Hydrated Ion Clusters in Hydrophobic Liquid: Equilibrium Distribution

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Hydrated Ion Clusters in Hydrophobic Liquid: Equilibrium Distribution, Kinetics and Implications Lingjian Wang, Nobuaki Kikkawa, and Akihiro Morita J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10740 • Publication Date (Web): 23 Jan 2018 Downloaded from http://pubs.acs.org on January 25, 2018

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The Journal of Physical Chemistry

Hydrated Ion Clusters in Hydrophobic Liquid: Equilibrium Distribution, Kinetics and Implications Lingjian Wang,† Nobuaki Kikkawa,†,¶ and Akihiro Morita∗,†,‡ Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan, and Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, Kyoto 615-8520, Japan E-mail: [email protected] Fax: +81 (0)22 7957716

∗

To whom correspondence should be addressed Tohoku University ‡ Kyoto University ¶ Current address: Toyota Central R&D, Inc., Nagakute 480-1118, Japan †

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Hydrophilic ions in oil phase tend to be hydrated in the presence of trace water, and form hydrated clusters. The present paper elucidates the equilibrium size distribution of hydrated ion clusters and the microscopic rates of adsorption and desorption of water with the help of molecular dynamics simulation. The size distribution is derived from reversible work of hydration, which is nearly constant over the hydration number except for small clusters. The intrinsic rate constants of adsorption and desorption are evaluated to be in several psec order after correcting the diffusion. The microscopic hydration properties of ion in the oil phase play key roles in chemical reactions involving both hydrophilic and hydrophobic reactants, as well as in the transport and reactivity of the ions in oil phase and at water/oil interface.

1

Introduction

Hydrophilic ions, either cations or anions, are stabilized in water phase, and usually less stable in a hydrophobic (organic liquid or oil) phase. In some occasions, nevertheless, hydrophilic ions dispersed in oil phase play important roles in chemistry. A typical example is seen in chemical reactions of hydrophilic ions in oil phase assisted by phase transfer catalysts. 1–4 Such situation of hydrophilic ions in oil often appears in ion transport process from water to oil phase, such as extraction, phase transfer catalyses, chemical sensors, 5–7 etc. During the transport from water to oil phase, the hydrophilic ions tend to accompany water molecules. 8–10 The hydration of the ions is thought to be one of the key factors to affect the interfacial ion transport and subsequent stability/reactivity of the ions in oil phase. 11–14 Water and ion hydration in non-aqueous media are widely relevant to organic chemistry. Many chemical reactions that take place in organic solvents are affected by the trace amount of water, and thus the control of water is an important issue in organic syntheses. 15 It is also noteworthy that recent attempts of reactions in the presence of water or on water have tried to utilize the effect of hydration, 16–18 though there is no unified theory to explain the effects 2


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on the reactivity and selectivity. The present paper focuses on the microscopic hydration status of the hydrophilic ions, including the equilibrium distribution of hydrated clusters and adsorption/desorption kinetics of hydration. Such microscopic investigation of ion hydration is of fundamental importance to expanding our understanding of non-aqueous chemistry. The present work is motivated by our recent attempts to pursue microscopic understanding of interfacial mass transfer mechanism. 10,19 Ion transport at water-oil interface has been investigated by using electrochemical means as well as spectroscopic ones. 20–26 However, the kinetics and mechanism of interfacial transport are still challenging to understand in a detailed microscopic manner. This challenge largely stems from difficulties of monitoring the interfacial processes on sufficiently fine spatial and temporal scales. To overcome experimental limitations of monitoring interfacial processes, molecular dynamics (MD) simulation is a powerful method. One of the epoch-making findings by MD simulation was reported by Ilan Benjamin in 1993, who discovered characteristic structural fluctuation of the interface associated with the transport of hydrophilic ions, called “water finger”. 8 When an ion moves from water to oil phase, some water molecules follow the ion and form a transient “finger-like” structure that connects the hydrated ion and the water phase. The transient structure was further discussed in relation to the ion transport process. 27,28 In our previous work, 10,19 we explicitly formulated two states of water finger “formed” and “broken” using a water-finger coordinate w based on the connectivity of hydrogen bond network. Then MD simulation revealed a hidden free energy barrier between the two states. When an ion transports from water to oil, the transition of water finger structure necessarily takes place with an activation barrier, and thereby elucidates the retarded kinetics of the interfacial ion transfer. 10,19 While the water finger structure has been studied extensively, the subsequent kinetics after the break of water finger remains mostly unexplored. After the water finger that connects the ion and bulk water is broken, usually there are still some water molecules attached to the ion in the form of a hydration shell. This structure is identified as a hydrated ion

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cluster. If the immediate size of hydrated clusters deviates from the equilibrium distribution, subsequent relaxation process (i.e. adsorption/desorption of hydrated water molecules) should take place. The hydration status should depend on the concentration of water in the oil phase, even if the concentration is quite low. It is experimentally suggested from the conductance measurement of ions in bulk oil phase that the size of hydrated clusters increases with increasing concentration of water. 14 However, retrieving the detailed size distribution of clusters is beyond the capability of experimental measurements. Straightforward MD is not useful either to reproduce the equilibrated cluster distribution, due to prohibitively large temporal and spatial requirement to obtain the equilibrium distribution. The workaround to this issue is a combination of classical statistical mechanics with free energy calculation using MD simulation. The intrinsic kinetics of adsorption/desorption of the hydrated clusters is further evaluated using direct MD simulation of collision events. The remainder of this paper is constructed as follows. Section 2 describes the theory of equilibrium distribution of the hydrated ion clusters and the kinetics of adsorption and desorption, and Sec. 3 describes the detailed procedure and conditions of the MD simulation. The results of the equilibrium and kinetic properties are presented and discussed in Sec. 4. Concluding remarks follow in Sec. 5.

2

Theory

In the oil phase with a specific water concentration, ions exist in the form of hydrated clusters with various hydration numbers. We formulate the equilibrium size distribution in Sec. 2.1, and the computational procedure based on the free energy calculation in Sec. 2.2. The kinetics of adsorption and desorption of water are discussed in Sec. 2.3.

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2.1

Size Distribution

The size distribution of hydrated ion clusters IWn consisting of an ion I and n water molecules W is given at an equilibrium condition of constant chemical potential µW of water. The probability distribution p(n) of the cluster IWn is represented with the grand canonical distribution as p(n) =

exp[−β{A(n) − nµW }] , Ξ

(1)

where β = 1/(kB T ) denotes the inverse of the Boltzmann constant kB times temperature T . Ξ is the grand canonical partition function

Ξ=

∞ ∑

exp[−β{A(n) − nµW }],

(2)

n=0

and acts as the normalization factor of Eq. (1). A(n) is the Helmholtz free energy of hydrated ion cluster IWn in oil, as described below. µW is the chemical potential of water in the oil phase, 29 µW = µ∗W + kB T ln(ρW Λ3W ),

(3)

where µ∗W is the solvation free energy of a water molecule in the bulk oil phase, and ρW is the number density of water in the oil. ΛW is the thermal de Broglie wavelength of water, √ ΛW = h/ 2πmW kB T , where mW is the mass of a water molecule and h is the Planck constant. Equation (3) indicates that the concentration of water ρW controls the chemical potential of water µW , which in turn changes the size distribution p(n) in Eq. (1). The free energy A(n) of the cluster IWn in the oil phase is the key quantity to determine the size distribution. It is evaluated with the reversible work for the following process of cluster formation, w(0 → n) :

[ (I)oil + n(W )vac −→ (IWn )oil ] ,

(4)

where (X)oil denotes the species X in the oil phase, and (X)vac denotes the hypothetical situation that the center of mass of X is located at a fixed position in the vacuum phase. 5



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(W )vac is regarded as a reference state of W with its translational degrees of freedom and intermolecular interactions eliminated. We also note in passing that µW in Eq. (3) corresponds to the process [(W )vac → (W )oil ], as will be treated in Sec. 3.1.4. The reversible work w(0 → n) is related to the free energy A(n) of the cluster IWn by exp[−βw(0 → n)] = exp[−βA(n)], n!

(5)

where A(n) takes account of indistinguishable n water molecules in the cluster IWn . The work w(0 → n) in Eq. (4) is calculated by successive processes from a naked ion (I)oil to the cluster with n water molecules (IWn )oil , w(0→1)

w(1→2)

w(2→3)

w(n−2→n−1)

w(n−1→n)

(I)oil −−−−→ (IW )oil −−−−→ (IW2 )oil −−−−→ · · · −−−−−−−−→ (IWn−1 )oil −−−−−−→ (IWn )oil . Thus w(0 → n) in Eq. (4) is given by w(0 → n) =

n ∑

w(j − 1 → j).

(6)

j=1

The work of adding one water molecule to the cluster,

w(n − 1 → n) :

[ (IWn−1 )oil + (W )vac −→ (IWn )oil ] ,

(7)

is calculated with MD simulation, on the basis of the scheme described in the next subsection. Once we obtain the set of {w(n − 1 → n)} for varying sizes n = 1, 2, 3, · · · , we can derive the free energy A(n) with any size n by Eqs. (5) and (6), and thereby derive the probability distribution p(n) at a given µW (or water concentration in the oil phase) by Eq. (1).

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2.2

Free energy calculation

Coupling parameter: The reversible work of Eq. (7) can be calculated by the overlapping distribution method 30 with intermediate states and bias potential. In the process of Eq. (7), we introduce a coupling parameter λ which connects the initial state (IWn−1 )oil + (W )vac and the final state (IWn )oil , and put m intermediate states between the initial (λ0 = 0) and final (λm+1 = 1) states, 0 = λ0 < λ1 < λ2 < · · · < λm < λm+1 = 1. Thus w(n − 1 → n) is given as w(n − 1 → n) =

m ∑

w(k) ,

(8)

k=0

where w(k) is the work from λ = λk to λ = λk+1 . The MD system at an arbitrary λ (0 ≤ λ ≤ 1) contains one ion I, n − 1 ordinary water molecules, one additional water molecule, and many oil (solvent) molecules. We can put the center of mass of the ion I at the origin without loosing generality in the following discussion. The center-of-mass coordinates of n − 1 ordinary water molecules are denoted to be r 1 , r 2 , · · · , r n−1 , the coordinates of the additional water is r n , and the coordinates of other degrees of freedom including the solvent molecules are collectively denoted with X. The coupling parameter λ scales the intermolecular interaction of the additional water molecule to the rest of the system. At the initial state (λ = 0) the additional water molecule has virtually no interaction to other molecules, while at the final state (λ = 1) the additional molecule is equivalent to the other n − 1 water molecules. Bias potential: The MD calculation of w(k) is performed with imposing an auxiliary bias potential, V bias (r n ; λ), to confine the additional water molecule around the ion I. The form of V bias will be given in Sec. 3.1. The bias potential guarantees the configuration sampling of hydrated ion clusters during MD trajectories by hindering the water molecule from straying away from the cluster. It is particularly necessary to treat a system of small λ

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(≈ 0), since the additional water molecule has small intermolecular interaction and readily escapes from the hydrated cluster. On the other hand, the bias potential is obviated at λ = 1, where the additional molecule becomes equivalent to others. Accordingly we set V bias (r n ; λ = 1) = 0. The effect of bias can be corrected afterward from calculated results of free energy as discussed below.

Overlapping distribution: The procedure of the overlapping distribution method with the bias potential is illustrated as follows. We deal with the reversible work w(k) from λk to λk+1 , where the initial (λ = λk ) and final (λ = λk+1 ) states are labeled with i and f , respectively. The configuration partition functions of the two states with and without the bias potential V bias are given by ∫ dr n dX exp[−βEi (r n , X)],

(9a)

dr n dX exp[−β{Ei (r n , X) + V bias (r n ; λk )}], ∫ = q(λ = λk+1 ) = C dr n dX exp[−βEf (r n , X)], ∫ ′ = q (λ = λk+1 ) = C dr n dX exp[−β{Ef (r n , X) + V bias (r n ; λk+1 )}],

(9b)

qi = q(λ = λk ) = C qi′ qf qf′

′

∫

= q (λ = λk ) = C

(9c) (9d)

where {r n , X} denotes the total coordinates of the whole system. Ei and Ef are the potential energies of the initial and final systems, respectively, without the bias potential. The prime ′ denotes the system with the bias potential. C is the prefactor due to the kinetic energy and symmetry number, which are common among the four cases. The reversible work w(k) is given by w

(k)

qf = −kB T ln qi

( ) qf (k) or exp[−βw ] = . qi

(10)

We derive w(k) through the MD sampling with the bias potential in the following, where we use simplified notation for the bias potential, Vibias = V bias (r n ; λk ) and Vfbias = V bias (r n ; λk+1 ). The probability of finding ∆E = Ef − Ei at each of the equilibrium states i and f with

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the bias is respectively given by ∫

dr n dX exp[−β{Ei + Vibias }] δ[Ef − Ei − ∆E] ∫ , dr n dX exp[−β{Ei + Vibias }] ∫ n dr dX exp[−β{Ef + Vfbias }] δ[Ef − Ei − ∆E] ∫ p′f (∆E) = , dr n dX exp[−β{Ef + Vfbias }] p′i (∆E)

=

(11a) (11b)

and the two probabilities are related by p′f (∆E) qf /qf′ qi′ = exp[β(w(k) − ∆E)]. exp[−β∆E] = p′i (∆E) qf′ qi /qi′

(12)

p′f (∆E) qf qi + ∆E − kB T ln ′ + kB T ln ′ . ′ pi (∆E) qf qi

(13)

Therefore, w(k) = kB T ln

The last two terms of Eq. (13) correct the effect of the bias potential, and are represented by

δ

bias

(i) = −kB T = −kB T

δ bias (f ) = −kB T = −kB T

∫ n dr dX exp[−β{Ei + Vibias }] exp[βVibias ] qi ∫ ln ′ = −kB T ln qi dr n dX exp[−β{Ei + Vibias }] ⟩ ⟨ (14a) ln exp[βVibias ] i′ , ∫ n dr dX exp[−β{Ef + Vfbias }] exp[βVfbias ] qf ∫ ln ′ = −kB T ln qf dr n dX exp[−β{Ef + Vfbias }] ⟩ ⟨ (14b) ln exp[βVfbias ] f ′ ,

where ⟨ ⟩i′ and ⟨ ⟩f ′ denote the thermal average at the initial (λ = λk ) and final (λ = λk+1 ) states with the respective bias potentials. If we tentatively omit the last two terms of Eq. (13), w(k) in Eq. (13) is approximated with w′(k) = kB T ln

p′f (∆E) + ∆E. p′i (∆E)

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(15)


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Corrections for bias potential: Then we consider the sum of w′(k) in Eq. (15) for k = 0, 1, · · · , m, w′ (n − 1 → n) =

m ∑

w′(k) .

(16)

k=0

The work w′ (n − 1 → n) in Eq. (16) corresponds to the following process with the bias potential, [

w′ (n − 1 → n) :

] (IWn−1 )oil + (W )vac w/bias −→ (IWn )oil ,

(17)

where the subscript w/bias denotes the environment with the bias potential. Note that the final state of Eq. (17) is free from the bias, because we set the bias potential to vanish at λ = 1. The difference of w′ (n − 1 → n) in Eq. (16) from w(n − 1 → n) in Eq. (8) is attributed to the last two terms of Eq. (13). The correction δw is given by δw = w(n − 1 → n) − w′ (n − 1 → n) { } ∑ qf qi −kB T ln ′ + kB T ln ′ = qf qi k { } m ∑ q(λ = λk+1 ) q(λ = λk ) = −kB T ln ′ + kB T ln ′ q (λ = λk+1 ) q (λ = λk ) k=0 = −kB T ln

q(λ = 1) q(λ = 0) + kB T ln ′ ′ q (λ = 1) q (λ = 0)

= δ bias (W )vac .

(18)

We note that −kB T ln[q ′ (λ = 1)/q(λ = 1)] = 0 in Eq. (18) since the bias potential vanishes at λ = 1. On the other hand, δ bias (W )vac corresponds to the process [(W )vac w/bias −→ (W )vac ], and thus

δ

bias

(W )vac

q ′ (λ = 0) = −kB T ln = −kB T ln q(λ = 0)

∫

dr n exp[−βV bias (r n ; λ = 0)] . Λ3

(19)

The corrections for the bias potential in Eq. (19) will be further discussed in Sec. 4.1. To

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summarize the above procedure, we can calculate w(n−1 → n) = w′ (n−1 → n)+δ bias (W )vac using Eqs. (16) and (19). By calculating w(n − 1 → n) with varying n, one can derive w(0 → n) by Eq. (6) and thereby the free energy A(n) with Eq. (5).

2.3

Kinetics of Hydration

The kinetics of adsorption and desorption processes is represented by kn−1→n

− ⇀ (IWn−1 )oil + (W )oil − ↽ −− −− −− − − (IWn )oil ,

(20)

kn→n−1

where kn−1→n and kn→n−1 are the adsorption and desorption rate constants, respectively. These rates constants are related to the ratio of equilibrium number densities of the above species by [IWn ] kn−1→n = , [IWn−1 ][W ] kn→n−1

(21)

where [X] denotes the number density of the species X. Equation (21) can be written using Eqs. (1) and (3) as p(n) exp[−β{A(n) − nµW }] kn−1→n = = kn→n−1 p(n − 1) · ρW exp[−β{A(n − 1) − (n − 1)µW }] · ρW exp[−βw(0 → n)] exp[βµW ] n! = · exp[−βw(0 → n − 1)] ρW (n − 1)! exp[−βw(n − 1 → n)] = · Λ3W exp[βµ∗W ]. n

(22)

The final result of Eq. (22) has been derived using Eqs. (5) and (3). The right-hand side of Eq. (22) will be calculated in Sec. 4.1. The ratio of adsorption (kn−1→n ) to desorption (kn→n−1 ) rate constants is thereby determined from the calculated equilibrium properties. However, each rate constant is hard to obtain by straightforward MD simulation, because of the following two reasons. First, the adsorption and desorption are regarded as rare events

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for MD standards, and are not amenable to sufficient statistical sampling during limited time scale of MD simulation. Second, evaluating the intrinsic rates of adsorption and desorption from the observed events would require the calibration of diffusion in the liquid phase. The phenomenological rate constant of mass transfer from the oil phase to the cluster k is given by the kinetic model of sequential steps, 1 1 1 = ads + diff , k k k

(23)

where k diff is the rate constant for diffusion of water in the bulk oil phase, and k ads is the intrinsic rate constant of adsorption. This equation has been commonly applied to many kinetic problems of sequential reactions, 31 including the Collins-Kimball model of diffusion-influenced reactions 32,33 or the resistance model in the heterogeneous atmospheric chemistry. 34,35 Equation (23) indicates that the phenomenological rate of mass transfer k could be influenced by the diffusion k diff , when the mass flux is driven by the non-equilibrium concentration gradient in the oil phase. On the other hand, k ads of Eq. (23) corresponds to kn−1→n in Eq. (22), the intrinsic rate constant to change the hydration number in the equilibrium condition. In the present analysis, we want to evaluate k ads or kn−1→n for the adsorption process. The MD simulation allows for evaluating the intrinsic adsorption rate constant kn−1→n by calculating the adsorptive cross section of (IWn−1 )oil of an impinging water molecule, ads σn−1 . The MD simulation of the cross section is illustrated in Figure 1. The collision is

carried out in the limit of small Knudsen number 31,34 in the condensed phase, which virtually eliminates the diffusion resistance. Though this situation may not be feasible in experimental condensed phase, it is readily realized in the MD simulation by turning off the interaction between the impinging water molecule and the oil molecules. The impinging water collides with the hydrated ion cluster which is embedded in the oil phase. We note that the hydrated ion cluster is fully solvated by the surrounding oil molecules, and the energy relaxation of

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the cluster is taken into account during the collision dynamics. The collision dynamics of the impinging water is investigated many times, and the fate of the impinging molecule is statistically treated to derive the adsorptive cross section. This treatment is regarded as a kinetic analogue of the Widom’s insertion method. 36 ads The details of the computational procedure of the adsorptive cross section σn−1 will be

described in Sec. 3.2. The rate constant of adsorption is then represented with the cross ads section σn−1 and the mean relative speed of the impinging water and the cluster v¯rel ,

√ ads rel ads kn−1→n = σn−1 v¯ = σn−1

8kB T , πµ

(24)

where µ is the reduced mass of the cluster and the water molecule. One could introduce the ads effective radius of the adsorptive cross section rn−1 by

( ads ads 2 σn−1 = π(rn−1 )

√

ads or rn−1 =

ads σn−1 π

) ,

(25)

ads 2 and represent Eq. (24) as kn−1→n = (4π(rn−1 ) )(¯ v rel /4) using the surface area of the cluster ads 2 (4π(rn−1 ) ). Once we obtain the adsorption rate constant kn−1→n in Eq. (24), the desorption

rate constant kn→n−1 can be evaluated by Eq. (22).

3

MD Simulation

In Sec. 3.1 we summarize the conditions of MD simulation and overlapping distribution method with intermediate states and bias potential, as described in Secs. 2.1 and 2.2, for calculating the distribution of hydrated ion clusters. Then the MD procedure to evaluate the kinetics of hydration is presented in Sec. 3.2. All the present MD calculations were carried out with our in-house code.

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3.1 3.1.1

Size Distribution MD Conditions

The reversible work w(n − 1 → n) in Sec. 2.2 is calculated by the MD simulation in the following conditions. We treat a hydrated cluster Cl– ·(H2 O)n dispersed in dichloromethane (DCM, CH2 Cl2 ) solution as a typical example. The hydration numbers n ranging from 0 to 10 are considered. For each case of n, the MD cell contains one Cl– , n H2 O and (599 − n) DCM molecules. The dimensions of the MD cell are set to 40 ˚ A × 40 ˚ A × 40 ˚ A so as to be consistent to the liquid density of DCM, 1.3266 g/cm3 , 37 and the three-dimensional periodic boundary conditions are adopted. We employ polarizable models for all the species in the present MD simulation. This is because polarizable models are more reliable for calculating the solvation of aqueous and nonaqueous molecules simultaneously. 10,19 For H2 O, the POL3 model is employed, 38 where the hydrogen atoms of water are deuterated to allow for a larger time step of MD simulation. For Cl– and CH2 Cl2 , Dang’s polarizable models 39 are used. The parameters of the force fields are shown in Table 1. Long-range electrostatic interactions are treated with the particle −1 mesh Ewald method. 40 The Ewald width parameter is 0.323 ˚ A . The order of interpolating

B-spline function for the particular mesh Ewald is 6, and the grid spacing is set to 0.7 ˚ A along each axis. We note that the MD cell contains no counter cation, and thus assume the charge neutrality with a uniform background. The MD simulation is performed under the N V T ensemble at T = 298.15 K. The temperature is controlled using the Nosè-Hoover chain of sequential 10 thermostats. 41,42 The masses of the thermostats are Q1 = N kB T τ 2 and Qj = kB T τ 2 (j = 2, · · · , 10), with the relaxation time τ to be 0.1 ps. The time evolution is carried out by the velocity Verlet algorithm 42,43 extended with the Nosè-Hoover chain. The time step is taken to be 2 fs, and the intramolecular configurations are fixed with the RATTLE algorithm. 42–44

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3.1.2

Bias potential

The bias potential V bias is introduced to confine the additional water molecule around the ion, as discussed in Sec. 2.2. We employ the following form of V bias for each size n of the cluster (IWn ),

( V

bias

(r; λ) = (1 − λ) · 10kB T

|r| rneff

)2 ,

(26)

where |r| is the center-of-mass distance of the water molecule from the ion. rneff is an effective range of the hydrated ion cluster (IWn ). The factor (1 − λ) scales the bias potential between λ = 0 and 1, and ensures the bias potential to vanish at λ = 1. The harmonic form of Eq. (26) is convenient for evaluation of δ bias (W )vac in Eq. (19). The range of the hydrated ion cluster rneff is defined as

rneff

 ( )1/3 3vwater  3  1.2 rion + n 4π =   reff n=4

(n ≥ 4) (27) (n < 4)

where rion = 1.67 ˚ A is the ionic radius of Cl– , 45 and vwater is the volume of a single water molecule calculated from the liquid density (18×10−6 m3 /mol). rneff for n < 4 is given as that eff for n = 4, rn=4 = 3.9 ˚ A, because the chloride ion has 4 water molecules in the first hydration

shell. The parameter rneff in Eq. (27) is roughly determined on the basis of effective cluster volume with some margin. The calculated free energies after correcting the bias potential are found to be rather insensitive to the parameter rneff . 3.1.3

Free energy calculation

The calculation of w(n − 1 → n) is carried out assuming 6 intermediate states. The coupling parameters for these intermediate states are set to λ1 = 0.0001, λ2 = 0.001, λ3 = 0.01, λ4 = 0.1, λ5 = 0.333, and λ6 = 0.666. Besides the intermediate states, the initial (λ0 = 0) and final (λ7 = 1) states are treated. Each value of the coupling parameters λk (k = 0, · · · , 7) scales down the Lennard-Jones parameter ϵ, charge q, and polarizability α at the sites of 15



the additional water molecule. The coupling parameter λ is also applied to change the bias potential of Eq. (26). In the case that λ is close to zero, the following two problems would arise. First, MD trajectories tend to be numerically unstable at small λ, where the additional molecule readily overlaps with its surrounding molecules. To maintain the numerical stability, we employ a modified distance rmod (r) for short-range interactions, as detailed in our previous paper. 10 This treatment effectively augments the site distance r in the force calculation at a short distance close to zero (r ≈ 0), and thereby avoids the artificially large force from overlapping sites. This treatment has no influence on the free energies of the end states at λ = 0 and λ = 1. Second, the proper N V T ensemble at small λ requires to use a more effective thermostat than the usual Nosè-Hoover. 46,47 The additional molecule is hard to be thermally equilibrated at small λ, due to the small intermolecular interactions. Therefore, we employ the Nosè-Hoover chain to account for this problem, as mentioned in Sec. 3.1.1. For each λk (k = 0, · · · , 7) at the cluster size n, 10 individual initial MD configurations are prepared and equilibrated for 200 ps by MD simulation. Subsequently the probability distributions in Eq. (11) are calculated for 200 ps for each initial structure. Thus the total sampling time to calculate w(n − 1 → n) is 200 ps × 10 × 8 = 16 ns for each size n. 3.1.4

Solvation of water molecule

We also calculate the solvation free energy of a water molecule µ∗W in the DCM oil phase. The value of µ∗W is necessary to estimate the chemical potential of water µW at a given water concentration ρW by Eq. (3) and to calculate Eq. (22). The calculation of µ∗W is carried out in the parallel manner to that of w(0 → 1) in Sec. 3.1.3, by eliminating the chloride ion from the MD cell. The MD conditions for calculating µ∗W are same as those of w(0 → 1) in this section except that the chloride ion is not included.

16


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3.2

Collision Dynamics

The MD scheme of the collision dynamics is illustrated in Figure 1. The present MD simulation investigates the thermal collision of an impinging water molecule to the hydrated ion cluster (IWn−1 )oil , and calculates the adsorption probability pads (b) as a function of collision parameter b. Then the cross section for the adsorption to (IWn−1 ) is derived from the integral,

∫ ads σn−1

=

∞

pads (b) 2πb db.

(28)

0

In the present simulation, hydration numbers of the core cluster n − 1 ranging from 0 to 9 are considered. The MD conditions used for the collision dynamics are the same as those for free energy calculation in Sec. 3.1.1 unless otherwise noted below. The MD scheme of the collision dynamics is illustrated in Figure 1. First, we randomly pick up 120 different MD snapshots for each hydration number n−1 from the MD trajectories in Sec. 3.1.1, and use these snapshots to generate initial configurations for the present collision simulation. We randomly choose a snapshot, set the chloride ion at the center of the periodic cell, (0, 0, 0), and add the impinging water molecule at (0, b, −20 ˚ A) with a collision parameter b and with an initial positive velocity sampled from the Maxwell-Boltzmann distribution along the z direction. The collision parameter b is set from 0 to 20 ˚ A with 2 ˚ A interval, and 240 independent trials of collisions are carried out for each value of b. During the MD simulation of the collision dynamics, interactions between the impinging water molecule and the oil molecules are turned off, and the site distances between the impinging molecule and the cluster are evaluated without adopting the periodic boundary conditions to avoid unwanted interaction between the colliding molecule and other replica of the cluster. For each trial of collision, the MD trajectory is traced for 50 ps to find the fate whether the impinging molecule is adsorbed or not. Technically a trial is judged as adsorbed when the closest distance between the impinging water and a constituent molecule of (IWn−1 ) cluster (either ion or water) continues to be smaller than 4 ˚ A for 20 ps.

17



pads (b) is obtained from the ratio of adsorbed events to all trials conducted for each value of b.

4

Results and Discussion

In this section we discuss the computational results of equilibrium distribution of hydrated ion clusters in Sec. 4.1 and the kinetics of hydration in Sec. 4.2.

4.1 4.1.1

Size Distribution Free energy of hydration

We have explained in Sec. 2 that the free energy of hydration is derived by calculating the work of adding one water molecule to the ion cluster, w(n − 1 → n) in Eq. (7). This work w(n − 1 → n) is calculated with the help of intermediate states and bias potential, as detailed in Sec. 2.2. Therefore, we first examine the performance of the intermediate states which are characterized with the coupling parameter λ = λ1 ∼ λ6 in Sec. 3.1.3. Figure 2 shows the intermediate reversible works w′(k) in Eq. (15) for the n = 0 to 1 process. In the overlapping distribution method, w′(k) should be constant over ∆E if the probability distributions of Eq. (11) well overlap each other. Figure 2 allows us to define a plateau for each line, indicating that the overlapping distribution is well performed. We also find that the calculation with small λ is technically more challenging and requires some cares described in Sec. 3.1.3. The calculated results of the reversible works of hydration are summarized in Table 2. We find that the work of adding one water, w(n − 1 → n), is nearly constant over the n value about ∼ −9 kcal/mol, except for a few small n values. The w(n − 1 → n) value for n = 1 or 2 is somewhat negatively larger than −9 kcal/mol. The tendency of larger solvation free energy for small clusters is consistent to the experimental results by Osakai and co-workers. 12 On the other hand, the solvation free energy of a single water molecule in 18


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DCM phase is calculated to be µ∗W = −2.39 kcal/mol. The chemical potential of water in the DCM phase, µW , is derived with Eq. (3), and the results of µW are shown in Table 3 with varying concentration ρW . We note that the saturated concentration of water in the liquid DCM is ρW = 1.3 × 102 mM at 298.15 K. 48 Using the results of w(0 → n) in Table 2 and µW in Table 3, one can derive the equilibrium distribution of hydrated ion clusters p(n) by Eqs. (1) and (5). The calculated distributions with varying water concentrations ρW are shown in Figure 3. The average hydration number of Cl– in liquid DCM at saturated water condition is estimated to be 0.43 in the present calculation. This is qualitatively comparable to previous experimental estimate of the hydration numbers of Cl– in other solvents, 3.3 in nitrobenzene, 2.4 in dichloroethane, and 1.0 in chloroform, 49 though the present estimate is somewhat smaller than these previous values. Figure 3 shows that the Cl− ion becomes less hydrated with deceasing water concentration in the liquid DCM.

4.1.2

Discussion

The results of free energy calculation in Table 2 indicate that w(n−1 → n) is nearly invariant for n ≃ 3 or larger. w(n−1 → n) should converge to an asymptotic value of condensation free energy of water for large n. Therefore, we may reasonably assume that w(n − 1 → n) = w is constant over wide range of n, and then the equilibrium size distribution and the kinetics can be understood with a simple model. Suppose that w(0 → n) in Eq. (6) is approximated to be w(0 → n) ≈ nw, the size distribution p(n) is represented with the Poisson distribution,

p(n) ≈

1 exp {−nβ(w − µW )} , Ξ n!

(1′ )

where Ξ = exp [exp {−β(w − µW )}]. The Poisson size distribution is analogous to the solvation clusters in supercritical fluid solution. 50 Consequently, the ratio of adsorption and

19



Page 20 of 38

desorption rate constants in Eq. (22) becomes kn−1→n p(n) exp{−β(w − µW )} Λ3 exp{−β(w − µ∗W )} = ≈ = W , kn→n−1 p(n − 1) · ρW n · ρW n

(22′ )

where Eq. (3) was employed for µW . Equation (22′ ) implies a qualitative picture that the adsorption rate of n − 1 → n is invariant to n while the desorption rate of n → n − 1 is proportional to n. We can see that this picture roughly holds for n ≃ 3 or larger in the next subsection. In our previous studies of ion transfer through water-DCM interface, we demonstrated that the average hydration number near the interface is significantly larger than that in the bulk oil. 10,19 The average hydration number just after the break of water finger in the oil phase was calculated to be about 10, which is larger than the equilibrium hydration number. Therefore, when the ion enters the oil phase across the interface, the nascent ion tends to have excessive water molecules. Quantitative evaluation of the evaporation kinetics is thus of essential significance in the dynamics of interfacial ion transfer. We discuss the kinetics of hydration in the following section.

4.2 4.2.1

Kinetics of Hydration Collision cross-section and the kinetics

The calculated adsorption probability pads (b) by the collision simulation in Sec. 3.2 are displayed in Figure 4 for n = 1 and n = 9 (i.e., the hydration number of the core cluster n − 1 = 0 and 8). The probability of adsorption ranges up to b ≈ 15 ˚ A, and the tail behavior ads is calculated from pads (b) of pads (b) is rather insensitive to the size n. The cross section σn−1 ads have a weak dependence on the using Eq. (28) for each n. The calculated values of σn−1

hydration number of the core cluster n − 1, and the dependence is well represented with the

20


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following regression line,

ads σn−1 ≃ 1.87 + 0.0454 · (n − 1) (unit : nm2 ).

(29)

We notice that the cross section is significantly larger than the geometric one. This is evident by comparing the effective radius of adsorption rnads in Eq. (25) with the size of the cluster rneff √ ads in Eq. (27). For example, Eq. (29) indicates that rn=4 = σ4ads /π = 8.1 ˚ A for n = 4, while eff rn=4 = 3.9 ˚ A. The apparently larger radius of rads is understood with the fairly long-range

electrostatic attraction of the core ion. Using the result of Eq. (29), the adsorption rate constant kn−1→n is obtained with Eq. (24). Then the desorption rate constant kn→n−1 is derived with Eq. (22). The results of the rate constants are listed in Table 4. We notice that the desorption rate constant kn→n−1 for n ≤ 3 is obviously smaller than others. This feature is understood from particularly strong binding energy for the small ion clusters, indicated with the negatively large w(n − 1 → n) values in Table 2.

4.2.2

Relaxation timescales of the adsorption/desorption process

The set of rate constants in Table 4 allows us to construct kinetic rate equations for the size distribution of the clusters, p(n, t), as a function of time t. The rate equations are represented in the matrix form, d P(t) = AP(t), dt

(30)

where P(t) denotes the size distribution of the clusters, 



p(0, t)      p(1, t)    P(t) =   ..   .     p(nmax , t)

21


(31)


Page 22 of 38

and A is the transition matrix,   −k ρ k 0 1→0  0→1 W     k0→1 ρW −(k1→2 ρW + k1→0 )  k2→1 0   A=  .. .. ..   . . . 0     0 knmax −1→nmax ρW −knmax →nmax −1

(32)

where nmax is set to 10 in the present case. The matrix A depends on the water concentration ρW in the oil phase. Equation (30) can be solved in principle by diagonalizing the matrix A. The matrix A has one zero eigenvalue 1/τ0 and other negative eigenvalues −1/τ1 > −1/τ2 > · · · > −1/τnmax . The zero eigenvalue 1/τ0 = 0 is related to the stationary state, and the corresponding eigenvector describes the equilibrium size distribution (shown in Figure 3). The other negative eigenvalues describe relaxation rates, and the corresponding eigenvectors denote relaxation modes. (Positive eigenvalues are not allowed due to the stability of the system.) The slowest relaxation time τ1 and its corresponding eigenvector are displayed in Table 5. The typical relaxation time is about 4 ∼ 6 psec in the present range of water concentration, and the principal relaxation modes mainly include the transition between n = 0 to 1. We note that these relaxation times reflect the intrinsic rates of adsorption and desorption, and should be considered as lower limit in actual situation including the diffusion resistance of water molecules. Direct experimental measurement of the intrinsic rates of adsorption and desorption may be difficult in oil phase. However, the experimental NMR measurement of water dynamics provides an interesting clue 12 that the rotational correlation times of water in hydrated ion clusters are comparable to the present relaxation times in psec order. This suggests that the rotational dynamics of water in the hydrated ion clusters is related to the adsorption and desorption kinetics, and further investigation is required. These relaxation kinetics becomes significant near the water-oil interface, as will be discussed in a forthcoming paper. 22


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5

Concluding Remarks

Hydrophilic ions tend to form hydrated clusters in oil phase, and the water molecules in the oil phase are known to play significant roles on stability and reactivity of the ions. However, detailed information of cluster distribution and kinetics is hard to retrieve by experimental means. The present work elucidated the equilibrium size distribution in the combination of free energy calculation by MD simulation and the statistical mechanics. The MD simulation is also used to reveal the intrinsic rates of adsorption and desorption of the hydrated ion clusters. The reversible work of hydration w(n − 1 → n) is nearly constant over n except for some small n values, and thus the particularly small desorption rates are predicted for the small clusters. These equilibrium and kinetic information about the hydrated ion clusters will be utilized to investigate detailed microscopic dynamics of ion transfer through liquid-liquid interfaces. Based on the present information, we will further investigate the role of evaporation kinetics of hydrated ion clusters through the ion transfer process. In the vicinity of the liquid-liquid interfaces, the dynamics of adsorption and desorption are coupled with the water finger formation and destruction, and a comprehensive picture of interfacial ion transfer should take account of these factors. Such studies are currently in progress in our group.

Acknowledgement This work was supported by the Grants-in-Aids (Nos. JP26288003, JP25104003) by Japan Society for the Promotion of Science (JSPS).

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(21) Dryfe, R. A. W. The Electrified Liquid-Liquid Interface. Adv. Chem. Phys. 2009, 141, 153–215. (22) Cai, C.; Tong, Y.; Mirkin, M. V. Probing Rapid Ion Transfer Across a Nanoscopic Liquid-Liquid Interface. J. Phys. Chem. B 2004, 108, 17872–17878. (23) Rodgers, P. J.; Amemiya, S. Cyclic Voltammetry at Micropipet Electrodes for the Study of Ion-Transfer Kinetics at Liquid/Liquid Interfaces. Anal. Chem. 2007, 79, 9276–9285. (24) Kakiuchi, T.; Takasu, Y.; Senda, M. Voltage-Scan Fluorometry of Rose Bengal Ion at the 1,2-Dichloroethane-Water Interface. Anal. Chem. 1992, 64, 3096–3100. (25) Kakiuchi, T.; Takasu, Y. Ion Selectivity of Voltage-Scan Fluorometry at the 1,2Dichloroethane/ Water Interface. J. Electroanal. Chem. 1994, 365, 293–297. (26) Ding, Z.; Wellington, R. G.; Brevet, P. F.; Girault, H. H. Spectroelectrochemical Studies of Ru(bpy)2+ 3 at the Water/1,2-Dichloroethane Interface. J. Phys. Chem. 1996, 100, 10658–10663. (27) Schweighofer, K. J.; Benjamin, I. Transfer of Small Ions across the Water/1,2Dichloroethane Interface. J. Phys. Chem. 1995, 99, 9974–9985. (28) Darvas, M.; Jorge, M.; Cordeiro, M. N. D. S.; Kantorovich, S. S.; Sega, M.; Jedlovszky, P. Calculation of the Intrinsic Solvation Free Energy Profile of an Ionic Penetrant Across a Liquid-Liquid Interface with Computer Simulations. J. Phys. Chem. B 2013, 117, 16148–16156. (29) Ben-Naim, A. Solvation Thermodynamics; Plenum: New York, 1987. (30) Bennett, C. H. Efficient Estimation of Free Energy Differences from Monte Carlo Data. J. Comp. Phys. 1976, 22, 245–268. (31) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 3rd ed.; Cambridge Univ. Press: Cambridge, 2009. 26


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(43) Allen, M. P.; Tildeslay, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (44) Andersen, H. C. RATTLE: A “Velocity” Version of the SHAKE Algorithm for Molecular Dynamics Calculations. J. Comp. Phys. 1983, 52, 24–34. (45) Shannon, R. D. Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Cryst. 1976, A32, 751–767. (46) Nosè, S. A Unified Formulation of the Constant Temperature Molecular Dynamics Methods. J. Chem. Phys. 1984, 81, 511–519. (47) Hoover, W. G. Canonical Dynamics: Equilibrium Phase-Space Distributions. Phys. Rev. A 1985, 31, 1695–1697. (48) Sazonov, V.; Shaw, D. IUPAC-NIST Solubility Database, Version 1.1. NIST Standard Reference Database 106. 2007. (49) Kenjo, T.; Diamond, R. M. Hydration of the Halide Ions in Certain Organic Solvents. J. Inorg. Nucl. Chem. 1974, 36, 183–188. (50) Morita, A.; Kajimoto, O. Solute-Solvent Interaction in Nonpolar Supercritical Fluid: A Clustering Model and Size Distribution. J. Phys. Chem. 1990, 94, 6420–6425.

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Figure 1: Illustration of collision dynamics. The hydrated ion cluster is drawn in the center of the cell, and the MD cell is filled with oil molecules of gray shade. The initial impinging water molecule is colored in purple, with collision parameter b and initial velocity (red arrow).

29



Figure 2: Calculated results of w′(k) (k = 0 ∼ 6) in Eq. (15) as a function of ∆E in the overlapping distribution method for n = 0 → 1 process. Note that w′(k) is the reversible work from λk to λk+1 , and the coupling parameters λk (k = 0 ∼ 7) are 0 (initial), 0.0001, 0.001, 0.01, 0.1, 0.333, 0.666, and 1 (final).

30


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Figure 3: Equilibrium distributions hydration numbers n of Cl− (H2 O)n cluster in liquid dichloromethane with various water concentrations ρW , (a) 130 mM, (b) 13 mM, (c) 1.3 mM, (d) 0.13 mM. The top values denote the average hydration numbers.

31



Figure 4: Calculated adsorption probability pads (b) as a function of the collision parameter b for n = 1 (blue) and 9 (orange). The circles and error bars are obtained by MD simulation, and the solid curves are fitted results.

32


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Table 1: (a) Force field parameters and (b) bond lengths in the molecular models. 38,39 (a) Molecule Site H2 O H O CH2 Cl2 C H Cl Cl– Cl

˚) ϵ (kcal/mol) σ (A 0 0 0.160 3.204 0.137 3.410 0.040 2.400 0.280 3.450 0.100 4.410

q (e) 0.3650 -0.7300 -0.2720 -0.0537 0.1897 -1.0000

(b) Molecule H2 O CH2 Cl2

Bond Length (˚ A) H-O 1.000 H-H 1.633 C-H 1.070 C-Cl 1.772 H-Cl 2.331 Cl-Cl 2.924

33


α (˚ A3 ) 0.170 0.528 0.878 0.135 1.910 3.690


Page 34 of 38

Table 2: Calculated reversible works of hydration. Unit: kcal/mol. The values in parentheses denote standard deviation. n w′ (n − 1 → n)(a) 1 −6.32 (0.28) 2 −5.62 (0.24) 3 −5.15 (0.31) 4 −4.60 (0.57) 5 −4.87 (0.39) 6 −4.74 (0.32) 7 −4.73 (0.39) 8 −4.47 (0.57) 9 −4.67 (0.64) 10 −4.65 (0.53) (a)

Eq. (16) or (17),

(b)

(b)

δ bias (W )vac −3.91 −3.91 −3.91 −3.91 −4.03 −4.12 −4.20 −4.28 −4.34 −4.40 Eq. (19),

(c)

w(n − 1 → n)(c) −10.23 (0.28) −9.53 (0.24) −9.06 (0.31) −8.52 (0.57) −8.90 (0.39) −8.86 (0.32) −8.93 (0.39) −8.75 (0.57) −9.01 (0.64) −9.05 (0.53)

Eq. (7),

34

(d)

w(0 → n)(d) −10.23 (0.28) −19.76 (0.37) −28.82 (0.48) −37.34 (0.75) −46.24 (0.85) −55.10 (0.91) −64.03 (0.98) −72.78 (1.14) −81.79 (1.31) −90.84 (1.41)

Eq. (4) or (6).


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Table 3: Calculated chemical potential of water in liquid dichloromethane µW with various concentrations ρW . Units: mM for ρW , kcal/mol for µW . ρW µW

130 13 1.3 −10.54 −11.90 −13.27

35


0.13 −14.63


Table 4: Adsorption rate constants kn−1→n (unit: M−1 s−1 ) and desorption rate constants kn→n−1 (unit: s−1 ) with various cluster size n. n 1 2 3 4 5 6 7 8 9 10

kn−1→n 8.1 × 1011 7.8 × 1011 7.8 × 1011 7.8 × 1011 7.9 × 1011 8.0 × 1011 8.1 × 1011 8.2 × 1011 8.3 × 1011 8.5 × 1011

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kn→n−1 1.8 × 1011 1.1 × 1012 3.7 × 1012 1.2 × 1013 8.1 × 1012 1.1 × 1013 1.1 × 1013 1.8 × 1013 1.3 × 1013 1.4 × 1013


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Page 37 of 38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 5: Estimated relaxation time τ1 and the corresponding eigenvectors for different water concentrations ρW . Units: mM for ρW , ps for τ1 . The eigenvectors are normalized, and the amplitudes smaller than 10−5 are denoted with 0.0 or omitted. ρW  τ1  n=0  1     2     3     ..   .  10

130  3.81  0.75  −0.66    −0.081    0.0024     ..   .  0.0

13  5.39  0.71  −0.70    −0.0078    0.00002      ..   . 0.0

37

1.3  5.63  0.71  −0.71    −0.0008    0.0      ..   . 0.0


0.13  5.65  0.71 −0.71    0.0     0.0     ..   .  0.0


Graphical TOC Entry

38


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Hydrated Ion Clusters in Hydrophobic Liquid: Equilibrium Distribution

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