the blob size, and 1<s<q(-4), corresponding to time scales in between the blob relaxation time and the relaxation time at scale q(-1), we find that the charge density fluctuations relax according to the power law g(q,s)similar to q(2)s(-1/2). This relaxation is qualitatively different from that of a neutral semidilute polymer solution. We expect our results to be valid for wave vectors q>0.1, where entanglements are unimportant.