J. Phys. Chem. B 2008, 112, 14911–14919
14911
Phase Behavior of Model Confined Fluids. Influence of Substrate-Fluid Interaction Strength Thomas W. Rosch and Jeffrey R. Errington* Department of Chemical and Biological Engineering, UniVersity at Buffalo, The State UniVersity of New York, Buffalo, New York 14260-4200 ReceiVed: May 19, 2008; ReVised Manuscript ReceiVed: September 17, 2008
We examine the relationship between the macroscopic phase behavior of nanoconfined fluids and the nature of microscopic interactions between a confining substrate and fluid. Two model slit-pore systems are explored using grand canonical transition-matrix Monte Carlo simulation. One system consists of a square-well fluid interacting with a square-well substrate, and the other contains an embedded point charge model of lysozyme interacting with a mica surface. Fluid phase diagrams are constructed for a broad range of substrate conditions. Our results indicate that one observes a maximum in the critical temperature of the fluid phase envelope upon variation of substrate strength for a given slit width. Both systems studied exhibit such maxima at intermediate wall strength. The physical rationale for this observation suggests that this behavior should be generally expected. We introduce two metrics that enable one to predict conditions that produce maxima in critical temperature. The first is related to the contact angle a fluid develops at a single confining substrate. The second is based upon virial coefficient information and requires knowledge of the substrate-fluid and fluid-fluid interaction potentials only. Introduction Confining a fluid to geometries characterized by nanoscopic length scales is known to affect its equilibrium phase behavior. The reduction in the number of fluid-fluid interactions, as well as the introduction of surface-fluid interactions, induces shifts in the location and shape of the phase diagram relative to the bulk system. The nature of microscopic interactions within a confined environment, i.e., how strongly particles attract or repel each other and the walls, in addition to the range at which these attractions and repulsions are felt, controls a fluid’s macroscopic behavior. From an engineering perspective it is desirable to quantitatively understand the relationship between the microscopic interactions within a system and its macroscopic phase behavior. With such information, one can tune molecular-level details of a system to produce a fluid with thermodynamic and/ or kinetic properties of interest. Molecular simulation provides a powerful means to enumerate these relationships. Adoption of a computational approach is particularly attractive when investigating the behavior of severely confined systems due to the challenges associated with employing experimental techniques at such length scales. In this work we use molecular simulation to systematically examine how the strength and range of the interaction between fluid molecules and a confining pore affect the phase behavior to two model systems. We also introduce two metrics that enable one to predict how the critical temperature of a fluid phase envelope evolves with substrate strength. In many industrial applications fluids necessarily exist in confining spaces. In order to optimize and scientifically design these applications, an intimate understanding of how these restrictive environments affect the overall properties of the fluid is needed. Both experimental and theoretical studies have examined the influence of confinement on a fluid’s phase diagram, with a general result that phase envelopes progressively * Corresponding author. E-mail:
[email protected].
shift to lower temperatures upon increasing the degree of confinement.1-3 Much less studied is the connection between the strength of the substrate-fluid interaction and the shape and location of fluid-phase boundaries. Scaling theory predictions from Fisher and Nakanishi3,4 suggest that the critical temperature of the fluid phase envelope decreases monotonically with increasing wall strength for a fixed degree of confinement. These authors also suggest that the depression in critical temperature approaches a limiting infinite-substrate-strength value as the magnitude of the wall-fluid interaction increases. Brovchenko et al.5 used molecular simulation to study a model in which water was confined within either slit or cylindrical pores with variable strength. They found qualitative agreement with scaling theory predictions; the critical temperature of the liquid-vapor phase envelope monotonically decreased with increasing substrate strength over the range of conditions examined. Zhang and Wang6 used density functional theory to examine a squarewell fluid within a cylindrical pore interacting with the fluid via a square-well potential. Their data suggested a distinctly nonmonotonic relationship between the critical temperature and wall-fluid interaction strength, with a maximum occurring at an intermediate value of the wall strength. Singh and Kwak7 employed Monte Carlo simulation to examine a similar squarewell system, focusing on a fluid confined within a slit pore. Their data also indicated that square-well fluid phase envelopes exhibit a maximum in critical temperature upon variation of substrate strength. The authors of the two square-well studies also described an interesting connection between maxima in critical temperature and the equilibrium structure of the saturated vapor phase. In this work we further examine how variations in the depth and range of the interaction potential between a substrate and fluid influence the phase behavior of nanoconfined systems. To probe this issue, we study two model systems. The first is similar to that examined by Singh and Kwak; fluid particles are confined to a slit pore and interact with both the walls and other fluid
10.1021/jp804419b CCC: $40.75 2008 American Chemical Society Published on Web 10/31/2008
14912 J. Phys. Chem. B, Vol. 112, No. 47, 2008 particles via square-well potentials. The attraction of working with this simple system stems from our ability to independently vary the substrate-fluid and fluid-fluid interactions. Here, we systematically vary both the depth and width of the substratefluid potential well while holding fixed the characteristics of the fluid. It is in this manner that we examine the extent to which modulation of the microscopic interaction between molecules and the confining pore walls affects the characteristics of the fluid-fluid phase diagram. Overall, we find that the critical temperature of the fluid phase envelope exhibits a maximum upon variation of the strength of the substrate-fluid interaction (characterized by either the well depth of width). To explore the generality of these findings, we examine a second model system that consists of a lysozyme solution confined between two mica surfaces.8 Carlsson and co-workers have used this protein model to study the adsorption behavior of lysozyme on a mica surface8 as well as the structure and behavior of lysozyme in bulk9 and in the presence of a polyelectrolyte.10 The model consists of hard sphere repulsion with spherically symmetric hydrophobic attraction and electrostatic interaction via a collection of embedded point charges. The pH of the solution dictates the distribution of charge within the protein, which is based upon an experimentally determined titration curve. The model mica surface is described by a hard wall with a collection of regularly spaced embedded point charges. We probe the extent to which substrate strength influences the fluid phase envelope via variation of the density of charge within the mica substrate. For each value of the protein charge (solution pH) examined, we observe a maximum in the critical temperature of the liquid-liquid phase envelope at intermediate substrate charge density, analogous to the maxima in critical temperature exhibited by the square-well system. Given the apparently general connection between substrate interaction strength and evolution of the fluid phase diagram through a maximum in critical temperature, we explore the possibility of using a simple metric to predict the location of these extremum points. We found two quantities to be particularly useful. Both metrics provide a measure of the strength of the substrate-fluid interaction relative to that for a pair of fluid particles. In the first case, we find that the contact angle a fluid droplet forms on one of the confining surfaces evaluates to a near-constant value at conditions related to maxima in the critical temperature. The second parameter explored is based upon virial coefficient information. Specifically, we examine a metric defined in terms of the canonical fluid second virial coefficient and an analogous effective wall-fluid virial coefficient. This metric is found to adopt a near-constant value at substrate strengths corresponding to maxima in the critical temperature. Given that computation of this quantity requires knowledge of interaction potentials only, the metric should serve as a useful guide in predicting the response of a confined fluid phase envelope upon variation of the substrate characteristics. Finally, we note that this approach is related to the use of the second virial coefficient to estimate the location of a fluid’s bulk critical temperature.11-14 In this case, one assumes that the second virial coefficient evaluates to a common value at the critical point, thus providing a means to approximate the critical temperature from information regarding the pair potential alone. Vliegenthart and Lekkerker have examined the validity of this approach for a series of spherically symmetric potentials,13 and we have recently explored its suitability in describing the protein model studied here.12 The rest of the paper is organized as follows: the next two sections describe the models, computational methods, and
Rosch and Errington relevant simulation details employed in this work. We then present several phase diagrams for confined square-well and protein systems and identify common trends in evolution of phase behavior with variation of substrate strength. Next, we discuss the physical rationale for the observed behavior and examine two metrics for predicting the response of a phase diagram upon modification of substrate characteristics. Finally, we end by summarizing the main results from this work. Model The first system we study consists of fluid particles that interact with each other via a square-well potential
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