Interchain Monomer Contact Probability in Two-Dimensional Polymer Solutions

Abstract

Using molecular dynamics simulation of a standard bead-spring model we investigate the density crossover scaling of strictly two-dimensional (d = 2) self-avoiding polymer chains focusing on properties related to the contact exponent θ2 set by the intrachain subchain size distribution. With R ∼ N ν being the size of chains of length N , the number nint of interchain monomer contacts per monomer is found to scale as nint ∼ 1/N νθ 2 with ν = 3/4 and θ2 = 19/12 for dilute solutions and ν = 1/d and θ2 = 3/4 for N ≫ g(ρ) ≈ 1/ρ 2 . Irrespective of the density ρ sufficiently long chains are thus found to consist of compact packings of blobs of fractal perimeter dimension dp = d − θ2 = 5/4.As first suggested by de Gennes [1], it is now generally accepted [2-13] that strictly two-dimensional (2D) selfavoiding polymer chains adopt compact and segregated conformations at high densities, i.e., the typical chain size R scales aswith N being the chain length, ρ the monomer number density, and d = 2 the spatial dimension. We assume here that monomer overlap and chain intersection are strictly forbidden [3]. Compactness obviously does not imply Gaussian chain statistics nor does segregation of chains impose disklike shapes minimizing the contour perimeter of the chains. Focusing on dense 2D melts it has been shown in fact [3,[11][12][13] that the irregular chain contours are characterized by a fractal perimeter of typical length L ∼ N n int ∼ R dp ∼ N dp/d withwhere n int stands for the fraction of monomers of a chain interacting with monomers of other chains. The fractal line dimension d p is set by Duplantier's contact exponent θ 2 = 3/4 characterizing the size distribution of inner chain segments [2]. Since the melt limit is experimentally difficult to realize for a strictly 2D layer [4][5][6], the aim of this Letter is to show that eq 2 holds more generally for all densities if the chains are sufficiently long. As described in Ref.[12] our numerical results are obtained by molecular dynamics simulations of monodisperse, linear and highly flexible chains using the wellknown Kremer-Grest bead-spring model [14]. LennardJones (LJ) units are used throughout this paper and temperature is set to unity. Note that monomer overlap and chain crossings are made impossible by the excluded volume interaction and the bonding potential; the chains are thus strictly 2D "self-avoiding walks" in the sense of Ref. [3]. While our previous studies [11][12][13] have focused on one melt density, ρ = 0.875, we scan now over a broad range of densities ρ as may be seen from Figure 1. Chains of length N = 16 up to N = 1024 monomers have been sampled for all ρ; for ρ = 0.125, ρ = 0.5 and ρ = 0.875 we have even computed systems with N = 2048 [15].

Strictly two-dimensional self-avoiding walks: Thermodynamic properties revisited

“…Polymer blends. As already argued elsewhere [6,15], the presented numerical results for the interchain interaction energy e int for homopolymer systems implies an enhanced compatibility for polymer blends confined to ultrathin films in qualitative agreement with recent experimental studies [24]. We remind that the critical temperature of unmixing T c should be proportional to the typical interaction energy Ne int between different chains.…”

“…As in ref. [15] where we have discussed various conformational properties of semidilute solutions and melts we scan over a broad range of densities ρ. All properties reported for ρ ≤ 0.25 have been sampled using MC [42].…”

“…[42] The reported MD data have been sampled over a period of about five years using different local and national computational resources which are difficult to compare. The configurations obtained in this limit have been already used and characterized in various previous publications [12][13][14][15][16]. The production of the MC data for smaller densities (ρ ≤ 0.25) performed to crosscheck and improve the MD simulations was much less expensive corresponding to production runs over a year using 8 cores of Intel Xeon E5410 processors.…”

“…It is thus incorrect to assume that excluded-volume effects are screened [10] as is approximately the case for three-dimensional melts [26]. Interestingly, compactness does not imply disklike shapes minimizing the chain perimeter length L. In fact the perimeter is found to be fractal with [6,[12][13][14][15] L ∼ Nn int ∼ R dp ∼ N dp/d with d p = d − θ 2 = 5/4 (2) where n int stands for the fraction of monomers interacting with other chains and the fractal line dimension d p is set by Duplantier's contact exponent θ 2 = 3/4 [5].…”

Swelling equilibrium of dentin adhesive polymers formed on the water–adhesive phase boundary: Experiments and micromechanical model

“…Results of small-angle neutron scattering have shown that polymer gels have similar characteristics as dense fluids of hard spheres [53]. Recent simulations agree with this granular or blob description of polymers [54]. Similar granular descriptions have also been associated with acoustic dispersion phenomena, wherein the grain size in materials such as polymers has been interpreted as the free path between acoustic scattering sites [55].…”