1 Supplementary Information: Mass Dependence of the Activation Enthalpy and Entropy of Unentangled Linear Alkane Chains Cheol Jeong and Jack F. Douglas Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (September 8, 2015) I. Free-Volume Models of the Molecular Friction Coefficient, ζ A number of “free volume” models [S1-S6] have been introduced to rationalize the T dependence of relaxation and diffusion in glass-forming liquids and free-volume models have been adapted to model polymer self-diffusion D and the shear viscosity of unentangled and entangled polymer melts, based on the Rouse and Reptation mean field models, respectively. [S5-S6] Leaving aside evidence described in the main text indicating that the Rouse model does not provide a quantitative description of our simulated unentangled alkane melts, we analyse the assumptions underlying free volume models that have been applied in modelling unentangled alkane melts. In the free volume approach of von Meerwall et al. [S7], the self-diffusion coefficient D of alkanes with molecular mass M was modelled as the product of three factors, D A exp[ Ea* / RT ] M 1 exp[ Bd / f (T , M )] (S1) where the first term is an Arrhenius factor borrowed from Eyring’s transition state theory (TST) [S8], the second M 1 factor is associated with the assumption that the chain segmental dynamics can be described by the Rouse model, and the last factor models the effect of ‘free-volume’. Although the factorization D, and the ‘friction coefficient’ ζ of the Rouse model, into Arrhenius and free-volume factors follows earlier reasoning by 2 Macedo and coworkers [S9], this assumption is not uniformly followed in modelling of polymer melts based on free-volume theory. In Macedo and coworkers, the free-volume contribution to D is modelled following a specific argument made by Bueche [S10] that attributes free volume to the presence of chain ‘free ends’. [S5, S10] (Variants of the free chain end argument have also been developed by Fox and Flory [S5] and Ferry and coworkers. [S6]) Based on the Bueche free-volume model, Macedo and coworkers [S9] estimate the fractional free-volume f (T , M ) in terms of the alkane melt density, (T , M ) , f (T , M ) f (T , ) 2Ve (T ) (T , M ) / M (S2) where f (T , ) is the free-volume fraction in the infinite mass limit and Ve (T ) is the free -volume contribution of the chain ‘free ends’. Formally, f (T , M ) vanishes at low temperatures and for n = 100 this extrapolated ‘critical condition’ arises for a T near T = 120 K. This finding is consistent with the relaxation time divergence temperature T0, indicated in Fig. 8 of the main article. Although Eq. (S1) involves adjustable parameters, von Meerwall and coworkers found that it provides a good fit to unentangled alkane melt D data where Ea* in Eq. (S1) [not to be confused with the activation energy of transition state theory given the incorporation of the free-volume factor in (S1)] is assumed to be M independent and the Rouse and the free chain end models are assumed to be applicable. We next consider the implications of this model for understanding the mass scaling of D in alkanes. Since the Arrhenius term is taken to be independent of M in this fitting scheme, the change in the mass scaling exponent for D derives from the free-volume factor, exp[ Bd / f (T , M )] . If we adopt the expression for f (T , M ) of von Meerwall and coworkers and further assume a power law scaling, D ~ M , then we find that the resulting effective mass scaling exponent depends rather strongly on M; this behaviour is illustrated in Fig. S1 for representative T values. Evidently, a stable power law scaling of D with M does not exist in this free-volume model for unentangled polymers. 3 -1.0 -1.5 303 K 343 K 383 K 423 K -2.0 -2.5 -3.0 1.0 1.5 2.0 2.5 log n FIG. S1. Effective mass scaling exponent from the free-volume model of von Meerwall and coworkers using Eqs. (4), (5a), (5b), and (6) of Ref. [S7]. Note that reproduces the estimates of van Meerwall and coworkers in the limit of unentangled chains [S7], i.e., β = 2.7, 2.4, 2.2, and 1.9 for 303 K, 343 K, 383 K, and 423 K, respectively. The red line and arrow indicate the mass range investigated by van Meerwall et al. [S7]. We note by comparison that the free-volume model of Pearson and coworkers [S11] does not include the Arrhenius factor A exp[ Ea / RT ] of von Meerwall and coworkers [S7] and that these authors focus rather on the effective activation energy of the relaxation time rather than . The variability of the effective activation energy in the model of Pearson and coworkers [S11], a quantity related to in the modelling of van Meerwall and coworkers, derives from M dependence of the characteristic temperature T0 4 and the thermal expansion coefficient of the polymer melt. Although von Meerwall and coworkers [S7] and Pearson and coworkers [S11] both assume a common Rouse factor M 1 in their expression for D, these models are otherwise built on different developments of free-volume theory. At first, we tried to adopt these conventional free-volume analyses, but ultimately we decided against them for the following reasons: (i) Our simulations of unentangled alkane chains indicate that the chains are highly non-Gaussian, but the Rouse theory assumes the chains are Gaussian. There is then a problem of consistency. (ii) Even within the realm of the free-volume model, we cannot rationalize the ad hoc combination [S7] of the Arrhenius and free volume factors on which the analysis of von Meerwall and coworkers is based. (iii) The application of the free volume model by Pearson et al. relies on the application of free-volume theory at temperatures above the onset temperature TA where D and of the alkane exhibit an Arrhenius temperature dependence. The applicability of this free-volume model at such elevated temperatures is unclear. (iv) The fitting of the D and data involves multiple fit parameters whose physical significance is uncertain. (v) A consistent friction coefficient cannot be obtained when multiple transport properties are considered. [S12] In our view, this finding provides further evidence that the use of the Rouse model is unsuitable for describing unentangled polymer melts. (vi) Recent simulations have shown that the chain free end argument for the density fails dramatically for star polymers. [S13] Despite these technical problems, however, the fitting of the alkane D data by von Meerwall et al. [S.7] does provide insight into the transition state theory activation energy, determined in the high temperature regime where this theory should be applicable. To appreciate this point, we note that the free-volume factor, exp[ Bd / f (T , M )] , originally introduced empirically by Doolittle [S2], reduces to a simpler power law in the liquid 5 density in the high temperature liquid regime. [S3] In particular, Batchinski’s original free-volume theory [S1] (on which Doolittle’s free-volume model was latter based) predicts D to have the high temperature scaling, D ~ (vs v0 ) / v0 (S3) where vs is the specific volume and v0 is the limiting volume at which fluidity extrapolates to zero. Eq. (S3), a high temperature variant of free-volume theory, holds remarkably well for a large number of liquids [S1] in a T range in which Arrhenius diffusion is applicable, i.e., T > TA. If we further adopt a simple two-state model of the liquid specific volume vs [S14] in which vs can be approximated by a low temperature constant, plus an Arrhenius term describing T dependence of vs with the excitation of ‘free-volume’ in the liquid upon heating. The difference (vs v0 ) / v0 in Eq. (S3) can then be expected to have an Arrhenius T dependence, as often observed in practice. We thus obtain a formal link between the Arrhenius relation for D described in the main text, and an expression derived from free-volume modelling. Based on this formal correspondence, we can infer that the analysis of von Meerwall and coworkers indicates that the alkane activation energy has two contributions - a constant intramolecular term associated with relaxation of the monomer units and a second term related to chain packing interactions (i.e., density), an intermolecular interaction contribution. This is exactly the interpretation of the TST activation energy of polymers advocated by Tabor [S15] and Bershtein et al. [S16], where the first term is identified with the barrier height of the bond rotational potential and the second intermolecular term is proportional to the cohesive interaction strength. This formal correspondence also explains long-standing observations linking the measurements of density and pressure to viscosity [S17] which ultimately led to a realization that the activation energy ΔHa should be related to the heat of vaporization. [S8] This alternative view of free-volume theory in relation to activated transport should provide an interesting starting point for further research into the molecular significance of the activation energy for D in polymer melts. 6 II. Non-Gaussian Conformations of Alkanes in the Melt The ratio R 2 /6 S 2 is a basic measure of the extent to which polymer chains follow Gaussian chain statistics since this quantity equals 1 for Gaussian random coils and other values quantify the extent of deviation from the flexible chain model. For the alkane melt data described in the main article, this ratio becomes significantly larger than 1 and exhibits a maximum for rod-like values of this ratio where the chains then buckle, n nbuck . Kratky and Porod. [S17] introduced the worm-like chain (WLC) model to describe the model the relative persistent conformations of short alkane chains. Yamakawa and coworkers [S18] derived analytic WLC model expressions for the average mean-squared end-to-end distance R 2 and the radius of gyration S 2 , R 2 2l p L 2l p 2 (1 e L/lp (S4) ) and S2 lp L 3 2l p 2 2 l p3 L 2 lp4 2 L (1 e L/lp ) (S5) where l p is the persistence length and L is the contour length of a linear chain. These expressions recover the asymptotic random coil and rod limit limits, lim L / l p lim L / l p 0 R 2 2l p L N l p2 n 2 l L N l p S2 p n 3 6 random coils (S6) rods. (S7) and lim L / lP 0 lim L / l p 0 R 2 L2 n 2 2 L 2 2 S n 12 7 FIG. S1. Average R 2 as a function of the number of bonds n 1 in alkanes. The dashed line indicates the fitting results using the wormlike chain model. [S19] For alkanes, L corresponds to the end-to-end distance of a fully extended chain so that L (n 1)b sin( / 2) where n is the number of monomers, b is covalent bond distance between backbone carbons, and θ is the angle between two neighbouring bonds. Assuming the molecular parameters, b = 1.53 Å and θ = 110̊, we take l p as a free parameter in Eq. (S4) to model the n of R 2 and S 2 . While the WLC describes R 2 well for all n, resulting in l p = 7.9 Å, it fails to describe S 2 quantitatively. Eq. (S7) underestimates S 2 because WLC model ignores local chain structure, but by incorporating excluded volume interactions into this semi-flexible chain model [S20] can an improved theoretical description can be obtained of the L dependence of S 2 in comparison to simulation estimates. 8 Supplementary Information References [S1] A. J. Batschinski, Z. Physik. Chem. 84, 643 (1913); R. H. Hildebrand, Science, 174, 490 (1971); R. H. Hildebrand and R. H. Lamoreax Proc. Nat. Acad. 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