Why Volume and Dynamics Decouple in Nanocomposite Matrices: Space that Cannot Be Accessed is Not Free.

Polymer nanocomposites have important material applications and are an ongoing focus of many molecular level investigations, however, puzzling experimental results exist. For example, specific volumes for some polymer nanocomposite matrices are 2% to 4% higher than for the neat polymer; in a pure polymer melt this would correspond to a pressure change of 40 to 100 MPa, and a decrease in isothermal segmental relaxation times of 3 to 5 orders of magnitude. However, the nanocomposite segmental dynamics do not show any speed up. We can explain this apparent uncoupling of dynamics from specific volume, and the key is to consider the system expansivity, i.e., the temperature dependence of the volumetric data, together with the concept of limiting volume at close liquid packing. Using pressure, volume, temperature data as a path to both, we are able to predict the effect of nanoadditives on the accessible, i.e., free, space in the material, which is critical for facilitating molecular rearrangements in dense systems. Our analysis explains why an increase in specific volume in a material may not always lead to faster segmental dynamics.

Spectroscopic ellipsometry as a route to thermodynamic characterization.

Strategies for synthesizing molecularly designed materials are expanding, but methods for their thermodynamic characterization are not. This shortfall presents a challenge to the goal of connecting local molecular structure with material properties and response. Fundamental thermodynamic quantities, including the thermal expansion coefficient, α, can serve as powerful inputs to models, yielding insight and predictive power for phenomena ranging from miscibility to dynamic relaxation. However, the usual routes for thermodynamic characterization often require a significant sample size (e.g. one gram), or challenging experimental set-ups (e.g. mercury as a confining fluid), or both. Here, we apply spectroscopic ellipsometry, which is an optical technique for thin film analysis, to obtain thermodynamic data. We clarify issues in the scientific literature concerning the connection between ellipsometric and volumetric thermal expansion coefficients for substances in both the glass and melt states. We analyze temperature-dependent data derived using both ellipsometry and macro-scale dilatometric techniques for ten different polymers. We find superb correlation between the α values obtained via the two techniques, after considering the effects of mechanical confinement by the substrate for a glassy thin film. We show how the ellipsometric α can serve as input to the locally correlated lattice theory to yield predictions for the percent free volume in each polymer as a function of temperature. We find that the ellipsometric α at the glass transition temperature, Tg, is not only material dependent, but it is linearly correlated with Tg itself. Spectroscopic ellipsometry, which requires only very small quantities of sample and is straightforward to perform, will significantly expand the range of systems for which thermodynamic properties can be characterized. It will thus advance our ability to use theory and modeling to predict the miscibility and dynamic relaxation of new materials. As such, ellipsometry will be able to underpin materials synthesis and property design.

The dynamics of freestanding films: predictions for poly(2-chlorostyrene) based on bulk pressure dependence and thoughtful sample averaging.

In this paper we model the segmental relaxation in poly(2-chlorostyrene) 18 nm freestanding films, using only data on bulk samples to characterize the system, and predict film relaxation times (τ) as a function of temperature that are in semi-quantitative agreement with film data. The ability to translate bulk characterization into film predictions is a direct result of our previous work connecting the effects of free surfaces in films with those of changing pressure in the bulk. Our approach combines the Locally Correlated Lattice (LCL) equation of state for prediction of free volume values (Vfree) at any given density (ρ), which are then used in the Cooperative Free Volume (CFV) rate model to predict τ(T, Vfree). A key feature of this work is that we calculate the locally averaged density profile as a function of distance from the surface, ρav(z), using the CFV-predicted lengthscale, Lcoop(z), over which rearranging molecular segments cooperate. As we have shown in the past, ρav(z) is significantly broader than the localized profile, ρ(z), which translates into a relaxation profile, τ(z), exhibiting a breadth that mirrors experimental and simulated results. In addition, we discuss the importance of averaging the log of position dependent relaxation times across a film sample (〈log τ(z)〉), as opposed to averaging the relaxation times, themselves, in order to best approximate a whole sample-averaged value that can be directly compared to experiment.

To Understand Film Dynamics Look to the Bulk.

We show that shifts in dynamics of confined systems relative to that of the bulk material originate in the properties of bulk alone, and exhibit the same form of behavior as when different bulk isobars are compared. For bulk material, pressure-dependent structural relaxation times follow τ(T,V)∝exp[f(T)×g(V)]. When two states (isobars) of the material, "1" and "2", are compared at the same temperature this leads to a form τ_{2}∝τ_{1}^{c}, where c=g[V_{2}(T)]/g[V_{1}(T)]. Using equation of state analysis and two models for P-dependent dynamics, we show that c is approximately T independent, and that it can be very simply expressed in terms of either the (free) volume above the close packed state (V_{free}) or the activation energy for cooperative motion. The effect of changing state through a shift in pressure (P_{1} to P_{2}) is thus mechanistically traceable to cooperativity changing with density, through V_{free}. The connection with confined dynamics follows when 1 and 2 are taken as bulk and film at ambient P, differing in density only due to the film surface. The general form for τ(T,V) also illuminates why samples in different states (film vs bulk, high P vs low) trend toward the same relaxation behavior at high T.

Substrate Roughness Speeds Up Segmental Dynamics of Thin Polymer Films.

We show that the segmental mobility of thin films of poly(4-chlorostyrene) prepared under nonequilibrium conditions gets enhanced in the proximity of rough substrates. This trend is in contrast to existing treatments of roughness which conclude it is a source of slower dynamics, and to measurements of thin films of poly(2-vinylpiridine), whose dynamics is roughness invariant. Our experimental evidence indicates the faster interfacial dynamics originate from a reduction in interfacial density, due to the noncomplete filling of substrate asperities. Importantly, our results agree with the same scaling that describes the density dependence of bulk materials, correlating segmental mobility to a term exponential in the specific volume, and with empirical relations linking an increase in glass transition temperature to larger interfacial energy.

The cooperative free volume rate model for segmental dynamics: Application to glass-forming liquids and connections with the density scaling approach⋆.

In this paper, we apply the cooperative free volume (CFV) rate model for pressure-dependent dynamics of glass-forming liquids and polymer melts. We analyze segmental relaxation times, [Formula: see text] , as a function of temperature (T and free volume ( [Formula: see text] , and make substantive comparisons with the density scaling approach. [Formula: see text] , the difference between the total volume (V and the volume at close-packing, is predicted independently of the dynamics for any temperature and pressure using the locally correlated lattice (LCL) equation-of-state (EOS) analysis of characteristic thermodynamic data. We discuss the underlying physical motivation in the CFV and density scaling models, and show that their key, respective, material parameters are connected, where the CFV b parameter and the density scaling [Formula: see text] parameter each characterize the relative sensitivity of dynamics to changes in T , vs. changes in V . We find [Formula: see text] , where [Formula: see text] is the value predicted by the LCL EOS at the ambient [Formula: see text] . In comparing the predictive power of the two models we highlight the CFV advantage in yielding a universal linear collapse of relaxation data using a minimal set of parameters, compared to the same parameter space yielding a changing functional form in the density scaling approach. Further, we demonstrate that in the low data limit, where there is not enough data to characterize the density scaling model, the CFV model may still be successfully applied, and we even use it to predict the correct [Formula: see text] parameter.

Explaining the T,V-dependent dynamics of glass forming liquids: The cooperative free volume model tested against new simulation results.

In this article, we derive a rate model, the "cooperative free volume" (CFV) model, to explain relaxation dynamics in terms of a system's free volume, Vfree, and its temperature, T, over widely varied pressure dependent conditions. In the CFV model, the rate a molecule moves a distance on the order of its own size is dependent on the cooperation of surrounding molecules to open up enough free space. To test CFV, we have generated extensive T,V dependent simulation data for structural relaxation times, τ, on a Kob and Andersen type Lennard-Jones (KA-LJ) fluid. The Vfree = V - Vhc values are obtained by estimating the limiting hard core volume, Vhc, through analysis of the KA-LJ PVT data. We provide the first simulation evidence that shows ln τ to be linearly proportional to 1/Vfree on isotherms, with T-dependent slopes, thus confirming our recent analysis of experimental systems. The linear relationship exhibited by the simulation data is further shown to occur at temperatures both above and below the transition to Arrhenius behavior. We also show that the gas kinetic T-dependent contribution is important in simulation results and that there can be a significant entropic contribution from lingering molecular hard-cores at high T. A key result is that non-Arrhenius relaxation behavior is always exhibited on isobars of the KA-LJ fluid, even at high T. The CFV model predicts all of this behavior over a surprisingly wide range of the KA-LJ T,V space, fitting it with just a single set of three parameters. The CFV approach leads to a framework wherein the number of cooperating particles, and thus, the process free energy of activation, is inversely proportional to Vfree, and this is the foundation for the form of the model's volume contribution, a form that we find to hold for all systems and at all temperatures.

Experimental and Modeling Comparison of the Dynamics of Capped and Freestanding Poly(2-chlorostyrene) Films.

Proximity to a nonrepulsive wall is commonly considered to cause slower dynamics, which should lead to greater relaxation times for capped thin polymer films than for bulk melts. To the contrary, here we demonstrate that Al-capped films of poly(2-chlorostyrene) exhibit enhanced dynamics with respect to the bulk, similar to analogous freestanding films. To quantitatively resolve the impact of interfaces on whole film dynamics, we analyzed the experimental data via the Cooperative Free Volume rate model. We found that the interfacial region adjacent to a cap contains an excess of free volume (relative to the bulk) about half of that proximate to a free surface. Employing a useful analogy between confinement and pressure effects, we estimated that the effect of capping an 18 nm freestanding film would be equivalent to applying a pressure increase of 19 MPa.

A Simple New Way To Account for Free Volume in Glassy Dynamics: Model-Free Estimation of the Close-Packed Volume from PVT Data.

In this article we focus on the important role of well-defined free volume (Vfree) in dictating the structural relaxation times, τ, of glass-forming liquids and polymer melts. Our definition of Vfree = V - Vhc, where V is the total system volume, means the use of Vfree depends on determination of Vhc, the system's volume in the limiting closely packed state. Rejecting the historically compromised use of Vfree as a dynamics-dependent fitting function, we have successfully applied a clear thermodynamics-based route to Vhc using the locally correlated lattice (LCL) model equation of state (EOS). However, in this work we go further and show that Vhc can be defined without the use of an equation of state by direct linear extrapolation of a V(T) high-pressure isobar down to zero temperature (T). The results from this route, tested on a dozen experimental systems, yield ln τ vs 1/Vfree isotherms that are linear with T-dependent slopes, consistent with the general ln τ ∼ f(T) × (1/Vfree) form of behavior we have previously described. This functional form also results by implementing a simple mechanistic explanation via the cooperative free volume (CFV) rate model, which assumes that dynamic relaxation is both thermally activated and that it requires molecular segmental cooperativity. With the degree of the latter, and thus the activation energy, being determined by the availability of free volume, the new route we demonstrate here for determination of Vfree expands the potential for understanding and predicting local dynamic relaxation in glass-forming materials.

Methods for calculating the entropy and free energy and their application to problems involving protein flexibility and ligand binding.

The Helmholtz free energy, F and the entropy, S are related thermodynamic quantities with a special importance in structural biology. We describe the difficulties in calculating these quantities and review recent methodological developments. Because protein flexibility is essential for function and ligand binding, we discuss the related problems involved in the definition, simulation, and free energy calculation of microstates (such as the alpha-helical region of a peptide). While the review is broad, a special emphasize is given to methods for calculating the absolute F (S), where our HSMC(D) method is described in some detail.

Chain fluids: contrasts of theoretical and simulation approaches, and comparison with experimental alkane properties.

In this work, we undertake a fundamental comparison of analogous lattice and continuum integral equation theories, with both held accountable to the results from Monte Carlo simulation and real experimental data on short chain molecules. Each integral equation method is applied to determine the system's microscopic intermolecular site-site distributions and the corresponding bulk thermodynamic properties. These properties and those from the MC simulations are then fitted to corresponding data on n-alkanes. Thus, in side-by-side comparisons we cover a number of fundamental contrasts: theory versus simulation, lattice-based theory versus continuum-based theory, and thermodynamic properties of model chain molecules versus the actual experimental properties of hydrocarbons. The observed behavior of the modeling methods is compared in terms of both the experimentally accessible physical properties (e.g., PVT and coexistence properties) and the more fundamental underlying quantities, such as free energies and model internal energies. We also discuss the various options for model parametrization and subsequent impact on the predicted physical properties. The results from this work are used (alone, with no additional fitting) in the article which follows, wherein we predict the experimental properties of alkane mixtures.

Fluid mixtures: contrasts of theoretical and simulation approaches, and comparison with experimental alkane properties.

In this work we compare lattice and continuum versions of the same theory, as derived in the previous paper (Paper I), to predict the behavior of simple alkane mixtures. In the course of doing this we use characteristic parameters obtained for the pure alkane fluids; no fits of mixture properties are involved. Our two sets of predictions are tested against experimental data and against new Monte Carlo simulation results. The experimental properties of interest include mixed pressure-volume-temperature surfaces, as well as a variety of coexistence diagrams characterizing mixed system liquid-vapor equilibria. We contrast the performance of the lattice and continuum approaches and discuss the results in the context of underlying model approximations as derived in Paper I.

Experimental Test of the Cooperative Free Volume Rate Model under 1D Confinement: The Interplay of Free Volume, Temperature, and Polymer Film Thickness in Driving Segmental Mobility.

We show that the Cooperative Free Volume (CFV) rate model, successful at modeling pressure-dependent dynamics, can be employed to describe the temperature and thickness dependence of the segmental time of polymers confined in thin films (1D confinement). The CFV model is based on an activation free energy that increases with the number of cooperating segments, which is determined by the system's free volume. Here, we apply the CFV model to new experimental results on the segmental relaxation of 1D confined poly(4-chlorostyrene), P4ClS, and find remarkable agreement over the whole temperature and thickness ranges investigated. This work further validates the robustness of the CFV model, which relates the effects of confinement on dynamics to pressure changes in the bulk, and supports the idea that confinement effects originate from local perturbations in density.

How Free Volume Does Influence the Dynamics of Glass Forming Liquids.

In this article we show that inverse free volume is a natural variable for analyzing relaxation data on glass-forming liquids, and that systems obey the general form, log(τ/τref) = (1/Vfree) × f(T), where f(T) is a function of temperature. We demonstrate for eight glass-forming liquids that when experimental relaxation times (log τ), captured over a broad pressure-volume-temperature (PVT) space, are plotted as a function of inverse free volume (1/Vfree), a fan-like set of straight line isotherms with T-dependent slopes ensues. The free volume is predicted independently of the dynamic results for each state point using PVT data and the Locally Correlated Lattice (LCL) equation of state. Taking f(T) ∝ 1/Tb, we show that, for each of the systems studied, only the single, system-dependent parameter, b, is required to collapse the fan of linear isotherms into a straight line. We conclude that log τ is a function of the combined variable, 1/(VfreeTb), and because it is linear, it allows us to write an explicit analytic expression for log τ that covers a broad PVT space.

Calculation of the Entropy of Lattice Polymer Models from Monte Carlo Trajectories.

While lattice models are used extensively for macromolecules (synthetic polymers proteins, etc), calculation of the absolute entropy, S, and the free energy, F, from a given Monte Carlo (MC) trajectory is not straightforward. Recently we have developed the hypothetical scanning MC (HSMC) method for calculating S and F of fluids. Here we extend HSMC to self-avoiding walks on a square lattice and discuss its wide applicability to complex polymer lattice models. HSMC is independent of existing techniques and thus constitutes an independent research tool; it provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation.

Free Volume in the Melt and How It Correlates with Experimental Glass Transition Temperatures: Results for a Large Set of Polymers.

There is a continuing, strong interest in making connections between the polymeric glass transition (Tg) and bulk properties. In this Letter we apply the Locally Correlated Lattice (LCL) model to study a group of 51 polymers and demonstrate two broad correlations. In the first, we show that the theoretically determined polymeric free volume in the melt, all at a single common T, P (425 K, 1 atm), correlates noticeably with the experimentally determined Tg values, and that this trend sharpens considerably when families of polymers are examined. Further, we show a strikingly linear correlation between the experimental Tg and the LCL model calculation for the percent free volume expected at the polymeric Tg. We suggest that this trend has a predictive value, acting as a boundary of T-dependent minimum-required free volume separating the melt and glassy regimes. Our theoretical estimates of free volume values at a polymer's Tg range between 4 and 16%, and their evident temperature dependence indicates an important role for temperature in glassification.

Thermodynamic treatment of polymer thin-film glasses.

We propose a new, simple, thermodynamically based model in order to study the effect of film thickness on the glass transition of a polymer. The model equation of state incorporates the effect of a free surface by accounting for missing interactions, and is parametrized using experimental data for bulk samples, leaving no freely adjustable parameters. To this end we focus on connecting model agreement in the bulk limit with two key physical properties: pressure-volume-temperature data and surface tension data. For the case of polystyrene we show that accounting for these effects, alone, yields results in surprisingly good agreement with experiment for the shift in glass transition in going from bulk to freestanding film. Insight from this simple model indicates that only a small increase in the specific volume of a thin film (compared to bulk) may drive the shift in the glass transition temperature. Herein we focus on a molecular weight independent regime. Looking ahead we consider the possible contributions of chain confinement to the thermodynamic properties of the film.

A simple approach to polymer mixture miscibility.

Polymeric mixtures are important materials, but the control and understanding of mixing behaviour poses problems. The original Flory-Huggins theoretical approach, using a lattice model to compute the statistical thermodynamics, provides the basic understanding of the thermodynamic processes involved but is deficient in describing most real systems, and has little or no predictive capability. We have developed an approach using a lattice integral equation theory, and in this paper we demonstrate that this not only describes well the literature data on polymer mixtures but allows new insights into the behaviour of polymers and their mixtures. The characteristic parameters obtained by fitting the data have been successfully shown to be transferable from one dataset to another, to be able to correctly predict behaviour outside the experimental range of the original data and to allow meaningful comparisons to be made between different polymer mixtures.

Minimalist explicit solvation models for surface loops in proteins.

We have performed molecular dynamics simulations of protein surface loops solvated by explicit water, where a prime focus of the study is the small numbers (e.g., ~100) of explicit water molecules employed. The models include only part of the protein (typically 500 - 1000 atoms), and the water molecules are restricted to a region surrounding the loop. In this study, the number of water molecules (N(w)) is systematically varied, and convergence with large N(w) is monitored to reveal N(w)(min), the minimum number required for the loop to exhibit realistic (fully hydrated) behavior. We have also studied protein surface coverage, as well as diffusion and residence times for water molecules as a function of N(w). A number of other modeling parameters are also tested. These include the number of environmental protein atoms explicitly considered in the model, as well as two ways to constrain the water molecules to the vicinity of the loop (where we find one of these methods to perform better when N(w) is small). The results (for RMSD and its fluctuations for four loops) are further compared to much larger, fully solvated systems (using ~10,000 water molecules under periodic boundary conditions and Ewald electrostatics), and to results for the GBSA implicit solvation model. We find that the loop backbone can stabilize with a surprisingly small number of water molecules (as low as 5 molecules per amino acid residue). The side chains of the loop require somewhat larger N(w), where the atomic fluctuations become too small if N(w) is further reduced. Thus, in general, we find adequate hydration to occur at roughly 12 water molecules per residue. This is an important result, because at this hydration level, computational times are comparable to those required for GBSA. Therefore these "minimalist explicit models" can provide a viable and potentially more accurate alternative. The importance of protein loop modeling is discussed in the context of these, and other, loop models, along with other challenges including the relevance of appropriate free energy simulation methodology for assessment of conformational stability.

Free volume hypothetical scanning molecular dynamics method for the absolute free energy of liquids.

The hypothetical scanning (HS) method is a general approach for calculating the absolute entropy, S, and free energy, F, by analyzing Boltzmann samples obtained by Monte Carlo (MC) or molecular dynamics (MD) techniques. With HS applied to a fluid, each configuration i of the sample is reconstructed by gradually placing the molecules in their positions at i using transition probabilities (TPs). With our recent version of HS, called HSMC-EV, each TP is calculated from MC simulations, where the simulated particles are excluded from the volume reconstructed in previous steps. In this paper we remove the excluded volume (EV) restriction, replacing it by a "free volume" (FV) approach. For liquid argon, HSMC-FV leads to an improvement in efficiency over HSMC-EV by a factor of 2-3. Importantly, the FV treatment greatly simplifies the HS implementation for liquids, allowing a much more natural application of the method for MD simulations. Given the success and popularity of MD, the present development of the HSMD method for liquids is an important advancement for HS methodology. Results for the HSMD-FV approach presented here agree well with our HSMC and thermodynamic integration results. The efficiency of HSMD-FV is equivalent to HSMC-EV. The potential use of HSMC(MD)-FV in protein systems with explicit water is discussed.