Article pubs.acs.org/JPCB
The Role of Solvent Heterogeneity in Determining the Dispersion Interaction between Nanoassemblies Jaehun Chun,*,† Christopher J. Mundy,‡ and Gregory K. Schenter‡ †
Nuclear Sciences Division and ‡Physical Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United States ABSTRACT: Understanding fundamental nanoassembly processes on intermediate scales between molecular and continuum scales requires an indepth analysis of the coupling between particle interactions and molecular details. This is because the discrete nature of the solvent becomes comparable to the characteristic length scales of assembly. Utilizing the spatial density response of a solvent to a surface in conjunction with the Clausius−Mossotti equation, we present a simple theory relating the discrete nature of solvent to dispersion interactions. Our study reveals that dispersion interactions are indeed sensitive to the spatial variation of solvent density, manifesting in dramatic deviations in van der Waals forces from the conventional formulation (e.g., with uniform solvent density). This study provides the first steps toward relating molecular scale principles, namely the detailed nature of solvent response to an interface, to the underlying hydration forces between surfaces.
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(i.e., Casimir−Polder), owing to a finite speed of electromagnetic waves at a larger separation, can be readily employed. A key underlying assumption in this theory is that a local dielectric response can be defined and characterized from that of the corresponding homogeneous bulk material. While a continuum-based Lifshitz’s formalism naturally incorporates many-body effects in dispersion forces, the formalism lacks a clear connection to the molecular nature of macroscopic bodies. There are two molecular regimes that are known to be important. One is associated with the details of molecule−molecule inhomogeneities that give rise to structure on the subnanometer scale. The other is the response of the molecular fluid to an interface that will yield inhomogeneous responses at nanometer distances. The current study is focused on the latter regime. A fundamental question pertaining to the latter condition is to what extent a homogeneous dielectric response can be utilized to correctly describe the dispersion forces between two macroscopic objects? Inhomogeneities are typically not important between two macroscopic bodies with O(1) μm size interacting at large distances (typically, about 100 nm). This can be explained by recognizing the fact that dispersion forces only become appreciable at a separation less than the size of macroscopic bodies. In other words, for conventional colloids, the dispersion force would be already enormous at O(1) nm where a description of the molecular details become important. However, it can be argued that the discrete nature of the dielectric solvent must be taken into account when characteristic size of macroscopic body, λs, becomes comparable to λd,
INTRODUCTION Dispersion forces or van der Waals forces are ubiquitous, being involved in most of physical, chemical, and biological phenomena such as crystal growth, nanoparticle self-assembly, DNA condensation, and lipid membrane interactions.1 These forces are generally attractive, although repulsive dispersion forces are still possible.2 A common way to approximate dispersion forces is to use a pairwise summation over all involved entities (e.g., atoms or molecules). For example, Hamaker used the pairwise summation, based on the London potential, to calculate the dispersion force between colloids.3 Many free energy calculations employing the Lennard-Jones potential (e.g., solvation free energy of small molecules and proteins) have successfully used the pairwise summation to account for dispersion force contributions. Moreover, simple pairwise formulations of dispersion interactions provide an efficient computational scheme and the neglect of many-body contributions to dispersion forces is often used as an approximation for isotropic condensed media such as liquids and solids. To this end, many-body corrections to classical empirical interaction potentials are usually formulated within the electrostatic formalism where corrections to point-charge models are developed in terms of self-consistent polarizable multipoles. These electrostatic corrections have been shown to be important in systems with broken translational symmetry. A macroscopic or continuum theory, developed by Lifshitz4 and Dzyaloshinski et al.,5 views dispersion forces in the context of quantum electromagnetic modes analogous to Planck’s approach for the blackbody radiation. The geometry and dielectric response of two macroscopic bodies lead to a relation of allowed electromagnetic modes corresponding to the geometric configuration, which in turn gives rise to the system’s free energy. In this formulation, a retardation effect © 2015 American Chemical Society
Received: December 16, 2014 Revised: March 2, 2015 Published: April 15, 2015 5873
DOI: 10.1021/jp512550c J. Phys. Chem. B 2015, 119, 5873−5881
Article
The Journal of Physical Chemistry B
constituent molecules.2 Therefore, the response of the solvent density to an interface is expected to lead to a spatially varying dielectric profile. To combine the phenomena associated with the local solvent density response and macroscopic dielectric response, we invoke the Clausius−Mossotti equation:
characteristic length scale associated with discrete nature of the solvent response to the nanocolloid or surface. To this end, Parsegian6 has shown that an error associated with neglecting the discrete nature of the dielectric medium is O(λd/L̃ ), where L̃ is a separation between macroscopic bodies. Therefore, one can expect that the molecular nature of the dispersion force cannot be neglected when λd ∼ O(L̃ ). This is exactly analogous to the fact that noncontinuum hydrodynamic interactions between aerosols become important in spite of a small mean free path of gas, when the mean free path is comparable to the separation between aerosols.7 We study this case as it is relevant and applicable to many interesting phenomena at the nanoscale. In the current work we will show the importance of solvent structure on the dispersion force between two flat surfaces using mica/water/mica, TiO2 (rutile)/water/TiO2 (rutile), and gold/water/gold systems as representative examples for dielectrics and metals. Our approach is to combine the unique solvent response in the vicinity of an interface with the Clausius−Mossotti equation and derive a Lifshitz theory in the presence of a inhomogeneous dielectric function.
ε−1 4π = ρα ε+2 3
(1)
where ε denotes the dielectric response of substance, ρ is the density of molecules in substance, and α is the molecular polarizability; both ε and α are dependent on frequency. Typically, eq 1 has been used with a constant density. However, assuming that the molecular polarizability is independent of both nearby molecules and a separation distance from the interface, the density response of the solvent molecules as a function of separation in the direction of the interface normal can be cast into the Clausius−Mossotti equation via a Lorentzian-type model of polarizability. Note that permanent dipoles existing in solvent are not expected to play a crucial role in contributing to the dielectric functions at the frequencies of interest for dispersion interactions, namely the far-infrared and higher frequencies. This is because the molecular dipoles in general cannot respond to rapidly oscillating electric fields. Furthermore, at separations of interest in this work (i.e., O(1) nm) possible contributions from the permanent dipoles to dispersion interactions are minimized because of contributions from low-frequency electromagnetic waves being more appreciable as the separation increases. Therefore, the use of eq 1 instead of the Debye’s formula11 is justified. It should be noted that a zero frequency contribution in dispersion interactions cannot be ignored; nevertheless, this approximation appears to be reasonable (see Discussion section for more details). Using the aforementioned Lorentzian-type oscillator model for the polarizability of water first suggested by Nir et al.,12 eq 1 can be reformulated incorporating a nonuniform density of solvent as
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FORMULATION OF THE DISPERSION INTERACTION To examine the effect of solvent structure on dispersion force, one needs to know the nature of the spatial density variation of solvent molecules that comprise the dielectric response between two ideally flat surfaces. To this end, we employ both experimentally and numerically determined response of water to the mica,8 TiO2 (rutile),9 and gold surfaces.10 Figure 1
ε(z , iξ) − 1 = β(z , iξ) = ε(z , iξ) + 2
N
∑ j=1
Cj(ρ(z)) 1 + (ξ /ωj)2 + γjξ /ωj 2 (2)
Here, z is a distance in the direction of the interface normal and ξ is a frequency. Note that i = (−1)1/2 and β(z,iξ) ≡ 4πρ(z)α(iξ)/3 by definition. In eq 2, Cj, ωj, and γj denote the coefficients associated with the oscillator strength, the absorption frequency, and the damping factor for the jth term, respectively. N is a number of terms in the polarizability model. Table 1 shows the parameters for the model with five absorption frequencies.12 The polarizability model gives rise to unreasonable dielectric responses based on eq 2 at zero frequency. Thus, the known zero frequency polarizability of
Figure 1. Water density near mica (black solid line), TiO2 (red solid line), and gold (green solid line) surfaces reproduced from Cheng et al.,8 Liu et al.,9 and Velasco-Velez et al.10 Bulk water has a density of 0.0331 1/Å3.
shows the spatial density variation of water near mica(001), TiO2 (rutile,110), and gold(111) surfaces under ambient conditions, obtained by high-resolution specular X-ray reflectivity, density functional theory (DFT), and ab initio molecular dynamics (AIMD) simulations, respectively. As expected, the water density in the direction of the interface normal oscillates. As seen in Figure 1, the density response is unique and significant near all surfaces considered here and decays to its bulk value of 0.0331 1/Å3 at approximately 1.2 nm (mica and gold) and 0.9 nm (TiO2), yielding clear definitions of λd ∼ 1.2 nm and λd ∼ 0.9 nm. A dielectric response of substance represents its capacity to affect the electric field through the orientational response of the
Table 1. Coefficients for the Oscillator Strength, the Absorption Frequencies, and the Damping Factors in the Polarizability Model of Water at 25 °C12
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N
Cj
1 2 3 4 5
0.154 0.186 0.004 0.020 0.202
ωj [rad/s] 3.64 1.32 3.10 6.59 1.90
× × × × ×
1013 1014 1014 1014 1016
γj [rad/s] 5.0 1.0 2.6 1.3 5.9
× × × × ×
1013 1014 1013 1014 1015
DOI: 10.1021/jp512550c J. Phys. Chem. B 2015, 119, 5873−5881
Article
The Journal of Physical Chemistry B water (i.e., α(0) = 1.45 × 10−24 cm3)13 is used with eq 2 in order to consistently include the effect of the local solvent density response at zero frequency. As an example, Figure 2 depicts the dielectric response for water as a function of the sampling frequency for van der Waals
Figure 3. A schematic of the system of two interacting materials in solution; L̃ and L denote a separation between surfaces and structured water layers, respectively. λd is a length scale for solvent structuring set to 1.2 nm for mica and gold surfaces and 0.9 nm for TiO2 surfaces. The boundary between bulk water and structured water layer is at za = L/2, and the boundary between structured water layer and surface is at za = L/2 + λd.
Figure 2. Dielectric response of water incorporating the solvent response in Figure 1 for different distances in the direction normal to the mica surface, namely z = 0.15, 0.25, 0.45, 0.65, and 0.85 nm. The solid black line represents dielectric response of bulk water based on a model proposed by Parsegian.2
Assuming the nonretarded formulation of Lifshitz theory and a negligible magnetic susceptibility, the van der Waals interaction between two flat surfaces with nonuniform solvent distribution (shown in Figure 3) can be analogously described by6,14
interactions (i.e., Matsubara frequency ξn = 4π2nkBT/h) with different separations in the direction of the interface normal, z, combined with eq 2 for the nonuniform density of water near mica shown in Figure 1. Here, kB, h, and T denote the Boltzmann’s constant, Planck’s constant, and system temperature, respectively. As seen in Figure 2, the dielectric response of water varies significantly and approach the bulk dielectric response of water as z becomes λd. Furthermore, it can be also deduced that, as the frequency increases, the variation of dielectric response determined by the nonuniform density becomes negligible because the water molecule cannot respond fast enough to high-frequency electromagnetic waves, and its dielectric response would become identical as frequency increases, irrespective of the nonuniform density. However, the main contribution on the dispersion interactions from the density variation still takes place at higher frequencies owing to the nature of sampling frequencies employed in dispersion interactions (i.e., the number of Matsubara frequency as a function of frequency range). To compute the van der Waals interactions in conjunction with the solvent response, one needs to incorporate an inhomogeneous dielectric response into Lifshitz theory. Parsegian and Weiss14 (also see Parsegian6) developed the formulation of the van der Waals interactions between two flat surfaces with an incompressible layer having an inhomogeneous dielectric susceptibility. The formulation was originally intended for calculating van der Waals interactions between thin-coated flat surfaces in an intervening medium. However, the formulation can be conceptually applied to investigate the role of the nonuniform solvent density to the point where there is overlap of the inhomogeneous regions. Thus, one can consider the nonuniform density as a thin layer producing an inhomogeneous dielectric susceptibility in a uniform solvent shown schematically in Figure 3.
G (L ; λ d ) =
kBT 2π
∞
∑′∫ n=0
0
∞
eff
ρ ̃ ln[1 − (Δ̅ Am )2 e−2ρ ̃L]
(3)
where ρ̃ denotes the magnitude of the wave vector. The prime in the summation indicates that the n = 0 term is to be multiplied by 1/2. In order to calculate the van der Waals interaction the evaluation of eff Δ̅ Am
L + Δ̅ am ( 2) ≡ L 1 + ua( 2 )Δ̅ am
ua
(4)
with L − εm ( 2) Δ̅ am = L εs( 2 ) + εm
εs
(5)
is needed. Here, εs and εm denote dielectric functions of the nonuniform aqueous layer and bulk, respectively. The quantity ua(L/2) can be obtained from solving dua(za) d ln[εs(za)] = 2ρ ̃ua(za) − [1 − ua 2(za)] dza 2dza
(6)
whose solution numerically begins at L/2 + λd: εin − εs ⎛L ⎞ ua⎜ + λa⎟ = ⎝2 ⎠ εin + εs
( L2 + λd) ( L2 + λd)
(7)
where εin denotes the dielectric function of the interface. Since no water exists at za = L/2 + λd shown in Figure 1, εs(L/2 + λd) becomes unity. 5875
DOI: 10.1021/jp512550c J. Phys. Chem. B 2015, 119, 5873−5881
Article
The Journal of Physical Chemistry B A Lorentzian formula can be used for the dielectric response of the interface. For mica, the Lorentzian formula by Chan and Richmond15 is εuv − 1 εin(iξ) = 1 + 1 + (ξ /ω0)2 + (ξ /g ) (8) where εuv (=2.45) is the near-visible dielectric permittivity, ω0 =2.38 × 1016 rad/s, and g = 4.89 × 1016 rad/s. This model exhibits a reasonable asymptotic behavior at a characteristic plasma frequency of mica (= 2.86 × 1016 rad/s). For TiO2, a similar Lorentzian formula by Bergström16 can be used to represent dielectric responses of the interface, namely c uv εin(iξ) = 1 + 1 + (ξ /ωuv )2 (9)
Figure 4. van der Waals interactions between 16 nm2 flat mica surfaces as a function of L̃ in the presence of a nonuniform solvent density (green diamonds) in kBT, where kB is Boltzmann’s constant and T is temperature. The red squares represent the result considering only the bulk dielectric response of water. The black solid line represents the conventional Lifshitz theory. The gray region represents L̃ = 2.4 nm where there will be overlap between nonuniform solvent distribution; namely, L̃ = 2λd where the present theory becomes invalid.
Here, cuv (=4.81) is the absorption strength in the UV frequency range and ωuv= 0.77 × 1016 rad/s is the characteristic UV absorption frequency. Furthermore, a Lorentzian formula for gold is given by Parsegian and Weiss17 fj
4
εin(iξ) = 1 +
∑ i=1
ωj̃ 2 + ξ 2 + gjξ
(10)
cases at ∼6−7 nm. The correct asymptotic behavior between all cases after ∼6−7 nm of separation clearly indicates that our mapping of the solvent response to the dielectric response using the Clausius−Mossotti equation is reasonable. Moreover, the asymptotic behavior is indeed consistent with the order of magnitude estimate by Parsegian6 associated with the neglect of a nonuniform density distribution being O(λd/L̃ ). Figure 4 suggests that the van der Waals interactions become appreciable (i.e., ∼kBT) at 3 nm due to the nonuniform density. The calculations with the nonuniform distribution are clearly limited to a separation beyond the overlapping nonuniform solvent density, namely L̃ = 2λd represented by the gray area in Figure 4. While the dispersion interaction at L
