Maximum entropy principle applied to semiflexible ring polymers

Abstract

Based on the success of the maximum entropy principle (MEP) in the study of semiflexible treelike polymers [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)], it is of much interest to establish MEP's potential for general semiflexible polymers which contain loops. Here, we embark on this endeavor by considering discrete semiflexible polymer rings in a Rouse-type scheme. Now, for treelike polymers a beads-and-bonds (i.e., a discrete) picture is essential for an easy inclusion of branching points. Moreover, one may envisage (similar to our former work [M. Dolgushev and A. Blumen, J. Chem. Phys. 131, 044905 (2009)]) to impose for each angle between two bonds a distinct stiffness condition. Working in this way leads already for a polymer ring to a complicated problem. Hence, we follow a reduced variational approach as applied earlier to polymer chains, in which a single Lagrange multiplier is used for each set of identical conditions imposed on topologically equivalent bonds and bonds' orientations. In this way, we obtain for the discrete ring an analytically closed form which involves Chebyshev polynomials. This expression turns out to lead to a series of solutions: Apart from the regular solution, several other solutions appear. One may be tempted to discard the other solutions, since for them the potential energy matrix is not positive definite. A more careful analysis based on topological features suggests, however, that such solutions can be assigned to rings displaying knots. Monte Carlo simulations which take excluded volume interactions into account agree with our interpretation.

Branched Semiflexible Polymers: Theoretical and Simulation Aspects

“…[127] Our own computer simulations, performed using the BFM in which excluded volume interactions are turned off, are in perfect agreement with the value -1/(N -1). [69] The limit of very high stiffness occurs at x ¼ t. We display the line x ¼ t in Figure 21 in gray. The intersection points (t k , x k ) of x ¼ t with the solutions of Equation 67 are shown through dots and are given by…”

“…A careful analysis [69] shows that the upper branch (dashed line) at t ¼ À1=ðNÀ1Þ corresponds to a fully flexible ring. This differs from the t ¼ 0 value for a free chain, but is very close to the value -1/N, reported for rings in ref.…”

“…We start by listing some of its properties. [69] For all t, except for t ¼ À1=ðNÀ1Þ, Equation 67 has (N -3)/2 roots for odd and (N -2)/2 roots for even N, respectively. As an example, Figure 21 displays for N ¼ 15 the solutions of Equation 67 as a function of t.…”

“…Equation 67 has been analyzed in detail, [69] and we defer its discussion to the part of the next section dealing with rings.…”

“…Recently, we reconsidered the problem of semiflexible polymers with loops in the MEP scheme by studying a basic example -the discrete semiflexible polymer ring. [69] It is worth noting that there exist other theoretical means to handle semiflexible rings, [39,[70][71][72] but being interested in extending the STP model, which is very general, we decided in favor of the MEP method. Remarkably, in this way some interesting consequences showed up: For a semiflexible ring, apart from an obviously physical solution, several other solutions emerged, which could be related to knotted ring topologies, as we will discuss in the following.…”