Internal distances in short polyelectrolytes: A Monte Carlo study

The new critical phenomenon of a coil-rod transition that was studied in our previous papers [J. Chem. Phys. 92, 4468 (1990); 93, 2736 (1990); 94, 3213 (1991); 97, 2119 (1992); J. Phys. Chem. 96, 5553 (1992)] is investigated further. The family of physical statistical bonds is expanded. An important additional bond, the linear statistical bond, is introduced. Parameters based on internal distance investigations are proposed and these are used to analyze the Monte Carlo data. One of the most important parameters is the average probability for a linear statistical bond at a site on the chain, ${\mathit{P}}_{\mathit{l}}$. Indications are reported that scaling behavior in polyelectrolyte chains exists only for chain lengths having the same kink fraction g. An important relation is shown between the average number of kinks, 〈${\mathit{n}}_{\mathrm{kink}}$〉, and D(2), the mean-square distance between the end beads of three adjacent beads in a cubic lattice: g=〈${\mathit{n}}_{\mathrm{kink}}$〉/N-2=1-[D(2)-2]/2=1-${\mathit{P}}_1$. In a previous article [J. Chem. Phys. 97, 2119 (1992)] we found that for a self-avoiding-walk chain, g is constant and equal to 0.77. This relation leads to new constants, ${\mathit{P}}_1$=0.23 and D(2)=2.46 squared cell units in addition to the constant mean straight length 〈${\mathit{l}}_{\mathit{s}}$〉=1.29 cell units, found in the above reference, to be connected to g.

The large size effect that was found in this reference is also demonstrated here, i.e., segments of a small chain tend to expand less than segments of a long chain. The blob concept is examined and it is shown that all interior segments of the chain are stretched by repulsive interactions. This is inconsistent with the basic assumption of the blob concept. Howver, because the change in short segments of a chain is small when compared to the large change in long segments, the continued use of the blob concept remains valid for analytic estimations of the whole chain length. The polyelectrolyte expansion is extended to describe polymer expansion in general. An empirical relation is proposed between the mean-square radius of gyration 〈${\mathit{S}}^2$〉 and the mean-square internal distance between the ends of a half chain D(N/2), and is expressed by D(N/2)=3〈${\mathit{S}}^2$〉. An additional empirical relation is shown: ${\mathit{P}}_1$=D(N/3)/(N/3${)}^2$ where D(N/3) is the mean-square internal distance between the ends of one-third of the chain. Verification is found for the existence of the physical statistical bonds that vary with internal distance. The existence of these bonds is extended to various internal distances and to systems with various types of charge.