Perspective pubs.acs.org/Macromolecules
Polymer/Nanoparticle Interactions: Bridging the Gap Yogendra Narayan Pandey,† George J. Papakonstantopoulos,‡ and Manolis Doxastakis*,† †
Department of Chemical and Biomolecular Engineering, University of Houston, Houston, Texas 77004, United States Global Materials Science Division, The Goodyear Tire & Rubber Company, Akron, Ohio 44316, United States
‡
ABSTRACT: Materials created by dispersing nanoparticles in a polymer matrix strive to meet the promise of enhanced and often unique properties at a reduced cost. The availability of structure− property relationships and predictive modeling are deemed necessary to tailor materials according to our needs. However, the road from detailed information at the atomistic level to macroscopic properties has been severely segmented due to diverse experimental, theoretical, and modeling methods employed to study polymer−particle mixtures, each with their own advantages and limitations. In this Perspective, we focus on seemingly simple polymer−nanoparticle mixtures where nanoparticles are bare or grafted with chains of the same chemical constitution as the matrix. We present a number of studies that attempt to quantitatively identify where complete miscibility is achieved. As we discuss, features pertaining to the nanoscale dimensions of particles continue to challenge our fundamental understanding on polymer−particle interactions. However, through a concerted approach of theory, experiments, and simulations, recent studies significantly expand our knowledge on the morphological behavior of these systems. Most importantly, our discussion demonstrates how new developments bridge knowledge of microscopic interactions with thermodynamic behavior, an achievement that has far more reaching implications in the area of polymer−particle mixtures.
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INTRODUCTION Polymer−particle mixtures represent an exemplary manifestation of the persistent and critical link between structure and property in synthetic materials offering improved mechanical, electrical, barrier, fire-retardant, and optical properties. Our knowledge accumulated over many decades of research in composite materials continues to be challenged by recent progress in manipulating structure at the nanoscale.1 Prescribing design rules to achieve the desired morphology is based on our fundamental understanding of polymer−particle interactions and our ability to predict thermodynamics a priori to the synthesis of the material. To this extent, molecular theories and modeling have made large contributions in recent years, and our purpose herein is to provide selected examples of exciting developments in characterizing such interactions that drive macroscopic behavior. In addition, results presented highlight persisting limitations and opportunities for future research. Several reports in the literature provide comprehensive reviews of developments in the area of nanocomposites1−5 or target selected subjects, such as modeling methods, pertaining to the interest and expertise of the readers and authors.6−10 In this Perspective, we present a selected small number of studies driven by the same aim as our own research endeavorsto advance fundamental knowledge and establish predictive routes that link chemical detail to macroscopic properties. More specifically, we report on several recent intriguing findings, mainly within the past three years, that attempt to address how polymer−particle interactions change as we transition from the colloidal limit (or flat surface) to the nanoscale, the impact of such effects on the miscibility point, and finally how can we © XXXX American Chemical Society
predict such interactions in the absence of experimental input. We tactically refrain from referring to important developments that pertain i.e. to particle shape, multicomponent matrices, self-assembly, dynamics, metastable states, and other features of particle mixtures; each of these subjects deserves their own independent survey and is beyond the scope of this Perspective. However, despite our focus on the miscibility point, the studies presented are anticipated to have far more reaching implications on our ability to tailor dispersion and properties of the resulting material.
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SIGNIFICANCE OF POLYMER−NANOPARTICLE INTERACTIONS Molecular theories have made paramount progress in predicting equilibrium phase behavior and macroscopic properties of polymer nanocomposites. A diverse range of theoretical methods that include integral equation theory, density functional theory, and self-consistent mean-field theory are continuously developed and applied to broaden our knowledge.11−18 As an example, complete phase diagrams, such as shown in Figure 1a, are capable of describing the thermodynamics of nanoparticles within a polymer matrix as a function of interfacial attraction strength in units of thermal energy. Explicit in these diagrams are the boundaries of a miscible phase between depletion-governed aggregation and bridging states. Received: March 1, 2013 Revised: May 14, 2013
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Figure 1. (a) Nanoparticle volume fraction at spinodal phase separation predicted by PRISM theory for hard spheres in a freely jointed chain polymer melt as a function of the strength of exponential interfacial attraction. The depletion and bridging induced phase separated regimes bracket a window of miscibility at intermediate interfacial cohesion strength. The type of polymer-mediated nanoparticle organization is schematically indicated. Reproduced with permission from ref 16. Copyright 2010 Elsevier. (b) Threedimensional representation of domains of high-shear modulus for a solid nanocomposite. Reproduced with permission from ref 20. Copyright 2007 the American Physical Society.
al., it is imperative to couple predictions with computer simulations to create necessary benchmark tests for the structure and properties that are free of unavoidable statistical mechanical approximations present in theoretical modeling.16 Computer simulations can characterize dispersion, fluctuations, and heterogeneities in mechanical properties, provided that judicious measures of interfacial segment−particle interactions are introduced. As an example, Papakonstantopoulos et al. have used molecular simulations to quantify mechanical heterogeneities in a polymer nanocomposite.20,21 They explicitly identified regions of high-shear modulus domains in filled and unfilled systems and the morphological behavior of these mechanical heterogeneities (Figure 1b) in addition to the distribution of the fillers in their systems. A propensity of high modulus domains was found close to fillers of favorable interactions to the polymer and also between such fillers. When dealing with such systems of even simplified models of particle−polymer mixtures, adequate sampling remains a challenge, and application of Monte Carlo strategies that assist to overcome connectivity limitations is critical for equilibration purposes.22−25 Clearly, the accuracy of theoretical predictions and computer simulations depends on our ability to quantitatively characterize interfacial interactions. In fact, even the presence and the extent of a polymer layer with altered characteristics from the bulk material continues to be a subject of experimental investigation. Often contradictory findings are reported depending on sample preparation, technique employed, and system studied.26−28 We briefly add that the optimum dispersion state for reinforcement does not necessarily translate to overall improved material characteristics.1 Most importantly, it was recently shown that for the same nanocomposite material spatial dispersions that optimize properties are sensitive to state (melt versus glass).29 Such findings stress our need for modeling that accurately accounts for the overall effective polymer−particle interactions (as they emerge by the interplay of enthalpic and entropic contributions) and provides unequivocal guidance to materials design. To support our view, we selectively discuss recent progress made on our understanding and modeling of such interactions at the nanoscale and how they impact the miscibility state point in a polymer−particle mixture.
A critical part of theoretical methods is the ability to quantitatively describe interfacial layers surrounding nanoparticles, regions with distinct properties from the bulk polymer phase. Kim et al. using contrast-matching small-angle neutron scattering characterized such interfacial layers by measuring partial collective structure factors in stable concentrated ternary solutions of short-chain polymers, nanoparticles, and solvent.19 It was demonstrated that predictions by the microscopic polymer reference interaction site model (PRISM) theory were in quantitative agreement with experimental data. PRISM theory employs a freely jointed chain and an exponential monomer−particle interfacial attraction as the only adjustable parameter to extract information on the overall spatial organization of the mixture. One intriguing prediction made was that in the case of particles of nanoscale size in the miscible region, lowering interfacial interactions toward the depletion− aggregation boundary should make the composite material stiffer.15,19 In general, theoretical modeling offers explicit knowledge of the introduced parameters and approximations made and is often the most versatile and efficient technique to rationalize experimental behavior. Nevertheless, echoing Hall et
INTERFACIAL INTERACTIONS AND GRAFTED NANOPARTICLE SYSTEMS Grafting is a common method to alter the interactions between the polymer matrix and the dispersed nanoparticles.1,30 In some cases, controlled grafting can be exploited not only to achieve miscibility but also to tailor the shape of aggregates formed in the immiscible regime of the phase diagram.31−33 Concepts of polymer brushes have long been explored by seminal contributions in the field of polymer science,34−46 and the reader interested in theoretical and simulation aspects is referred to a comprehensive review by Binder and Milchev.47 We consciously restrict ourselves to recent findings on the “autophobic” effect as it applies in nanoparticle systems. In the special case where grafted chains and polymer matrix are of identical chemistry, no disparity in enthalpic interactions between segments of these components exists. At intermediate grafting density, the phase behavior of the nanoparticle− polymer mixture is governed by the mixing entropy of free chains exploring the additional volume within the grafted layer and the elastic free energy cost of stretching for the grafted chains. The key parameter is then the ratio between the length
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of grafted chains (N) and free chains (P): R = N/P.48,49 When free chains are shorter, they are able to penetrate into the grafted layer leading to a “wet” brush. As a result, particles can be well-dispersed into the matrix. In contrast, for short grafted chains, free polymer molecules are expelled, the brush becomes “dry”, and particles aggregate. The wetting region is bound by two grafting density limits: a lower σ* and a higher σ**. A theoretical phase diagram for this “autophobicity” as calculated for a flat surface (or particles with low curvature−colloidal limit) is shown in Figure 2a with σ* scaling as ∝ P1/2 and σ**, characterizing the boundary of a second-order autophobic transition, similar to ∝ P−1/2 or ∝ P−3/2.44,48 The application of this “wet-to-dry” transition to control the dispersion of silica nanoparticles grafted with polystyrene (PS) chains in a PS matrix was recently studied by Chevigny et al.49−51 A combination of neutron scattering with contrast variation was employed to extract the conformation of the grafted brushes in the matrix while X-ray scattering and microscopy assessed the dispersion state. While it was evident that tuning the length of the grafted chains provides a unique tool to tailor dispersion, the exact boundaries of this transition did not conform to theoretical predictions. Nanoparticles were well-dispersed for values of R = N/P > 0.24, suggesting that aggregation required much longer free polymer chains. This mismatch was attributed to processing kinetics, chains polydispersity effects, or surface curvature.49 At the nanoparticle limit, the increased curvature of the grafted particle reduces the crowding of grafted chains, effectively assisting free molecules to wet the brush. Selfconsistent field theoretical calculations by Trombly and Ganesan studied the limit of nanoparticles showing explicit changes in the width of the grafted polymer−free polymer interface as a function of R and curvature.13 Calculations supported that curved systems are less sensitive to grafting densities than what may be expected from considerations pertaining to flat surfaces. In addition, the matrix molecular weight has to be increased further than the corresponding flat surface in order to induce a dewetting transition. These advantages of highly curved nanoparticles were also brought forward in a molecular dynamics simulation by Smith and Bedrov, employing a “bead−spring” model for short grafted chains (N = 10) and free polymer molecules (P = 10−140).54 Significantly longer bead−spring models (above the entanglement threshold) were studied by Kalb and co-workers55 employing connectivity-altering algorithms (N = 501, P = 1000) and variable grafting density. Surprisingly, after detailed comparison to theoretical scaling laws, no signature of dewetting phenomena was observed, a feature attributed to the finite size of melt chains studied which displayed comparable size to the interface width. In addition, ends of tethered chains presented a higher probability to reside in proximity to the nanoparticle surface compared to earlier theoretical predictions.55 The latter finding provides an example of the necessity to complement theoretical predictions with accurate numerical calculations. A subsequent study with similar models (N = 30, 60, P = 10−140) supported that while the interface for a single nanoparticle does not provide any signature of autophobic dewetting in a regime where such a feature is expected, matrix-induced depletion attractions from free chains on a pair of nanoparticles dominate the free energy profile at intermediate separations.56 Thus, while aggregation remains entropic in origin, it is essential to consider collective affinities to form clusters in the presence of long free polymer
Figure 2. (a) Theoretical phase diagram in terms of the grafting density σ and the length of the chains of the polymer melt P for a constant graft length of N = 200.48 Reproduced with permission from ref 48. (b) Phase diagram for 10 nm PS-g-silica nanoparticles dispersed in PS matrices as determined by X-ray scattering and transmission electron microscopy. Reproduced with permission from ref 52. (c) Phase diagram for suspensions of 10 nm PEG-g-silica nanoparticles in polymeric hosts. Black symbols represent well-dispersed systems, red symbols denote completely phase-separated systems, and orange symbols denote partially phase-separated systems. Lines correspond to theoretical predictions for dewetting based on flat surfaces. Reproduced with permission from ref 53.
chains. This is even more significant at low-grafting densities where it was recently shown that grafted particles can selfassemble to anisotropic shapes.31 At this limit, favorable particle−particle interactions compete with the polymer brush C
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also worth noting that our past work has demonstrated how we can apply preferential sampling schemes, focusing computational effort in proximity to the surface and enhancing further sampling.24 This technique is particularly beneficial when large systems are considered, necessary to avoid finite-size effects due to periodic boundary conditions.
free energy; redistribution of the brush favors aggregation at the point diametrically opposite to the first contact. As a result, interactions between the particles become effectively anisotropic. Formation of self-assembled structures provides an additional level of control to the structure and properties of the polymer−particle mixture.29 We believe that a quantitative characterization of such many-body effects by calculations of free energy changes to form shapes in an explicit polymer matrix would contribute to our understanding of the miscibility limit and identification of metastables states. New additional experimental studies continue to supply data on the thermodynamics of grafted nanoparticles in a polymer matrix forming complete phase diagrams. Sunday et al. employed X-ray scattering and microscopy to study silica nanoparticles grafted with PS chains (PS-g-silica) in a PS matrix.52 As shown in Figure 2b, the P/N ratio for certain grafting densities could reach values up to 4.3 before aggregation was observed. The power laws extracted follow σ* ∝ P0.54 and σ** ∝ P−0.71. A different report for poly(ethylene glycol) (PEG) tethered silica nanoparticles dispersed in PEG hosts53 presented a phase diagram that bears some similarities to the data by Sunday et al. Specifically, Srivastava et al. argued that while indeed the P/N ratio can be as high as 5 maintaining a well-dispersed state, above a certain threshold (required to ensure screening of attractive forces) dispersion becomes independent of grafting density. However, a low particle diameter to size of grafted chains (D/Rg) ratio was necessary in order to maximize curvature effects at the small particle limit. Furthermore, by summarizing the results from previous studies (Figure 2c), it was argued that the phase diagram is independent of polymer chemistry.53 We note that theoretical modeling has stressed the role of particle size and, in addition, new simulations suggest that polydispersity effects can alter these boundaries.57,58 Accounting for polymer chemistry remains a daunting task for computer simulations. Studies are limited to a single or a pair of particles at most, immersed in oligomers.59,60 Ndoro studied atomistic models of PS-g-silica (N = P = 20), confirming geometric effects (higher available volume around a small nanoparticle) as discussed earlier.61 Peters et al. performed atomistic simulations with silica particles grafted with linear hydrocarbon chains (N = 9, 17, 35) in explicit solvents (P = 10, 24, 48 and a branched solvent),62 and Ghanbari et al. built coarse-grained (CG) models of PS and silica to extend the work of Ndoro to N = 80 and P = 20− 160.63 The last two studies have shown evidence of free chain migration out of the interfacial region which do support a “wetto-dry” transition. We clarify though that lengths reported in these works correspond to repeat units and not Kuhn segments. So the chains are shorter than in the systems employed by Kalb and co-workers.55 Brute-force atomistic modeling will remain severely limited within the foreseeable future in addressing truly polymeric systems. Long relaxation times restrict sampling in the vicinity of the initial state and hinder our ability to quantify thermodynamics of mixing. However, chemically specific CG models provide significant promise depending on the ability of simplified representations to reproduce the target structure including interfacial layers. Coupling such models with advanced simulation methods can expand the range of molecular weights studied. For example, Spyriouni and coworkers modeled bulk CG PS systems up to P = 100064 through connectivity-altering Monte Carlo (MC) methods. It is
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POLYMER-SEGMENT/PARTICLE INTERACTIONS: ACCOUNTING FOR CHEMICAL DETAIL Polymer chemistry relies on an ever-growing gallery of chemical repeat units to synthesize materials with desired properties. The majority of the theoretical and modeling results presented earlier require input or introduce approximations at the level of polymer-segment/particle interactions. Thus, a question that pertains to our discussion is whether interfacial phenomena between nanoparticles and polymers can be predicted without experimental characterization. Toward this aim, the field of quantum mechanics has made significant progress in the past decade.65 We anticipate that overcoming challenges associated with translating results from electronic calculations to polymer/ particle interactions will be in the center of future modeling efforts. We refer readers to a recent article that provides an excellent overview of the challenges faced66 and detail selected important points. Quantum calculations targeting interactions between small compounds and surfaces predict different absolute energies depending on the method employed. Nevertheless, it is the relative energies between adsorption sites and the monomer states that are important. Such energies need to be reproduced by classical force fields introduced in molecular simulations. This is verified by performing distancedependent scans as well as lateral- and orientation-dependent evaluations.66 These tests are critical to ensure that a representative ensemble of configurations as a function of a multitude of order parameters is retrieved by force-field simulations. Figure 3 depicts interaction energies between a single PS monomer and a gold surface as calculated by density functional theory.67 Johnston and Harmandaris stressed the need to introduce a Morse functional form in classical modeling to simultaneously capture both distance and orientation dependent interaction energies. These potentials were subsequently employed in classical molecular dynamics simulations to study the properties of short polystyrene chains confined between two gold surfaces.67 It is noted that a building-block approach, where interactions of a specific segment extracted by electronic calculations are treated in an additive approach to model macromolecules on surfaces,68−71 remains highly nontrivial. A large challenge in these methods is the interplay of enthalpic contributions with nonlocal entropic effects that are difficult to account for in original electronic calculations.66,72 We note, therefore, that translating chemically specific segment interactions to polymeric molecules is an area that requires particular development. Challenges discussed in bridging quantum calculations to classical atomistic force fields pertain to methods available to systematically derive CG models.73,74 However, significant progress has already been made, and it is now feasible to build polymer−segment interactions based on input from atomistic simulations with classical force fields. As mentioned earlier, PSg-silica nanoparticles in a PS matrix were studied by Ghanbari and co-workers employing such a scheme.63 In their study, Si atoms served as the centers of interaction sites with a polymer CG segment and pairwise interaction potentials were determined through the iterative Boltzmann inversion (IBI) D
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or the COM of the particle is considered. In fact, the affinity of polymer segments to track the curved nanoparticle surface depends on particle curvature.24 As we will show herein, we can extract coarse-grain curvature-dependent potentials based on atomistic simulations with flat surfaces that are computationally less demanding. Interactions with implicit flat surfaces can be computed by integration of pairwise potentials over the atoms constituting the solid slab; in the case of Lennard-Jones interactions this results to 10−4 (single solid layer) or 9−3 (multilayer) power law expressions for the repulsive and attractive part of the potential.81 However, when the interaction of the COM of a group of atoms with a flat surface is considered, conformational characteristics alter the shape of the corresponding effective potential.82 The need to properly account for changes in the ensemble of configurations of the group of atoms that are now represented by a single site is also manifested when considering changes in the resolution of the CG model. Such features in theoretical modeling of melts in proximity to surfaces were addressed by the work of de Pablo and co-workers.83,84 Khounlavong et al. suggested a recalculation of the effective interactions with the nanoparticle which not surprisingly leads to different (but still accurate) effective potentials.77 Given the growing application of chemically specific CG models, we decided to independently assess the ability to extract polymer-segment/nanoparticle potentials based on atomistic simulations on a flat surface. Our motivation was to evaluate if features beyond the total density profile (as conformations of chains in contact with the surface) are captured accurately and whether we can efficiently predict changes with particle curvature. Following the literature, we performed bulk simulations with a fully atomistic model of cis1,4-polyisoprene (PI)85,86 and employed the IBI method (with pressure corrections as described by Wang et al.87) to extract CG potentials for polymer segments centered along the bond connecting consecutive repeat units (interactions were cutoff at 1.5 nm).75,88−90 All simulations were performed with a 24 repeat units PI to avoid chain-end effects present for lower molecular weights.91,92 The next step of our quest included atomistic molecular dynamics simulations of PI on an amorphous silica slab and around a silica particle of radius 2 nm referred to as SIL-2.0 (as in our previous work24). Simulations were performed with 400 chains of 24 repeat units) for the slab (141 032 atoms) and 800 chains of equal length for the single particle system (253 854 atoms) at T = 413 K and P = 1 bar (semi-isotropic coupling for slab) using the software Gromacs.93 Note that the generation of 50 ns trajectories for these large systems (necessary to avoid finite size effects) requires several months of computational time even with highly efficient software as Gromacs. Subsequently, atomistic trajectories were mapped to CG configurations created with the fully atomistic potential. Figure 4a presents snapshots of atomistic simulations where chains in contact with the surface or the particle are depicted based on the CG representation (single site for each repeat unit). The distribution of CG beads as a function of distance from the surface extracted from the atomistic trajectories is shown in Figure 4b (solid lines) together with a decomposition to trains, tails, and loops based on a distance criterion of 0.4 nm to define a bead−surface contact. The second step of our procedure was to extract a wall−CG bead potential via IBI targeting only the total density prof ile (black line) and maintaining the same CG bead−CG bead and intramolecular polymer interactions determined from
Figure 3. Adsorption energy of PS monomer as a function of the distance from a gold surface. The density functional theory results (symbols) are compared to the results from the pair potentials (lines) obtained from an optimization procedure using (a) Lennard-Jones and (b) Morse nonbonded potentials. Reproduced with permission from ref 67. Copyright 2012 The Royal Society of Chemistry.
method.75 A thorough test of this technique for bulk systems appeared in the literature recently.76 Note that in the study by Ghanbari and co-workers challenges in matching the bare nanoparticle−polymer density persisted with CG polymer segments forming density layers of higher packing than the corresponding results from atomistic simulations.63 Moving back to atomistic representations resulted to a better agreement to atomistic models, supporting that discrepancies are due to fundamental differences between an atomistic and the spherical CG representation of beads in a PS segment. In another recent study, Khounlavong et al. determined the polymer-segment/ particle effective potential that acts between the center of mass (COM) of a nanoparticle and a CG polymer bead.77 This method presents additional computational advantages due to reduction to a single interaction site for each solid nanoparticle.78 Simulations with the extracted CG representations of polymer revealed the need for additional correction terms to the nanoparticle−nanoparticle potential as a side effect of the CG polymer representation. Similar findings were reported by Hong et al. for bare and grafted silica nanoparticles dispersed in poly(ethylene oxide) oligomers.79 Thus, it is clear that further development and testing of the ability of coarse-graining to capture confinement effects is needed. For nanoconfinement between flat surfaces, a test of IBI with polyamide-6,6 between graphene was presented by Eslami and co-workers.80 While there were some deviations in the total density profile, the peaks observed in the atomistic simulations were captured by the simpler CG model. However, the surface was again represented by a collection of CG beads as in the study of Ghanbari and co-workers. While retaining interactions sites for surface atoms could result to potentials that are curvatureindependent, this is not necessarily the case when a single interaction potential as a function of distance from the surface E
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Figure 4. (a) Rendered images from atomistic simulations of cis-1,4-PI on silica. Chains in contact with silica are explicitly depicted by spheres according to the CG scheme utilized. (b) Radial distribution functions of CG beads and their decomposition as a function of distance from the slab surface. The solid lines represent the distributions obtained from the atomistic trajectories mapped to CG sites, and symbols represent the data from CG NVT simulations based upon derived effective potentials. (c) CG potentials extracted for different radii. For the slab system, IBI and atomistic simulations were performed while for particles potentials were determined based on effective interactions of polymer segments with the flat surface. (d) Same as (b) with CG simulations of the SIL-2.0 particle system based on an estimate of the particle−polymer bead interaction.
COM separation shown for different radii in Figure 4c. These potentials can now serve to model single nanoparticles immersed in CG representations of the polymer matrix and extract density profiles as performed previously for the flat surface. Figure 4d shows that our CG model, which accounts for chemical stiffness through bonded, angular, and dihedral potential terms, is able to capture conformations in proximity to the particle in excellent agreement with independent fully atomistic simulations. Deviations are observed only at short separations between the particle COM and the polymer beads, an anticipated outcome given that roughness is increasingly important for smaller particles24 and is not accounted for explicitly in the single pairwise potential employed. Finally, we investigated whether such models can capture features beyond the mean density profile at a specific separation. As discussed earlier, this is a critical test when evaluating the performance of classical force fields with respect to data from quantum calculations.66 Using our simple contact criterion, we can examine changes in free energy along the surface by calculating the probability to form a train segment of consecutive CG beads of length S, Ptr(S). Simulations with bead−spring models recently demonstrated that the train size distribution depends on the strength of the interaction between polymer segments and the surface.96 Figure 5a shows that these
bulk polymer simulations. The resulting numerical potential is shown in Figure 4c with final density profiles contrasted to CG beads profiles from atomistic simulations shown in Figure 4b as symbols. A major finding of our study was that subsequent CG simulations with the derived potentials reproduced almost quantitatively the conformational characteristics (trains, tails, and loops) of the polymer layer in contact with the surface. While it remains to be tested whether this is the case for other polymers as well, it is particularly encouraging that CG simulations meet successfully this challenge. We anticipate that the quantitative representation of the total density profile of the melt suffices for melt systems; this CG scheme has been applied for PS solutions,94 and certainly future research needs to focus on blends and copolymers. Within the main theme of our article we decided to examine further the predictive ability to construct polymer-segment/ particle effective interactions for curved nanoparticles. An explicit per surface site pairwise interaction can be reconstructed using the CG−slab potential calculated from the IBI and the number density of silica sites using a procedure outlined by Nielsen et al.82,95 Numerical integration of the per site interaction over the nanoparticle volume within the cutoff distance allows the estimation of a single, pairwise, spherical nanoparticle−polymer CG segment potential as a function of F
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previously unaccounted for. To support our view, we chose to present herein studies in recent years that advanced our knowledge regarding the effect of nanoscale dimensions on interfacial interactions and complete miscibility in seemingly simple systems, nanoparticles, grafted or bare, within a homopolymer matrix. Control over dispersion was brought forward by a series of experimental studies that are now able to construct phase diagrams and probe material structure with unprecedented detail. Experiments continue to explore and propose structure− property relationships necessary to design improved materials.29,98−101 Theoretical modeling continues to offer the most versatile and efficient route toward providing predictions and establishing reasoning for experimental findings. However, given the underlying approximations, computer simulations need to continue to test and challenge the accuracy of theory. Our aim was to present that there is still progress to be made on quantitatively characterizing the underlying factors that drive the observed behavior. We have provided herein several examples on the miscibility limits of grafted-nanoparticles immersed in a matrix of similar chemical constitution to justify this point of view. While we acknowledge this critical role of modeling, it is also important to recognize the prospect of hierarchical materials design. Quantum calculations can now offer detailed information on the interactions between polymer segments in proximity to the surface, even for systems that are yet to be realized in the laboratory. It is anticipated that an increasing number of studies will focus on translating data from such calculations to classical force fields with several challenges to be met. Force-field-based atomistic modeling suffices to probe ensembles of conformations in a specific environment and major progress in systematically derived efficient coarse-grain representations has already been made. It appears that it is now feasible to employ such specific models to capture the structure of interfacial layers at the nanoscale. An ambitious aim for the future is to systematically develop a library of interaction potentials for specific polymer−surface chemistry which will consolidate the efforts in predictive modeling of polymer− nanoparticle interactions. This will allow comprehensive tests on the performance of these models by the community. Futhermore, theoretical modeling will have the opportunity to foster on effective interactions extracted and provide predictions for the phase behavior without experimental input. Multiscale modeling has long been conceived as the only viable route toward predicting properties of nanocomposites;10,102 the recent progress in fundamental understanding of polymer−particle interactions is now bridging the gaps and completing the realization of this ambitious aim.
Figure 5. (a) Probability distributions for train segments from CG simulations and atomistic simulations. (b) Normalized RMS bound layer thickness as a function of particle size relative to polymer Kuhn segment length. Data on particles immersed in polyethylene melts are taken from our previous study.24
probabilities depend on curvature as we described in our previous atomistic simulations.24 Most importantly, the CG model developed herein for PI/silica systems appears to accurately capture the distributions both for the slab and the SIL-2.0 system. To the best of our knowledge, this is the first time where not only mean density profiles are matched, but fluctuations on the surface are described faithfully and throughout different curvatures. As a final result, Figure 5b shows that the interfacial layer thickness (normalized to the flat surface result) quantified for polyethylene on silica through atomistic simulations reported previously24 presents excellent agreement with our new CG PI data when the latter are scaled with the corresponding Kuhn length (0.82 nm97). These results provide a first indication that the scaling of the layer follows a universal behavior when polymer stiffness is accounted for. Overall, application of systematic CG techniques presents significant potential to capture curvature effects in polymer− nanoparticle interactions. We believe that further research with different polymer chemical architectures, surfaces, many particles, blends, and copolymers is required to establish the capabilities and limitations of these simpler effective potentials in capturing the behavior of polymer−particle mixtures.
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OUTLOOK Progress made in polymer synthesis and characterization methods reveals that we are not far from tailoring dispersion and morphology of nanoparticles within a polymer matrix.1 Despite extensive fundamental knowledge accumulated over many decades on the thermodynamics of these systems, as we move to the nanoscale, further challenges appear with features
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (M.D.). Notes
The authors declare no competing financial interest. G
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Biographies
Manolis Doxastakis received his Diploma in Chemical Engineering from the National Technical University of Athens in 1996. He received a master’s in Materials Science and Ph.D. in Chemical Engineering from the University of Patras in 2002. He worked with Professors Doros N. Theodorou and George Fytas designing and performing simulations and experiments on polymer melts, blends, and copolymers. He moved to the University of Wisconsin in Madison, working as a postdoctoral fellow with Professor Juan J. de Pablo in 2003, investigating polymer−particle mixtures and lipid membranes. Since 2007, he is an assistant professor in Chemical and Biomolecular Engineering at the University of Houston, studying interfacial phenomena with molecular modeling.
Yogendra Narayan Pandey is a senior graduate student at the department of Chemical and Biomolecular Engineering at the University of Houston. Yogendra defended his PhD in April 2013. He worked with Professor Manolis Doxastakis on modeling and simulation of materials, e.g., polymer−nanoparticle systems, polymer thin films, and zeolites. Earlier, he received Bachelor of Technology (B. Tech.) in Chemical Engineering and Technology from Institute of
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Technology, Banaras Hindu University, Varanasi, India, in 2005. Before joining graduate school in Fall 2008, he worked as a software
ACKNOWLEDGMENTS We acknowledge financial support by the National Science Foundation under Grant CBET-1067356 and The Goodyear Tire and Rubber Company.
developer (2005−2008) with leading information technology companies in India.
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George Papakonstantopoulos received his Diploma in Chemical Engineering from the National Technical University of Athens in 2002. He received his PhD in Chemical Engineering in 2007 from the University of WisconsinMadison. He worked with Professors Juan J. de Pablo and Paul F. Nealey in the areas of polymeric nanocomposites and block copolymer self-assembly using molecular modeling techniques. He worked for Arkema Inc. (2007−2010) in the Analytical and Systems Research department as a Scientist, mainly investigating the modification of the mechanical behavior of polymeric materials with the aid of molecular modeling and microscopy techniques. Currently he works for the Goodyear Tire and Rubber Company as a Senior Scientist in the area of Multiscale Modeling of polymeric composites and polymer blends. H
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