Surface thermodynamics at the nanoscale

Fluids confined in regions with nanoscale dimensions are of great interest for a number of applications. In biology, important cellular phenomena are controlled by transport through cell membrane nanopores.^1,2 Delivery of hydrophobic pharmaceuticals to aqueous environments relies on encapsulation in nanometer-sized micelles.³ There is great interest in understanding the behavior of oil and gas in nanoporous shale reservoirs.⁴ Nanoscale aerosols are important in diverse fields from cloud physics^5,6 to pathogen transmission.⁷ Charge transport through the fluid portion of nanostructured fluid–solid hybrid materials determines the performance of novel energy storage devices.^8,9 Novel nanofluidic devices are being exploited to probe new aspects of fluids.^10–12 The words “nanopore” and “nanofluidics”—hardly appearing in 1995—appear in the title, abstract, or keywords of 1605 and 2871 papers published in 2020, respectively (Scopus). There are fundamental scientific questions arising from this intense interest in nanoscale fluids, and one of the most exciting questions is whether or not Gibbsian thermodynamics should be expected to hold at the nanoscale. The field is at an exciting scientific nexus of (a) the ability to make nanoscale experimental measurements,^2,10–15 (b) the reaching up to the nanoscale of molecular simulations,^16–23 and (c) the reaching down to the nanoscale of Gibbsian thermodynamics.^{2,15,19,21,24–28}

The definitions of $1/R1+1/R2$ as appearing in Eqs. (2)–(4), with the direction of curvature conventions as written in the equations, are given in Table I for a variety of geometries with constant interfacial curvature.

Nonideal and/or multicomponent forms of the Kelvin equation [Eq. (2)],^{16,24,30,40–45} the Gibbs–Thomson equation [Eq. (3)],^46–48 and the Ostwald–Freundlich Equation [Eq. (4)]^39,47 have been given by several authors.

The objective of this Perspective is to explore whether these Gibbsian thermodynamics equations [Eqs. (2)–(4)] pertaining to equilibrium conditions of systems with curved fluid interfaces should be expected to be valid for nanoscale interfacial radii of curvature.

Thermodynamics is an extremely powerful framework that describes the average behavior of a collection of molecules. The averaging may be done explicitly such as by computing an ensemble average with statistical thermodynamics procedures, or the averaging may be implicit in that thermodynamic quantities such as pressure in and of themselves describe the average behavior of a collection of molecules. Composite-system thermodynamics describes the equilibrium of a collection of simple systems, each simple system having constant values of intensive properties throughout. Important questions are as follows: What is the lower limit of length scale at which relationships between macroscopic thermodynamic quantities derived by composite-system thermodynamics no longer hold, and of specific interest to this Perspective, is it possible that the combined equilibrium conditions describing the impact of a curved fluid interface on the adjoining bulk phase properties can hold for fluids confined at the nanoscale? It is important to delve deeper into two important concepts: (i) ensemble averaging and the requirements for a thermodynamic system, and (ii) the constructs of the Gibbs dividing surface and the Gibbs surface of tension.

That a collection of molecules arrives at thermodynamic equilibrium requires (a) that there be a number of molecules and (b) that the molecules be able to exchange energy. Obviously, a single molecule cannot meet this requirement nor can a large number of uncontained molecules so dilute that they never interact. However, what about a few molecules? Some authors presuppose that the required number of molecules is very large; for example, Hill mentioned 10²⁰ molecules (p. 1, Ref. 49). However, it is becoming apparent that the number of molecules required may be pretty small, on the order of thousands,¹⁷ or hundreds,¹⁸ or even fewer.¹⁹ Also, it is important to note that at equilibrium of molecules that meet these thermodynamic requirements (that a number of molecules exchange energy), ensemble averaging over a region in space at one instant in time is equivalent to ensemble averaging over time at one point in space. This is an important principle in statistical thermodynamics: that an ensemble is a collection of systems, each representing the system of interest at a different allowable microstate, and that averaging over the ensemble of systems (a snapshot at one instant in time of the probabilities describing the system) is equivalent to the time average of the system (pp. 3–4 of Ref. 49). Therefore, the region of space over which we describe a macroscopic thermodynamic property can be very small if we are interested in the behavior of the system averaged over a long enough time scale. A nice visual representation of this time averaging can be seen in molecular dynamics simulations of the contact angle that a nanodroplet makes with a surface.^17,18 In the supplementary video of Ref. 18, molecules can be seen bouncing around, but a contact angle can be observed by averaging the position of the fluid molecules over time. There is support for this molecular scale contact angle satisfying the Gibbsian-composite-system-thermodynamics-consistent Young equation relating the contact angle to the liquid–vapor, solid–liquid, and solid–vapor interfacial tension values.¹⁷ 

Another support for Gibbsian composite-system thermodynamics being valid at very small length scales can be found in adsorption equilibrium. An adsorbed layer of molecules can be described with Gibbsian thermodynamics, even when it is only one molecule thick. As long as the molecules can either thermalize by exchanging energy with each other or can come to equilibrium with molecules in an adjoining bulk phase, thermodynamic equilibrium equations such as the Langmuir adsorption isotherm,^30,50 other adsorption isotherms,³⁰ and the Gibbs adsorption equation^29,30 may be used to describe this sub-nanometer thick region.

Another important concept in the question of what is the smallest number of molecules that can meet the thermodynamic requirements (that a number of molecules exchange energy) is that molecules need not interact directly with each other to exchange energy if they come to equilibrium with an adjacent phase (e.g., a gas or a solid surface) that is also in equilibrium with the next fluid molecule, thereby enabling the required energy exchange. Hence, even fluid molecules located individually on a surface or individually moving through a nanopore can be described by thermodynamic equilibrium if they are in equilibrium with an adjoining phase.

It is known that surfaces may have an increased affinity for fluid molecules resulting in extra fluid molecules at the surface compared with the bulk fluid (positive adsorption) or a decreased affinity for fluid molecules resulting in a deficit of fluid molecules at the surface compared with the bulk fluid (negative adsorption). Molecular modeling of fluid in nanopores shows that the density of fluid is not homogeneous across the pore; there may be increased¹⁶ or decreased density²⁰ near the pore wall. At first glance, one may think that this inhomogeneity of the bulk phase would be in contradiction to macroscopic thermodynamic treatments, but this is not necessarily true. The prescience of Gibbs in carefully constructing his dividing surface concepts means that Gibbsian composite-system thermodynamics does not presuppose homogeneity of a fluid near a phase boundary and, in fact, specifically incorporates it. The dividing surface concept introduced by Gibbs is illustrated in Fig. 1. A real system has properties, such as density, that far away from the interface take on their bulk values but that vary near the interface. Gibbs represented the real system (to the left of the equal sign in Fig. 1) with two important parts: the constant bulk phase properties up to a dividing surface that has area but no volume (the first graph to the right of the equal sign in Fig. 1) plus surface excess properties that capture the excess internal energy, entropy, and numbers of moles in the variation of properties near the interface (the second graph to the right of the equal sign in Fig. 1). The excess properties are assigned to an interfacial phase with area but no volume that is a separate phase of the composite system. The thermodynamic equations of this phase follow from multivariable calculus in the same way that the bulk thermodynamic equations do.

Some might mistakenly conceptualize Gibbs’s approach as only the first graph to the right of the equal sign in Fig. 1 and therefore might assume that the dividing surface approach would not be applicable to systems with interface effects that vary far from the interface compared to the scale of the system or the interface curvature, as would be the case at the nanoscale, but actually Gibbs’s approach anticipates inhomogeneity near the interface and quantitatively represents it with surface excess properties. The only thing needed to completely specify the system is the decision of where to place the dividing surface within the interfacial region. There are two common conventions: the Gibbs dividing surface and the Gibbs surface of tension. The Gibbs dividing surface approach uniquely places the dividing surface for a flat interface such that there are no surface excess molecules of one component (for example, $N1jk=0$⁠). The Gibbs surface of tension approach uniquely places the dividing surface for a curved interface such that the interfacial tension σ^jk is independent of curvature. This Gibbs surface of tension approach is particularly useful for systems of nanoscale curvature since it means that the macroscopically obtained interfacial tension values can be used. There are other conventions of use to specific circumstances.^51,52 See the perspective by Radke for insight into the thermodynamic relationships between different conventions for planar interfaces.⁵³ 

Although Gibbsian composite-system thermodynamics is consistent with including this surface adsorption, there are still many questions remaining about its application at the nanoscale. To apply Gibbsian composite-system thermodynamics to a given circumstance, one needs equations of state for each phase including bulk phases and surface phases. Equations (2)–(4) have contained within them assumptions that vapor phases are ideal gases, that solid and liquid phases are incompressible substances, and that multicomponent solutions are ideal, dilute solutions; more advanced versions of the Kelvin equation, the Gibbs–Thomson equation, or the Ostwald–Freundlich equation have other assumptions about the bulk phase equations of state. Equations of state for adsorbed phases are still needed for many applications. Even with accurate equations of state for adsorbed phases, there are other complexities such as how to relate the position of the dividing surface (based on Gibbs dividing surface or Gibbs surface of tension) to a measurable surface position such as a pore diameter. This is confounded at the nanoscale by experimental pore dimension often being acquired indirectly from fitting thermodynamic equations with their embedded assumptions to experimental data. Several authors are contributing to understanding these remaining questions by correlating molecular modeling of adsorption with applicability of thermodynamic phase equilibrium¹⁶ or by correlating experimental adsorption measurements with thermodynamic adsorption isotherm equations.²⁸ 

As discussed above, there is reason to believe that Gibbsian composite-system thermodynamics, in general, and specifically the Kelvin, Gibbs–Thomson, and Ostwald–Freundlich equations that describe the impact of fluid interface curvature on fluid phase equilibrium, might be valid at much smaller interfacial radii of curvature than previously appreciated. Several authors point out discrepancies between these equations and data; for a review with regard to the Kelvin equation (the most researched of the three equations), see Ref. 16 where discrepancies between the Kelvin equation and molecular modeling or experiment begin to appear at interfacial radii of curvature below 10 nm. Disagreement can be for many reasons, which are not immediately clear within the research given the difficulty of experimental measurements and molecular modeling at this scale and the remaining unknown questions of what effects are, and are not, included when one uses a thermodynamic equation with its embedded equation-of-state assumptions. Cases of agreement may be more instructive. Below, I highlight a few key papers where agreement between the thermodynamic equations and experimental data or molecular modeling was found to astonishingly small interfacial radii of curvature.

Fifty years ago, Fisher and Israelachivili began the careful experimental validation of the Kelvin equation at the nanoscale, showing that menisci of cyclohexane condensed between crossed mica cylinders obeyed the Kelvin equation down to interfacial radii of curvature of 4 nm (radii that are only 8 times the molecular diameter).^25,26 In 2018, Zhong et al. measured the capillary condensation of n-propane in silicon nanochannels and showed agreement with the Kelvin equation for nanochannels with half-width of 4 nm.¹² In 2013, Zandavi and Ward showed that experimental measurements of capillary condensation of toluene, heptane, or octane in nanoporous silica were consistent with the Kelvin equation down to pore radii of 1.3 nm.²⁸ In 2014, Factorovitch et al. computed the vapor pressure over mW water nanodroplets from molecular dynamics and showed that it agreed with that predicted by the Kelvin equation for water molecular clusters as small as 0.7 nm in radius; these clusters contained only 37 water molecules.¹⁹ Importantly, these clusters are so small that they are not instantaneously spherical or of homogeneous bulk phase density, but their time averages over 100 ns are spherical and have the expected bulk phase density.¹⁹ 

Shardt and Elliott developed multicomponent forms of the Kelvin equation for ideal and nonideal mixtures and numerical methods to predict the interface-curvature-affected multicomponent phase diagrams.^24,45 They predicted the composition-dependent adsorption and desorption dew temperatures of nitrogen/argon mixtures in 2-nm-radius pores²⁴ in agreement with the independent measurements of Alam et al.¹³ Their predictions relied on a new semi-empirical equation for mixture surface tensions as a function of temperature and composition.^54,55

Twenty-five years ago, Hirama et al. presented experimental validation of the Gibbs–Thomson equation for melting points of water in porous Vycor glasses, with pore radii down to 1.8 nm.¹⁵ In 2012, Johnston and Molinero performed molecular dynamics simulations of mW water nanoparticles that were in agreement with the Gibbs–Thomson equation down to radii of 1.05 nm.²¹ In 2019, Karlsson, Bravslavsky, and Elliott used the Gibbs–Thomson equation to infer the protein–water–ice contact angle from the freezing point depression caused by ice growing through nanoscale gaps between proteins, and surprisingly, a single contact angle of 88.0° ± 1.3° was found even though the experiments were done for three different proteins, in three different laboratories, at three different length scales.² Drori, Davies, and Bravslavsky had measured the temperature at which ice grows between hyperactive antifreeze proteins from Tenebrio molitor adsorbed on an ice crystal with pore radii of 3.5–17.5 nm.¹⁴ Acker, Elliott, and McGann had measured the temperature at which ice grows from cell to cell through gap junction pores in Madine−Darby canine kidney cells with pore radii of 0.4–1.2 nm.⁴⁶ Higgins and Karlsson had measured the temperature at which ice grows through tight junction pores between adjacent mouse insulinoma cells with pore radii of 0.33–0.4 nm.⁵⁶ In as much as the applicability of the Gibbs–Thomson equation is not in doubt for the larger pores between the adsorbed antifreeze proteins in the Drori, Davies, and Braslavsky experiment, these results give strong support for the validity of the Gibbs–Thomson equation to describe the curvature-induced freezing point depression measured in the Higgins and Karlsson experiment and the Acker, Elliott, and McGann experiment for the growth of ice through biological membrane protein pores with radii from 0.33 to 1.2 nm.² A pore with a radius of 0.33 nm would fit only 2.4 water molecules across its diameter.² 

Pouralhosseini et al. developed a depletion flocculation model to describe the phase behavior of asphaltene/toluene/polystyrene mixtures.²⁷ Two functions were fit from experimental data: the fraction of asphaltene molecules residing in colloidal asphaltene aggregates and the average size of such aggregates, both as functions of asphaltene volume fraction in the mixtures. For the single phase region, the obtained functions were found to be in agreement with the Ostwald–Freundlich equation down to asphaltene aggregate radii of 4 nm.²⁷ 

The form of the Gibbs–Thomson equation shown in Eq. (3) is for the freezing of a pure liquid. The form of the Ostwald–Freundlich equation shown in Eq. (4) is for precipitation of a solute from an ideal, dilute solution. Liu et al. developed multicomponent, nonideal forms of the Gibbs–Thomson equation and the Ostwald–Freundlich equation and showed that these equations were identical except for which component of a two-component solution was solidifying; identifying the solvent as the solidifying component resulted in a multicomponent, nonideal Gibbs–Thomson equation, and identifying the solute as the solidifying component resulted in a nonideal Ostwald–Freundlich equation.⁴⁷ These equations could be used to describe how curvature lowers the freezing liquidus line and how curvature lowers the precipitating liquidus line in a multicomponent solution.⁴⁷ The freezing liquidus and precipitating liquidus meet at the eutectic point, an important point since no liquid solvent can occur at temperatures below the eutectic temperature. Importantly, the depression of the freezing liquidus and depression of the precipitating liquidus result in interfacial-curvature-induced depression of the eutectic temperature, an effect that becomes significant only at nanoscale interfacial curvatures.⁴⁷ This means that if a solution is cooled in a nanopore, the confinement can prevent both freezing and precipitating because neither the critical ice nucleus defined by the Gibbs–Thomson equation nor the critical solute precipitate nucleus defined by the Ostwald–Freundlich equation can form. Li et al. showed that a sodium chloride solution confined in nanoporous polyampholyte hydrogel maintained liquid water and ionic conductivity at −30 °C and confirmed by solid-state nuclear magnetic resonance (NMR) the existence of mobile, amorphous water at temperatures as low as −54 °C (well below the bulk eutectic temperature of sodium chloride solutions of −21 °C).⁹ The nanoporous polyampholyte hydrogel had nanometer-sized polymer globules that were segregated in non-frozen regions.⁹ Although Li et al. did not measure the interglobule distance, it would need to be on the order of a few nanometers to prevent eutectic freezing at the temperatures mobile, amorphous water was observed.

The Kelvin, Gibbs–Thomson, and Ostwald–Freundlich equations are all based on combining chemical and thermal equilibrium with the mechanical equilibrium Young–Laplace equation. The direct validity of the Young–Laplace equation at the nanoscale has also been investigated. Malek et al. showed that the pressure inside TIP4P/2005 water nanodroplets computed from molecular dynamics simulations was in agreement with the Young–Laplace equation down to radii of 1.75 and 1 nm, respectively, in Refs. 22 and 23.

Three important thermodynamic combined equilibrium equations describe the impact of fluid interface curvature. The Kelvin equation describes the curvature-induced change in vapor pressure, the Gibbs–Thomson equation describes the curvature-induced freezing point depression, and the Ostwald–Freundlich equation describes the increase in solubility caused by the curvature of a solid solute. The underpinnings of Gibbsian composite-system thermodynamics, the fundamental framework from which these equations can be derived, are robust to surprisingly small phase dimensions. Experimental measurements and molecular modeling have given new tools to study the lower limits of applicability of these equations. The applicability of these equations to fluid interfaces with radii of curvature above 10 nm is not in doubt.¹⁶ There is much evidence for the validity of these equations in certain circumstances for interfacial radii of curvature of 1–4 nm.^{2,12,15,19,21–28} At the very smallest scale, molecular dynamics simulations of the vapor pressure above water nanodroplets have been accurately described with the Kelvin equation down to interfacial radii of curvature of 0.7 nm,¹⁹ and experimental measurements of the freezing point depression for ice growth through biological protein pores have been described accurately with the Gibbs–Thomson equation down to pore radii of 0.33–0.4 nm.² 

These results are changing the fundamental understanding of the physics of fluids. Alone a single one of these results might be a curiosity, but taken together as collected for this Perspective, they illuminate the strength of Gibbsian composite-system thermodynamics to surprisingly small length scales. This is important because both experimental measurements and molecular simulations must be redone for each circumstance at great expense, whereas thermodynamic equations can be used efficiently for understanding and design. It is unlikely that Gibbsian composite-system thermodynamics will be found to be valid uncorrected at much smaller length scales than those described. There are many systems of interest about which our understanding will be greatly enhanced by applying all three of experimental study, molecular simulation, and Gibbsian composite-system thermodynamics to the same system at the same conditions under the same assumptions. In the future, researchers will need to include additional effects (e.g., interface charge and field effects) into these equations and develop equations of state for more complicated adsorbed phases to allow Gibbsian composite-system thermodynamics to describe an even broader array of complicated systems at the nanoscale.

J.A.W. Elliott holds a Canada Research Chair in Thermodynamics and acknowledges a Discovery Grant (No. RGPIN-2016-05502) from the Natural Sciences and Engineering Research Council (NSERC) of Canada.