Effective Polarizability Models - ACS Publications - American

Article Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Effective Polarizability Models Johannes Fiedler,*,† Priyadarshini Thiyam,*,‡,§ Anurag Kurumbail,∥ Friedrich A. Burger,† Michael Walter,†,⊥,# Clas Persson,‡,§ Iver Brevik,∥ Drew F. Parsons,*,¶ Mathias Boström,*,∥ and Stefan Y. Buhmann*,†,∇ †

Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany Department of Materials Science and Engineering, KTH, Royal Institute of Technology, SE-100 44 Stockholm, Sweden § Centre for Materials Science and Nanotechnology, Department of Physics, University of Oslo, P.O. Box 1048, Blindern, NO-0316 Oslo, Norway ∥ Department of Energy and Process Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway ⊥ FIT Freiburg Centre for Interactive Materials and Bioinspired Technologies, Georges-Köhler-Allee 105, 79110 Freiburg, Germany # Fraunhofer IWM, Wöhlerstrasse 11, D-79108 Freiburg i. Br., Germany ¶ School of Engineering and IT, Murdoch University, 90 South Street, Murdoch, WA 6150, Australia ∇ Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universität Freiburg, Albertstrasse 19, 79104 Freiburg, Germany ‡

ABSTRACT: Theories for the effective polarizability of a small particle in a medium are presented using different levels of approximation: we consider the virtual cavity, real cavity, and the hard-sphere models as well as a continuous interpolation of the latter two. We present the respective hard-sphere and cavity radii as obtained from density-functional simulations as well as the resulting effective polarizabilities at discrete Matsubara frequencies. This enables us to account for macroscopic media in van der Waals interactions between molecules in water and their Casimir−Polder interaction with an interface.

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INTRODUCTION The optical behavior of a small object dissolved in a medium is of interest for a large number of investigations, such as optofluids,1 medium-assisted density-functional theory (DFT),2 nanomedicine,3 hydrogen storage,4 bio-organics,5 and photodynamic therapy.6 The impact of a cavity around the particles is large compared to the typically studied situations where the particles are considered without an environment. In this fundamental study we derive different models for the effective polarizability of small particles in a medium. The level of accuracy for six different approximations will be discussed and our results exploited to calculate Casimir−Polder forces in a medium. The new theoretical results are then applied to greenhouse gas molecules,7 and other gas molecules, dissolved in water. After water vapor and carbon dioxide, methane and nitrous oxide are the most important long-lived greenhouse gases in the atmosphere. Release of greenhouse gases from surfaces in water is influenced by the Casimir−Polder interaction. To estimate the Casimir−Polder binding gas particles to surfaces, one needs to have accurate effective polarizabilities of the greenhouse gas molecules in water. As input parameters in our models, we require the frequencydependent polarizability of the molecules in a vacuum, hardsphere radii, cavity radii in water, and the dielectric function of water. We use ab initio quantum chemical calculations to accurately calculate the frequency-dependent polarizabilities of different gas molecules in a vacuum. The result is given as a set © XXXX American Chemical Society

of parametrized functions, which enables easy access in future investigations that require these molecular polarizabilities. We also provide the theory for determining the molecular radii and cavity radii of different greenhouse gas molecules in water. These parametrized functions and tabulated data for radii, together with the new theories and tabulated dielectric function of water for discrete Matsubara frequencies, enable us to calculate the effective polarizability of gas molecules in water. This is a key quantity needed in future studies of binding or release of gas molecules near solid−water interfaces. As an illustrative example we consider the nonretarded van der Waals potential of two molecules in water and their Casimir−Polder potential near a perfect metal surface to illustrate the impact of the different cavity models. Therefore, we treat the environment as a continuous medium as a kind of ensemble average of its constitutions. Due to this reason we focus on single-particle interactions, as they are valid for dilute gases. Further investigations on few- and many-particle interaction embedded in a cavity can be performed additionally.8−11 This enables us to compare how sensitive the results are to different approximations done when the effective polarizability is evaluated. It turns out that the theoretical sensitivity can be quite large; even the sign of the force on molecules close to a Received: October 13, 2017 Revised: November 21, 2017 Published: November 29, 2017 A

DOI: 10.1021/acs.jpca.7b10159 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

Table 1. Weights (αj [10−42 A2 s4 kg−1] in SI-Units Which Transform to CGS Units via [αSI] = 4πε010−30 Å3) and Characteristic Frequencies (ωj [1016 rad s−1] in SI-Units, Which Transform to CGS Units via [ωSI] = 2πe/ℏ eV) for Five-Mode London Fits of the Dynamic Polarizabilities of Four Greenhouse Gas Molecules CH4, CO2, N2O, and O3 and Other Atmospheric Gas Molecules mode 1 CH4 CO2 N2 O O3 O2 N2 CO NO2 H2S

mode 2

mode 3

mode 4

mode 5

α1

ω1

α2

ω2

α3

ω3

α4

ω4

α5

ω5

89.3 131.7 121.5 72 38 90 49.1 16.5 55.1

1.75 1.95 1.64 0.89 1.37 2.14 1.43 0.63 1.12

137.5 116 146.2 130 85.4 10 120 124 99

2.56 3.14 54.9 2.28 2.78 3.27 2.46 1.65 3.25

41.2 41.5 54.9 98.7 41.1 29.9 41.7 134.1 251.6

4.42 6.2 5.59 4.32 5.42 5.92 4.95 3.57 1.86

2.78 6.43 8.05 21 8.16 3.55 6.89 28.8 2.33

10 13.3 12.3 9.82 10.9 13.2 11 8.39 9.58

0.18 0.43 0.41 0.89 0.76 0.23 0.39 1.19 0.98

48.3 51.5 53.9 36.5 29.5 60.5 45.7 31.6 34.2

Figure 1. (a) Sketch of Onsager’s real cavity model. A particle (red dot) at position rA is embedded in a medium with dielectricity ε(ω) (grey area) surrounded by a spherical vacuum cavity (white area). The scattering process from an external point r (green dot) is separated into the propagation to and back from the particle. (b) Sketch of Onsager’s real cavity model for finite size particles. A spherical particle with radius R and the dielectric function εs is embedded in a medium with ε(ω) (grey area) surrounded by a spherical vacuum cavity (white area) with radius RC. The scattering process from an external point r′ (red dot) to another point r (green dot).

basis set, aug-cc-pVQZ.16 The geometry of each molecule was optimized by energy minimization before evaluating its polarizability. The full anisotropic polarizability tensor was evaluated, and anisotropic effects are known to have an impact on Casimir−Polder interactions.17,18 However, for simplicity we use the isotropic average, α = (αxx + αyy + αzz)/3, in this manuscript. Quantum chemical calculations of dynamic polarizabilities were performed at the Matsubara frequencies iξn = i2πkBTn/ℏ for T = 298.15 K with n = 0, 1, ..., 2100. It is convenient to represent the polarizability at an arbitrary imaginary frequency iξ by fitting the polarizability modes to an oscillator model

metal surface can change if we go from one effective polarizability model to another. There is thus a need for experimental information to identify the best model.

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FREE-SPACE POLARIZABILITY OF GAS MOLECULES The Casimir−Polder energy is determined from the effective polarizability of a molecule in a medium at imaginary frequencies iξ. Each of the models of effective polarizability considered in this manuscript describes a different representation of the transformation induced by the medium on the underlying dynamic polarizability α(iξn) of the given molecule in a vacuum (the free-space polarizability). We have computed the free-space polarizability of each molecule by quantum chemical methods. The closed-shell gas molecules, CH4, CO2, N2O, O3, N2, CO, and H2S were calculated at the coupled cluster singles and doubles (CCSD) level of theory12 using Molpro.13 The open-shell (paramagnetic) molecules O2 and NO2 were calculated using Turbomole14 at a density functional theory (DFT) level with a hybrid PBE0 functional.15 Ground states involved higher spin states, O2 having spin multiplicity 3 (two unpaired electrons), NO2 with multiplicity 2 (one unpaired electron). For both CCSD and DFT/PBE0 calculations we employed an augmented correlation-consistent

α(iξ) =

∑ j

αj 1 + (ξ /ωj)2

(1)

A five-mode fit has previously been found to describe the dynamic polarizability accurately to a 0.02% relative error.19 The adjusted parameters for a five-mode model fitted to agree with the free-space polarizability obtained from ab initio calculations are given in Table 1. B


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CASIMIR−POLDER AND VAN DER WAALS POTENTIAL The Casimir−Polder force arises when a small particle comes close to a dielectric surface.20 It is caused by the fluctuations of the ground-state electromagnetic field.21 Performing the field quantisation and applying second-order perturbation theory to the dipole-electric field Hamiltonian Ĥ = −d̂·Ê for the ground state of the field and particle, one finds the Casimir−Polder potential acting on a particle located at rA21,22

which has to be read from right to left, and is due to a virtual photon which is created at particle A, propagates to particle B, where it interacts with its polarizability, and is backscattered to particle A. Again, the sum (integral) over all two-photon exchanges results in the van der Waals potential. According to the nonretarded Casimir−Polder potential, the nonretarded Green function21 G(rA,rB,ω) = −

∞

UCP(rA) = μ0 kBT ∑ ′ξn 2α(iξn) tr[G(S)(rA,rA,iξn)] n=0

UvdW(r ) = −

ω2 ε(r,ω) G(r,r′,ω) = δ(r − r′) c2

with the dielectric function εH describing the electric response of the half-space. A perfectly conducting plate requires the limit εH(ω) → ∞. Due to the separation of spatial and frequency dependencies of the scattering Green function, the Casimir− Polder potential, eq 2, simplifies to the well-known result z3

n=0

αA(iξn) αB(iξn) ε 2(iξn)

(9)

(10)

In the second picture, we reverse the optical paths and describe the scattering effect as an effective reflection at a sphere, which means that the corresponding initial and final points are located outside the sphere; see Figure 1b. The propagation is again represented by the free propagation in a bulk medium. Thus, we can write

(5)

In analogy, the van der Waals potential describing the interaction between two neutral, but polarizable particles can be derived via the fourth-order perturbation of the two-particle dipole-electric field Hamiltonian Ĥ = −d̂A·Ê − d̂B·Ê , which will be performed for ground-state fields and particles. It results in21,22

Gcav (r,r′,ω) = G(r,R,ω)·α*(ω) · G(R,r′,ω)

(11)

where R is a point at the spherical cavity’s surface and the excess polarizability α*(ω). In the following we analyze different models for such excess polarizabilities. To apply eqs 5 and 8, the free-space polarizability has to be exchanged by the excess polarizability25

∞

UvdW(rA,rB) = −kBTμ0 2 ∑ ′ ξn 4αA(iξn) αB(iξn) n=0

× tr[G(rA,rB,iξn) ·G(rB,rA,iξn)]

8π ε0

α*(iξ) = R(iξ) α(iξ)

∞

α(iξn) kBT ∑′ 8πε0 n = 0 ε(iξn)

2

∞

′ 2∑

Due to the linearity of the Casimir−Polder potential, eq 2, in the polarizability and the scattering Green function, an excess polarizability can be defined as

(4)

C3 =

3kBT

Gcav (r,r′,ω) = R(ω) G(r,r′,ω)

εH(ω) − ε(ω) c2 diag(1,1,2) 32πω ε(ω)z 3 εH(ω) + ε(ω) 2

C3

r

6

C6 =

where r denotes the distance between particle A and B. The advantage of these formulas is the separation into particle properties α(iξ) and the scattering processes G(S)(rA,rA,iξ). To describe the Casimir−Polder interaction in media, the scattering Green function changes its properties with respect to the dielectric function of the medium and to the geometrical shape of the cavity around the particle, which gives rise to additional resonances due to the cavity modes. Two different pictures for the consideration of the scattering processes will be used in describing the influence of a cavity effectively. In the first case, the particle is centered in the cavity and the scattering through the cavity’s boundary is considered; see Figure 1a. Assuming that the scattering processes between the surface and the particle−cavity system are dominated by a single scattering effect (we neglect multiple scattering effects), the Green function factorizes into a bulk propagation G (here: in water) and an effective transmission coefficient R(ω), which has to be found, through the cavity’s surface, which includes the properties of the cavity and yields24

with the relative permittivity ε(r,ω). The dyadic Green function can be separated into the free propagation through the bulk medium G(0) and the scattering part G(S), G = G(0) + G(S). The Casimir−Polder potential, eq 2, shall be understood due to a virtual photon with frequency iξ being created by the particle at rA and backscattered at the dielectric surface as expressed by the scattering Green function. The strength of its fraction to the Casimir−Polder potential is weighted by the polarizability of the particle. Because the fluctuations of the fields ensue at all frequencies, a superposition of all scattering processes results in the Casimir−Polder potential. Assuming that the particle is located in the nonretarded regime in front of planar surface, the Casimir−Polder potential reduces to the established C3-potential. This situation is described by the scattering Green function for a planar twolayer system23

UCP(z) = −

C6

(8)

(3)

G(r,r,ω) =

(7)

with ϱ = rB − rA, ϱ = |ϱ|, eϱ = ϱ/ϱ, and k2(ω) = ε(ω)ω2/c2, for the propagation through a bulk medium can be applied. This results in

(2)

with the Boltzmann constant kB, the vacuum permeability μ0, the particle polarizability α(iξ) at imaginary frequencies iξ, the primed sum means that the zeroth term has to be weighted by 1 /2 and the scattering part of the dyadic Green’s function G(S), which satisfies the vector Helmholtz equation ∇ × ∇ × G(r,r′,ω) −

1 (I − 3eϱ ⊗ eϱ) 4πk 2(ω)ϱ3

α(iξ) → α*(iξ)

(6) C

(12) DOI: 10.1021/acs.jpca.7b10159 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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⎛ ε(ω) + E′ = ⎜ ⎝ 3

POLARIZABILITY OF PARTICLES EMBEDDED IN A MEDIUM In the following we introduce the established models estimating the effective electric polarizability of a particle embedded in a

α

local-field corrected, eq 19

( ε +3 2 ) α 2 ( 1 +3ε2ε ) α

Onsager, eq 24

⟨Ê ′(r,ω) ⊗ Ê ′† (r′,ω′)⟩

α* =

free space

⎛ ε(ω) + 2 ⎞2 † ⎟ ⟨Ê (r,ω) ⊗ Ê (r′,ω′)⟩ =⎜ ⎝ ⎠ 3

2

hard sphere, eq 33

ε −ε 4πε0R s3 ε s+ 2ε s

finite size, eq 36

αC* + α

(

⎛ ε(ω) + 2 ⎞2 ℏ ω 2 Im G(r,r′,ω)δ(ω − ω′) ⎟ =⎜ ⎝ ⎠ πε0 c 2 3

)

⎛ ε(ω) + 2 ⎞2 α*(ω) = ⎜ ⎟ α(ω) ≡ αvirt ⎝ ⎠ 3

G(2S)(rA,rA,ω) = R(ω) G(S) bulk (rA,rA,ω)

(20)

G(S) bulk

where denotes the scattering Green function for a bulk medium with the permittivity of the outer layer and the transmission coefficient24,30 R(ω) =

i zs[j1 (zs)[zh1(1)(z)]′

− ε(ω)[zsj1 (zs)]′h1(1)(z)] (21)

with the spherical Bessel and first kind Hankel functions jn and h(1) n , respectively, which yields the exact excess polarizability

(14)

α*(ω) = −α(ω)

and that the polarization is the response to the application of the local electric field to the sphere

1 zs 2

⎛ ⎞2 1 ⎟ × ⎜⎜ ⎟ (1) (1) ′ − ε ω ′ j ( z )[ zh ( z )] ( )[ z j ( z )] h ( z ) ⎝1 s ⎠ 1 s1 s 1

(15)

with the number density η of atoms inside the sphere and the polarizability, one finds the polarizability of a sphere as

ε(ω) − 1 α = 4πε0R3 ε(ω) + 2

(19)

which we denote by αvirt. Onsager’s Real Cavity Model. Onsager’s real cavity model considers a spherical vacuum bubble around a particle at position rA embedded in a medium with permittivity ε(ω).27 Figure 1a illustrates the arrangement. The scattering Green function for a spherical two-layered system with final and source points in the outer layer can be found in refs 28 and 29. The boundary conditions entering the reflection coefficients read as z = kRC and zs = ksRC with the cavity radius RC and the absolute values of the wave vector inside and outside of the sphere ks and k, respectively. Considering a single scattering event starting outside the cavity toward its center and back scattering, these processes can be described with the Born series expansion.22 The scattering Green function can be expressed as24

with the polarization of the sphere P. Using the relation that the polarization is the electric response of an externally applied electric field

P = αη E′

(18)

Hence, the local-field-corrected excess polarizability can be written as

2 1 3ε 1 + 2ε 1 + αC*α / (8π 2ε02R C6ε)

medium. First, we start with the virtual cavity model, which results in the Clausius−Mossotti relation. We continue with the real cavity model, where we consider two different configurations. One describes the effective polarizability by treating the particle as a point-like object which leads to the Onsager’s real cavity model. In analogy, the hard-sphere model corresponds to Onsager’s real cavity model and uses a spatially spread out dielectric function over the complete cavity volume. This yields the vanishing of the vacuum layer. An interpolation between both real cavity models modeling finite-size particles is also considered. Clausius−Mossotti Relation and Virtual Cavity Model. The Clausius−Mossotti relation describes the relation between the microscopic quantity polarizability and the macroscopic quantity dielectric function. It is also known as the virtual cavity model, as or local-field corrections, because it describes the increase of the electric field in the presence of a dielectric sphere with radius R. By considering the fields inside and outside the sphere E′ and E, respectively, one finds that the local field around the sphere increases by26 1 E′ = E + P 3ε0 (13)

P = ε0[ε(ω) − 1]E

(17)

This results in that the self-correlation between the local electric field21

Table 2. Summary of the Polarizability Models model

2⎞ ⎟E ⎠

(22)

Applying the Taylor series expansion assuming that the cavity radius RC is small compared to the relevant wavelengths, RC ≪ k−1, ks−1, to this transmission coefficient results in

(16)

This equation is the Clausius−Mossotti relation and means that the electric response, expressed by the polarizability α, of a sphere with radius R is given by the product of its volume and the Mie reflection. Furthermore, the local-field correction due to the presence of a spherical object, eq 13, together with the polarization, eq 14, leads us to write the local electric field at the sphere as

R(ω) ≈

3ε(ω) 1 + 2ε(ω) −

2 3 ε(ω)[10ε 2(ω) − 9ε(ω) − 1] ⎛ ωR C ⎞ ⎜ ⎟ ⎝ c ⎠ 10 [1 + 2ε(ω)]2

(23) D


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The Journal of Physical Chemistry A and leads to the famous excess polarizability for small radius RC ⎛ 3ε(ω) ⎞2 α*(ω) = α(ω)⎜ ⎟ ≡ αOns ⎝ 1 + 2ε(ω) ⎠

α*(ω) = −i

3ε(ω) ⎡ 3ε(ω) ⎢ 1 + 2ε(ω) ⎢⎣ 1 + 2ε(ω)

⎛ 3 ε(ω)[10ε 2(ω) − 9ε(ω) − 1] ⎞⎛ ωR C ⎞2 ⎤ ⎟ ⎥ − 2⎜ ⎟⎜ [1 + 2ε(ω)]2 ⎝ 10 ⎠⎝ c ⎠ ⎥⎦

r32̃ ≈ B1N =

(25)

shows that this approximation works for typical cavity radii. It has an agreement of more than 99% compared with the exact solution. The corresponding frequency of the cavity mode can be estimated as ωc = c/RC, with the speed of light c. Typical values for the cavity radius are on the order of several Angstroms which result in a cavity mode of the order of 1018 rad s−1, where typical materials are transparent. Onsager’s Real Cavity Model for Finite-Size Particles. The Onsager real cavity model for spatially extended particles can be described by a spherical three-layer system.31 The inner layer represents the particle, the outer layer the surrounding medium, and the second layer between both denotes the vacuum cavity. The description of this arrangement is similar to Onsager’s real cavity model. Again, a scattering process at the outer boundary has to be described that follows the same scattering Green function for a spherically layered system29 whereas the reflection coefficients need to be exchanged by the one for a three-layered system which read28,32

r32̃ = r32 +

t 23r21t32 1 − r23r21

+

(εs + 2)(2ε + 1) + 2(εs − 1)(1 − ε)R /R C

αs = 4πε0R3

εs − 1 εs + 2

εiμi + 1 Hn(1)′(ki + 1a) Hn(1)(kia)

(27)

εiμi + 1 Jn (kia)

Hn(1)′(ki + 1a)

εi

εiμi + 1 Jn (kia) Jn ′(ki + 1a) εi + 1μi Hn(1)(ki + 1a) Jn ′(kia)

(29) iεi εiμi + 1 Jn (kia)

Hn(1)′(ki + 1a)

μi εi + 1

−

(35)

(36)

which we denote by αfs and the dressed polarizability αs*. The performed approximation agrees with the exact values with a confidence of more than 99% for small molecules. Due to the multiple reflections occurring here the quality of the series expansion strongly depends on the radii R and RC. It can be imagined that, for larger objects, such as a fullerene, this approximation might fail because of the increase of the multiple reflection term. Table 2 summarizes the different effective polarizability models. We note the appearance of the prefactor ε in the effective polarizabilities of the sphere in eq 33 (and for the cavity in eq 35), suggesting that the effective polarizability of a molecule is generally larger in a more polar medium. This contrasts with the effective polarizability presented by Netz,33 which does not contain the prefactor ε. This apparent contradiction arises

(28) εiμi + 1 Jn (kia) Hn(1)′(ki + 1a) −

1−ε 1 + 2ε

= αC* + αs* ≡ αfs

εi + 1μi Hn(1)(ki + 1a) Jn ′(kia)

εi + 1μi Jn (ki + 1a) Jn ′(kia) −

(34)

⎛ 3ε ⎞2 1 ⎟ αS*+ C = αC* + αs⎜ ⎝ 2ε + 1 ⎠ 1 + αC*αs/(8π 2ε02R C6ε)

μi + 1

−

⎤ ⎥ ⎦

Substituting eqs 34 and 35 in eq 33 the excess polarizability for the three-layered system simplifies to


iεi + 1

3

and the excess one for the cavity αC* = 4πε0εR C3

εiμi + 1 Jn (kia) Hn(1)′(ki + 1a) −

ti + 1, i =

3

which is the Mie coefficient and is denoted αHS, meaning that the hard-sphere polarizability arises when the vacuum layer vanishes. Using this result, one can define excess polarizabilities in the three-layer model. One is the free-space polarizability for the sphere surrounded by vacuum

ri , i + 1 =

ri + 1, i =

9εR3(εs − 1)/(2ε + 1)

(32)

which denotes the multiple reflection coefficients at the second boundary with the transmission and reflection coefficient tij and rij, respectively, between the i-th and j-th layer24,28,32

ti , i + 1 =

3 2i ⎛⎜ ω⎞ ⎡ 1−ε εμ ⎟ ⎢R C3 ⎝ ⎠ 3 c ⎣ 1 + 2ε

where R and RC denote the radii of the particle and the cavity, respectively. In analogy to the virtual cavity model, it can be considered as the electromagnetic scattering at a sphere with radius Rs and permittivity εs embedded in a medium ε. Considering the electric fields through a dielectric sphere with εs and radius Rs embedded in a medium ε, one finds an excess polarizability (Rs → RC) ε −ε α* = 4πε0εR s 3 s ≡ αHS εs + 2ε (33)

(26)

εi + 1μi Hn(1)(ki + 1a) Hn(1)′(kia) −

(31)

Note that the dielectric function of the inner sphere is connected to the corresponding polarizability via the Clausius− Mossotti relation, eq 16. Assuming a small radius of the cavity and of the particle, the vector wave functions reduce to the first order (n = 1), because all other terms vanish for a centered particle. The reflection coefficient for a purely electric field, eq 26, can be expand into a series expansion for small radii R and RC

(24)

which is denoted by αOns. For small but finite radii, a Taylor expansion α*(ω) = α(ω)

6πε0 c 3 r32̃ ε ω3


(30) (1) with Jn(x) = xjn(x) and H(1) n (x) = xhn (x) and a denotes the position of the boundary between the i-th and (i + 1)-th layer. Again, the comparison of the exact Green function for the spherically layered system with that for the bulk propagation, eq 11, results in the excess polarizability that reads in general

E


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radius can be expected to depend on temperature, as observed in solvation radii determined from measurement of ion conductivity.45 Table 3 lists the cavity volumes VC and corresponding radii RC = (3VC/4π)1/3 that were obtained from the cavity definition in the recently developed continuum solvent model implemented in the GPAW package.2,46 The generally nonspherical cavity with a smooth boundary is obtained from an effective repulsive potential that describes the interaction between the continuum of the water solvent and the solute molecule. This potential leads to an effective solvent distribution function 0 ≤ g ≤ 1 that can be related to measurable partial molar volumes at the limit of infinite dilution through the compressibility equation.47 Using this connection to fit the effective potential, only a single parameter is needed to predict the volumes of various test molecules in water in good agreement to experiment.2 The cavity volume VC is obtained by an integration over the solvent excluded volume 1 − g. The hard-sphere radii R describe the effective spherical radius for the volume occupied by the electron cloud of the molecules,48 defined as the volume within which the electron density exceeds 0.001 electrons/bohr3 (evaluated using Gaussian49). The cavity radii RC, by contrast, correspond to the position at which the electron density of the solvent molecules becomes significant (i.e., the position in space where the dielectric medium starts to respond to an external field).43,44 The success of polarizable continuum models in the description of solvent effects, ranging from ground-state to excited-state properties, shows that considering the solvent as a continuum is appropriate down to the atomic scale. Dynamic effects arising from averaging over the solvent molecules’ movements can be conveniently included into the effective continuum.

because of alternative possible definitions of what the effective polarizability means. The effective polarizability may be thought of as the linear relationship between an external electric field and the dipole (embedded in a medium) induced in a molecule by that field.26 The polarization field P, thought of as the field generated by that induced dipole, has the same value regardless of the definition of the induced dipole. In the picture we have presented here, we have defined the effective polarizability such that the polarization field corresponds to an induced dipole located inside the medium with ε. Netz, by contrast, adopted a definition of the effective polarizability that generates the same polarization field as if the corresponding induced dipole was located in a vacuum, rather than in the medium. In other words, the definition depends on whether we take the effective polarizability to refer to an effective induced dipole in medium or in a vacuum. The difference matters in the evaluation of the van der Waals energy, whether the latter treats the medium explicitly (cf. the appearance of ε in the van der Waals parameters eq 5 or eq 8) in section below or whether, as in Netz’s approach, the van der Waals energy is evaluated as if the field is in a vacuum. In our case (effective induced dipole in a medium), ε in the effective polarizability cancels with that in denominator of the van der Waals parameters. In Netz’s case (effective induced dipole in a vacuum), the factors of ε are never present. The net van der Waals energy is the same either way (the two definitions were incorrectly mixed in ref 19, resulting in an underestimate of van der Waals parameters). We suggest that the approach of effective-dipole-in-medium provides a more natural or more general definition of the effective polarizability, because in the three-layered system ε cannot be simply lifted out from eq 36.

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CAVITY RADIUS FOR GAS MOLECULES IN WATER To apply the model of eq 36, we need to estimate R and RC for the molecules under investigation. The definition of the cavity

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APPLICATIONS: EFFECTIVE POLARIZABILITIES FOR GAS MOLECULES IN WATER The effective polarizabilities determine the strength and sign for van der Waals potentials and Casimir−Polder potentials of gas molecules dissolved in water near surfaces or in bulk water. The free-space polarizabilities and the resulting effective polarizabilities for two different dissolved gas molecules (using the models described in previous sections) are shown in Figure 2. It is the expectation that the effective polarizability of a molecule in a medium is less than its corresponding free-space polarizability because it reflects the difference between the dielectric function in a sphere and the surrounding media (water). This is not observed in the simpler models αvirt, eq 19 and αOns, eq 24. The most sophisticated model αfs, eq 36, accounts for a small polarizable sphere (with a size modeled by the hard-sphere radius R) inside a vacuum bubble (with radius modeled by the cavity radius RC). Here the presence of a vacuum layer means that the effective polarizability may become negative in some frequency regions. This is obviously so, because a simple vacuum bubble is less polarizable than the surrounding water. We observe that the closer the values for the hard-sphere radius and cavity radius are, the less negative the effective polarizability will be. To illustrate the impact of the different results, we determine the expected van der Waals and Casimir−Polder parameters for the different molecules. Note that the Casimir−Polder (C3) parameters in Table 3 correspond to more attraction in free space than any of the C3 parameters using the effective polarizability models in water.50

Table 3. Cavity Volume VC (in General Not Spherical), Cavity Radii RC (Spherical Approximation), and HardSphere Radii R of Four Greenhouse Gases CH4, CO2, N2O, and O3 and Other Atmospheric Gas Molecules gas

VC (Å3)

RC (Å)

R (Å)

CH4 CO2 N2O O3 O2 N2 CO NO2 H2S

47.00 62.28 58.83 58.79 43.84 44.98 51.31 58.33 52.73

2.239 2.459 2.413 2.412 2.187 2.206 2.305 2.406 2.326

1.655 1.723 1.991 1.984 1.300 1.409 1.966 1.986 2.030

radius RC is under discussion in the literature34 and is often based on fits to solvation energies.35,36 More generally, it is possible to compute the probability distribution p(RC,T) of cavities with radius RC in a solvent at temperature T,37−40 with the cavity energy for a given radius equal to −kT ln p(RC,T). The cavity radius for a given ion may be extracted from the ionsolvent radial distribution function g(r), or more specifically the ion-oxygen rdf in the case of water.41,42 The position of the first peak in g(r) has been used to determine ion solvation energies43 and partial molar volumes.44 It follows from the temperature dependence of g(r) that, in general, the cavity F


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Figure 2. Polarizabilities (green) and effective polarizabilities for (a) N2O and (b) H2S. We compare the local-field-corrected model, following eq 19, αvirt (blue), the Onsager real cavity model, following eq 24, αOns (orange), the hard-sphere model αHS, eq 33 (black), and Onsager’s real cavity model for finite-size particles, eq 36, αfs (red). In addition, the corresponding cavity modes ωC = c/RC and the hard-sphere modes with respect to the molecule radius ωM = c/R are drawn as straight lines.

Table 4. C3 Coefficients in 10−49 J m3 for the Different Gases in a Vacuum,a in a Medium without a Cavity,b with the Local-Field Correction (Virtual Cavity),c with Onsager’s Real Cavity,d for the Real Cavity with Finite Size of the Particles,e and with the Hard-Sphere Cavity Modelf

Table 5. C6 Coefficients in 10−79 J m6 for the Different Gases in a Vacuum,a in a Medium without a Cavity,b with the Local-Field Correction (Virtual Cavity),c with Onsager’s Real Cavity,d for the Real Cavity with Finite Size of the Particles,e and with the Hard-Sphere Cavity Modelf

gas

vacuum

medium

loc corr

Onsager

finite size

hard sphere

gas

vacuum

medium

loc corr

Onsager

finite size

hard sphere


2.69 3.61 3.81 3.75 2.28 2.38 2.44 3.56 3.35

2.01 2.8 2.94 2.92 1.79 1.85 1.87 2.76 2.45

2.56 3.46 3.65 3.6 2.19 2.28 2.34 3.41 3.21

2.36 3.23 3.4 3.36 2.05 2.13 2.17 3.18 2.91

−0.78 −0.96 −0.54 −0.58 −0.91 −0.9 −1.29 −0.73 −0.57

3.07 5.71 2.29 2.25 7.59 5.72 0.1 1.89 1.05


116.71 161.5 188.44 168.7 58.02 70.29 79.39 155.27 215.27

48.95 71.88 82.58 75.01 26.61 31.52 34.47 68.87 85.37

155.66 205.2 245.01 224.46 72.78 88.03 104.2 203.93 311.73

79.04 112.31 130.07 117.06 40.88 49.07 54.56 107.68 141.94

6.35 12.42 7 7.85 11.16 9.33 16.33 8.95 15.25

230.33 452.31 94.63 75.4 678.26 448.07 7.84 59.28 92.75

a Eq 5 with ε(ω) = 1 and the free-space polarizability. bEq 5 with ε(ω) = εwater(ω) and the free-space polarizability. cUsing eq 5 with ε(ω) = εwater(ω), the polarizability follows from eq 19. dUsing eq 5 with ε(ω) = εwater(ω) and the polarizability follows from eq 24. eUsing eq 5 with ε(ω) = εwater(ω) and the polarizability follows from eq 36. fEq 33.

a Eq 8 with ε(ω) = 1 and the free-space polarizability. bEq 8 with ε(ω) = εwater(ω) and the free-space polarizability. cUsing eq 8 with ε(ω) = εwater(ω) and the polarizability follows from eq 19. dUsing eq 8 with ε(ω) = εwater(ω) and the polarizability follows from eq 24. eUsing eq 8 with ε(ω) = εwater(ω) and the polarizability follows from eq 36. fEq 33.

The reason is partly the large repulsive contribution from the zero-frequency Matsubara term due to the high value for the dielectric constant of water. However, it is not only the zeroth frequency that gives repulsion. Figure 2 shows the effective polarizabilities for methane and nitrous oxide. On the basis of the three-layer model, two subclasses of molecules can be found: one with a purely negative effective polarizability, where CH4, CO2, O2, O3, N2, CO, and CO2 belong, and one with also positive contributions N2O and H2S. This is caused by the optical density of water. In eq 36, αC* is always negative and the dressed polarizability αs* can dominate it to generate a positive result. This is a direct consequence of the ratio of the optical densities for the considered materials. One needs to account for many frequencies where the polarizability is either positive (attractive) or negative (repulsive). A repulsive dispersion force based on the same argumentation was found in previous works by Elbaum and Schick.51 It turns out that the Onsager’s real cavity model for finite-size particles even predicts repulsion between a molecule in water and a perfect metal surface. This is in contrast to the other models in water. We suggest to test the

validity of the different effective polarizability models experimentally. For instance, the different predicted Casimir− Polder forces can be experimentally verified by balancing them against gravity. On the basis of the C3 coefficients given in Table 4, a solution of H2S can be brought horizontally toward a metal surface. Due to the repulsive force, expected for the finite-size and the hard-sphere model, the molecules will stabilize at a certain distance due to the equilibrium of the Casimir−Polder and the gravitational forces. The results for the van der Waals (C6) parameters are no less interesting and differ by an order of magnitude between the different models. Interactions between equal molecules and between unequal molecular pairs are given in Tables 5 and 6, respectively.

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IMPACT OF PARTICLE SIZE The polarizabilities in this model vary strongly between different molecules: although the polarizability for H2S is only negative for high frequencies, the polarizability of CH4 is also negative for the first two Matsubara frequencies. One main difference between H2S and CH4 can be found in Table 3. H2S G


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Table 6. C6 Coefficients in 10−79 J m6 for the Different Combinations of Interacting Particles Using the Three-Layer Model, Eq 36 CH4 CO2 N2 O O3 O2 N2 CO NO2 H2S

CH4

CO2

N2O

O3

O2

N2

CO

NO2

H2S

6.35

7.65 12.4

6.4 8.28 7

6.4 9.51 7.13 7.85

5.85 11.3 6.36 8.09 11.2

5.99 10.6 6.42 7.76 10.1 9.33

7.84 13.7 8.09 9.94 13.3 12.2 16.3

6.77 10.4 7.41 8.32 9.15 8.69 11.2 8.95

7.33 4.39 6.91 5.27 0.76 2.05 2.83 4.89 15.2

Figure 3. Excess polarizability in the real cavity model for finite-size particles (36) for a toy-methane molecule with the polarizability given by eq 37. The radius of the toy-methane sphere is varied between the original radius and the cavity radius, given in Table 3. One sees a transition from partially negative to purely positive polarizabilities.

When the effective polarizability of small particles is calculateda subject closely related to Casimir Polder and van der Waals potentialsthe presence of a dielectric environment implies complications that are at present not very well understood. In the present paper we have approached this problem from a phenomenological viewpoint, making use of various parameter-based models. In essence they consist of (1) the virtual cavity model, implying the Clausius−Mossotti relation; (2) the real cavity model, which in itself can be divided into two subclasses, the first treating the particle as a point-like object surrounded by a spherical cavity volume that denotes the original Onsager cavity model, and the being a finite-size model of the particle that is now assumed to be surrounded by an annular cavity volume and is an extension of the Onsager model. Its limit when the particle radius reaches the cavity radius is the hard-sphere model. We calculated the interaction in terms of scattering Green functions for a spherical three-layer system. The Green function technique was applied to several greenhouse gas molecules as examples. Effective polarizabilities for gas molecules in water were calculated. To illustrate the sensitivity of the formalism with respect to the input parameters, we analyzed also a real cavity model for finitesize particles for a toy-methane model. As a striking demonstration of the sensitivity of the formalism, we found that molecules can be attracted to, or be pushed away from, a metal surface depending on which effective polarizability model is used. There is thus an obvious need for testing these parameter-dependent theories against experiments to improve the modeling of van der Waals interactions in media.

has a larger sphere radius compared to the cavity radius. Thus, when the sphere radius of CH4 is enlarged, the polarizability should also become positive for the first two Matsubara frequencies. To verify this, we calculate the leading order excess polarizability eq 36 for different sphere radii. The permittivity of the CH4 sphere is fixed using eq 34 with the original sphere radius R = 1.655 Å, and the sphere radius R̃ is then varied between the original one and the cavity radius RC = 2.239 Å (for R = RC, one obtains the hard-sphere polarizability eq 33 with R → RC). This means we use a radius-dependent polarizability for the methane sphere, given by αs(R̃ ) = 4πε0R̃ 3

εs(R ) − 1 εs(R ) + 2

(37)

By varying the radius, one indeed sees a transition to positive excess polarizabilities for small frequencies in Figure 3. Furthermore, the excess polarizability becomes positive for all frequencies for R̃ ≥ 2.033 Å.

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CONCLUSIONS We have demonstrated that accurate estimates of the effective polarizability in a medium, and the related Casimir−Polder and van der Waals potentials, may require a new theory. Such a theory has been worked out in the present paper. However, even more urgent is to test predictions of the real cavity model for finite-size particles in a medium. We predict that molecules may be either attracted or pushed away from a metal surface depending on the effective polarizability model. Testing these predictions should pave the way for improved modeling of van der Waals interactions in a medium. H


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A decisive model cannot be made on this theoretical level. We note that only the finite-size and hard-sphere models are able to produce repulsive forces, and such forces have indeed been observed in some cases for the related Casimir force.52 Further, the finite-size model is the most detailed description as it takes into account both the finite particle size and the exclusion volume. To improve this model, the hard cavity boundary has to be adapted to realistic cavity profiles and the constraint to centered particles has to be discarded. On the basis of the results, an experimental improvement is given by measuring the expected van der Waals and Casimir−Polder forces. However, deeper theoretical investigations are required to transform the results to measurable spectra, such as refractive indices or molecular extinctions. To describe realistic scenarios, further investigations are required with respect to the anisotropy of particles, which results in nonspherical cavities, and to the cavity’s boundary as described by a continuous profile.2

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AUTHOR INFORMATION

Corresponding Authors

*J. Fiedler. E-mail: johannes.fi[email protected]. *P. Thiyam. E-mail: [email protected]. *D. F. Parsons. E-mail: [email protected]. *M. Boström. E-mail: [email protected]. *S. Y. Buhmann. E-mail: [email protected]. de. ORCID

Johannes Fiedler: 0000-0002-2179-0625 Michael Walter: 0000-0001-6679-2491 Drew F. Parsons: 0000-0002-3956-6031 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge support from the Research Council of Norway (Project 250346). This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. We gratefully acknowledge support by the German Research Council (grant BU1803/3-1, S.Y.B. and J.F.) the Research Innovation Fund by the University of Freiburg (S.Y.B., J.F. and M.W.) and the Freiburg Institute for Advanced Studies (S.Y.B.).

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