ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия C, 2013, том 55, № 7, с. 845-865

УДК 541.64.539.199


© 2013 г. Hsiao-Ping Hsu", Wolfgang Paul4, and Kurt Binder"

a Institut für Physik, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Staudinger Weg 7, Germany b Theoretische Physik, Martin-Luther Universität Halle-Wittenberg von Seckendorf Platz 1, 06120 Halle, Germany e-mail: hsu@uni-mainz.de

Abstract—The persistence length of macromolecules is one of their basic characteristics, describing their intrinsic local stiffness. However, it is difficult to extract this length from physical properties of the polymers, different recipes may give answers that disagree with each other. Monte Carlo simulations are used to elucidate this problem, giving a comparative discussion of two lattice models, the self-avoiding walk model extended by a bond bending energy, and bottle-brush polymers described by the bond fluctuation model. The conditions are discussed under which a description of such macromolecules by Kratky-Porod worm-like chains holds, and the question to what extent the persistence length depends on external conditions (such as solvent quality) is considered. The scattering function of semiflexible polymers is discussed in detail, a comparison to various analytic treatments is given, and an outlook to experimental work is presented.

DOI: 10.7868/S050754751306007X


Flexibility of chain molecules (or lack of flexibility, respectively) is one of their most basic general properties [1—5]. It affects the use of macromolecules as building entities of soft materials, and controls some aspects of the functions of biopolymers in a biological context. Thus, it is important to understand its origin in terms of the macromolecular chemical architecture, and the extent to which it depends on external conditions (temperature, solvent quality if the polymer is in solution, as well as polymer concentration), and one therefore needs to be able to characterize macromolecular flexibility or stiffness precisely. The quantity that is supposed to describe the local intrinsic stiffness of a polymer is termed "persistence length" and often it is introduced (e.g. [4, 5]) as a length describing the exponential decay of orientational correlations of segments with the length of the piece of the chain separating them. Thus, let us consider a linear macromolecule composed of segments vectors {a^ i = 1,...,N}, all having the same bond length € ь(^ =

€ b, if we wish to allow for thermal fluctuations of the length of these segments). Then it is assumed that the correlation of two segments i, j, that are 5 = | i - j\ steps along the chain apart, varies as

(cos Щ) = (a • dj) / (af) = exp(-s€ ь/€ Д

s ^ да,

where i is the persistence length.

In fact, Eq. (1) holds for models of linear polymer chains that strictly follow Gaussian statistics (for large distances between monomeric units), however, Eq. (1) is not true for real polymers, irrespective of the considered conditions: for dilute solutions and good solvent conditions one rather finds a power law behavior [6]

(cos e(s)) « s, (2)

p = 2(1 -v), 1« s « N. Here v is the well-known Flory exponent, describing

the scaling of the end-to-end distance R = ^ i a,

with the number N of segments, ^R<x N2v, with v - 3/5 (more precisely [7], v = 0.588) in d = 3 dimensions [1—5]. Polymer chains in dense melts do show a scaling of the end-to-end distance as predicted

by Gaussian statistics, ('r<x N (i.e., v takes the

mean-field value v MF = 1/2), and hence it was widely believed, that Eq. (1) is useful for polymer chains under melt conditions. However, recent analytical and numerical work [8, 9] has shown that this assertion is completely wrong, and there also holds a power law decay, though with a different exponent,

(cos 0(s)) <x s~3/2, 1 < s < N. (3)

More recently, it was also found by approximate analytical arguments [10], and verified in extensive simulations [11] that Eq. (3) also holds for chains in dilute

(cos 0(5» (cos 0(5»

Fig. 1. (a) Semi—log plot and (b) Log—log plot of (cos 9(s)) versus s as obtained from Monte Carlo simulations (as described in [11]) using the pruned-enriched Rosenbluth method (PERM algorithm [12]) for a self-avoiding walk with nearest-neighbor attraction £, under Theta point conditions. N = (1) 6400, (2) 3200, (3) 1600, (4) 800 and (5) 400. The full curve in (a) and straight


line in (b) represents the relation (cos 9(s)) = 0.16s .

solutions at the Theta point. In practice, since asymptotic power laws such as Eqs. (2), (3) hold only in the intermediate regime 1 < s < N and hence one must consider the limit N ^ to, one easily could be misled if data for (cos 0(s)) are considered for insufficiently long chains. As an example Fig. 1 presents simulation results for the simple self-avoiding walk (SAW) model on the simple cubic (sc) lattice, where an attractive energy s between neighboring occupied sites (representing the effective monomers of the chain) occurs and the temperature is chosen as kBT / s = 3.717 which is known to reproduce Theta point conditions for this model [12]. One can see clearly that the data for N —> to and s > 10 do approach Eq. (3), but for finite N systematic deviations from Eq. (3) clearly are visible already for s = N/10. On the semi-log plot, for rather short chains one might be tempted to apply a fit of an exponential decay proportional to exp(-si b/i p) to the data for rather large 5, but resulting estimates for i /i b are not meaningful at all: for the considered model, the chain is fully flexible, any reasonable estimate for i /i b that describes the local intrinsic stiffness of the chain should be (i) of order unity (see Fig. 1a, i /ib ~ 0.94), and (ii) independent of N. Both conditions are dramatically violated, of course, if estimates for i /i b were extracted from fits to an exponential decay in this way.

Since the intrinsic stiffness of a chain is a local property of a macromolecule, one might alternatively try the recipe to either fit Eq. (1) in the regime of small s to the data, or assume that Eq. (1) holds for s = 1 already and hence

i p/i b = -1/ ln((cos 0(1))). (4)

This recipe works in simple cases, such as the SAW model where an energy sb associated with bond bending is added (every kink of the walk by ±90° on the sc lattice costs s b), see Fig. 2, but it fails for molecules with more complex chemical architecture, such as bottle-brush molecules [13—16]. The dramatic failure of Eq. (4) for bottle-brush polymers is understood in terms of their multiscale structure (Fig. 3): The side chains lead to a stiffness of the backbone on a mesos-copic scale, even if on the local scale of nearest-neighbor bonds the backbone is still rather flexible. The question of understanding this stiffening of bottle-brush polymers because of their grafted linear side chains [11, 13—38] or grafted branched objects [39— 42] is an issue of longstanding debate in the literature.

Complex polymer architecture is only one out of many reasons which make the analysis of bond orien-tational correlations based on Eqs. (1) or (4) problematic. In dilute solutions we expect that a nontrivial crossover occurs when the solvent quality is marginal, i.e. close to the Theta point a large size £,T of "thermal blobs" [43] exists, such that for values of s along the backbone of the chain corresponding to distances r(s) > t,T one expects that excluded volume effects are visible and hence Eq. (2) should hold. For semidilute solutions [43], on the other hand, in the good solvent regime the inverse effect occurs: there exists a screening length %(c) depending on the polymer concentration c (also called size of "concentration blobs" [43]), such that excluded volume effects are pronounced for r(s) < %(c) but are absent for r(s) > %(c). Then Eq. (3) holds for the latter case and Eq. (2) for the former, for rather flexible chains. If the chains are semiflexible, in favorable cases (e.g., for simple chemical architecture of the polymers) we might observe Eq. (1) for 1 < s < s* where s* depends on the local intrinsic stiff-

(cos 0(5» 100



□ 1 * 2 o 3 x 4




(cos 0(5»


80 5





80 5







Fig. 2. Semi—log plot of (cos 9(s)) vs. s for a semiflexible version of (a) a SAW model on the sc lattice, cf. text, and (b) the bond-fluctuation model of bottle-brush polymers under very good solvent conditions [11, 17]. Part (a) refers to the chains of length N = = 50000, and several choices of the parameter qj = exp(-sj/kgT) controlling the chain stiffness, namely qj = (1) 0.02, (2) 0.05, (3) 0.1, (4) 0.2 and (5) 0.4. Using Eq. (4), the straight lines indicate the exponential decay exp(-si j/i p) for the choices of qj. Part (b) refers to the case of bottle-brush polymers where every effective monomer of the backbone has one side chain of length Ns = 24 grafted to it, and several choices of backbone chain length Nb = (1) 259, (2) 131 and (3) 67. Here i p/i j = = -1/ ln((cos 9(1))) has been extracted from the chain backbone only.

ness of the chain, which we wish to characterize by i . Then the question arises whether s* is smaller than any of the other crossover chemical distances (due to marginal solvent quality, described by a Flory-Huggins parameter % with (1 /2 - x) ^ 1 [43, 44], or due to nonzero c) or not. The conclusion of this discussion is that the behavior of bond orientational correlations !sai • aj) is subtle, and not always suitable to obtain straightforwardly information on the intrinsic stiffness of macro-molecules; as a further caveat we mention that in general it is also not true that this correlation depends on the relative distance s = |i - j| only: it matters also, if one of the sites is close to a chain end.

Another popular definition is the local persistence length i p(i) defined as [1, 2]

ip(i)/ib = (at ■ R)/(a2). (5

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