Structure, Thermodynamics, and Position-Dependent Diffusivity in

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Structure, Thermodynamics, and Position-Dependent Diffusivity in Fluids with Sinusoidal Density Variations Jonathan A. Bollinger, Avni Jain, and Thomas M. Truskett* McKetta Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: Molecular dynamics simulations and a stochastic method based on the Fokker−Planck equation are used to explore the consequences of inhomogeneous density profiles on the thermodynamic and dynamic properties of the hard-sphere fluid and supercooled liquid water. Effects of the inhomogeneity length scale are systematically considered via the imposition of sinusoidal density profiles of various wavelengths. For longwavelength density profiles, bulk-like relationships between local structure, thermodynamics, and diffusivity are observed as expected. However, for both systems, a crossover in behavior occurs as a function of wavelength, with qualitatively different correlations between the local static and dynamic quantities emerging as density variations approach the scale of a particle diameter. Irrespective of the density variation wavelength, average diffusivities of hard-sphere fluids in the inhomogeneous and homogeneous directions are coupled and approximately correlate with the volume available for insertion of another particle. Unfortunately, a quantitatively reliable static predictor of positiondependent dynamics has yet to be identified for even the simplest of inhomogeneous fluids.

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INTRODUCTION

perhaps provide mechanistic insights into, the local dynamic properties in such systems. In this Letter, we use molecular dynamics (MD) simulations together with a stochastic method based on the Fokker−Planck equation to provide detailed information about the relationships between position-dependent static and dynamic quantities for inhomogeneous fluids. Because inhomogeneities can emerge on a variety of length scales, we examine fluids exhibiting periodic, one-dimensional (1D) sinusoidal density profiles of various wavelengths. By construction, these model systems are simpler to assess than those of other commonly studied inhomogeneous liquids such as confined fluids (which exhibit specific interfacial and finite-size effects due to the presence of solid boundaries) or sheared fluids (which display nonequilibrium effects), but still capture the essential physics of inhomogeneity in a general way. We first consider the hard-sphere fluid, a canonical reference model for the structure and dynamics of dense simple liquids that exhibits negative correlations between density and diffusivity.3 We also examine a model for supercooled liquid water, which is an interesting system to consider because of the distinctive properties that it exhibits due to its tetrahedral hydrogen-bond network;4 e.g., its diffusivity anomalously increases upon isothermal compression along supercooled isotherms.5

When a fluid is subjected to an external field φ(r), it becomes inhomogeneous; i.e., its structural properties such as the oneparticle density ρ(r) and its dynamic properties (e.g., relaxation times, diffusivity tensor, etc.) exhibit dependencies on position r. Such inhomogeneities are ubiquitous, with examples ranging from the slowly varying properties of particulate suspensions in gravitational, magnetic, or centrifugal fields to the stronger spatial variations characteristic of liquid molecules near an interface with another phase. As a result, there is interest in developing an understanding of how to accurately model and predict these position dependencies across a wide range of length scales, which in turn could enable the design of new material systems with novel properties. Unfortunately, despite the recent progress in developing theories that predict the static properties of inhomogeneous fluids,1 very little is known about how to estimate the corresponding position-dependent dynamics.2 If density variations in an inhomogeneous fluid are sufficiently weak, then the local dynamic properties can be estimated by simply inputting ρ(r) into known relationships that describe how the bulk isotropic fluid’s dynamic properties depend on density. However, this local density approximation will break down for fluids displaying stronger density inhomogeneity (e.g., structuring near an interface with a solid) and will necessarily fail for systems where ρ(r) takes on values larger than can be realized in the isotropic fluid. A natural question is whether there are other position-dependent models or structural metrics that more reliably anticipate, and © 2014 American Chemical Society

Received: May 5, 2014 Revised: June 30, 2014 Published: July 1, 2014 8247

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We extract position-dependent diffusivity for the HS fluid in the inhomogeneous dimension, Dz(z), from MD trajectories using a previously published mean first-passage time (MFPT) method applied to the steady-state Fokker−Planck equation.21,22 This method requires nonperiodic boundary conditions in the z-dimension, so we generate external potentials that stitch together the FMT-derived potentials and confining walls, and apply them in MD simulations to systems of N = 2400 particles. The confining walls are separated by distances large enough to ensure that they do not affect static or dynamic properties in the “core” system of interest, a criterion straightforward to verify in this system. Calculating Dz(z) profiles requires extensive trajectory data. To facilitate MD parallelization and the use of relatively large time steps, we perform canonical ensemble simulations using GROMACS 4.5.523 with a time step of 0.001, fixing T = 1.0 with a Nose−Hoover thermostat (τ = 1.0). Diffusion coefficients in the homogeneous plane, Dxy, are calculated using particle mean-squared displacements from separate simulations of N = 2400 particles. Periodic boundary conditions are applied in all three dimensions, making it unnecessary to account for particle proximity to walls. We perform these simulations with a custom MD program in the microcanonical ensemble, integrating the equations of motion with a time step of 0.0002 using the velocity-Verlet algorithm.24 To model supercooled liquid water, we carry out canonical ensemble MD simulations of N = 1600 TIP4P/200525 molecules at T = 270 K. Simulations are performed using a time-step of 2.5 fs and a Nosé−Hoover thermostat. To impose oscillatory ρ(z) profiles, we use external potentials of the form φosc(z) = Aw[1 + cos(2πz/λw)], where Aw and λw are the amplitude and wavelength, respectively. For a desired λw, we manually tune Aw to produce density profiles ρ(z) oscillating within a target density range. We calculate the local orientational order parameter qtet(z) and Dz(z) profiles, the latter using the MFPT analysis described above, from the configurations generated by the MD trajectories. Detailed protocols for computing Dz(z), Dxy, φosc(z), s(2)(z), and qtet(z) are presented in the Supporting Information (SI).

We examine the relationships between the position-dependent diffusivity Dz(z) in the inhomogeneous direction and various position-dependent static quantities for these models, including density ρ(z), the one-body direct correlation function c(1)(z), the two-body contribution to the excess entropy (relative to ideal gas) s(2)(z), and, for supercooled water, an orientational order parameter qtet(z) that probes the tetrahedral structure of the hydrogen-bond network. We also investigate the predictions of a heuristic local average density model (LADM) for diffusivity motivated by analogous treatments of the free energy density in classical density functional theory.6 For hard-sphere fluids, c(1)(z) and s(2)(z) characterize local contributions to “available space” and “available states” in the system,7 and the corresponding global versions have been shown, under confinement, to strongly correlate with the oscillatory pore width dependence of diffusivity parallel to confining boundaries.7−9 More broadly, excess entropy has been found empirically to correlate with appropriately nondimensionalized transport coefficients for a variety of fluids in the bulk10−13 as well as those describing average dynamics in the homogeneous directions under confinement.7,14,15 As we report below, our simulation results demonstrate the emergence of nontrivial relationships between local static quantities and position-dependent diffusivity in fluids with different scales of structural inhomogeneity. Despite their ability to successfully predict dynamic trends in certain limits, we show that neither the LADM approximation nor the known static-dynamic correlations of isotropic fluids (with inhomogeneous structure as an input) can provide reliable predictions of local and average diffusivity across a wide variety of density profiles, highlighting that a “mechanistic” understanding of such processes is still lacking.

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COMPUTATIONAL METHODS

RESULTS AND DISCUSSION

We begin our discussion by examining results for the inhomogeneous HS fluids. Figure 1 compares local density ρ(z) and diffusivity Dz(z) for various wavelengths λ to explore how different length scales of density variations impact correlations between equilibrium structure and mobility. The average density ρavg and the range of ρ(z) are kept constant across systems to isolate the influence of λ. We choose ρavg to correspond to a bulk packing fraction ϕ = (π/6)ρ = 0.30. At long wavelengths (e.g., λ ≥ 6), local diffusivity simply decreases with increasing density (and vice versa), as expected based on the physics of the bulk HS fluid. However, for density variations on the scale of a particle diameter (e.g., λ ≃ 1), Dz(z) is clearly positively correlated with ρ(z), as previously observed in confined systems exhibiting similar inhomogeneities.2,26 Remarkably, this qualitative crossover in behavior passes through wavelengths (e.g., λ = 2.5) for which Dz(z) is spatially homogeneous (i.e., flat profile) despite substantial oscillations in ρ(z). To understand why this result is striking, consider that mean collision frequencies in the bulk HS fluid at densities corresponding to the ρ(z) extrema differ by approximately an order of magnitude. In the SI, we demonstrate that this qualitative crossover is similarly observed for dense inhomogeneous systems of 2D hard disks (Figure S3). The crossover in the correlation between Dz(z) and ρ(z) is presented directly in Figure 2, where it is shown to be a smooth transition as a function of λ. The inset shows that the crossover is qualitatively captured by an existing LADM approximation6,27 (recently modified by17,28) defined as DLADM(z) = [ ρ ̅ σ (z)/ ρ ̅ τ (z)]Dρbulk( ρ ̅ τ (z)), which assumes that transport

To model hard-sphere (HS) particles with a continuous representation, we adopt the following steeply repulsive Weeks−Chandler− Andersen (WCA) pair potential:16 φ(r) = 4ε([σ/r]48 − [σ/r]24) + ϵ for r ≤ 21/24σ and φ(r) = 0 for r > 21/24σ, where r is the interparticle separation. Particles are subjected to one-body external potentials, φosc(z), derived via Fundamental Measure Theory (FMT)1 to produce 1D sinusoidal density profiles ρ(z) = ρavg − (1/2)A cos(2πz/λ), where the average density is ρavg = (2/λ)∫ λ/2 0 ρ(z)dz, A is the oscillation amplitude, and λ is the wavelength (similar to previous studies17,18). Unless otherwise indicated, we report quantities for this system that are implicitly nondimensionalized by appropriate combinations of the characteristic length scale, σ, and energy scale, ϵ (ε = kBT, where kB is Boltzmann’s constant and T is temperature). We calculate several static properties of these inhomogeneous HS fluids that previous studies indicate might correlate well with their dynamics,2,7,8 including the one-body direct correlation function c(1)(z), the excess entropy per particle sex, and the local two-body contribution to the excess entropy per particle s(2)(z). Note that c(1)(z) = ln p0(z), where p0(z) is the average areal fraction of the plane located at z that can accommodate an identical hard sphere “added” to the system without overlapping existing particles.19 In other words, c(1)(z) provides quantitative information about the local available space in the system. On the other hand, sex characterizes the global reduction of fluid entropy due to static interparticle correlations (i.e., relative to an ideal gas exhibiting an equivalent ρ(z) profile), and s(2)(z) quantifies the local excess entropy contribution due to pair correlations. The latter reduces to s(2) = −(1/2)ρkB∫ {g(r)lng(r)− [g(r)− 1]}dr for isotropic systems.20 The quantities c(1)(z) and sex are calculated from FMT, which is known to accurately describe the structure and thermodynamics of inhomogeneous HS fluids for the densities considered here,1,7 while s(2)(z) is calculated using configurations from MD particle trajectories. 8248

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Figure 2. Position-dependent diffusivity in the inhomogeneous z direction, Dz(z), plotted versus density, ρ(z), for 3D HS systems at ρavg = 0.573, ρ(z) = [0.287, 0.860], and λ = 24, 12, 6, 3, 2.5, 2, 1.5, 1, and 0.75 (black circles to gold diamonds, consistent with Figure 1). Error bars are on the scale of the symbols. Heuristic predictions DLADM(z) (see text) are shown in the inset (same symbols). Data for the bulk 3D HS fluid are shown as dashed lines.

Figure 1. Position-dependent density, ρ(z), and diffusivity in the inhomogeneous z direction, Dz(z), for 3D HS fluids exhibiting sinusoidal density profiles with ρavg = 0.573, ρ(z) = [0.287, 0.860], and wavelengths λ = 12, 6, 2.5, 1, and 0.75. Error bars are on the scale of the symbols.

quantities can be estimated in a fashion similar to free energy densities Φ within the weighted density approximation (WDA) of density functional theory: ΦWDA{ρ(r), r} = [ ρ ̅ σ (r)/ τ 19 ρ ̅ τ (r)]Φbulk We calculate weighted densities ρ ̅ σ (z) ρ ( ρ ̅ (r)). τ and ρ ̅ (z) using models previously applied to transport properties in the LADM formalism,6,17,27,28 including the Tarazona, generalized van der Waals (GVDW), and generalized hard-rod (GHR) models.3 The inset shows results from the Tarazona model, which approximately predicts the transition between the long- and short-wavelength correlations (although not shown here, we find that predictions based on the GVDW model also qualitatively capture the transition; those based on the simplistic GHR model do not). However, a direct comparison between the measured Dz(z) and predicted DLADM(z) values, shown in Figures S4 and S5, highlights relatively poor local predictive quality for 2 ≤ λ ≤ 6. The LADM approximation also fails to converge to the correct average diffusivity as λ → 0 (see Figure 4 discussion). In Figure 3, we examine how Dz(z) correlates with exp{c(1)(z)} and s(2)(z) (top inset), which quantify the local fractional available volume and two-body excess entropy, respectively. Average available volume and excess entropy

Figure 3. Position-dependent diffusivity in the inhomogeneous z direction, Dz(z), plotted versus fractional available volume, exp{c(1)(z)}, and (top inset) two-body excess entropy, s(2)(z), for 3D inhomongenous HS fluids at ρavg = 0.573, ρ(z) = [0.287, 0.860], and λ = 24, 12, 6, 3, 2.5, 2, 1.5, 1, and 0.75 (black circles to gold diamonds; consistent with Figure 2). Error bars are on the scale of the symbols. In the bottom inset, s(2)(z) is plotted versus exp{c(1)(z)}. Data for the bulk 3D HS fluid are shown as dashed lines.

positively correlate with each other and with average fluid diffusivity in isotropic directions of HS fluids.2,7,8 First, we note that s(2)(z) versus exp{c(1)(z)} data for all wavelengths collapse onto a common line exhibiting strong positive correlation (bottom inset). Significantly, we observe in Figure 3 that Dz(z) exhibits highly nontrivial λ-dependent correlations with exp{c(1)(z)} and s(2)(z), where Dz(z) can display positive correlations with exp{c(1)(z)} and s(2)(z) (e.g., λ = 12); Dz(z) can remain constant while exp{c(1)(z)} and s(2)(z) vary considerably (e.g., λ = 2.5); and Dz(z) can vary considerably 8249

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Dρbulk(ρavg)∫ 0λ/2[ρ(z)]2/ρavgdz. The dimensionless ρ ̅ σ (z) = ρ ̅ τ (z) factor can be removed to resolve this problem, but then Davg LADM no longer captures the local diffusivity crossover (as in Figure 2).] The most successful approach shown in bulk (1) Figure 4, Davg c(1) = ⟨Dc(1) (c (z, λ))⟩ρ, is based on averaging local diffusivity profiles generated by substituting the wavelengthdependent c(1)(z) profiles into the bulk HS relationship between diffusivity and fractional available volume. This approach qualitatively captures all extremes while converging to the correct limits, and builds on an intuitive physical connection, namely, that diffusion might be conceptualized as particles probing unoccupied space.2,7 Based on Figure 3, it can be deduced that predictions of similar quality would result from knowledge of s(2)(z), a two-point position-dependent quantity considerably more expensive than c(1)(z) to obtain across a broad parameter space. We note, however, that the success of these local measures-with regard to describing spatiallyaveraged dynamics-should be considered cautiously in terms of mechanistic interpretation given their failure to directly predict local diffusivities (see Figure 3). The final prediction, bulk ex Davg sex = Dsex (s (λ)), is based on substituting wavelengthdependent molar excess entropy, sex, values into the bulk HS relationship between diffusivity and excess entropy. This approach qualitatively captures the short-wavelength oscillations in the Figure 4 data, but converges to a quantitatively incorrect long-wavelength prediction. This is not surprising because sex is a global, rather than local, quantity and thus, for long wavelengths, does not necessarily weight positiondependent static correlations in a way that is predictive of averaged local diffusivities. To consider whether similar structural-dynamic behaviors emerge in complex fluids capable of exhibiting “anomalous” bulk diffusion trends, Figure 5 shows local density ρ(z) and diffusivity Dz(z) for TIP4P/2005 water as a function of λw, where the density range 0.9 ≤ ρ(z) ≤ 1.1 g/cm3 and temperature T = 270 K correspond to an isotherm along which diffusivity anomalously increases with compression in the bulk.30 Note that bulk-like anomalous behavior is still evident at λw = 4.0 nm (roughly corresponding to 14 water diameters), where Dz(z) is positively correlated with ρ(z). However, as λw is decreased below 1.0 nm, Dz(z) becomes negatively correlated with ρ(z), effectively reversing the diffusion anomaly by imposing variations of the scale of a water molecule.22 Figure 6 highlights the crossover from positive to negative correlation between Dz(z) and ρ(z) with decreasing λw, where this transition is complementary to the crossover for HS shown in Figure 2. The Figure 6 inset shows Dz(z) versus qtet(z), where the latter approximately quantifies the strength of the local hydrogen bond network, which one might intuitively expect to negatively correlate with molecular diffusivity. While this expectation is borne out at longer wavelengths (e.g., λ = 4.0 nm), qtet(z) becomes spatially homogeneous at small λw, while D z(z) still displays appreciable variations. Thus, local tetrahedral ordering associated with hydrogen-bond network effects does not appear to drive Dz(z) variations for all wavelengths λw, and very different physics are governing dynamics in strongly inhomogeneous states. More broadly, the wavelength-driven crossovers of the relationships between Dz(z) and ρ(z) are similar for the supercooled water and HS systems, that is, both occur at λw ≃ λ ≃ 3 particle diameters, which points to a generic, yet incompletely understood, transition in the correlations between

while exp{c(1)(z)} and s(2)(z) converge on the values associated with the bulk fluid at ρavg (e.g., λ ≤ 1). To our knowledge, such a diversity of local static-dynamic correlations has not been previously observed, and demonstrates that correlations for isotropic fluids do not generalize in a straightforward way to describe inhomogeneous particle motions. In Figure 4, we examine the λ-dependency of average diffusivities Davg = ⟨Dz(z)⟩ρ = 2λ∫ λ/2 Dz(z)ρ(z) dz/ρavg and z 0

Figure 4. Average diffusivities, Davg (red triangles) and Dxy (blue z squares), versus inverse wavelength λ−1 for 3D HS fluids at ρavg = 0.573, ρ(z) = [0.287, 0.860]. Model predictions Davg LADM (dashed-dotted avg line), Davg c(1) (solid line), Dsex (dashed line) are calculated using knowledge of various structural quantities (see text). Note that as λ−1 → ∞, we recover a bulk isotropic fluid with a density of ρavg. Davg z was only calculated out to λ−1 ≤ 4/3 due to the increasing computational burden of collecting MD trajectory data necessary as λ−1 → ∞.

Dxy, which characterize some global aspects of particle singleparticle mobility (note that to characterize certain transport quantities in the inhomogeneous direction within the Smoluchowski model, e.g., diffusion-controlled chemical reaction times, one alternatively requires integrals over z1 position-dependent resistivity to diffusion, ∫ z0 [Dz(z)ρ−1 29 avg (z)] dz ). Here, Dz and Dxy are apparently coupled and depend nonmonotonically on inverse wavelength λ−1 (using λ−1 highlights the short-wavelength oscillations). Both Davg z and Dxy exhibit relative maxima at λ−1 ≃ 1, which approximately corresponds to stacked “sheets” of particles and recalls previous work that optimized particle mobility along slit-pores by imposing strongly layered ρ(z).14 In contrast, the diffusivities exhibit minima at λ−1 ≃ 1/3, which suggests that crossover wavelengths (i.e. λ ≃ 3) inherently frustrate single-particle motions. Figure 4 also shows three “model” predictions for average diffusivity Davg, the first using the DLADM(z) profiles, and the latter two using available volumes and excess entropies calculated with FMT, respectively. The first prediction, Davg LADM ⟨DLADM(z)⟩ρ, qualitatively captures the oscillatory dependence on λ−1, but fails to converge to the correct short wavelength limit, which is the diffusivity Dbulk ρ (ρavg) corresponding to a bulk HS fluid at ρavg. [As λ−1 → ∞, Davg LADM does not converge to the σ correct limit, Dbulk ρ (ρavg), due to the behavior of its ρ ̅ (z) = τ ρ ̅ (z) prefactor. The Tarazona and GVDW models give ρ ̅ σ (z) =σ(z), and at short wavelengths, ρ ̅ τ (z) → ρavg. Thus, Davg LADM → 8250

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OUTLOOK The results in this Letter highlight, despite considerable recent progress, the lack of a physically intuitive model that quantitatively relates equilibrium inhomogeneous fluid structure and position-dependent particle mobility. While transport coefficients in isotropic fluids and along homogeneous coordinates of confined fluids correlate with thermodynamic quantities for a diverse array of scenarios, we find that, even for simple fluids with precisely imposed density variations, static quantities cannot be used to reliably predict local and average diffusivity. We look forward to addressing whether alternative theoretical or empirical models (e.g., other WDAs for LADM) can provide insight into the connections between externally induced structural variations and local and average dynamic responses reported here. It tentatively appears that any models that capture the qualitative crossover in diffusive motions as a function of λ might be applicable to both simple and complex fluids, including those that exhibit anomalous bulk fluid relationships between diffusivity and compression. More investigations must be conducted to ascertain whether such crossovers extend to other equilibrium fluid models such as those with significant attractions or penetrable cores, and to nonequilibrium systems including, e.g., carefully manipulated shear conditions.

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ASSOCIATED CONTENT

S Supporting Information *

Detailed protocols for computing Dz(z), Dxy, φosc(z), s(2)(z), and qtet(z); results for 2D hard-disk fluids; and additional discussion of LADM predictions for local diffusivity. This material is available free of charge via the Internet at http:// pubs.acs.org.

Figure 5. Density ρ(z) (g/cm3) and position-dependent diffusivity in the inhomogeneous z direction Dz(z) (105 cm2/s) for TIP4P/2005 water at T = 270 K tuned to exhibit sinusoidal inhomogeneities within 0.9 ≤ ρ(z) ≤ 1.1 g/cm3 for λw = 4, 2, 0.6, and 0.3 nm (or λwσw ≃ 14, 7, 2, and 1, respectively). Error bars are on the scale of the symbols.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank James Carmer for helpful discussions. This work was supported by the Gulf of Mexico Research Initiative and the Robert A. Welch Foundation (F-1696). We also acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources for this study.

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REFERENCES

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Figure 6. Dz(z) plotted versus density ρ(z) and local tetrahedral order qtet(z) (inset) for supercooled water at T = 270 K tuned to exhibit sinusoidal inhomogeneities within 0.9 ≤ ρ(z) ≤ 1.1 g/cm3 for λw = 4, 2, 1, 0.6, and 0.3 nm (red triangles to blue triangles). Error bars are on the scale of the symbols. Data for bulk water are shown as dashed lines. Note that, by definition, qtet ranges from 0 in an ideal gas to 1 in a perfect tetrahedral network.

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