ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия C, 2013, том 55, № 7, с. 990-1020
УДК 541.64:539.2
STRICTLY TWO-DIMENSIONAL SELF-AVOIDING WALKS: DENSITY CROSSOVER SCALING
© 2013 г. N. Schulmann", H. Meyer", T. Kreeré, A. Cavalloc, A. Johner", J. Baschnagel", and J. P. Wittmer"
aInstitut Charles Sadron, Université de Strasbourg & CNRS, 23 rue du Loess, P 84047, 67034 Strasbourg Cedex 2, France bLeibniz-Institut for Polymerforschung Dresden e.V., Postfach 120411, 01005Dresden, Germany cDirtimento di Fisica, Universita degli Studi di Salerno, via Ponte don Melillo, 1-84084 Fisciano, Italy
e-mail: joachim.wittmer@ics-cnrs.unistra.fr
Abstract — The density crossover scaling of thermodynamic and conformational properties of solutions and melts of self-avoiding and highly flexible polymer chains without chain intersections confined to strictly two dimensions (d = 2 ) is investigated by means of molecular dynamics and Monte Carlo simulations of a standard coarse-grained bead-spring model. We focus on properties related to the contact exponent 02 set by the
intrachain subchain size distribution. With R ~ Nv being the size of chains of length N and p the monomer density, the interaction energy eint between monomers from different chains and the corresponding number
v9
nint of interchain contacts per monomer are found to scale as eint ~ nint ~ 1/N 2 with v = 3/4 and 02 = 19/12
2
for dilute solutions and v = 1/d and 02 = 3/4 for N > g(p) « 1/ p . Irrespective of p, long chains thus become
compact packings of blobs of contour length L ~ Nnint ~ Rdp with dp = d - 92 = 5/4 being the fractal line dimension. Due to the generalized Porod scattering of the compact chains, the Kratky representation of the intramolecular form factor F(q) reveals a non-monotonous behavior approaching with increasing chain length
d 9
and density a power-law slope F(q)q /p « 1/(qR) 2 in the intermediate regime of the wavevector q. The specific intermolecular contact probability is argued to imply an enhanced compatibility for polymer blends confined to ultrathin films. We comment briefly on finite persistence length effects.
DOI: 10.7868/S0507547513070131
INTRODUCTION
Background. Three-dimensional (3D) bulk phases of dense homopolymer solutions and blends [1—6] are known to be well-described by standard perturbation calculations [2, 7, 8]. Provided that the length N of the (both linear and monodisperse) polymer chains and the monomer number density p are sufficiently large, the polymers adopt thus (to leading order) Gaussian chain statistics, i.e. the chain size R scales as1
R2 ~ N2v (1 - p*/p) for p > p* (1)
with d = 3 being the spatial dimension, v = 1/2 the inverse fractal dimension of the chains [9], often called
"Flory's exponent", and p* ~ N/Rdv the so-called
1 Notations: If we compute a quantity exactly, including all numerical coefficients, we use an equals sign, i.e., we write A = B. If we state only a scaling law, ignoring all numerical coefficients, but keeping all dimensional factors, we use the symbol «. If we want to stress only the power law involved, we use the symbol Although we focus on d = 2-dimensional systems we often indicate the general spatial dimension d and the general Flory exponent v to show the structure of the relations. The dilute limit of a property considered is often characterized by an index 0.
"overlap density" or "self-density" [1, 4, 7]. (All pref-actors are omitted for clarity.) The underlined term in Eq. (1) indicates a weak scale-free correction with respect to the Gaussian reference. It arises due to an interplay of the chain connectivity and the incompress-ibility of the solution on large scales which is perfectly described by one-loop perturbation calculations [7]. The success of the perturbation approach for dense systems in d = 3 is expected from the fact that large scale properties are dominated by the interactions of many polymers, with each of these interactions only having a small (both static and dynamical) effect [1]. Since dense 3D chains are open objects [9], all monomers of a chain have (essentially) the same probability to be in contact with monomers of other chains, i.e. the fraction nint of monomers in interchain contact is known to be essentially chain length independent [1]
nint ~ P (1 - p*/p) for p > p* (2)
where the underlined term indicates the small N-de-pendent correction due to the already mentioned long-range correlations. It has been shown to be of im-
450
Dilute density limit
0 450
Fig. 1. In this work we consider self-avoiding polymers in strictly two dimensions (d = 2) without chain intersections focusing on the density crossover scaling. The dilute limit (p ^ 0) is characterized by sampling one single chain (M = 1) in a huge box of lin-
4
ear dimension Lbox = 10 using a mix of local and slithering snake MC moves [7]. Main panel: Snapshots of two chains of length N = 2048. As expected from theory [1], the chains are shown to reveal swollen chain statistics with power-law exponents v = vo = 3/4 and 02 = 02,0 = 19/12. The typical root-mean square end-to-end distance Re(N), the radius of gyration Rg (N) and the corresponding dilute statistical segment lengths be,o and bgo for asymptotically long chains are indicated. We use b0 = bgo = 0.37 as a natural length scale to make our numerical model-depending results comparable to experimental data. Re = 299.2, R„ = 112.0. Inset: Short subchain showing that the monomers do not overlap and that the chains do not intersect.
portance for the precise characterization of the phase transition of polymer blends [10].
General motivation. Obviously, the number of chains a reference chain interacts with depends on the spatial dimension d as implied by the strong-overlap condition [1]
p*/p « N/pRd ~ N1_dv < 1, (3)
i.e. for to leading order Gaussian chains (v = 1/2) the perturbation approach becomes questionable below d = 3. Due to the increasing experimental interest in mechanical and rheological properties of nanoscale systems in general [11] and on dense polymer solutions in reduced effective dimensions d < 3 in particular [12—30], one is naturally led to question theoretically [31—39] and computationally [40—56] the perturbation results. Especially polymer solutions and melts confined to effectively two-dimensional (2D) thin films and layers are of significant technological relevance with opportunities ranging from tribology to biology [12, 13, 16, 19]. In the present review we shall consider such ultrathin films focusing on numerical results [40—46, 51—56] obtained on both self-avoiding and flexible homopolymers confined to strictly d = 2
dimensions. We assume that chain overlap and intersection are strictly forbidden as shown in the snapshot presented in Fig. 1 obtained by Monte Carlo (MC) simulation [57—59] of a standard bead-spring model [60-63].
Compactness of the chains. As first suggested by de Gennes [1], it is now generally accepted [14, 15, 3335, 40-42, 46, 51-53] (with the notable exception of Refs. [17, 24, 43]) that such "self-avoiding walks" (SAWs) adopt compact conformations at sufficiently high densities, i.e., the chain size R scales as
R « (N/p)v, where v = 1/d = 1/2. (4)
Interestingly, according to Eq. (4) the chain size is
assumed to be set alone by the distance dcm ~ (p/N)1/d between the centers of mass of the chains and does thus not depend on local monomeric parameters. As already stated, we assume that chain intersections are strictly forbidden. This must be distinguished from systems of so-called "self-avoiding trails" (SATs) which are characterized by a finite chain intersection probability [35]. Relaxing thus the self-avoidance constraint, SATs have been argued to belong to a different universality class revealing mean-field-type statistics
0 450 300 400
Fig. 2. Semidilute regime for number density p = 0.125, chain length N = 2048, chain number M = 48 and linear box dimension Zbox a 887. The numbers refer to a chain index used for computational purposes. The dashed spheres represent a hard disk of uniform mass distribution having a radius of gyration Rg equal to the semidilute blob size 2, a 31 of the given density. As shown in panel (a) some chains are still rather elongated (e.g., chain 10 or 30) and the swollen chain statistics remains relevant on small scales (r < On larger scales the chains are shown to adopt (on average) compact configurations with power-law exponents v = 1/d and 02 = 3/4. The square in the first panel is redrawn with higher magnification in panel (b). The small spheres (not to scale) correspond to monomers interacting with monomers from other chains. As investigated in Sec. "CONFORMATIONAL (REAL SPACE) PROPERTIES — Interchain Binary Monomer Contacts", the fraction «¡nt of such monomers remains small, even at higher densities, if the chain length is large. We argue in Sec. "CRITICAL TEMPERATURE OF DEMIXING OF POLYMER BLENDS" that this low interchain monomer contact probability implies an enhanced miscibility of polymer blends confined to ultrathin films.
with rather strong logarithmic N-corrections [32, 35]. An experimental relevant example for SATs is provided by polymer melts confined to thin films of finite width H allowing the overlap of the chains in the direction perpendicular to the walls. At variance to Eq. (4) such films reveal swollen chain statistics with (omitting density dependent factors) [35]
R2 ~ N (1 + log(N)/H),
(5)
perimeter of the (sub)chains. As may be seen from the snapshots presented in Figs. 2 and 3, the chains adopt instead rather irregular shapes [46—54]. Focusing on dense 2D melts it has been shown both theoretically [35] and numerically [51—54] that the irregular chain contours are characterized by a perimeter L scaling as
L ~ Nnu
Ra
N
dp/d
nint ~ (1 + log(N)/H)2. (6)
These predictions are consistent with computational results obtained by means of MC simulation of the bond-fluctuation model [49, 64] and by molecular dynamics (MD)
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