Counterintuitive Gas Transport through Polymeric Nanocomposite

Nov 21, 2014 - Key Laboratory of Leather Chemistry and Engineering of Ministry of Education and. ‡ .... influences gas transport through the nanocom...
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Counterintuitive Gas Transport through Polymeric Nanocomposite Membrane: Insights from Molecular Dynamics Simulations Yi Chen,† Maolin Jia,† Hui Xu,† Yang Cao,‡ and Haojun Fan*,† †

Key Laboratory of Leather Chemistry and Engineering of Ministry of Education and ‡Key Laboratory of Bio-Resource and Eco-Environment of Ministry of Education, Sichuan University, Chengdu 610065, P. R. China ABSTRACT: Completely contrary to classical composite theory, the gas permeability of certain rigid polymers was frequently found to increase upon addition of nonporous, nanoscale inorganic particles. Until now, the underlying mechanism remains elusive. In this study, polycaprolactone−TiO2 nanocomposite model was computationally constructed to clarify this issue. The molecular dynamics simulation results indicated that such counterintuitive behaviors arose from an extra region with depleted matrix phase and, hence, higher free volume at the polymer−filler interface. Owing to its inaccessibility and delicacy, this interfacial region was indiscernible by experimental means. However, it could be qualitatively visualized and quantitatively measured by simulated density field and density profile, respectively. By reproducing the thermodynamic property of polycaprolactone, we also conducted, for the first time, a comparative simulation of how polymer chains that shared the same primary structure but differed in rigidity behaved differently when packing around the highly curved nanoparticle surface. Such discrepancy was further found to correlate well with opposite trends in gas self-diffusivity in the resultant polycaprolactone−TiO2 nanocomposite model. Based on these results, the molecular mechanism leading to the formation of the interphase whose properties differed significantly from the bulk polymer was proposed. Coupled with previous data experimentally obtained, the present study offered a generic framework for understanding the molecular basis of interfacial architecture in polymeric nanocomposites, which was crucial in designing membrane devices with tailored permeability for specific applications, covering from breathable leather coating, through ultrahigh barrier blood sacs in ventricular assist devices, up to advanced gas separation membrane not subject to the empirical permeability/selectivity trade-off.

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INTRODUCTION Over the past decade, there have been many alleged observations of enhanced gas permeability in certain rigid polymers after incorporation of nonporous, nanoscale inorganic particles.1−20 These observations, however, seem to contradict qualitative expectation based on classical composite theory. In general, the presence of nonporous fillers reduces the permeability of a polymer. This barrier effect is attributable to a reduced amount of polymer through which transport may occur and a more tortuous diffusion path that the penetrants are forced to experience when they traverse the composite.21,22 To explain these counterintuitive phenomena, some researchers1−6 postulated an extra, delicate region with higher free volume at the polymer−filler interface. Compared with the bulk polymer, this interfacial region opened relatively low-resistance diffusion path in the composite, dominating over the volume filling and tortuosity effects to allow faster gas transport. However, despite the plausibility of such hypothesis, previous investigations1−6,13,15,19,20 using commercial nanoparticles containing aggregates and conventional detection techniques with limited sensitivity failed to obtain explicit evidence supporting the objective existence of this molecular-scale or submolecular-scale interphase. More importantly, the molecular © XXXX American Chemical Society

mechanism underlying the formation of such interfacial region whose properties differ significantly from the bulk polymer remains elusive. Therefore, this hypothesis has been constantly challenged by skeptics since its conception, who rationally argued the impact of interstitial cavities enclosed in nanoaggregates or merely membrane defects.11−15,17−20 In the context of such ongoing controversy, an organic− inorganic hybrid technique was employed in our previous study 23 to in situ generate TiO 2 nanoparticles in a thermosensitive polyurethane (TSPU) solution. The rigidity of the TSPU chain varies significantly around a switch temperature (Ts). Cooling below the Ts freezes the TSPU chain to its rigid state, while heating up above the Ts releases the frozen chain to its flexible state. By carefully designing hybrid conditions, nanoaggregate- and defect-free TSPU nanocomposite membranes could be successfully obtained after solvent removal. Without possible interference, it was found that the gas permeability of the resultant nanocomposites correlated with the solution casting temperature (Tsc) and, Received: July 25, 2014 Revised: November 2, 2014

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converged to unity beyond the cutoff. Throughout the simulations, Andersen thermostat and Berendsen barostat algorithms were employed to maintain a constant temperature and pressure (1 × 105 Pa), respectively. The time step was set to 1 fs for all dynamics runs. As already established, it is the phase transition of the soft segment (melting transition of polycaprolactone in this study) that determines the rigidity of the whole TSPU chain.29−34 For a rather drastic simplification therefore, the crystalline and, hence, gas-impermeable hard segment could be judiciously ignored in modeling the matrix phase of TSPU nanocomposite. Accordingly, one polycaprolactone chain (PCL10000) consisting of 88 repeat units (1587 atoms) was built in a head-to-tail orientation with random torsion. The number-average molecular weight of this fully atomistic model was approximately 10 000 g/mol, consistent with that of the PCL monomer we previously adopted for TSPU synthesis.23 For geometry optimization, the energy of this model was smart minimized, using three iterative algorithms (steepest descent, conjugate gradient, and quasi-Newton methods) in a cascading way, with a convergence level of 0.001 kcal/(mol Å). Subsequently, three PCL chains were enclosed in a cubic cell by the amorphous cell module at 300 K, based on the “selfavoiding” random-walk method developed by Theodorou and Suter.35 To improve the efficiency of construction, the option of ramping cell density from an initial low value of 0.60 g/cm3 was employed. Periodic boundary conditions were applied in all three directions to suppress finite size effect. After construction, the simulation cell was subject to energy minimization using the smart minimizer algorithm until complete convergence, followed by full equilibration successively using NVT (T = 300 K) and NPT (T = 300 K) ensembles. Before the NVT run, random initial velocities and, hence, kinetic energies to all atoms were assigned in accordance with a Maxwell−Boltzmann distribution at 300 K. Full equilibration was validated by a stable density fluctuation as a function of simulation time. Following the NPT run, a stepwise procedure involving three annealing cycles was applied, by which the system was heated to 1000 K and then cooled back to 300 K at intervals of 10 K. At each temperature interval, a 3 ns NPT dynamics was carried out. Initial atomic coordinates, velocities, and cell length for subsequent temperature MD run were provided by the final conditions of the previous temperature run. By periodical annealing, the PCL packing model was relaxed to reach a global minimum, producing a well-equilibrated structure referred to henceforth as PCL-300. For comparison, PCL-300 was reheated to 350 K at intervals of 10 K by using the NPT ensemble. The resultant structure was denoted as PCL-350. For all dynamics runs, full atomic trajectories were recorded every 5 ps, and the last 50 ps of simulation at each temperature interval was used for statistical analysis. To construct PCL nanocomposite, a spherical TiO 2 nanocluster with a radius of 5.0 Å was first built from the primitive unit cell of rutile TiO2 structure provided by Materials Studio 5.5. This initial TiO2 model was annealed at 8500 K for 500 ps and then cooled back to 300 K at intervals of 100 K to form amorphous structure. Annealing-induced phase transformation was validated by comparing the radial distribution functions of atom pairs with those in previous simulation report.36 Afterward, one amorphous TiO2 nanoparticle was embedded into PCL-300 and PCL-350. As solvent evaporates from a filled polymer solution, the volume of the assembly shrinks due to solvent loss. During this process, polymer chains

hence, rigidity of the TSPU chains during packing. When the TSPU chains were frozen into rigid state as solvent evaporated (Tsc < Ts), gas permeability coefficients of the nanocomposite membrane were found to increase with increasing TiO2 content. In contrast, the combination of flexible TSPU chains (Tsc > Ts) and tiny nanoparticles resulted in gas transport behavior consistent with the prediction of classical composite theory. To explain such discrepancy, it was assumed that rigid polymer chains, unlike their flexible counterparts, could not pack efficiently around the nanoparticle surface, leading to an extra interfacial region with higher free volume that allowed more facile diffusion jumps. Unfortunately, probably due to overlapping of neighboring interfacial regions or indefinite boundary between the high-free-volume region and the bulk, we failed to discriminate these delicate or morphologically ambiguous interphases out of the nanocomposite, despite successful characterization by density measurement that found negative deviation of actual density of TSPU nanocomposites from additive model and positron annihilation lifetime spectroscopy (PALS) demonstrating subtly increased free volume in TSPU nanocomposites relative to pure control. As continuous work, for the first time, a comparative study of how the rigidity of polymer chains, controlled by a reproduced thermodynamic phase transition, influenced the way they packed around the nanoparticle surface and, thereby, gas diffusivity in the nanocomposite was conducted in the present study using molecular dynamics (MD) simulations. This computational method has been widely adopted to quantify microscopic morphology or molecular dynamics experimentally immeasurable with the current technological level.24 As a proofof-principle, this report, together with our previous experimental data, 23,25−27 aims to provide insights of wide applicability into the molecular basis of how a high-free-volume region shapes at the polymer−filler interface and consequently influences gas transport through the nanocomposite. Up until now, these issues are still disputable but recognized crucial in designing membrane devices with tailored permeability for specific applications.

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SIMULATION DETAILS MD simulations were performed using Materials Studio 5.5, a commercial software package developed by Accelrys Inc. (San Diego, CA), installed on an Intel dual-processor XEON 32 bit workstation. The condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS) force field was chosen to describe interatomic interactions. This ab initio force field enables accurate and simultaneous prediction of structural, conformational, and thermodynamic properties for a broad range of materials.28 The nonbond terms in COMPASS potential functions include a Lennard-Jones 9−6 potential type for van der Waals term and a Coulombic function for electrostatic interaction. For computational inexpensiveness, a cutoff was introduced to neglect nonbond interactions for atom pairs separated by a distance longer than 14.5 Å. This cutoff distance was always smaller than half of the cell length (ca. 30−37 Å). To mitigate energy discontinuities at the cutoff, a quintic switching function was used with a spline width of 1.0 Å. A buffer width of 0.5 Å was explicitly specified in generating the neighbor list. Charge groups were defined in terms of chemical functional groups, over which the nonbond cutoff was applied. This was carried out to prevent artifacts due to splitting of dipoles. Long-range correction was applied to the energy, assuming that the radial distribution functions B

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herein acceptably reproduced the condensed-phase properties of PCL10000. Furthermore, PCL is known as a semicrystalline aliphatic polyester with a molecular-weight-dependent melting point (Tm).41 In the present study, simulated annealing, an efficient global optimization algorithm, was used to induce regular alignment of PCL segments to form crystallites. In Figure 1, the specific volume of the PCL10000 packing model

gradually approach the nanoparticle and ultimately pack on its surface. In an attempt to simulate this physical process closely, a buffer region was created between the PCL chains and the nanoparticle embedded before any dynamics calculation. To this end, the length of the PCL packing model was enlarged by 35%, keeping the position of the PCL chains constant. Then, a solvent surface with a solvent probe radius of 10.0 Å was created based on a coarse grid resolution. The region into which the nanoparticle was embedded was restricted to the empty space enclosed by the solvent surface. Subsequently, the nanocomposite was subject to a geometry optimization, followed by a NPT (T = 300 K for PCL-300; T = 350 K for PCL-350) dynamics until a stable density fluctuation. The equilibrated nanocomposite models were referred to as PNT300 and PNT-350, respectively. Diffusion coefficients of H2, CO2, O2, N2, and CH4 in PCL and nanocomposite models were determined according to a previously published protocol.37 Briefly, five gas molecules of species α were randomly inserted into the model. The assembly was then equilibrated for 3 ns using NPT (T = 300 K for PCL300 and PNT-300; T = 350 K for PCL-350 and PNT-350) ensemble. After that, successive positions of a diffusant were computed as a function of time by solving Newton’s equations that determined the “random walk” trajectory of the diffusant. Self-diffusion coefficients Dα could then be calculated from the slope of the MSD (t) plot according to Einstein relation:38 1 d Dα = lim 6Nα t →∞ dt

Figure 1. Plot of specific volume versus temperature for PCL10000 packing model. The specific volume was calculated from the cooling procedure of the third simulated annealing cycle. DSC thermogram of actual PCL10000 (Daicel Chemical Industries, Ltd., Osaka, Japan) is also present for comparison. It was obtained by using a NETZSCH DSC 200 PC analyzer (Germany) at a cooling rate of 10 K/min under the protection of high-purity nitrogen. The dashed lines are drawn to guide the eyes.

Nα

∑ ⟨|R i(t ) − R i(0)|2 ⟩ i=1

(1)

where Nα is the number of diffusants of species α, Ri(t) − Ri(0) stands for the vector distance traveled by diffusant i over the time interval t, and ⟨|Ri(t) − Ri(0)|2⟩ represents the averaged mean-square displacement (MSD) of diffusant i. The angular brackets denote averaging over all time origins. To prevent initial nonlinear oscillation, as well as quality decline of statistics over a long time, the MSD(t) plot with t in the range of 500− 2250 ps was used in self-diffusivity calculation.

is plotted against temperature. Over the temperature range of 330−340 K, the plot displayed a readily discernible discontinuity in thermal expansivity, which described well the first-order nature of a melting transition that involved a latent heat. For comparison, the differential scanning calorimetric (DSC) thermogram of actual PCL10000 is also present in Figure 1. Clearly, the simulated Tm was quite close to experimental value, 344 K, determined as the extrapolated onset of the crystallization exotherm. These results demonstrated the feasibility of reproducing the thermodynamic property of PCL10000 by using simulated annealing. Annealing-induced crystallographic alignment of PCL10000 segments could be further validated by comparing the simulated X-ray diffraction patterns at different temperature. As illustrated in Figure 2, no intense diffraction peaks could be detected in the PCL packing model until the system was cooled below 340 K. This observation confirmed that some PCL segments in the model were able to align into regular structure by periodical annealing. Such phase transition occurred over the temperature range from 330 to 340 K, well consistent with previous specific volume result. As the temperature decreased from 330 K, these two crystalline peaks were found to gradually shift toward higher values. Especially at 300 K, they increased to 21.4° and 23.7°, which agreed well with experimental data.42 Generally, intense X-ray diffractions at 21.4° and 23.7° were ascribed to the (110) and (200) planes of the orthorhombic crystal structure in semicrystalline PCL, respectively.42 Fairly good agreement between experimental and simulated data validated again the accuracy of the modeling methodology used in the present study in reproducing the thermodynamic property of PCL10000.

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RESULTS AND DISCUSSION To investigate the impact of PCL chain rigidity on their packing efficiency around nanoparticle, accurate simulation of the melting transition of semicrystalline PCL is crucial. Validation of the equilibrated PCL10000 packing model was first carried out by comparing specific properties experimentally obtained with simulated values. As summarized in Table 1, the density, cohesive energy density, and solubility parameter of PCL-300 were found to be 1.137 ± 0.005 g/cm3, (3.35 ± 0.06) × 108 J/ m3, and 18.31 ± 0.163 (J/cm3)0.5, respectively. These data closely matched the counterparts from experiment, 39,40 indicating the force field and modeling methodology employed Table 1. Simulated and Experimental Density (ρ), Cohesive Energy Density (CED), and Solubility Parameter (σ) of PCL10000 a

simulated experimental

ρ (g/cm3)

CED × 108 (J/m3)

σ ((J/cm3)0.5)

1.137 ± 0.005 1.146b

3.35 ± 0.06 3.5439,40

18.31 ± 0.163 18.8239,40

a

The simulated data were derived from PCL-300. bSpecified by Daicel Chemical Industries, Ltd. (Osaka, Japan), the manufacturer from which PCL10000 monomer was obtained for TSPU synthesis in our previous study.23 C

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crystallites, which imposed constraints on rotation of bonds constituting the PCL backbone. Under such circumstance, the PCL chains should have difficulty in changing conformation in response to an external force. This conclusion, as would be shown later, was crucial in interpreting the unusual way rigid PCL chains packed in the vicinity of the TiO2 nanoparticle. When the PCL model was heated up to 350 K, it was found that all distribution profiles in Figure 3 contained more peaks and tended to be smooth compared with the counterparts at 300 K. This shift indicated that as the crystallites disappeared, the PCL chains were not rigid anymore; rather, they displayed a high degree of freedom to rotate about backbone bonds. In a polymeric composite, such flexible nature allowed the polymer chains to writhe and continually change conformation, intimately wrapping around irregular contour or surface geometry without creating imperfection at the interface. Another indicator of PCL chain rigidity across the Tm was MSD of the chain. In general, rigid polymer chain exhibits low MSD, and vice versa.45 The MSD(t) plots of PCL chains over the cooling procedure of the third simulated annealing cycle are shown in Figure 4. Clearly, the MSD(t) plot experienced a discontinuous drop as cooled from 340 to 330 K. This reflected significantly decreased mobility or rigidification of PCL chains caused by partial crystallization. Quantitatively, the selfdiffusion coefficients of PCL chains could be calculated from the slope of the MSD(t) plot using the Einstein equation. The calculated results are plotted in Figure 4. Below the simulated Tm, the PCL chains showed self-diffusion coefficients lower than 9.2 × 10−9 cm2/s because of restricted segmental mobility. Once the temperature exceeded 330 K, the crystallites melt as demonstrated previously by simulated X-ray diffraction patterns. Unrestricted segmental rotation enabled continuous conformational changes, leading to significantly accelerated Brownian motion and thereby a rather flexible PCL chain. In Figure 4, this transition was manifested by sharply increased self-diffusion coefficients up to (4.27−4.72) × 10−8 cm2/s, which were 1 order of magnitude larger than those below the T m. In previous reports,1−6 counterintuitive permeability enhancement in some nanoparticle-filled polymers was ascribed to an envisioned high-free-volume interphase extending from the nanoparticle surface that ultimately merged with the bulk polymer. Owing to its inaccessibility and delicate structure, the morphologies and fundamental properties of this interfacial region were indiscernible using conventional detection techniques. Previously, PALS and 129Xe NMR spectroscopy were used for quantification of the interphase.1−6,13,15,19,20,23 It turned out that such interfacial region, if any, only contributed

Figure 2. Simulated X-ray diffraction patterns of PCL10000 packing model as a function of temperature. The patterns were computationally obtained from the cooling procedure of the last simulated annealing cycle using the reflex module.

Typically, the rigidity of a semicrystalline polymer chain varies significantly below and above its Tm.43,44 Cooling below the Tm induces the formation of crystallites, the regularity within which allows close segmental packing and, hence, exceptionally high intersegmental interactions. As a result, the segments involved are frozen at strictly defined positions, with little freedom to rotate about backbone bonds. In addition, partial crystallization imposes topological constraints on mobility of the neighboring amorphous segments as well, further contributing to rigidification of the whole polymer chain. Upon heating through the Tm, the regularly aligned segments in crystallites fall out of the crystallite structure, releasing constraints on adjacent amorphous segments. Accordingly, the rigid polymer chain turns into a flexible one, which is capable of free rotation. In MD simulations, the degree of rotational freedom, or rigidity, of a polymer chain can be characterized by determining the torsion angle distribution of the backbone bonds. The torsion angle distribution of some rotatable single bonds in PCL10000 backbone is illustrated in Figure 3. At 300 K, the distribution profiles for bonds a, b, and c were all found lack of variety, showing a unimodal or a bimodal pattern. This observation could be ascribed to the formation of

Figure 3. Torsion angle distribution of rotatable single bonds in PCL backbone. The statistics were obtained from PCL-300 and PCL-350, based on an extra NPT (T = 300 K for PCL-300; 350 K for PCL-350) dynamics for a duration of 1000 ps after full equilibration. D

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Figure 4. MSD and self-diffusion coefficients of PCL10000 as cooled from 360 to 290 K in the third simulated annealing cycle. To exclude initial anomalous diffusion regime as well as quality decline of statistics over a long time, the MSD(t) plot with t in the range of 1000−2500 ps was used in self-diffusivity calculation.

Figure 5. FFVact accessible to Connolly probes of different radius in PCL and nanocomposite models. The insets show the relative FFVact, calculated as the ratio of FFVact in nanocomposite model to that in pure PCL control. The dashed lines represent theoretical prediction based on additive model, which was estimated from the volumetric composition and known FFVact of pure components.

to overall increased free volume in the matrix phase, rather than be detected as an entity independent of the bulk polymer. In the absence of explicit experimental evidence, controversy remains as to whether an extra high-free-volume interphase really exists in a seemingly binary polymer/filler system. In particular, some researchers argued that either those interstitial cavities enclosed in nanoaggregates or membrane defects should be responsible for the counterintuitively increased permeability in polymeric nanocomposites, implying the socalled interphase might not exist at all. In our previous study,23 the important role played by polymer chain rigidity during packing in determining the microstructure at the polymer− nanoparticle interface, and hence, gas permeability of the nanocomposite was highlighted. This correlation was established by comparative experiments, utilizing the thermal sensitivity of segmented TSPU chains. By solution casting at different temperature, TSPU chains that shared the same primary structure but differed in rigidity were allowed to pack around in situ generated TiO2 nanoparticles to form nanocomposite membranes. As tested by a series of carefully devised experiments, the resultant membranes were identified as nanoaggregate- and defect-free. In this interference-free nanocomposite, completely opposite trends in gas permeability were observed, depending on the TSPU chain rigidity as solvent evaporated. Based on these experimental results, it was assumed that because of the rigid nature of TSPU chains at low temperature, close packing achieved in the bulk polymer was

disturbed in the vicinity of the nanoparticle, resulting in a highfree-volume and, thus, diffusivity-promoting interfacial region. On the contrary, flexible TSPU chains were able to pack around the spherical nanoparticle as efficiently as in the bulk. In this case, the filler particles only acted as an obstacle to penetrant diffusion, leading to decreased gas permeability. With such encouraging progress, the instant challenge we are facing is how to discriminate the delicate interphase out of the nanocomposite. Density measurement and PALS analysis provided some useful information,23 but not in an explicit way. In the present work, continuous efforts were devoted to quantitatively probing the interfacial architecture in TSPU nanocomposite membranes by MD simulations. From a molecular perspective, we explored for the first time that how the rigidity of polymer chains, controlled by a simulated melting transition, influenced their packing efficiency around the nanoparticle and, thereby, gas diffusivity in the nanocomposite. The mechanism proposed herein was expected to be generic to explain these frequently observed high permeability in the field of polymeric nanocomposites. Assuming volume additivity, the fractional free volume (FFV) in a binary polymer/filler system is supposed to decrease as nonporous filler particles substitute parts of polymer matrix that contain free volume elements. However, the actual FFV (FFVact) may deviate from theoretical prediction if the polymer and filler combine in a manner that alters their original properties. Figure 5 illustrates the FFVact in PCL and E

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Figure 6. Density field of PCL component in PCL and nanocomposite models. For clarity, the PCL chains are displayed as ultramarine or turquoise lines, corresponding to their rigidity. Those atoms constituting the TiO2 nanoparticle are rendered as spheres with radii identical to the van der Waals radii of the atoms.

slice that paralleled to the bc-plane while intersecting with centroid of the nanoparticle was included. According to the map legend, the PCL component in PNT-350 exhibited similar density distribution to that in unfilled control, which fluctuated slightly and randomly across the simulation cell. In the vicinity of the nanoparticle, no unusual densification or less compact area was observed, indicating that flexible PCL chains in PNT350 packed in the simulation cell as if the nanoparticle were not present. They wrapped around the nanoparticle surface as ideally as envisioned by classical composite theory. Under such circumstance, only volume substitution took effect, leading to decreased FFVact. In PNT-300, however, it was found that the TiO2 nanoparticle was encapsulated by an extra spherical shell with lower PCL density than the bulk region. Compared with PNT-350, local PCL depletion in PNT-300 indicated that rigid PCL chains failed to pack around the nanoparticle as efficiently as in the bulk, leading to an extra-low-density region at the PCL−TiO2 interface. Free volume enhancement in this area appeared to be so significant that the volume substitution effect was more than offset. In addition to density field, the one-dimensional density profiles of PCL and TiO2 components were also generated, comparative analysis over which allowed quantitative evaluation of the interfacial morphologies practically inaccessible by experimental means. As can be seen in Figure 7, in both nanocomposite models, a sudden increase in TiO2 density from zero was evident, which could be used to determine the periphery of the filler particle. In PNT-350, flexible PCL in the bulk exhibited a stable density fluctuation around a mean value almost equivalent to that in PCL-350, indicating unaffected PCL chain packing. Upon contacting the periphery of the filler particle, the density of the PCL component declined immediately, which was simply because the filler substituted parts of PCL. This result coincided well with previous observation by density field that no densification or less compact area existed at the flexible PCL-TiO2 interface. In contrast to PNT-350, the density of PCL component in PNT300 started to decline at a distance of about 2.3 Å away from the TiO2 surface. In particular, the PCL density in close proximity to the nanoparticle surface was found to be

TiO2-filled nanocomposite models, determined as the ratio of free volume accessible to diffusants of interest to the total volume of the model. The accessible free volume was defined as the empty space enclosed by a Connolly surface. The radius of the Connolly probe varied from 1.445 to 1.900 Å, corresponding to kinetic radius of different gas molecules. According to the data in Figure 5, the FFVact in both PCL-300 and PCL-350 decreased with increasing Connolly radius, consistent with the fact that large probes are inaccessible to relatively narrow cavities. In PNT-350, the combination of PCL and TiO2 components resulted in lower FFVact than unfilled control; the relative FFVact was approximated as 0.975 regardless of the Connolly radius, closely matching theoretical value based on additive model. However, this was not the case for PNT-300 where the PCL chains were in rigid state as they packed around the nanoparticle. Independent of the probe size, the FFVact in PNT-300 tended to increase compared with control, showing positive deviation from additive model. For example, the FFVact accessible to O2 was found to be 1.86% in PNT-300, higher than the ideal additive value (1.21%) and also the counterpart in PCL-300 (1.24%). Before embedded into the simulation cell, the nanoparticle model proved inaccessible to any Connolly probe being considered. Thus, there was no chance the particle would introduce any extra free volume into the nanocomposite itself. In that case, the packing of rigid PCL chains must have been perturbed by TiO2 nanoparticle, which dominated over the volume substitution effect to augment FFV. From the inset in Figure 5, it was also noted that the positive deviation became progressively significant with increasing Connolly radius. The FFVact accessible to a relatively large probe, such as CH4, increased more than that of a smaller probe, such as CO2. This finding was in good agreement with established principle that any free volume change in a polymer will have a more significant influence on accessibility of large molecule than that of the smaller one.46 To trace the origin of extra free volume in PNT-300, spatial distribution of PCL component in the nanocomposite model was qualitatively visualized using density field. Figure 6 presents the density fields of PCL component in PCL and TiO2-filled nanocomposite models. For easy access to the interior views, a F

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Scheme 1. Schematic Illustration of How the Rigidity of Polymer Chains Influenced the Way They Packed around the Nanoparticle Surface

Figure 7. One-dimensional density profiles of PCL and TiO2 components along the c-axis in (a) PNT-300 and (b) PNT-350. The thickness of those slabs defined for density calculation was set to 0.25 Å. For clarity, the density (ρ) of each component was normalized by the bulk density (ρ0) of pure component.

matrix at a fixed volume fraction increases as particle size decreases.4 Correspondingly, the particle surface area also increases substantially, resulting in much more potential polymer−filler interfacial area. Then, the total volume of those high-free-volume interfacial regions, if any, will be present in greater amount around smaller particles, leading to more rapid gas transport through the composite. Despite its simplicity, this correlation was found at least qualitatively correct to interpret the extent of gas permeability enhancement in certain polymeric nanocomposite systems.1,4,47 However, there still remains one unknown question: does the nanoparticle size also correlate with the thickness of the high-freevolume interphase? Based on the discussion above, it is the inability of rigid polymer chains to wrap around the highly curved surface of nanoparticle that leads to polymer depletion near the surface of the filler particle. Thus, the thickness of the depleted layer should depend inversely on the nanoparticle size, with smaller filler particle having thicker depleted layer around it. Theoretically, this is because smaller filler particle offers a surface of larger curvature, around which rigid polymer chains will encounter much more difficulties in packing closely. Furthermore, if such correlation exists, then there must be a threshold particle size, above which the surface curvature of the filler becomes so insignificant that even rigid polymer can wrap around it as intimately as in the bulk. Indeed, the determination of this threshold particle size is as important as demonstrating the objective existence of an extra high-free-volume region at the polymer−filler interface. The latter rationalizes those counterintuitive gas transport behaviors frequently observed, while the former indicates that rigid polymer chain alone may not be sufficient to guarantee an interfacial layer with depleted matrix phase. Probably, the size of the filler particle is. Unfortunately, conventional experimental methods are helpless in the face of these unknown issues of crucial importance, since a sandwiched interphase of extremely small size is hardly discernible using modern nanostructural and nanochemical analysis techniques. Therefore, to the best of our knowledge, the answers to these unknowns were not involved in all relevant prior published work. However, MD simulation provides a convenient and efficient way to address this challenge. As demonstrated earlier, the interfacial morphologies in a nanocomposite can be qualitatively visualized by density field and quantitatively measured based on the density profile. This enables us to specifically explore the dependence of interphase thickness on the nanoparticle size. With such crucial correlation, how a high-free-volume interfacial layer forms will be clearer. Meanwhile, the establishment of a mathematical

approximately 0.758 g/cm3, much less than that of the bulk region (ρ ≈ 1.137 g/cm3). To understand the origin of such extra interphase in PNT-300, both the rigidity of PCL chains and the surface curvature of the nanoparticle should be taken into account. First, one should keep in mind that a buffer region was created between the PCL chains and TiO2 nanoparticle embedded before any dynamics calculation. Once the dynamics calculation started, the volume of the nascent nanocomposite cell tended to shrink spontaneously, driving the system to its new equilibrium state. During this process, individual PCL chains must be able to relax on the same time scale to reach their new equilibrium state with decreased intermolecular distance; otherwise a segmental stress arose. For rigid PCL chains, relaxation via conformational changes was difficult. Thus, they could only relax by diffusing, although slowly, into the buffer region at the initial stage. As the buffer region eliminated, diffusion-induced relaxation became incrementally difficult such that the time scale for chain relaxation mismatched that of stress generation. Thereafter, even small volume shrinkage would lead to huge stress at the segmental level. When approaching quite close to the nanoparticle, rigid PCL chains were unable to wrap around the highly curved surface of tiny nanoparticle by conformational changes. Meanwhile, huge segmental stress accumulated also prevented them from bending or twisting to accommodate the filler particle. As a result, an extra interfacial region with disrupted PCL packing shaped (see Scheme 1). Now consider the counterpart model PNT-350 where the PCL chains were in flexible state. Throughout the dynamics simulation, any segmental stress generated in flexible PCL chains could be relaxed over a short time scale. This was achieved by either always prompt conformational changes or diffusion of PCL chains into the buffer region initially. Even when the PCL chains arrived at the nanoparticle surface, they were still so flexible that they could change conformation easily to adapt to the highly curved nanoparticle surface. In this case, the matrix and dispersed phases maintained intimate contact in the resultant nanocomposite, suppressing the formation of an extra interfacial layer. Previous investigations1,4,47 correlated the nanoparticle size with the total polymer volume of influence and, hence, the extent of gas permeability enhancement in the nanocomposite. In general, the number of filler particles inserted into a polymer G

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Figure 8. Gas self-diffusion coefficients in PCL and nanocomposite models. The insets show the relative gas self-diffusion coefficients, calculated as the ratio of self-diffusion coefficient in nanocomposite model to that in pure PCL control. The dashed lines are drawn to guide the eyes.

influence on the accessibility of large Connolly probe than that of the smaller one.

model that quantitatively predicts the degree of gas permeability deviation from classical composite theory may be possible. More information regarding this idea will be addressed in a follow-up paper. In traditional filled polymer systems, the presence of nonporous particles typically reduces the permeability of the polymer due to reduced polymer volume and increased diffusion tortuosity available for gas transport.21,22 Figure 8 presents the self-diffusion coefficients of H2, CO2, O2, N2, and CH4 in PCL and TiO2-filled nanocomposite models. Similar to experimental observations in dense polymer membranes, gas diffusion coefficients in both PCL-300 and PCL-350 decreased with increasing kinetic radius of the diffusants, showing a sizeselective feature. The combination of PCL and TiO2 nanoparticle resulted in completely opposite trends in gas diffusivity, depending on the interfacial morphologies. In PNT-350, the polymer−filler interface only represented an area of increased resistance to mass diffusion. Correspondingly, the self-diffusion coefficients of all five gas species in PNT-350 were found to be lower than the counterparts in unfilled control. From the inset in Figure 8, it was also observed that the diffusion coefficients of different gases in PNT-350 decreased by almost the same extent compared with control. According to classical composite theory,4 the gas diffusion coefficient in a composite, Df, is the diffusion coefficient in the unfilled polymer, D, divided by a tortuosity factor τ, defined as the ratio of the average diffusion path length in the composite to that in unfilled control. In PNT-350, the τ, determined only by the geometry and volume of the nanoparticle, should remain the same regardless of the diffusant size. Consequently, the same extent of decrease in diffusivity, or unchanged size selectivity, relative to unfilled control was observed. However, as displayed in Figure 8, all five gas species diffused faster in PNT-300 than in unfilled control. For example, the CH4 self-diffusion coefficient in PNT-300 was 1.91 × 10−7 cm2/s, more than 5 times that of the counterpart in PCL-300. This departure from traditional filled polymer behavior was attributed to increased free volume at the PCLTiO2 interface that lowered the activation energy for molecules to make a diffusive jump. In addition, the diffusion coefficients of different gases increased to different degrees in the presence of the nanoparticle. This finding was more clearly shown in the inset of Figure 8. It was found that the diffusivity of larger diffusant increased more than that of smaller one. This result was in good agreement with previous finding that the free volume enhancement in PNT-300 had a more significant

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CONCLUSIONS The melting transition of semicrystalline PCL10000 could be acceptably reproduced by a fully atomistic molecular modeling using the COMPASS force field and simulated annealing. Below and above the simulated Tm, the rigidity of the PCL chain model varied significantly as the real material did, which significantly influenced their packing efficiency around a nanoscale TiO2 particle and, hence, gas permeability of the resultant nanocomposite. Above the simulated Tm, the flexible nature of PCL chain model enabled them to wrap around the nanoparticle intimately; the net effect on gas transport was a more tortuous diffusion path and less free volume available for diffusion, which made the composite less permeable relative to unfilled control. As the temperature decreased below Tm, the inability of rigid PCL chains to change conformation easily, as well as the segmental stress arose from volume shrinkage, prevented the PCL chains from packing efficiently around the highly curved nanoparticle surface. This would lead to an interfacial region with depleted matrix phase that outweighed the volume filling and tortuosity effects to allow faster gas diffusion. With more experimental and simulated data on a wider set of polymers, the general applicability of the abovementioned correlation, particularly its potential usefulness for screening and designing membrane devices with a target or predetermined permeability, could be assessed.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (H.F.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge financial support from National Natural Science Foundation of China (21206096), New Teachers’ Fund for Doctor Stations, Ministry of Education of China (20120181120116), and Fundamental Research Funds for the Central Universities, China.

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