Monte Carlo Simulation of Short Chain Branched Polyolefins

Article pubs.acs.org/Macromolecules

Monte Carlo Simulation of Short Chain Branched Polyolefins: Structure and Properties Krzysztof Moorthi* Materials Science Laboratory, Mitsui Chemicals, Inc., 580-32 Nagaura, Sodegaura City 299-0265, Japan

Kazunori Kamio Mitsui Chemicals Analysis and Consulting Service, 580-32 Nagaura, Sodegaura City 299-0265, Japan

Javier Ramos Department of Macromolecular Physics, Instituto de Estructura de la Materia, CSIC, Serrano 113 bis, 28006 Madrid, Spain

Doros N. Theodorou Department of Materials Science and Engineering, School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou St., Zografou Campus, 15780 Athens, Greece ABSTRACT: The effect of higher α-olefin comonomer on physical properties of short chain branched (SCB) polyethylene (PE) melts at 450 K has been studied using connectivity altering Monte Carlo simulations. The calculated chain dimensions per molecular mass scale with backbone weight fraction, ϕ, as ⟨S2⟩/M ∼ ϕ1.27±0.03 for the radius of gyration S and ⟨R2⟩/M ∼ ϕ1.27±0.03 for the end-to-end distance R, in very good agreement with the experiment-based result ⟨R2⟩/M ∼ ϕ1.30. The observed dependence is consistent with the decrease in the fraction of trans states along the backbone. The entanglement tube diameter, app, computed for SCB melts scales as ⟨app⟩ ∼ ϕ−0.46±0.01, which is close to the result for model concentrated (⟨R2⟩ = const) PE solutions created by deleting randomly chosen chains from equilibrated melt configurations of linear PE (⟨app⟩ ∼ ϕ−0.41±0.01). The latter result agrees very well with the scaling based on rheological experiments on concentrated hydrogenated polybutadiene (HPB)/C24H50 solutions at 413 K (⟨app⟩ ∼ ϕ−0.45). The tube diameter in model athermal PE solutions scales as ⟨app⟩ ∼ ϕ−0.6±0.03, in excellent agreement with the scaling based on the neutron spinecho experiments on athermal HPB/C19D40 solutions at 509 K (⟨app⟩ ∼ ϕ−0.6). The computed scaling relationships for both SCB melts and model PE solutions are close to the binary contact model (app ∼ ϕ−0.5) and disagree with the packing model (app ∼ ϕ−1.27). The solubility parameters calculated for poly(ethylene-co-1-butene) (PEB) melts are in excellent agreement with relative solubility parameters based on SANS analysis of appropriate SCB blends, which scale as δ ∼ ϕ0.18. The SANS-derived relative solubility parameters for poly(ethylene-co-1-hexene) (PEH) and poly(ethylene-1-octene) (PEO) systems scale more weakly (δ ∼ ϕ0.1) and suggest breakdown of a universal correlation. This pattern is followed by simulated relative solubility parameters.

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INTRODUCTION Many short chain branched (SCB) copolymers of ethene and higher α-olefins, such as 1-butene (PEB), 1-hexene (PEH), and 1-octene (PEO), are versatile plastics.1 Their solid and melt physical properties can be tailored for specific purposes by controlling the concentration and type of α-olefin. Ultimate properties of such materials are influenced by crystallinity. Nevertheless, failure of semicrystalline polymers is strongly affected by entanglements.2−4 Also, melt properties are affected by entanglements, and the study of such properties as plateau moduli, compliances, and zero shear viscosities within the © 2012 American Chemical Society

reptation concept has been a classical subject of polymer physics.5−8 These and other properties have been intensively studied in the context of SCB polyolefins.9−11,44,56−62,70,71 A detailed knowledge of how entanglements relate to chemical composition and chain dimensions and how they evolve under strain in solids and in melts would greatly help in optimizing and developing new polymeric materials. It could Received: June 26, 2012 Revised: August 25, 2012 Published: October 1, 2012 8453

dx.doi.org/10.1021/ma301322v | Macromolecules 2012, 45, 8453−8466

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Table 1. Polymers Studied, Their Composition, Molecular Weight per Backbone Bond, mb, Density, ρ, at T = 450 K and p = 1 atm, and Number of Independent Systems system name PE PEB038 PEB198 PEB398 PEH038 PEH120 PEH198 PEH298 PEH398 PEO038 PEO118 PEO198 PEO298 PED038 PED120 PED198

comonomer

comonomer mole fraction x

ϕa

mb [g/mol]g

density ρ [g/cm3]

no. of systems

1-butene 1-butene 1-butene 1-hexene 1-hexene 1-hexene 1-hexene 1-hexene 1-octene 1-octene 1-octene 1-octene 1-decene 1-decene 1-decene

0.000 0.038 0.198 0.398 0.038 0.120 0.198 0.298 0.398 0.038 0.118 0.198 0.298 0.038 0.120 0.198

1.000 0.963 0.835 0.715 0.929 0.806 0.716 0.627 0.557 0.898 0.739 0.627 0.528 0.868 0.676 0.558

14.04 14.58 16.82 19.63 15.11 17.41 19.60 22.41 25.22 15.64 19.01 22.38 26.60 16.18 20.78 25.16

0.775d 0.776d 0.781d 0.787d 0.776d 0.778e 0.779d 0.7805e 0.782d 0.775d 0.7765e 0.778d 0.779f 0.775d 0.7765e 0.778d

2b 2b 2b 2b 2b 2b 2b 2b 2b 2b 2b + 2c 2b + 2c 2b + 1c 2b 2b 2c

a

Equation 2. b20-chain systems. c12-chain systems. dMD simulated value. eInterpolated value. fInterpolated value using MD-derived density for PEO398 (ρ = 0.780 g/cm3). gReference 9; mb = M/999, where M is molar mass.

agreement with an analogous experiment-based relationship (⟨R2⟩/M ∼ ϕ1.30).9 One may think about SCB polymer configurations as sets of backbones immersed in a “solvent” formed by their short-chain branches as suggested, for example, by eq 1. To explore this analogy, we have generated model concentrated7 (⟨R2⟩ = const) PE solutions by deleting randomly chosen chains from well-equilibrated configurations of molten linear PE. A similar approach has been also applied for creating model athermal7,30 (⟨R2⟩ ∼ ϕ−1/4 up to ϕ = 1) solutions. In these solutions, the role of ϕ is played by the fraction of carbon atoms remaining after the deletion. The tube diameter scaling relationships for model concentrated solutions, athermal solutions, and SCB melts, resulting from primitive path analyses (PPA), are all found to be close to the binary contact model scaling.33−35 An important method for regulating mechanical properties of polymers is blending them. Although the identification of factors affecting miscibility in polyolefin blends is often notoriously difficult, simple rules originating from the regular solution model have been proposed for about 80% of SCB polyolefin blends.8,10,36,37 It is, therefore, of great interest to compare simulation-based solubility parameters of SCB melts with analogous quantities derived from sophisticated SANS measurements on polyolefin blends.8,10,36,37 The relative solubility parameters calculated in this work for PEB melts agree well with the relative solubility parameters derived from the SANS analysis of PEB blends,36 which scale as δ ∼ ϕ0.18. The SANS-based relative solubility parameters of PEH and PEO systems scale somewhat more weakly (δ ∼ ϕ0.1). The simulated relative solubility parameters follow this pattern qualitatively.

also provide insights for tuning mesoscopic models of entangled polymers. However, the molecular origin of entanglements in flexible polymers and their relationship to chemical structure is still a matter of active research.12−19,32,49,50 One difficulty is illustrated by the requirement of two independent length scales: the Kuhn length, lK, and the contour length density L/Vch, with ρ, M, L, and Vch = M/ρ denoting the mass density of the polymer melt, the molar mass of a chain, the contour length of a chain, and the volume occupied by the segments of a chain, respectively, in order to formulate a scaling model for the plateau modulus, G0N.20 ⎛ L ⎞ GN0 lK 3 = f ⎜ lK 2⎟ kBT ⎝ Vch ⎠

(1)

In general, formulation of a scaling model based on eq 1 requires additional assumptions at the molecular level, which might be difficult to verify. In the case of SCB melts the amount of branches can be varied at constant backbone length or, equivalently, volume Vbb occupied by backbone segments, and the length scales introduced above vary differently with the backbone volume fraction, ϕ = Vbb/Vch: L/Vch ∼ ϕ1 and lK ∼ ϕα with 0 < α < 1. Studies related to entanglement formation gradually become practical using molecular simulation methods. Connectivity altering Monte Carlo methods permit equilibrating polymer melts at the atomistic level,21,22 and with the advent of powerful algorithms applicable to long-chain branched23,24,72 and SCB polyethylenes,24,25 the unwieldy task of equilibrating branched polymers appears to be possible, too. Ingenious algorithms to determine primitive paths from melt geometry13,15,28,29 and including full dynamical effects16,22,27,65 have become available. In this paper, a connectivity altering Monte Carlo method extended to SCB melts,25 combined with the Z1 and CReTA algorithms, has been applied to inspect how chain dimensions and key microscopic quantities related to entanglements, such as the tube diameter, scale with backbone concentration in SCB melts. Good equilibration of the melts has been achieved. The short chain branches reduce chain dimensions, and the resulting scaling relationship ⟨R2⟩/M ∼ ϕ 1.27 is in very good

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MOLECULAR SYSTEM AND METHODS Table 1 specifies model melts used in this study. The average length of the backbone, ⟨Nbb⟩, is 1000 carbon atoms for all models. In the following, the notation PEXν will be used to refer to the various SCB PE melts studied. X will denote the type of comonomer; X = B, H, O, and D for 1-butene (B = 2), 1-hexene (B = 4), 1-octene (B = 6), and 1-decene (B = 8), respectively, where B is branch length. The integer number ν = 8454


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x⟨Nbb⟩, where x is the mole fraction of SCB comonomer, will indicate the mean number of comonomer units in a chain. The backbone volume fraction is given by ϕ=

1 1 + xB /2

(2)

For PE solutions, the xB/2 term is replaced by the fraction of carbon atoms removed from the system per remaining backbone carbon. The mass per backbone bond9 values, mb, are also included in Table 1. The branches were placed randomly along the backbone, and both head-to-head and head-to-tail additions were allowed with equal probability. The number of primary chains in the simulation cell is either 20 or 12. The smaller cells have been used mainly for the systems containing higher amounts of branches in order to reduce calculation time. Also for smaller cells, the cell size is at least 1.5 times larger than the averaged radius of gyration of the polymer, ⟨S2⟩1/2. The TraPPE united atom model39,40 has been used to model interactions in simulated systems. Molecular dynamics (MD) simulations have been performed at 450 K under isobaric conditions (p = 1 atm). Monte Carlo (MC) simulations have been performed at 450 K under isochoric conditions at densities determined by MD simulations. Preparation of Initial Melt Structures and Melt Densities from MD Simulations. The initial structures for MC simulations have been prepared as follows. For the PEX038, PEX198, and PEX398 systems the initial melt density has been set to 0.5 g/cm3, and a MC protocol for generating initial melt structures built in the Connectivity Altering MC program25 has been executed. The resulting low-density cells have been equilibrated via molecular dynamics runs in the NPT ensemble at 450 K and 1 atm. Such runs have been performed using the LAMMPS2009 (large-scale atomistic/molecular massively parallel simulator) program41 with a velocity-Verlet integrator, Nosé−Hoover thermostat, and Nosé−Hoover barostat. The cutoff for nonbonded interactions in MD simulations has been set to 14 Å, and an analytical tail correction has been applied.39,40 The TraPPE force field has been augmented with a stiff spring constant, K = 450 kcal/(mol Å2), and an equilibrium bond length set at 1.54 Å in order to simulate variable C−C bond lengths. The coupling constants for thermostat and barostat were 0.01 and 0.001 fs, respectively. The runs have been executed on eight cores. The length of the production MD run has been set to 10 ns. Three independent runs for each simulated system have been executed. The resulting averaged densities are denoted as “MD simulated values” in Table 1. As the density variation within the SCB mole fraction range studied is rather modest (1−2%, cf. Table 1 and Figure 1), for the systems with intermediate SCB mole fractions (PEX118, PEX120, PEX298) the initial configurations have been prepared by executing the MC protocol for chain placement at density values obtained by interpolation of the simulated results for PEX038, PEX198, and PEX398 systems, which are denoted as “interpolated values” in Table 1. For linear PE, the calculated density at 450 K and p = 1 atm is 0.775 g/cm3, close to the experimental value of 0.766 g/cm3 for C1000 PE at 450 K.63 The value calculated (0.775 g/cm3) is in excellent agreement with other simulation results based on the TraPPE force field: MD in the NPT ensemble (0.773 g/cm3)26 and MC in the NnPTμ* ensemble at 450 K and p = 1 atm (0.780 g/cm3).19 For SCB melts the densities simulated by MD increase with comonomer mole fraction. The available

Figure 1. Calculated densities at 450 K and p = 1 atm for the copolymers studied (solid symbols) and experimental densities for PE at 450 K63 and PEB copolymers at 413 K11 (open symbols) as a function of comonomer mole fraction.

experimental data for PEB at 413 K11 are in agreement with the predicted trend (Figure 1). Monte Carlo Simulations. The initial configurations described above were subjected to connectivity altering Monte Carlo equilibration. A connectivity altering MC program for simulating SCB random copolymers25 has been generalized by one of the authors (J.R.) to handle branches of arbitrary length. The details of algorithms were presented earlier.23,25 The connectivity altering MC runs have been performed with a mix of reptation (5%), monomer flip (10%), SCB concerted rotation (10%), SCB end-bridge (50%), branch point (10%), configurational bias (CB) SCB regrowth (7%), and CB chain end (END) regrowth (8%) moves in the semigrand-canonical ensemble at constant volume (NnVTμ* ensemble). In order to reduce computational time, the constant volume simulations used a shorter cutoff of 10.1 Å with the tail correction as described in ref 74. Comparisons of preliminary MC simulations based on 14 and 10.1 Å cutoffs have suggested negligible effect on chain dimensions and small (ca. 2−3%) difference in cohesive energy density. Fixed C−C bonds of 1.54 Å have been used in the MC simulations. The molar mass polydispersity index has been set at 1.053. Statistical Analysis of Primitive Paths. The statistical analysis of primitive paths in the copolymer melts has been performed using the programs CReTA15 and Z1.29 Both programs construct primitive paths by fixing the ends of chains in space and minimizing their contour lengths under the constraint of uncrossability. The CReTA program employs a MC algorithm of contour straightening moves.15 In the course of the procedure, the chain diameter is reduced to a preset value of 0.5 Å. The program fully handles periodic boundary conditions, also with respect to entanglements, which is important in the case of small cells containing very long chains. The program calculates the lengths of all primitive paths and the spacings between entanglements (measured as numbers of atoms and as lengths). The Z1 program uses a deterministic, geometric contour reduction algorithm.18,42 It returns primitive paths of finite but very small thickness. The program calculates the length of all primitive paths in the systems and the number of entanglements per chain, Ztopo, as well as the distances between them. Also, this program handles periodic boundary conditions.29,42 The minimum line thickness has been set to 0.0001 Å. The above programs have been applied to perform primitive path analysis (PPA) in four distinct types of systems: linear PE 8455


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Table 2. Percentage Acceptance Ratios of MC Moves, Average Run Lengths, τrun, Correlation Times (τcor, Eq 3), and the Average Number of Successful Scissions per Backbone Bond, X, for Systems Studied system

REP

FLIP

SCB-CONROT

SCB-EB

PE PEB038 PEB198 PEB398 PEH038 PEH120 PEH198 PEH298 PEH398 PEO038 PEO118 PEO198 PEO298 PED038 PED120 PED198

20.5 19.9 17.5 11.7 19.7 19.0 16.0 14.0 9.5 19.6 17.8 16.1 15.5 19.6 17.8 17.2

77.0 73.2 58.1 44.4 70.7 58.7 49.9 41.2 34.4 67.7 53.5 43.5 34.5 66 49 38.7

8.3 8.1 5.3 3.3 8.0 6.4 4.8 3.1 2.1 7.6 6.2 4.7 2.7 7.8 6.3 4.5

0.17 0.096 0.045 0.011 0.095 0.054 0.049 0.013 0.009 0.093 0.060 0.053 0.011 0.095 0.055 0.036

BP 0.70 0.77 0.60 0.20 0.13 0.13 0.13 0.18 0.05 0.05 0.05 0.04 0.03 0.02 0.02

CB-SCB

CB-END

τrun (109 MC steps)

τcor (106 MC steps)

X

38.7 38.3 38.1 13.4 13.5 13.7 13.7 13.4 4.9 4.9 5.0 5.0 1.8 1.8 1.8

26.5 26.0 23.5 19.1 26.4 24.8 22.5 21.9 18.2 26.4 24.5 24.0 23.6 25.8 24.2 23.4

3.5 2.5 5 5.5 5 4.5 7 7.5 7.5 4.5 5.5 6 8 3.5 4.2 6

2 2.5 6.3 61 2.5 6.2 13 111 130 3 6 12 150 3.5 13 18

149 60 56 15 119 61 86 24 17 105 103 99 28 83 58 90

melt, model concentrated and athermal PE solutions, and SCB melts. In the case of linear PE melt, the configurations were reduced without any special preprocessing. For model concentrated7 PE solutions (⟨R2⟩ = const for ϕ > ϕ** where ϕ** is the upper limit of semidilute solutions taken here as ϕ** = 0.15), from each MC configuration of linear PE melt, 20 configurations of ϕ = 0.95, and 190 configurations of ϕ = 0.9 have been prepared by systematically removing all possible single chains (pairs of chains). For each remaining concentration (ϕ < 0.9), 200 solution configurations from one melt configuration have been prepared by removing randomly a prescribed number of chains from the linear melt. These configurations were subjected to PPA without prior equilibration. PE athermal7,30 solutions (ϕ** = 1) are prepared in an analogous way, but the chain deletion is based on a reweighting such that the scaling, ⟨R2⟩ ∼ ϕ−1/4 is approximately satisfied (⟨R2⟩ ∼ ϕ−0.23±0.03) under the ⟨Nbb⟩ = const condition. The short chain branches in SCB melts have been assumed to act as phantom chains during entanglement formation. For this reason, they were removed before performing the PPA.

Figure 2. Decay of the end-to-end unit vector autocorrelation function with simulation time (in millions of MC steps) as a function of comonomer mole fraction.

number of MC steps for most of the systems studied here. In line with the acceptance ratio trends, the decay of the autocorrelation functions slows down with increasing degree of branching and branch size. Table 2 presents the correlation times, τcor, calculated according to43

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RESULTS AND DISCUSSION Monte Carlo Equilibration. Table 2 presents the acceptance ratios of the MC moves for the systems studied. The acceptance ratios tend to decrease with branch content. The least sensitive to the changes in branch content are CB-SCB, CB-END, and branch point moves. The lowest values of acceptance ratios are observed for the end-bridging and branch point moves. The acceptance ratio of the end-bridging move is sensitive to both branch content and branch length. For example, for linear PE it is 0.17%, close to the acceptance ratio reported by others;19,25 however, for systems with many branches it decreases to 0.01%. Table 2 lists also the number of scissions an average backbone bond undergoes during the simulation run. Typically, this number is higher than 50, and only for the systems with highest amount of branches it decreases to about 10−20. The trends and the acceptance ratio values are similar to those reported earlier.25 The acceptance ratios for the smaller systems containing 12 chains did not differ significantly from the acceptance ratios for 20-chain systems. Figure 2 presents plots of the end-to-end unit vector autocorrelation functions (EVACF) ⟨u(0)·u(t)⟩ against the

τcor =

∫0

tb

⟨u(0) ·u(t )⟩ dt

(3)

where tb, the length of data block for analysis, was set to 1500 million MC steps for SCB systems. The correlation times increase in nonlinear fashion with branch content and size: they are below 10 million MC steps for linear PE and systems containing small amounts of branches, for example, PEX038; about 10−20 million steps for systems with moderate branch content; and of the order of 102 million MC steps for systems with the largest amount of branches. In order to partially compensate for the slowdown in the EVACF decay for systems containing many branches, multiple independent systems, system-dependent run lengths of the order of 2.5−8 billion MC steps, and reduced simulation cell sizes were employed (Table 2). The decay of the EVACF is a necessary but not sufficient condition for melt equilibration. In order to characterize equilibration of structural features from Kuhn length up to the end-to-end distance, plots of the subchain mean-squared endto-end distance divided by the number of carbon atoms in the chain backbone, ⟨Rn2⟩/n, as a function of the subchain 8456


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backbone length n have been studied. The semilogarithmic plots75 of ⟨Rn2⟩/n vs n for PEB systems (Figure 3) increase

characteristic of ideal chain statistics is satisfied to a high degree of accuracy up to n = 1000. The extrapolation of these plots to n → ∞ (1/n → 0) permits calculation of the characteristic ratio in the limit of infinite chain length, C∞.19,48 The calculated C∞ values for PEB systems decrease monotonically with branch content (Table 3). A representative PEB198 chain taken from an equilibrated trajectory is presented in Figure 3b. Figure 4

Figure 4. Subchain mean-squared end-to-end distance per backbone carbon atom ⟨Rn2⟩/n as a function of subchain length (number of backbone carbons n) for the PEO systems studied.

presents analogous plots for PEO systems. Also for these systems, the ⟨Rn2⟩/n values form plateaus. Again, the characteristic ratio C∞ values decrease monotonically with branch content. Small differences between PEO198 and PEO298 values may suggest increasing influence of branch− branch steric repulsion. For PEH and PED systems, the ⟨Rn2⟩/n plots exhibit similar features, suggesting very good equilibration of these systems too (figures not shown). Chain Dimensions and Conformational Properties. Linear Polyethylene. Table 3 presents the values of the meansquared radii of gyration, ⟨S2⟩, at 450 K, for the systems studied. For linear PE the calculated value, ⟨S2⟩ = 3340 Å2,

Figure 3. (a) Subchain mean-squared end-to-end distance per backbone carbon atom ⟨Rn2⟩/n as a function of backbone length (number of backbone carbons n) for the PEB systems studied. (b) A representative PEB198 chain containing 1034 backbone atoms taken from an equilibrated trajectory.

monotonically and form plateaus for n > 102. The plots48 of ⟨Rn2⟩/n vs 1/n (inset in Figure 3) are close to linear near the origin. Both results suggest that the relationship ⟨Rn2⟩ ∝ n1

Table 3. Calculated Mean-Squared Radii of Gyration, ⟨S2⟩, Their Standard Deviations, σ(⟨S2⟩), Ratios ⟨R2⟩/⟨S2⟩, and ⟨R2⟩/M, Characteristic Ratios, C∞, and Tube Diameters, app, at 450 K system

⟨S2⟩ [Å2]

σ(⟨S2⟩) [Å2]

⟨R2⟩/⟨S2⟩

⟨R2⟩/M [Å2 g−1 mol]

C∞

appb [Å]

appc [Å]

PE PEB038 PEB198 PEB398 PEH038 PEH120 PEH198 PEH298 PEH398 PEO038 PEO118 PEO198 PEO298 PED038 PED120 PED198 exponenta value

3340 3245 2994 2785 3219 3010 2919 2833 2784 3235 3012 2941 2847 3225 2968 2790 α 0.27 ± 0.03

8 11 17 41 14 23 20 56 54 18 28 30 58 15 28 32

6.06 6.04 6.10 6.13 6.07 6.03 6.05 6.09 6.04 6.05 6.06 6.02 6.12 6.07 6.01 6.03

1.44 1.35 1.09 0.87 1.30 1.05 0.90 0.77 0.67 1.26 0.96 0.79 0.66 1.21 0.86 0.67 1+α 1.27 ± 0.03

8.50 8.30 7.67 7.21 8.23 7.67 7.45 7.29 7.03 8.26 7.75 7.50 7.35 8.26 7.53 7.09 α 0.27 ± 0.04

41.0 41.6 43.5 48.5 42.1 43.4 46.1 50.7 52.4 42.3 46.0 49.6 56.4 43.2 46.9 52.0 α−β −0.46 ± 0.03

40.7 41.1 42.5 46.9 41.1 42.4 44.5 49.8 51.2 41.8 44.6 48 56 42.4 46.2 49.8 α−β −0.44 ± 0.03

Empirical scaling exponents determined by fitting to ⟨S2⟩, C∞(α); ⟨R2⟩/M(1 + α); app(α−β). bCalculated using Z1 code. cCalculated using CReTA code. a

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modified TraPPE force field27 provide an excellent representation for the combined experimental data set (Figure 5). The calculated ⟨R2⟩/M values (Figure 5) decrease with branch content and fall approximately on a common curve, which scales as ⟨R2⟩/M ∼ ϕ1.27±0.03 and is in very good agreement with the scaling ⟨R2⟩/M ∼ ϕ1.30 determined on the basis of a large set of experimental data by Fetters et al.9,11 Other quantities characterizing chain dimensions, namely the mean-squared radius of gyration, ⟨S2⟩ (calculated taking into account all atoms in the chain), and the characteristic ratio, C∞, yield the same value of the scaling exponents, 0.27 (Table 3). The unscaled chain dimensions (here taken as the meansquared radius of gyration, ⟨S2⟩) tend to fall on a common curve, too (Figure 6). However, for ϕ > 0.88 the chain

agrees well with the experimental value (3238 Å2 at 413 K).45 It also agrees well with simulation results obtained using the same force field (3299 Å2 at 450 K).26 The original TraPPE force field appears to overestimate somewhat the population of trans states along the backbone; our result is about 4% higher than the ⟨S2⟩ value pertaining to the modified TraPPE force field73 (3187 Å2 at 450 K for C1000).26 The characteristic ratio computed from our simulations for linear PE, C∞ = 8.50, is in good agreement with the values calculated using the original TraPPE force field, 8.34 at 450 K,26 and the modified TraPPE force field, 8.06 at 450 K26 and 8.26 at 450 K.19 Our result is in excellent agreement with the value obtained using PRISM calculations (8.47), which use the TraPPE force field for intramolecular structure factor calculations.47 However, both the characteristic ratio, C∞ (8.50), and ⟨R2⟩/M (1.44 Å2 mol/g) are by about 15% higher than the experimental values11 7.4 and 1.25 Å2 mol/g at 413 K, respectively. The overestimation of backbone trans states due to our use of the original TraPPE force field plays some role; nevertheless, the combined PEB experimental data set9,11,36 in Figure 5 appears to extrapolate to a somewhat higher ⟨R2⟩/M value. The ratio ⟨R2⟩/⟨S2⟩ calculated here for C1000 linear PE is 6.06 (Table 3), very close to the ideal chain value.

Figure 6. Average of the mean-squared radius of gyration, ⟨S2⟩, for all systems considered (black square: linear PE; purple squares: PEB; green triangles: PEH; blue circles: PEO; red circles: PED) as a function of the backbone weight fraction, ϕ. Solid line represents ⟨S2⟩ = 3264ϕ0.27. Open blue diamonds represent ⟨S2⟩ values calculated from the hindered rotation model (eq 4) with the adjusted C−C−C angle cos θ = 119°.

dimensions appear to be independent of the branch length, which suggests absence of side chain−side chain interactions in SCB copolymers with small amounts of branches. The branches do not cause ratios ⟨R2⟩/⟨S2⟩ for copolymers to deviate significantly from the ideal chain statistics value of 6 with the exception of PEB398 and PEO298 systems, for which these ratios are 6.13 and 6.12, respectively (Table 3). Positive departures from the ideal chain value of 6 are typical for short chains.48 In general, the data in Figure 6 indicate that the meansquared radii of gyration decrease with branch content. This suggests that branches affect the population of torsional angles in the backbone. A chain model with symmetrically hindered rotation48 could be invoked to link chain dimensions with backbone conformational properties:

Figure 5. Mean-squared end-to-end distances divided by polymer number-average molar mass, ⟨R2⟩/M, as functions of the backbone weight fraction ϕ. Solid symbols: values calculated in this work; open squares: experimental9,11,36 values for PE and PEB systems; open black circle: MC result for PE;19 open purple diamonds: PRISM47 results for PE and PEB random copolymers; green asterisks: PRISM47 results for PEH random copolymers, open green diamonds: MC results27 for PEH systems. Solid line scales as ⟨R2⟩/M ∼ ϕ1.30.

⎡ 1 − cos θ ⎤⎡ 1 − ⟨cos ψ ⟩ ⎤ ⟨Nbb⟩ − 1 2 l ⟨S2⟩ = ⎢ ⎢ ⎥ ⎣ 1 + cos θ ⎥⎦⎣ 1 + ⟨cos ψ ⟩ ⎦ 6

(4)

where ψ is the backbone torsional angle, θ is the bond angle, ⟨Nbb⟩ − 1 is the number of backbone bonds, and l is the backbone bond length. Strictly speaking, eq 4 is not applicable to our chains, for it assumes independent contributions to the conformational energy from individual torsion angles along the backbone, ignoring any couplings between successive torsion angles. Nevertheless, ⟨cos ψ⟩ can serve to describe conformational changes brought about by short-chain branching through a single number. Figure 7 presents the dependence of the

SCB Polyethylenes. Figure 5 presents the plot of the calculated ⟨R2⟩/M ratios for all copolymers studied against backbone weight fraction, ϕ. For ϕ < 0.8, very good agreement between the calculated and experimental data11 is observed. For 0.8 < ϕ < 1 there is some scatter in experimental data;9,11,36 nevertheless, our results are in agreement with data of Reichart et al.36 The ⟨R2⟩/M values calculated by PRISM47 are in excellent agreement with our results for 0.5 < ϕ < 1 (Figure 5). The ⟨R2⟩/M values for PEH systems calculated using the 8458


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function of the backbone weight fraction of copolymers, ϕ. For linear PE, the ⟨Lpp⟩ value resulting from the Z1 program (491 Å) is close to the value of 467.3 Å obtained for C1000 at 450 K.19 The difference (ca. 4%) observed between these two values is probably caused by differences in the force field, as discussed earlier. The CReTA analysis yields ⟨Lpp⟩ = 498 Å for PE, in very good agreement with the value calculated by the Z1 code (491 Å). The extrapolation of CReTA results to zero thickness has been recommended in order to achieve full consistency with primitive path definition.15 Such extrapolations have not been carried out in this study, however. Within the range of the backbone weight fractions 0.4 < ϕ < 1, the primitive path lengths increase monotonically with ϕ, which indicates that copolymer melts containing more SCB comonomer are less entangled. The ⟨Lpp⟩ values derived from Z1 analysis fall approximately on a single curve when plotted against ϕ, which scales as ⟨Lpp⟩ ∼ ϕ0.73±0.02. The primitive path length values obtained from CReTA analysis exhibit very similar properties as Z1-based values but are systematically higher by few percent, similarly as in the case of PE. The regression line based on CReTA results scales as ⟨Lpp⟩ ∼ ϕ0.69±0.02. Thus, both Z1 and CReTA analyses lead to practically the same scaling results. Also in Figure 8, the average values of primitive paths returned by the Z1 code for model concentrated (⟨R2⟩ = const) and athermal (⟨R2⟩ ∼ ϕ−0.23±0.03) PE solutions are presented. For concentrated solutions primitive path length scales with backbone fraction as ⟨Lpp⟩ ∼ ϕ0.41±0.01 within the 0.15 < ϕ < 1 range. The latter relationship depends somewhat on concentration; it is weaker for high backbone weight fractions, 0.5 < ϕ < 1, ⟨Lpp⟩ ∼ ϕ0.35±0.01, and stronger for more dilute solutions, ⟨Lpp⟩ ∼ ϕ0.45±0.01 (0.15 < ϕ < 0.5). For the athermal PE solutions the primitive path length scales within the 0.15 < ϕ < 1 range as ⟨Lpp⟩ ∼ ϕ0.37±0.01, very similarly to the concentrated solutions. On the other hand, the scaling of primitive path length for SCB melts (⟨Lpp⟩ ∼ ϕ0.73) is stronger, which suggests differences in primitive path networks of SCB melts and PE solutions. Tube Diameters. The tube diameter is a measure of distance between entanglements; it is of fundamental significance for understanding the viscoelastic properties of polymer melts. According to the theory of Doi and Edwards,6 the tube diameter is equivalent to the Kuhn length of the primitive path.

Figure 7. Absolute value of the average cosine of torsional angles |⟨cos ψ⟩| as a function of backbone weight fraction, ϕ. The trans conformational state is assigned a value of ψ = 180°.

absolute value of the cosine of the torsion angle, |⟨cos ψ⟩|, as extracted from the actual distributions of backbone torsion angles, on the backbone weight fraction ϕ. All data, including |⟨cos ψ⟩| values for ϕ > 0.9, fall on a single curve, which scales as |⟨cos ψ⟩| ∼ ϕ0.22±0.01 and results in ⟨S2⟩ ∼ ϕ0.25 when substituted into eq 4. The branches cause a decrease in the population of trans states along the backbone and therefore a decrease in |⟨cos ψ⟩⟩|. Figure 6 presents also mean radii of gyration, ⟨S2⟩, calculated using eq 4 with ⟨cos ψ⟩ taken from the actual torsion angle distributions (Figure 7) and with the C−C−C bond angle θ = 119° adjusted to minimize the standard deviation between the MC-based and eq 4-based ⟨S2⟩ values. Estimates from eq 4 are compared against the correlation ⟨S2⟩/Å2 = 3264ϕ0.27 extracted directly from fitting radius of gyration results obtained from the simulation. Both plots scale in a very similar way and suggest that ⟨S2⟩ changes are consistent with ⟨cos ψ⟩ variation. Entanglements. Primitive Path Length. The average length of the primitive path, ⟨Lpp⟩, is a key microscopic quantity resulting from PPA. It is independent of any specific entanglement model; unfortunately, it is not directly accessible experimentally. Figure 8 presents the calculated ⟨Lpp⟩ as a

⟨a pp⟩ =

⟨R2⟩ = ⟨Lpp⟩

Me⟨R2⟩ ∝ ϕα − β M

(5)

In eq 5, α and β are the scaling exponents with ϕ for ⟨R2⟩ and ⟨Lpp⟩, respectively, and Me is the molar mass between entanglements. The tube diameter is accessible experimentally via neutron spin-echo (NSE)31,49−51 and rheological32,46 measurements. For linear PE at 450 K, we calculate ⟨app⟩ = 41 Å (Table 3), which is very close to the value of 45 Å derived from NSE measurements on PE melts at 509 K.51 The tube diameter increases within the 450−500 K by ca. 3 Å as determined computationally.19 By adding this correction term, we obtain ⟨app⟩(509 K) = 44 Å, in excellent agreement with experiment.51 The tube diameter can be also estimated from molecular dynamics simulations.16,22,27,65 For PE melts at 450 K, the analysis of chain diffusion leads to the value of 32−33 Å,22,27,65 somewhat lower than calculated here (41 Å). The calculated tube diameter for slightly branched PEB038 (41.6 Å at 450 K) is in excellent agreement with the experimental data for PEB-2 melt (PEB040 in our notation) at 446 K: 39.9 Å at

Figure 8. Average primitive path length, ⟨Lpp⟩, calculated using the Z1 program (full symbols) and the CReTA program (open symbols) as a function of backbone weight fraction, ϕ. The black solid line (Z1 results for SCB melts) scales as ⟨Lpp⟩ ∼ ϕ0.73. The green line (CReTA results for SCB melts) scales as ⟨Lpp⟩ ∼ ϕ0.69. The red solid line (Z1 results for model PE concentrated solutions) scales as ⟨Lpp⟩ ∼ ϕ0.41. 8459


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446 K.31 Thus, for linear PE and slightly branched PEB038 melts, Monte Carlo simulations combined with geometric PPA yield tube diameter values, which are in excellent agreement with available experimental data. For PEH038 and PEH120 melts, the calculated tube diameters, 42.1 and 43.4 Å, respectively, are in agreement with the results of molecular dynamics simulations 36.4 and 44.8 Å, respectively.27 Molecular dynamics simulations16,22,27,65 entail an entirely independent method of calculating tube diameters; thus, the agreement observed is highly encouraging. The probability distribution function of tube diameter, p(app), for PE resulting from Z1 analysis exhibits a maximum at ca. app = 15 Å, far from ⟨app⟩ = 41 Å, and is in excellent agreement with the analogous function reported by Foteinopoulou et al.19 Upon increasing the SCB comonomer content, the maximum in the p(app) for SCB copolymers decreases and the tail intensity increases, which indicates a gradual increase of the average tube diameter. Figure 9 presents probability

particularly the tail at high app values, which is most meaningful physically. This result suggests that the quantities app/⟨app⟩ and NES/⟨NES⟩ have very similar statistical properties. A detailed analysis of distributions resulting from PPA will be presented elsewhere. The assumptions regarding the nature of the entangled state affect tube scaling considerably. Table 4 compares several scaling predictions, which take into account the conformation− concentration effect, ⟨R2⟩ ∼ ϕα, for SCB melts. Consequently, the Kuhn length scales as lK = ⟨R2⟩/L ∼ ϕα and the spatial Kuhn segment density ρK= nK/V = ⟨R2⟩/(lK2V) = ⟨R2⟩ϕ/ (lK2Vbb) ∼ ϕ1−α. The binary contact model (I)33−35 attributes entanglements to a fixed number of binary contacts per chain and predicts ⟨app⟩ ∼ ϕ−0.5, independently of the concentration−conformation effect. If binary contacts are assumed to be fixed in a certain volume surrounding the chain, then, for ⟨R2⟩ = const, ⟨app⟩ ∼ ϕ−2/3 (model II).52 However, for SCB melts, ⟨R2⟩ ∼ ϕ0.27±0.03, and model (II) predicts a somewhat stronger scaling ⟨app⟩ ∼ ϕ−0.76. The packing model (III) for flexible melts53−55 assumes that the entangled state arises when the weight of polymer corresponding to the volume pervaded by a chain is twice the chain’s weight. This model predicts much stronger dependence of the SCB tube diameter on the backbone weight fraction, ⟨app⟩ ∼ ϕ−1.27. We compare the above predictions with the scaling of the tube diameter derived from PPA (Figure 10). For our model concentrated PE solutions the tube diameter computed within the 0.15 < ϕ < 1 range scales as ⟨app⟩ ∼ ϕ−0.41±0.01, which agrees very well with the result of rheological measurements for concentrated PEB-2 solutions in C24H50 at 413 K (⟨app⟩ ∼ ϕ−0.45).32 The regression of two combined rheological data sets at 403 K32,46 yields a very similar result, ⟨app⟩ ∼ ϕ−0.5.32 In these systems, the upper semidilute limit is ϕ** < 0.25;8,32 thus, the comparison with the result for concentrated solutions is appropriate. For model athermal PE solutions, the computed tube diameter scales as ⟨app⟩ ∼ ϕ−0.6±0.03, which is in excellent agreement with the result for athermal solutions of PEB-2 in deuterated nonadecane at 509 K based on neutron spin-echo experiments31 (⟨app⟩ ∼ ϕ−0.6). Thus, the PPA-based tube diameter scaling relationships support the binary contact model. The variation of the PPA-based tube diameter scaling exponent for PE solutions when upper semidilute limit varies is mostly due to conformation−concentration effect (cf. Figures 8 and 10). For SCB melts, the tube diameter values (Table 3) increase with branch amount, which indicates that the melts of more branched copolymers are less entangled. The tube diameters derived from Z1 and CReTA analyses for SCB melts scale as ⟨app⟩ ∼ ϕ−0.46±0.03 and ⟨app⟩ ∼ ϕ−0.44±0.03, respectively. These results are close to those observed for simulated and experimental tube diameters in solutions and also to the binary

Figure 9. Probability distribution functions of tube diameter, app, for PE, PEB, and PEO systems. The solid black line is calculated according to eq 6.

distribution functions, p(y), with y = app/⟨app⟩, for PE and several SCB copolymers. These functions collapse approximately on a single curve, similarly as the distribution P(NES/ ⟨NES⟩) of strand lengths between entanglements along the chain, NES, resulting from CReTA analyses.15,22,68,76 An analytical equation for this distribution has been proposed15 P(n) =

bc (e−bn − e−cn) c−b

(6)

where n = NES/⟨NES⟩ and b (=1.30) and c (=3.78) are coefficients determined in ref 15. By substituting n = y, eq 6 with parameters based on CReTA PPA results, is found here to represent very well the Z1-derived tube diameter distribution,

Table 4. Scaling of the Tube Diameter, app, and Plateau Modulus G0N, with Respect to Polymer Volume Fraction, ϕ, According to Several Theoriesa ⟨app⟩

model b

(I) packing model (II) Colby−Rubinsteinc (III) binary contactd (IV) std modele + PPA (V) ref 17 + PPA

lK/w lK/w2/3 lK/w1/2 ⟨R2⟩/⟨Lpp⟩ ⟨R2⟩/⟨Lpp⟩

exponent in app ∝ ϕy −(1 + α) −(2 + α)/3 −1/2 α−β α−β

y

G0N/kBT

exponent in G0N ∝ ϕz

z

z(α=0)

−1.27 −0.76 −0.50 −0.46 −0.46

lK−3w3 lK−3w7/3 lK−3w2 lK−3w(lK/⟨app⟩)2 lK−3w7/5(lK/⟨app⟩)8/5

3(1 + α) (7 + 5α)/3 2+α 1 − α + 2β 1.4 − 0.2α + 1.6β

3.81 2.78 2.27 2.19 2.51

3.00 2.33 2.00 2.46 2.57

Kuhn length is denoted by lK, Kuhn segment density by ρK and w = ρKlK3. Scaling exponents for ⟨R2⟩ and ⟨app⟩ are taken from Table 3. bReferences 53−55. cReference 52. dReferences 33−35. eReference 6. a

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above). The observed differences from linear melts, where the packing model is followed closely,13−15,17 could be due to the fact that our copolymers contain explicit short chain branches. Plateau Moduli. Tube diameters are most commonly extracted from plateau moduli, macroscopic quantities characterizing elastic properties of melts. The standard model6 yields ⎡ ⎤2 GN0 lK 3 4 3 ⎢ lK ⎥ = (ρK lK ) kBT 5 ⎢⎣ ⟨a pp⟩ ⎥⎦

(7)

Plateau moduli for models I−III are obtained via eq 7: ∼ ϕ3, G0N ∼ ϕ7/3, and G0N ∼ ϕ2, respectively, at the theta state (Table 4, column 8). When the conformation−concentration effect is operable, lK ∼ ϕα, 0 < α < 1, the scaling exponents for models I−III increase: G0N ∼ ϕ3(1+α), G0N ∼ ϕ(7+5α)/3, and G0N ∼ ϕ2+α, respectively. This effect is particularly large for the packing model, which predicts G0N ∼ ϕ3.81 for SCB melts. On the other hand, for a general tube diameter scaling, app ∼ ϕα−β, eq 7 yields G0N ∼ ϕ1−α+2β. The latter relationship gives G0N ∼ ϕ2.19 for SCB melts. Recently, model V, which is capable of a unified description of tightly entangled solutions and loosely entangled melts, has been proposed.17 This model yields G0N

Figure 10. Simulated tube diameters (Z1 code), ⟨app⟩, as a function of backbone weight fraction, ϕ, for concentrated (⟨R2⟩ = const) PE solutions (solid blue diamonds), athermal (⟨R2⟩ ∼ ϕ−0.23±0.03) PE solutions (solid red diamonds), and SCB melts (solid circles, triangles and squares) at 450 K. Experimental51 tube diameters for PE melt at 509 K (open black square), for PEB-2 (PEB040) melt31 at 446 K (open purple square), and for PEB-2 (PEB040)/deuterated nonadecane solutions31 at 509 K (red asterisks and blue crosses). The red line scales as ⟨app⟩ ∼ ϕ−0.6. The purple line is based on hydrogenated polybutadiene solutions in n-alkane solvents32 at 413 K and scales as ⟨app⟩ ∼ ϕ−0.45. The tube diameters of our SCB melts scale practically the same way (⟨app⟩ ∼ ϕ−0.46). The blue line scales as ⟨app⟩ ∼ ϕ−0.41. All three lines are bound at ⟨app⟩(ϕ=1) = 40.4 Å in order to facilitate comparison.

⎡ ⎤8/5 GN0 lK 3 2/5 3 7/5 ⎢ lK ⎥ = cGcξ (ρK lK ) kBT ⎢⎣ ⟨a pp⟩ ⎥⎦

(8a)

where cGcξ2/5 is a single adjustable parameter (0.2). Equations 7 and 8a are equivalent when tube diameter conforms to the packing model ⟨app⟩ ∼ lK/w but differ otherwise as eq 8a predicts a scaling G0N ∼ ϕ(7−α+8β)/5. The plateau moduli for SCB melts, calculated according to eq 8a with tube diameters obtained from our PPA, scale as G0N ∼ ϕ2.51. Figure 12 presents

contact model scaling. Although the concentration−conformation effects in our model PE solutions (⟨R2⟩ = const; ⟨R2⟩ ∼ ϕ−0.23±0.03) are qualitatively different from those in SCB melts (⟨R2⟩ ∼ ϕ0.27±0.03), the differences in tube diameter scaling exponents are small (Figures 10 and 11).

Figure 12. Comparison of the plateau moduli, GN0, for SCB copolymers calculated from simulation lK, ρK, and ⟨app⟩ values using (a) model IV,6 G0N ∼ ϕ2.19 (open squares); (b) model V,17 G0N ∼ ϕ2.51 (solid points) with (c) experimental data56−62 (asterisks). The black line represents best fit to experimental data G0N/[MPa] = 1.91ϕ2.67, and the purple line represents G0N/[MPa] = 2ϕ2.62 (eq 8b).

Figure 11. Tube diameter scaling exponent, α − β, as a function of chain dimension scaling exponent, α.

The calculated tube diameters show clear deviations from packing model predictions for SCB melts (⟨app⟩ ∼ ϕ−1.27 vs PPA-derived ⟨app⟩ ∼ ϕ−0.46±0.03), for concentrated (⟨app⟩ ∼ ϕ−1 vs PPA-derived ⟨app⟩ ∼ ϕ−0.41±0.01), and for athermal PE solutions (⟨app⟩ ∼ ϕ−0.77 vs PPA-derived ⟨app⟩ ∼ ϕ−0.60±0.03) (Figure 11). Deviations from the packing model have been observed in PPA-based tube diameters for loosely entangled solutions with the largest deviations seen for the lowest polymer concentrations, ρKlK3 < 1.17 For model PE solutions we observe analogous deviations within the range 1 < ρKlK3 < 9 (consistently with experimental evidence31,32,46 discussed

calculated and experimental values of plateau moduli for systems compared by Vega et al.:57 PE,61,62 PEB,56 PEH,57−59 and PEO.60 The regression of these data yields G0N/[MPa] = 1.91ϕ2.67±0.16, which is close to the PPA-based scaling predicted by model V (Table 4). For melts satisfying ⟨R2⟩ ∼ ϕ0.3 and conforming strictly to the binary contact model, ⟨app⟩ ∼ ϕ−0.5, model V predicts G0N ∼ ϕ2.62. Combining this with the 8461


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recommended value of plateau modulus for PE,61 the following relationship for plateau moduli of these SCB melts could be proposed: G N0 /[MPa] = 2ϕ2.62

K15 and 2.8 at 300 K and 2.5 at 600 K19). The higher than unity Ztopo/Zcoil ratio indicates that the primitive length is not a random walk at the scale of the internodal strand length.15 Radial Distribution Functions. Figure 15 presents the evolution of the total pair radial distribution functions for PEB

(8b)

Model V with tube diameters conforming exactly to the binary contact model (eq 8b) represents well the experimental plateau moduli for PEB, PEH, and PEO melts considered here (Figure 12). In general, the dependence of chain dimensions on concentration (“concentration−conformation effect”) affects, sometimes strongly, the scaling relationships for tube diameters and plateau moduli. Kinks. The number of kinks per chain derived from PPA, Ztopo, differs from the number of kinks per chain calculated on the basis of rheological models, Zcoil.15 Figure 13 presents a plot Figure 15. Total radial pair distribution function for PEB copolymers.

copolymers. An increase in 1-butene content is accompanied by intensity decrease of the first intermolecular peak at ca. 5.1 Å. This suggests that the number of intermolecular contacts decreases, presumably due to decrease of chain dimensions. Simultaneously, the intensity of an intramolecular peak in the vicinity of 3.2 Å, which is characteristic of gauche conformations, increases. The total pair distribution functions, g(r), for other copolymers exhibit similar features. The denser intramolecular packing is consistent with the trend in Figure 1 showing density increase with increase of SCB comonomer content. It could arise due to coiling of backbones (increase of gauche fraction along backbones, cf. Figure 7) and a crowding effect caused by branch points. Figure 16 presents plots of the intermolecular radial distribution functions for the systems studied. The intensity

Figure 13. Plot of the average number of kinks per chain calculated by Z1 program for all systems studied as a function of backbone weight fraction.

of the average number of kinks per chain, Ztopo, as a function of backbone weight fraction, ϕ. This quantity scales as Ztopo ∼ ϕ1.28±0.05. The latter scaling is in very good agreement with the prediction from the standard model,6 which for the number of entanglements along the chain yields Zcoil =

⟨Lpp⟩ ⟨app⟩

=

⟨Lpp⟩2 ⟨R2⟩

(9)

and therefore, Zcoil ∼ ϕ ∼ ϕ . The average value of the ratio Ztopo/Zcoil (Figure 14), with Ztopo being equal to the mean number of strands per primitive path defined by the nodes of the topological analysis and Zcoil being equal to the number of Kuhn steps in a primitive path (eq 9), is about 2.6. This is in excellent agreement with the values reported for PE (2.6 at 450 2β−α

1.19

Figure 16. Intermolecular radial distribution functions, ginter(r), as functions of radial distance, comonomer type, and comonomer content for representative SCB copolymer melts.

of all intermolecular peaks decreases with branch weight fraction, consistently with the observations made in discussing total radial distribution functions (Figure 15). Even for the maximum distance studied (r = 40 Å), ginter(r) < 1, indicating large intermolecular correlation holes. In order to quantify this effect, we determine the effective correlation lengths, ξinter, according to g inter (r ) − 1 = −

Figure 14. Ratios of geometrical to random coil kinks, Ztopo/Zcoil, as a function of the backbone weight fraction. 8462

3vm πrlK 2

⎛ r ⎞ exp⎜ − ⎟ ⎝ ξinter ⎠

(10)


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where vm is an excluded volume parameter. The fitting of correlation lengths to the complete range of data produces ξinter values of the order of ⟨S2⟩1/2. However, in order to reduce scatter, correlation lengths have been determined by restricting the fitting of ginter(r) to the range 17−40 Å. This reduced the values of correlation lengths to about 40−50 Å. The correlation lengths tend to decrease with branch amount and scale as ξinter ∼ ϕ0.13±0.07, in agreement with ⟨S2⟩1/2 ∼ ϕ0.13. This suggests that the correlation lengths of the intermolecular radial distribution functions are controlled by chain dimensions. Solubility Parameters. First, we investigate the accuracy of the cohesive energy and solubility parameter (SP) calculations and consider lower alkanes: n-hexane and n-hexadecane (Table 5). Within the investigated temperature range (298−473 K) the

concentration. The simulated solubility parameters follow the same trend. The extrapolation of SP values in Table 5 to 1/N → 0 at 473 K yields δ = 13.62 MPa0.5 for PE, which is close66 to 14.8 MPa0.5. We note also excellent agreement between the simulated solubility parameter68 for a-PS, 14.5 MPa0.5, and the value reported in ref 66, 15.2 MPa0.5. Figure 17 presents the cohesive energy density for all systems studied here as a function of backbone weight fraction (solid

Table 5. Comparison of Simulated and Experimental Solubility Parameters for Several Lower n-Alkanes, PE, and PS substance n-hexane (C6) n-hexadecane (C16)

n-tetratetracontane (C44) PE(C1000) PS

temp (K) 298 298 423 473 473 450 473 500

δcalcd (MPa0.5) a

14.1 15.4a 13.0a 12.1a 13.1a 13.7a 13.6a 14.5b

δexptl (MPa0.5) 14.9d 16.4e 14.0c

Figure 17. Total nonbonded energy density, NBD (open blue squares), cohesive energy density, CED (solid symbols), and intramolecular nonbonded energy density, IED (open symbols; values shifted by +100 MPa), as functions of backbone weight fraction at 450 K.

14.8c 15.2 (at 473 K)c

a

This work. bReference 68. cReference 66. dReference 69. eReference 64.

symbols). The CED values fall approximately on a common curve, and the overall scaling with backbone weight fraction is CED ∼ ϕ0.3±0.02. On the other hand, the intramolecular nonbonded interaction energies decrease with backbone weight fraction as IED ∼ ϕ−0.6±0.04. The trend in CED is probably due to less favorable intermolecular packing with increasing branching, as discussed earlier (Figure 15). The trend in IED is probably related to the more coiled conformations adopted by highly branched chains, which create the opportunity for attractions between different parts of the same chain. The total nonbonded energy, NBD, slightly increases with backbone weight fraction and scales as NBD ∼ ϕ0.09±0.005. The decrease in CED and in NBD with branch content occurs despite the fact that the copolymer density increases with branch content (Figure 1). The cohesive energy density has been related to the packing length, lp= lK/w = M/(ρ⟨R2⟩) (M = molar mass, ρ = mass density), using a simplified variant of PRISM theory.38 The following equation with linear PE as reference compound represents well experimental solubility parameters for a number of polyolefins:38

solubility parameters, δ, calculated for n-hexane and nhexadecane are in very good agreement with SP values based on calorimetrically determined molar vaporization enthalpies ΔvapH δ = CED1/2 =

Ucoh ≅ V

Δ vapH − RT V

(11)

where CED is the cohesive energy density, Ucoh is the cohesive energy per mole, and V the molar volume. Compared to experimental data, the calculated SP values for these compounds are lower only by ca. 1 MPa0.5 (Table 5). This suggests that the present MC simulations combined with the TraPPE-UA force field represent very well cohesive energy for (C6−C16) n-alkanes for which direct measurements of vaporization enthalpy are available. However, polymers are nonvolatile and SP are often estimated on the basis of internal pressure, Pint, which is determined from pVT measurements: Pint =

⎛ ∂U ⎞ ⎛ ∂S ⎞ αP ⎜ ⎟ = T⎜ ⎟ − p = T −p ⎝ ∂V ⎠T ⎝ ∂V ⎠T κT

⎛ ρ ⎞2 1 + 0.176lp,PE CEDi = CEDPE ⎜⎜ i ⎟⎟ ⎝ ρ0 ⎠ 1 + 0.176lp, i

(12)

with αP and κT being the thermal expansion coefficient and the compressibility, respectively. The internal pressure is then assumed equal to CED: Pint = CED. The ratios Pint/CED generally depend on the material, temperature, and pressure, however, and the latter assumption is not well satisfied for polymer fluids.66,67 Thus, a more sophisticated method for extracting estimates of cohesive energies from pVT measurements has been proposed.66 The experimentally derived solubility parameters decrease in approximately linear fashion with inverse alkane chain length N, which is most probably caused by free volume effects due to the variation in chain end

(13)

The cohesive energy densities predicted using eq 13 (open squares) are in very good agreement with MC-based CED values (Figure 18). This suggests that MC simulations capture correctly the trends observed in experimentally determined CED’s. Extensive SANS studies of SCB copolymer blends made in order to determine factors affecting blend miscibility suggest that about 80% of SCB copolymer blends appear to conform to the regular mixture model.8,10,37 Thus, attempts have been 8463


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Upon increasing of the comonomer content, the chain dimensions decrease because the population of trans states along the backbone decreases. The calculated chain dimensions per molecular mass scale as ⟨S2⟩/M ∼ ϕ 1.27±0.03 for the radius of gyration S and ⟨R2⟩/M ∼ ϕ 1.27±0.03 for the end-to-end distance R, where ϕ is the backbone weight fraction, in very good agreement with the experiment-based result ⟨R2⟩/M ∼ ϕ 1.30.9,11 For SCB melts, the tube diameter values increase with branch amount, which indicates that the melts of more branched copolymers are less entangled. The computed average values of tube diameters for PE and SCB melts agree very well with the available experimental31,51 and computational19 results. The distributions of tube diameters p(app/⟨app⟩) for PE and SCB melts fall approximately on a single curve, which is well described by the exponential form15 for the strand length distribution P(NES/⟨NES⟩). This demonstrates high degree of statistical similarity of the app/⟨app⟩ and NES/⟨NES⟩ quantities. The tube diameters resulting from primitive path analysis are close to tube diameters calculated using an entirely dynamical approach.16,22,27,65 Thus, the effect of thermal fluctuations on tube diameters appears to be moderate. One may think of SCB melts as backbones immersed in a “solvent” consisting of the branches as suggested, for example, by the scaling relationship eq 1. To investigate this analogy, we have generated model solutions of linear PE by randomly deleting chains from well-equilibrated PE melt configurations. In these solutions, ϕ is identified with the fraction of carbon atoms remaining after the deletion. The computed tube diameters for our model concentrated solutions scale as ⟨app⟩ ∼ ϕ−0.41±0.01, in excellent agreement with the scaling obtained in rheology experiments32 for PEB-2/C24H50 solutions at 413 K (app ∼ ϕ−0.45), with upper semidilute limit ϕ** < 0.25. The calculated tube diameters for our model athermal solutions scale as ⟨app⟩ ∼ ϕ−0.60±0.03, in excellent agreement with the scaling based on the NSE experiments31 for the athermal PEB2/C19D40 solutions studied at 509 K (app ∼ ϕ−0.6). Thus, the combination of connectivity altering Monte Carlo and geometric methods of primitive path analysis applied to the model PE solutions yields scaling relationships in very good agreement with experiment. The scaling of tube diameters determined for SCB melts (⟨app⟩ ∼ ϕ−0.46±0.03) is close to the scaling derived for model concentrated PE solutions (⟨app⟩ ∼ ϕ−0.41±0.01). Perhaps branches, which are phantom objects at the length scale of entanglements, but are of nonzero volume, affect the tube diameter in a similar way as a solvent. The tube diameter scaling relationships derived from primitive path analysis both for model concentrated and athermal PE solutions (⟨a pp⟩ ∼ ϕ −0.41±0.01 and ⟨app ⟩ ∼ ϕ −0.6±0.03 , respectively) and for SCB melts (⟨app⟩ ∼ ϕ−0.46±0.03) are all consistent with the binary contact model scaling (⟨app⟩ ∼ ϕ−0.5).33−35 Scaling considerations invoking the binary contact model suggest that plateau moduli for SCB melts could scale as G0N = 2ϕ2.62, which is close to the experiment-based G0N ∼ ϕ2.67±0.16. The reduction of chain dimensions with branching is accompanied by a decrease in the number of intermolecular contacts, which reduces the cohesive energy density. The simulated relative solubility parameters for PEB are in very good agreement with the SANS-based relative solubility parameters (Figure 19), which scale as δ − δref ∼ ϕ0.18. On the other hand, the SANS-based relative solubility parameters for PEH and PEO scale more weakly with the backbone weight

Figure 18. Comparison of the cohesive energy densities calculated by MC and predicted (eq 13).

made to interpret SANS-derived χ parameters in terms of the solubility parameter concept. However, the SANS-derived solubility parameters, δ − δref, are relative quantities. In order to determine the δref parameter, we fit experimental δ − δref parameters for PEB copolymers36,37 to the equation δ − δref = Δδ + δ PE(ϕc − 1)

(14)

where Δδ and c are adjustable parameters and δPE is the solubility parameter for linear PE, taken here as 13.7 MPa0.5. The regression yields Δδ = 1.57 MPa0.5, and c = 0.18; thus the δref = 12.13 MPa0.5. The adjustment of eq 14 to the MC-based PEB solubility parameters yields δref = 12.1 MPa0.5. The MCderived PEB relative solubility parameters agree well with the regression line of eq 14 (Figure 19), and both data sets are

Figure 19. Comparison of the calculated relative solubility parameters, δ − δref, for all systems studied at 450 K (solid symbols) with SANSbased values36,37 at 440 K (open symbols). The solid line has been fitted to the SANS-based PEB data and represents the equation δ − δref = 1.57 + 13.7(ϕ0.18 − 1).

consistent with respect to the δref value of 12.1 MPa0.5. On the other hand, the SANS-based relative SP for PEH and PEO scale more weakly with the backbone weight fraction (δ − δref ∼ ϕ0.1) and signal departures from the universal curve for PEB systems. The MC-based relative SP follow the experimental trend but scale somewhat more strongly than the SANS-based SP values (Figure 19).

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CONCLUSIONS A variety of short chain branched melts have been equilibrated using the connectivity altering Monte Carlo method.25 The results permit to determine the effect of α-olefin comonomer on chain dimensions, entanglement tube diameters and related quantities, and solubility parameters. 8464


Macromolecules

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fraction. Simulated SP parameters capture qualitatively trends in SANS-derived solubility parameters.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS K.M. is indebted to Prof. S. K. Sukumaran and Prof. J. Takimoto for their hospitality and discussion during his visit at Yamagata University, to Prof. S. K. Sukumaran for his suggestion to study solutions, and to Dr. T. Kakigano and Dr. Y. Yamamoto for encouragement and support. We thank Prof. M. Kröger for providing the Z1 code and Dr. C. Tzoumanekas for providing the CReTA code. Mitsui Chemicals, Inc., is thanked for the permission to publish this work. J.R. acknowledges financial support through the Ramon y Cajal program, Contract RYC-2011-09585.

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Monte Carlo Simulation of Short Chain Branched Polyolefins

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