Letter pubs.acs.org/macroletters
Molecular Simulations of Solute Transport in Polymer Melts Kai Zhang and Sanat K. Kumar* Department of Chemical Engineering, Columbia University, New York, New York 10027, United States S Supporting Information *
ABSTRACT: Polymer membranes are typically used to separate gas mixtures on the basis of molecular size differences (“sieving”). The gas purity is known to be inversely proportional to the membrane flux, and the slope of this plot in glassy polymers is empirically found to be determined by the sizes of the gas molecules being separated, λ = (dB/dA)2 − 1. Despite potential mechanistic differences, the separation performance of rubbery polymers is often discussed in the same framework as their glassy counterparts. Here we perform molecular dynamics simulations of spherical solutes in coarse-grained high-density, high temperature polymer melts to gain a molecular understanding of their transport and separation behavior. We find that the diffusion coefficient follows an exponential law D ∼ e−ad. Since this dependence results in λ = dB/dA − 1, these findings do not provide a direct understanding of the experimentally deduced slope of the Robeson plot. field simulations are capable of reproducing the experimentally measured gas diffusion coefficient,26−30 the key modes of polymer dynamics that couple with solute diffusion are hidden in the complexity of atomic details (see the Supporting Information). Thus, it is unknown if a true upper bound exists and if it does what controls the slope of this plot. Here, we employ molecular dynamics (MD) simulations to study the permeability of model solutes in coarse-grained polymer melts. Our goal is to delineate the general mechanisms that underpin these phenomena. We find that, in flexible melts, the slope of the Robeson plot is ≈dB/dA − 1 for small solutes at low concentration (“tracer” limit), consistent with our finding that D ∼ e−ad. For trace amounts of large solutes, a power law dependence D ∼ 1/d3 is found;31 this is expected following the scaling ideas of Brochard Wyart and de Gennes.32 Thus, slopes even smaller than ≈dB/dA − 1 are found. The Freeman prediction λ = (dB/dA)2 − 1 can only be realized for gas-like solutes in the tracer limit when bond rigidity is introduced into polymer chains. We therefore conclude that the experimental form of λ does not represent the purely sieving ability of the membranes when small solutes are concerned, but instead reflects sieving coupled to some other effect (such as van der Waals attractions, chain stiffness, etc.), which is unresolved at this time. We consider spherical solutes dissolved in coarse-grained polymer melts. Polymer chains, each containing Np = 120 monomers, maintain their connectivity via finite extensible nonlinear elastic (FENE) bonds (Model I).33 We also examine the effects of bond rigidity by adding a stiff harmonic bond
T
he efficient and selective transport of small gases, for example, H2, O2, CO2, and CH4, across membranes is critical to many clean energy technologies. Polymeric membranes have excellent sieving ability coupled to lightweight, low-cost, and easy processability making them useful in this context.1−5 For a penetrant pair, membrane performance is characterized by the permeability of species A (PA) and the selectivity with respect to B (αAB = PA/PB). In the solutiondiffusion model, where the penetrant dissolves on the upstream membrane surface and diffuses through, P = D × S, where D is the diffusivity and S is the solubility. Membranes possessing both high permeability and selectivity are desirable.6 However, experimental αAB versus PA for a given pair of gases through different membrane materials exhibits a trade-off relation between these two quantities (“Robeson plot”), bounded from above by the “Robeson upper bound”.6−10 Understanding what factors affect the location of this upper bound is a central goal of membrane design. Current work on many gas pairs suggests that the upper bound follows log αAB = −λ log PA + κ, where A is the more permeable species.6 For gases with diameters dA and dB (dA < dB), there is experimental evidence that λ = (dB/dA)2 − 1. This expression can be obtained by using kinetic theory with two assumptions:11 (i) diffusion is a thermally activated process with D ∼ e−Ea/kBT,12 where kB is the Boltzmann constant and T is the temperature; (ii) the activation energy Ea ∼ d2,13 which follows by assuming that the motion of gas molecules of size d involves the opening of a tunnel with cross sectional area ∼d2.13 The actual d dependence of Ea is still under debate,14−21 since other models such as the free-volume theory suggest that Ea ∼ d3.22−25 The contradictions among these different theories suggest that a fundamental understanding of the factors controlling solute motion in polymer membranes is missing. While all-atom force © XXXX American Chemical Society
Received: May 3, 2017 Accepted: July 25, 2017
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Figure 1. (a) Total (p) and partial solute pressure (ps = p − p0) in the polymer phase as a function of solute concentration cs for σs = 1.28 and cm = 1.0. (b) Solubility of gas particles of sizes d = σs = 0.16−2.2 at various monomer densities cm = 0.70−1.0 (from top to bottom) for polymer Model I. Inset: Sd3 as a function of d approaches constants as d increases. Solid lines are derived from scaled particle theory.43
first equality arises by Taylor expanding p about the pure polymer liquid and truncating the expansion at first order. This truncation is justified by the linearity of a plot of p versus cs in Figure 1a. For large densities and large solutes we can reasonably argue that vs̅ ∼ d3, implying that Sd3 should be approximately constant ∝ κT in this limit. This dependence also follows from the simple idea of creating a cavity large enough to hold the solute of interest, an idea that is embodied in scaled particle theory40,43 and forms the basis of our understanding of unrelated phenomena such as the hydrophobic effect.44 We now calculate the self-diffusion coefficient D using Ns = 1000 solute particles.45 To obtain D, and to understand the factors controlling its behavior, we calculate the mean square displacement (MSD) ⟨Δr2⟩, the fourth moment of the displacement ⟨Δr4⟩, and the non-Gaussian parameter α2 = 3⟨Δr4⟩/(5⟨Δr2⟩2) − 1, as a function of time t. The chains are long enough (Np = 120) that they are entangled;33 hence, the dynamics of inner monomers enters the ∼t1/4 regime at the longest times (see the Supporting Information). As known from past work,33 the MSD averaged over all the chain monomers only show a ∼t1/2 over the same times (Figure 2a). Since small solutes with σs ≲ 1.0 occupy less than 2% system volume, chain dynamics are not affected by their presence. Larger solutes, on the other hand, suppress chain motion especially at these high concentrations (Figure 2a). However, there is no signature of transient caging that is seen in the case of typical, glass-forming liquids. Indeed, for all purposes the background polymer is a “normal” liquid which is far from its glass transition. Previous work has shown that these models of polymer liquids with cm ≈ 1 exhibit a glass transition at Tg ≈ 0.45.46 In contrast to the polymer motion, it is clear that all of the solutes reach their diffusive limit, ⟨Δr2⟩ ∼ t over the time scales shown (Figure 2b). Previous studies find that solute dynamics in glassy hosts involves transient trapping in polymer voids, with rare hopping between voids.30,47,48 It can be seen from Figure 2b that this glass-like “caging effect”49 is more prominent for larger solutes (σs ≳ 2.0). To quantitatively capture the onset of the trapping of solute particles, we calculate α2(t), whose peak time t* corresponds to the time at which the solute motion is maximally non-Gaussian and subdiffusive (Figure 2c).50 For Model I at cm = 1.0, the measured t* as a function of solute size clearly shows a transition in the vicinity of σs ≈ 1.7, indicating a crossover from the fast (t* ∼ 100) to a slow (t* ∼ 104) relaxation regime (see
angle potential (Model II).34 Nonbonded interactions are described by the repulsive Lennard-Jones (LJ) potential.35 While we utilize pure solutes in each simulation, we compare the behavior of pairs of solutes with a fixed diameter ratio of 0.8:36 namely, with sizes (dA, dB) = (σA, σB) in the range of (0.16, 0.2)−(1.76, 2.2) (in units of monomer size, σ). If each chain monomer is considered as a Kuhn blob, the size d of real gases, such as N2 and CO2, is ≲0.3.21 Solutes with d ≳ 0.3, thus, do not correspond to gases that are normally encountered in separation applications. We prepare Nc = 500 polymer chains in a cubic simulation box of fixed volume V at various monomer number densities cm ≡ NcNp/V, ranging from 0.70 to 1.0. The relatively dense polymer matrix guarantees that solute transport should follow the solution-diffusion model rather than the Knudsen mechanism.37 The MD simulations are performed under constant T = 1.0 using the LAMMPS package38 (see the Supporting Information for more details). The typical means to measure the solubility of a solute in a polymer, S, is to use the equation cs = Spv, where cs is the solute concentration in the polymer phase, and pv is the solute’s gas phase partial pressure at equilibrium. (S is inversely related to the Henry’s law constant.) There are some interesting issues with using this relationship for systems with purely repulsive interactions.1,39−42 Since the pressure corresponding to a high density repulsive polymer phase is large, p0 ≈ 10 in reduced units (Figure 1a), we consider an inert, nonadsorbing gas phase at pressure p0 in coexistence with the repulsive polymer liquid. To this we add a small amount of the solute; due to pressure equality at phase coexistence, it follows that the partial pressure of the solute in the gas phase must be ps ≡ p − p0 (p is the pressure of the melt with the solute) from which we deduce that cs = Sps. We dissolve Ns = 200, 400, 600, 800, and 1000 solute particles in the polymer matrix and calculate ps. In Figure 1b we show the resulting S as a function of d at various monomer densities cm for Model I. As expected, the S decreases as the solute size d increases, dropping by about 2 orders of magnitude as d changes from 0.16 to 2.2. Similarly, increasing the polymer density also causes S to decrease. At each density, for d approaching 2, we find that Sd3 is approximately constant, a result that follows from the fact that p − p0
( ) and v ̅ = ( ) cs
=V
s
∂p ∂Ns
∂V ∂Ns
T , V , Np
=
vs̅ , κT
where κT is the compressibility
is the solute’s partial molar volume. The
T , V , Np
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and construct the Robeson plots for all the penetrant pairs (A, B) at a fixed size ratio dA/dB = 0.8. In this construction, the variation of membrane types is achieved by changing the monomer density cm.6 For Model I, the resulting Robeson plots are collected in Figure 3, where each symbol corresponds to a
Figure 3. Robeson plots for gas pairs of sizes (σA,σB) with fixed size ratio dA/dB = 0.8 for Model I. For each pair, 16 data points are shown, corresponding to polymers of density cm = 0.70−1.0, with higher cm on the left (smaller P). The blue-dotted/red-dashed line fits all the data points above/below PA ≈ 6 × 10−5, respectively. Two triangles denote slope magnitudes derived from the ∼d and the ∼d2 (Freeman) laws. Inset: the fitted Robeson plot slope λ for each pair of gases as a function of the larger gas size dB. The values of the solid triangles and blue-dotted/red-dashed lines in the main panel are also shown for comparison.
gas pair (A, B) and the scan of 16 polymer densities cm is shown for each pair. We find that all the data points for smaller solutes (dB ≤ 1.8) and the low-cm data points for larger solutes (dB ≥ 2.0) roughly collapse on a universal line, with λ ≈ 0.2, which is close to dB/dA − 1 = 0.25. The high-cm data points for larger solutes (dB ≥ 2.0) have a λ ≈ 0.56, suggesting the concentration of solutes could affect the form of the separation law. Correspondingly, the fitted λ for each solute pair rises up toward higher value as dB passes 2.0 (inset of Figure 3). This crossover behavior as solute size σs and polymer density cm increases suggests that two different transport mechanisms might be responsible for the two regimes. In the most popular theory for gas transport, Freeman’s theory,11 it is inherently assumed that the solute motion in the glassy matrix is an activated process. Under this condition, the slope λ is defined by the functional form of the diffusivity D versus d, log D = −adn + b, where a and b are fitting parameters, and n = 2 is rooted in the argument that the activation energy Ea scales with the cross sectional area of the hopping tunnel. We find that, while n = 1 fits better for smaller solutes σs < 1.7, n = 2 fits better for larger ones σs > 1.7 (Figure 4). Several models have been proposed to explain the dependence of Ea (and D) on d.19,51 In the Brandt model, which suggests that a cavity is formed by local chain distortion, Ea is decomposed into two geometric contributions: (1) the intermolecular energy Ei due to the repulsion between a locally distorted chain and its neighboring chains; (2) the intramolecular energy Eb due to chain bending.14 Others claim that attractive forces play a role in Ei.52 Since our Model I has purely repulsive interactions, the n = 1 law in the small solute limit is better explained by small local distortion effects15 or geometric obstruction.53 In high
Figure 2. Mean square displacement (MSD) Δr2 of (a) polymer chains and (b) solutes of size σs = 0.2−2.2 (from top to bottom) at cm = 1.0 and T = 1.0 for Model I. (c) Corresponding non-Gaussian parameter α2 for solute diffusion. For each σs < 1.8, two overlapping curves are shown, which are obtained from the short-time and the long-time measurements, respectively. Inset: the peak time t* of α2 as a function of σs.
the inset of Figure 2c). Thus, due to their high concentration, solute molecules with diameter nearly twice the size of monomers show caged dynamics, even though the polymer matrix by itself is far from its glassy regime. In the tracer limit, no obvious caging dynamics appears even for these large solutes.31 With the measured S and D, we can calculate the permeability P for each solute of size σs = 0.16−2.2 dissolved in the polymer melt with density in the range cm = 0.70−1.0, 866
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Figure 5. Robeson plots for gas pairs of sizes (σA, σB) with fixed size ratio dA/dB = 0.8 for Model II at polymer density cm = 0.70−1.0. The two solid triangles and two fitted lines for Model I in Figure 3 are also shown for comparison.
Figure 4. Diffusion coefficient D as a function of gas diameter d = σs = 0.16−2.2 (left) and d2 (right) at cm = 0.70−1.0 (from top to bottom). Fitting lines with the law logD = −adn + b, via n = 1 for σs < 1.7 and n = 2 for σs > 1.7 are also shown.
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temperature polymer melts, the motion of small gas penetrants can still be Arrhenius-type D ∼ e−Ea/kBT.26 It is thus plausible that the motion of small solutes follows the same D ∼ e−ad law over a certain range of temperature. For the high concentration 2 of large solutes in this work, an apparent D ∼ e−ad law is found in nonglassy matrices, because the volume fraction of solutes is no longer negligible. For a trace amount of large solutes, polymer segments behave as a viscous medium and D follows a power law instead.31,54,55 Because chain distortion is postulated to play a role, we test the effect of chain rigidity by adding a bond angle potential into the polymer model (Model II), which effectively makes the material closer to being glassy. After calculating S, D, and P with the same protocol as for Model I, we construct the Robeson plots of three gas pairs (σA, σB) = (0.16, 0.2), (0.48, 0.6), (0.8, 1.0) for polymer systems of density cm = 0.70−1.0. (For the (0.8, 1.0) pair, the result is only obtained for cm up to 0.94, due to the slow diffusion at higher cm.) Compared with the previous Model I results, the crossover from n = 1 to n = 2 regime occurs at much smaller solute size dB ≈ 1.0 in the tracer limit (Figure 5). Evidently, the bond angle energy increases the chain rigidity and the tendency for solute particles to be trapped. The 2 crossover from D ∼ e−ad to D ∼ e−ad therefore appears at a threshold penetrant size dc, which decreases as the glass transition density is approached. In this Letter, we show that under conditions where pure size sieving mechanism should apply, the diffusion of small solutes 2 in flexible polymer melts does not obey the idea that D ∼ e−ad , −ad but instead follows the scaling law D ∼ e . A recent theoretical study suggests that penetrant diffusion in glassy polymers also exhibits D ∼ e−ad, at least for small solutes d < 1.0.56 However, mechanistic differences for penetrant diffusion are often thought to exist between glassy and rubbery 2 polymers.11 Here, the absence of the D ∼ e−ad , which is only valid when the spacing between polymer chains is on the order of or smaller than the size of the penetrant,11 may be related to the intrinsic cavity size set by a coarse-grained polymer model. Since the Freeman theory itself does not concern about the chemical details of polymers, more accurate models are needed to fully understand the problem.
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00339. Table S1: Summary of molecular simulations of gas permeation in polymers; Simulation details; and Figure S1 (PDF).
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Kai Zhang: 0000-0001-9700-7943 Sanat K. Kumar: 0000-0002-6690-2221 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We acknowledge primary financial support from the National Science Foundation under Grant DMR-1507030. REFERENCES
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