Correlation functions of main-chain polymer nematics constrained by tensorial and vectorial conservation laws

Abstract

We present and analyze correlation functions of a main-chain polymer nematic in a continuum worm-like chain description for two types of constraints formalized by the tensorial and vectorial conservation laws, both originating in the microscopic chain integrity, i.e., the connectivity of the polymer chains. In particular, our aim is to identify the features of the correlation functions that are most susceptible to the differences between the two constraints. Besides the density and director autocorrelations in both the tensorial and vectorial cases, we calculate also the density-director correlation functions, the latter being a direct signature of the presence of a specific constraint. Its amplitude is connected to the strength of the constraint and is zero if none of the constraints are present, i.e., for a standard non-polymeric nematic. Generally, the correlation functions with the constraints differ substantially from the correlation functions in the non-polymeric case, if the constraints are strong which in practice requires long chains. Moreover, for the tensorial conservation law to be well distinguishable from the vectorial one, the chain persistence length should be much smaller than the total length of the chain, so that hairpins (chain backfolding) are numerous and the polar order is small.

Generalized conservation law for main-chain polymer nematics

“…In this model, the free energy cost of the sources is purely entropic and is expressed in a form of Eq. (13) or alike. A similar free energy contribution can be set up also for ∇ · g. Within this simple model of the sources, the source densities do not appear explicitly as additional variables, since their quadratic free energy contributions can be expressed directly via the conservation laws Eqs.…”

Splay–density coupling in semiflexible main-chain nematic polymers with hairpins

“…In the same manner, we compute [12] also the longitudinal director fluctuation D L (q) = δn L (q)δn L (−q) /N , with the theoretical expression [2,7]…”

“…The exact nature and form of this conservation law proposed independently by de Gennes and Meyer, received recently a renewed scrutiny [6] that uncovered its deeper structure and important consequences missed in the previous analysis. In fact, its consequences trickle all the way down to fundamental macroscopic, observable properties such as structure factors and coarse-grained order parameters [7] as in, e.g., the ordered and/or confined phases of DNA [8][9][10]. The fundamental issue that we address in this contribution is the way this conservation law enters the coarse-grained Ornstein-Zernicke free energy description of a nematic polymer with arbitrary chain backfolding and, specifically, the magnitude of the corresponding phenomenological coupling strength [11].…”

“…( 3) for the recovered polar order. Neglecting variations of nematic order modulus and expanding ρ = ρ 0 + δρ(q), n = n 0 + δn(q) around equilibrium values ρ 0 , n 0 , the free-energy [2,7] contribution of a Fourier mode q = (q ⊥ , q ) in the volume V is F q = f (q)/V and…”

“…The structure factor S(q) = δρ(q)δρ(−q) /N (where N is the total number of 0 segments) corresponding to Eq. ( 10) is then found to be [2,7]…”