Molecular Model of Self Diffusion in Polar Organic Liquids

Article pubs.acs.org/JPCB

Molecular Model of Self Diffusion in Polar Organic Liquids: Implications for Conductivity and Fluidity in Polar Organic Liquids and Electrolytes Roger Frech* and Matt Petrowsky Department of Chemistry and Biochemistry, University of Oklahoma, 101 Stephenson Parkway, Norman, Oklahoma, 73019 United States S Supporting Information *

ABSTRACT: Decades of studying isothermal and temperature-dependent mass and charge transport in polar organic liquids and electrolytes have resulted in two mutually incompatible models and the failure to develop a general molecular level picture. The hydrodynamic model describes conductivity, diffusion, and dielectric relaxation in terms of viscosity, while the inadequacy of the thermal activation model leads to empirical descriptions and fitting procedures whose adjustable parameters have little or no physical significance. We recently demonstrated that transport data can be characterized with a high degree of accuracy and self-consistency using the compensated Arrhenius formalism (CAF), where the transport property of interest assumes an Arrhenius-like form that also includes a dielectric constant dependence in the exponential prefactor. Here, we provide the molecular-level basis for the CAF by first modifying transition state theory, emphasizing the coupling of the diffusing molecule’s motion with the dynamical motion of the surrounding matrix. We then explicitly include the polarization energy contribution from the dipolar medium. The polarization energy is related to molecular and system properties through the dipole moment and dipole density, respectively. The energy barrier for transport is coupled to the polarization energy, and we show that accounting for the role of the polarization energy leads naturally to the dielectric constant dependence in the exponential prefactor.

1. INTRODUCTION At present, mass and charge transport in polar liquids and polar liquid electrolytes are described in two very different ways: (1) a thermally activated process characterized by an activation energy Ea and (2) a hydrodynamic model in which the motion of molecules or ions is opposed by a resistive force determined by the solution viscosity, η. Descriptions of temperaturedependent transport data of liquids in terms of a thermally activated process are often unsatisfactory, requiring the use of empirical equations1−4 and thus providing no insight into the molecular-level nature of the transport process. The hydrodynamic model, utilizing the Stokes equation, is often used to describe isothermal conductivity,5 self-diffusion coefficient,6 and the dielectric relaxation time7 in terms of the viscosity. However, the use of one macroscopic solution property to describe other macroscopic solution properties cannot lead to any particular molecular level insight into the transport processes involved. Further, these equations, although intuitively appealing, often provide inadequate descriptions of experimental data.8−10 These two general pictures of transport, the thermally activated model and the hydrodynamic model, are incompatible and do not lead to any degree of self-consistency in describing the various kinds of mass and charge transport phenomena in polar liquids and polar liquid electrolytes. A direct consequence of this inconsistency is the lack of a general molecular-level model that adequately describes both isothermal and temper© 2014 American Chemical Society

ature-dependent transport phenomena in these important systems. We recently developed a new approach to study the temperature dependence of mass and charge transport in polar liquids, specifically the ionic conductivity,11 selfdiffusion,12 viscosity,13 and dielectric relaxation14 with corresponding transport coefficients σ, D, η, and kdr. Mass and charge transport phenomena have the same general mathematical form: the rate of flow (flux) is proportional to the gradient driving the flow. The proportionality constant is the transport coefficient, and the temperature dependence of the transport flux is the temperature dependence of the transport coefficient. Our approach emphasizes the critical role of the solution static dielectric constant, rather than the solution viscosity; the solution viscosity is simply another transport coefficient on equal footing with the conductivity, diffusion coefficient, and dielectric relaxation rate constant. We describe the temperature dependence of the reciprocal viscosity, conventionally defined as the fluidity, with coefficient f = 1/η. Our designation of dielectric relaxation as a transport phenomenon will be discussed later; the corresponding transport coefficient kdr is the dielectric relaxation rate constant (reciprocal of dielectric relaxation time). Received: September 5, 2013 Revised: January 27, 2014 Published: February 6, 2014 2422

dx.doi.org/10.1021/jp408899y | J. Phys. Chem. B 2014, 118, 2422−2432

The Journal of Physical Chemistry B

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equations that have been introduced.19 The early application of transition state theory to viscosity is relevant here.20 The Vogel−Tammann−Fulcher (VTF) equation2−4 was originally developed to describe the viscosity of glass-forming liquids and is widely used in other liquid systems.21 The Williams− Landel−Ferry (WLF) equation,1 also originally developed to describe mechanical and electrical relaxation in glass-forming liquids, is used only occasionally in studies of viscosity in polymers and oligomers.22 There is also extensive literature describing the temperature dependence of the ionic conductivity in the liquid state. The VTF equation was first applied to temperature-dependent conductivity data in aqueous electrolytes.23 Subsequently it has seen widespread use to describe the non-Arrhenius conductivity behavior observed in liquid electrolytes24 and polymer electrolytes.25 Temperature-dependent conductivities of liquid and polymer electrolytes have also been analyzed using the WLF equation.26,27 The temperature dependence of the self-diffusion coefficient D in liquids has been studied much less extensively than either the conductivity or the viscosity, due to the relatively late development of convenient NMR spin echo techniques.28 While simple Arrhenius behavior describes diffusion in a fair number of polar liquids, the temperature dependence of D is often described using the VTF equation in liquids,29 polymers and polymer electrolytes,30 although other empirical equations have also been used.29 Dielectric relaxation is included here as a transport phenomenon because the reorientation of a dipolar molecule in a static applied electric field is driven by the gradient of the electrostatic interaction energy between the dipole and the field, tending to minimize the angle between the dipole moment and the field. A more sophisticated discussion of dielectric relaxation in terms of the reorientation of molecular dipoles in an external field is given by Fröhlich.31 A variety of equations have also been used to describe the temperature dependence of dielectric relaxation. Early studies of the dielectric relaxation rate constant were based on transition state theory,20 although much of the later work utilized the VTF equation,32 the WLF equation,1,33 and others.34 2.2. Compensated Arrhenius Formalism. As an alternative to the purely empirical equations used to describe the temperature dependence of mass and charge transport, we developed the CAF initially focusing on ionic conductivity in a variety of organic liquid electrolytes.11 Subsequently, we extended this approach to self-diffusion in organic liquids and organic liquid electrolytes.12,35 We have also shown that this procedure can be applied to dielectric relaxation rate constants, with the caveat that the data were taken from the literature and limited in scope to a few solvent families: linear alcohols, bromoalkanes, nitriles, and acetates.14 Very recently we showed that this method could also be successfully applied to fluidity.13 Our development of the CAF starts with two fundamental postulates.11 First, the temperature dependence of some transport coefficient (e.g., self-diffusion coefficient D, ionic conductivity σ, dielectric relaxation rate constant kdr, and fluidity f) can be accurately described by an Arrhenius-like equation in which the exponential prefactor depends on the temperature T and the static dielectric constant, εs. Our inclusion of εs is based on studies that show that transport coefficients depend on εs as well as on T.36−38 Here we will use the self-diffusion coefficient, since this is the primary focus of this paper

We treat all four transport coefficients in exactly the same manner, resulting in the same mathematical description of their temperature dependence. Our approach, termed the compensated Arrhenius formalism (CAF), contains no adjustable parameters. The CAF describes transport only in terms of molecular and system properties; we show in this paper that consideration of those properties leads naturally to a molecular model of transport in polar liquids and electrolytes. This model is based on a modification of transition state theory, focusing on the electrostatic potential energy of the dipolar interactions. The molecular picture of self-diffusion presented here rejects the conventional view based on hydrodynamic theory, which adequately describes transport phenomena on the macroscopic scale. For example, the Stokes equation characterizes the resistive drag of a ball bearing falling in a vat of oil. However, it has been well documented that transport phenomena such as diffusion15 and dielectric relaxation10 are not accurately described by hydrodynamic equations when the solute molecule is the same size or smaller than the solvent molecules. For example, Nelson and Smyth have shown that relaxation times calculated from Debye’s equation can vary by orders of magnitude from those observed experimentally for molecules that are comparable in size.10 However, the agreement between theory and experiment became better as the solute-to-solvent size ratio increased, and hydrodynamic equations were shown to work fairly well when the solute size was substantially larger than that of the solvent. Hydrodynamic theory is derived assuming a fluid continuum that is applicable when the solute is much larger than the solvent, but molecular motion is discontinuous when the particles are comparable in size.5 Our picture of self-diffusion describes the discontinuous motion that occurs on the molecular level in a pure, polar liquid.

2. BACKGROUND 2.1. Present Status of Transport Models. Many studies of transport phenomena have focused on the temperature dependence of the process, and this is also the focus of the CAF. In some liquids and liquid electrolytes, a thermally activated process seems to provide an adequate picture of temperature-dependent mass and charge transport.16,17 In such systems, the transport coefficient A can be described by a simple Arrhenius equation, A(T) = A0 exp(−Ea/RT), where Ea is the activation energy and A0 is the temperature-independent exponential prefactor. However, in many polar liquids and polar liquid electrolytes a plot of ln(A) vs 1/T exhibits significant deviations from the predicted linearity. Consequently, an accurate description of the temperature dependence of transport coefficient A(T) requires the use of empirical equations that provide no insight into the molecular-level nature of the transport process. There is extensive literature describing the application of numerous such equations to the temperature dependence of transport coefficients. These applications are occasionally accompanied by attempts to assign physical significance to one or two of the fitting parameters. In the next few paragraphs we give a few representative examples of these applications. The temperature dependence of the viscosity has been studied for over a century using an astonishing number of empirical equations to describe the data, although very little physical insight has resulted from these efforts. In an early review article, Brush attempted to summarize these equations and the various competing theories that had been developed.18 Other authors have also listed some of the numerous empirical 2423



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Figure 1. Conductivity prefactors σ0 and self-diffusion coefficient prefactors D0 plotted against static dielectric constant values. Conductivity data are for 0.0055 M solutions of tetrabutylammonium trifluoromethanesulfonate (TbaTf) in ketones CH3CO(CH2)nCH3 (n = 2−7), acetates CH3CO2(CH2)nCH3 (n = 3−5, 7, 9), and nitriles CH3(CH2)nCN (n = 5−8, 10).

D(T , εs) = D0(T , εs) exp( −Ea /RT )

resulting values of Ea fall within a very narrow range. We note that choosing a different value for the reference temperature Tr results in values of Ea that fall within the same very narrow range. Once the values of Ea are known from the scaling procedure, the exponential prefactor D0 is calculated from eq 2 by dividing D by the Boltzmann factor exp(−E̅a/RT), where E̅ a is the average of the values of Ea for the solvent family. Since values of Ea and therefore E̅a are determined directly from the scaling procedure, these values cannot be regarded as adjustable fitting parameters. In our studies of not only self-diffusion, but also conductivity, fluidity and dielectric relaxation, we observed that the exponential prefactor for each family of solvents and salt solutions had a strikingly similar dependence on the static dielectric constant, a dependence that appeared to be exponential:

(1)

where Ea is the activation energy. Our second postulate states that the temperature dependence of the exponential prefactor is given entirely by the temperature dependence of the static dielectric constant contained in the prefactor. In other words, eq 1 may be written D(T , εs) = D0(εs(T )) exp( −Ea /RT )

(2)

The activation energy can be determined by scaling out the dielectric constant dependence in the prefactor, and this procedure has been previously described in detail.11,39 To briefly summarize the scaling procedure, the temperaturedependent value of the self-diffusion coefficient (eq 2) is divided by Dr, where Dr (Tr , εs) = D0(εs(Tr)) exp( −Ea /RTr)

(3)

Dr is the value of D at the same value of εs as in eq 2 and is experimentally determined from an isothermal reference curve at a reference temperature Tr. The reference curve is constructed from values of the diffusion coefficient measured isothermally at Tr for each member of a solvent family differing only in the length of the alkyl chain. Dividing eq 2 by eq 3 results in D(T , εs) D (ε (T )) exp(−Ea /RT ) = 0 s Dr (Tr , εs) D0(εs(Tr)) exp(−Ea /RTr)

D0(εs) ∝ exp(Bεs)

where B is a constant. Self-diffusion coefficients of pure ketones, nitriles and acetates, conductivities of electrolytes based on these liquids, and the static dielectric constants of these liquids and electrolytes were measured from 5 to 85 °C in 10 °C increments. The exponential prefactors obtained from these data are plotted as a function of εs in Figure 1. The selfdiffusion data and conductivity data from which the prefactors were calculated were taken from our previously published studies of acetates,40 ketones,41 and nitriles.42 The error in the measured conductivity values is about 2%, whereas the average error in the self-diffusion data is about 5%. Detailed descriptions of the experimental methods used to obtain the data and estimations of the associated errors in the activation energies derived from these data are given in the above cited references. The striking similarity of the curves suggests that they have the same functional form. We found that a simple exponential function provided the best fit for the prefactors of self-diffusion, conductivity, fluidity, and dielectric relaxation for all systems studied to date. An example is illustrated by the solid lines in

(4)

Since εs depends on the chain length, we choose the value of Dr(Tr) where the condition εs(Tr) = εs(T) is satisfied and the two exponential prefactors cancel. Taking the logarithm of both sides of the above equation leads to the compensated Arrhenius equation: ⎛ D(T , εs) ⎞ E E ln⎜ ⎟=− a + a ε ( , ) D T RT RT ⎝ r r s ⎠ r

(6)

(5)

A plot of the left-hand side of eq 5 against 1/T yields the value of the activation energy Ea. When this procedure is repeated for each member of a given solvent family, the 2424



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Figure 1, which show exponential growth functions fit to each set of data. In this paper we show why this simple exponential dependence occurs in self-diffusion data and present an underlying molecular level picture of transport leading to this dependence.

collective movement of the group of N molecules. The positions and orientations of the immediate, surrounding molecules differ slightly between the initial and final states, however the energy of the final state is essentially identical to that of the initial state. It is important to note that the higher energy of the transition state originates in the 1/2N(N − 1) intermolecular interactions of the dipolar molecules in the region defining the immediate potential energy environment of the target molecule. In the language of transition state theory, the set of N molecules in the higher energy transition state may be regarded as an activated complex in which the higher energy originates in the slight decrease in local density and underlying intermolecular interactions. We define the concentration of activated complexes as c† and the concentration of the remaining molecules in the remaining volume of the liquid in which no jump is occurring as c0. Of course the boundaries of a region comprising an activated complex are fluid and after the N molecules relax back into a locally stable configuration, they become part of the molecules counted in the concentration c0 until some become part of another activated complex. With the assumption that the regions in which transitions are occurring and the rest of the molecules in the liquid are in thermodynamic equilibrium with steady state concentrations of c† and c0, respectively, an equilibrium constant on a concentration basis Kc can be written simply as

3. RESULTS AND DISCUSSION 3.1. Modified Transition State Theory. We begin with the temperature dependence of self-diffusion coefficients. Eyring and co-workers pioneered the use of transition state theory to describe a variety of transport properties such as diffusion,43 ionic conductivity,44 viscosity,43 and dielectric relaxation.45 We note that the earlier work by Eyring and coworkers referred to this model as “absolute reaction rate theory”. Here we examine diffusion in a pure dipolar liquid, modifying a picture introduced by Eyring, Kauzmann, and others.43,46 Consider a region in the liquid containing N dipolar molecules that are in a transient state of local equilibrium. The region contains a sufficient number of molecules so the potential energy environment of a target molecule t in the interior of the region roughly approximates its potential energy environment in the bulk. Thermal fluctuations lead to transport of the target molecule through a series of thermally activated, discrete jumps. A jump occurs during a process in which the group of N molecules that comprise the local potential energy environment of the target molecule undergoes a transition from one transient, locally stable configuration to another, passing through a high energy transition state created by thermal fluctuations in the system. In the initial state, the target molecule is confined by its local potential energy environment and its molecular motion consists of intramolecular, librational, and translatory vibrations. We adopt Kauzmann’s picture of a transition state where thermal fluctuations create a less rigid and more open environment.46 It seems reasonable to assume that this environment is not homogeneous at a molecular scale. In particular, the fluctuations may open a transient channel in which a translatory vibration in the direction of the channel can momentarily become a pure translational motion. In this direction the fluctuation-driven reduced local density of molecules serves as a concentration gradient. After a small translational movement of the target molecule, the neighboring molecules then relax back to a locally stable configuration, which can be regarded as the final state. This process is illustrated in Figure 2. The passage of the target molecule from its position in the initial state to its position in the final state is described here as a jump, with the understanding that such a jump requires the

K c = c †/ c 0

(7)

In the language of transition state theory as described by Eyring, the rate of reaction (here the rate of diffusion) is equal to the concentration of activated complex c† divided by the average time τ to cross the barrier. In his original treatment,20,47 τ was set equal to the length δ across the top of the barrier divided by the average rate of crossing ⟨dx/dt⟩ = (kBT/2πm)1/2 where m is the mass of the molecule. Then the rate of diffusion is ⎛ k T ⎞1/2 1 rate = c †/τ = c †⎜ B ⎟ ⎝ 2πm ⎠ δ

(8)

Using a statistical mechanical treatment of transition state theory,20,47 we write the equilibrium constant between the activated complex and the initial state as

Kc =

F† F (i)

(i)

e−ΔE0 / RT

(9a)

†

where F and F are the partition functions for the initial state and transition state of the system, respectively, and ΔE0 scales the various partition functions to the same energy zero on a per mole basis. Recall that in the process of a jump, a very low frequency translatory vibration of target molecule t in the initial state becomes a pure translational motion in the transition state. Therefore the translatory vibrational partition function of molecule t in the initial state can be factored out from F(i) in eq 9a. Similarly, the translational partition function of molecule t can be separated from the partition function of the transition state. The equilibrium constant is now Kc =

Figure 2. Schematic representation of the target molecule and immediate surroundings during the transition from the initial to the final state as described in the text.

Ft†(translation) Ft(i)(vibration)

K ′c

(9b)

where Kc′ is the equilibrium constant with a vibrational and a translational contribution of molecule t factored out from the 2425



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The compensated Arrhenius equation differs from the simple Arrhenius equation by accounting for an additional temperature dependence not present in the latter. It is important to note that the CAF can only be applied to polar liquids and liquid electrolytes, suggesting that the additional temperature dependence is related to the presence of the permanent electric dipoles. We have attempted to use the CAF to analyze diffusion data in a series of alkanes with absolutely no success (unpublished data). We now show that this additional temperature dependence originates in the electrostatic dipole−dipole interaction potential energy, which is the work done to assemble the dipoles in a given configuration from infinite separation and can be identified as the (Helmholtz) free energy. This discussion will ignore the small difference between the Helmholtz free energy and the Gibbs free energy for a process in the liquid state. Therefore, for a particular configuration of the molecules in the system, we write the electrostatic dipole−dipole interaction potential energy as

initial state and the transition state, respectively. The translational partition function can be written as Ft†(translation) = (2πmkBT )1/2 δ /h

(9c)

The quantity δ is a rough measure of the distance across the top of the transition state barrier where the motion of molecule t is a pure translation. Similarly, the partition function for the translatory vibration of molecule t in the initial state is Fti(vibration) = (1 − e−hνt / kBT )−1

(9d)

where νt is the translatory vibrational frequency. Then the equilibrium constant in eq 9b is Kc =

(2πmkBT )1/2 δ /h (1 − e−hνt / kBT )−1

K ′c

(9e)

This result is now substituted into eq 7, and the resulting expression for c† is substituted into eq 8 to yield rate = c 0K ′c

kBT 1 h (1 − e−hνt / kBT )−1

† ΔGelec,dd =

(10)

The translatory vibration is a very low frequency mode. The far-infrared spectrum of liquid polar molecules exhibits a very broad band with a maximum typically between 10 to 100 cm−1.48,49 Jain and Walker have pointed out the similarity of this spectral feature to translational lattice mode absorption in solids, based on the temperature dependence of the band maximum.48 Therefore since hνt ≪ kBT for the translatory vibration of the target molecule, the vibrational partition function can be written as approximately equal to kBT/hνt. The equation for the rate then becomes rate = c 0νt K ′c = c 0k

(11)

(12)

where λ is the average distance traversed by molecule t in the transition from the initial to the final state. This expression differs from the result obtained by Eyring et al. in two important points. The factor νt is the frequency of the translatory vibrational mode of the target molecule t in the initial state; it is this motion that becomes pure translational motion in the transition state. This vibrational motion of t in the initial state and the purely translatory motion of t in the transition state are explicitly excluded from the equilibrium constant K′c. Since the initial state and the transition state are in thermodynamic equilibrium, we use the usual relationship between the equilibrium constant Kc′ and the standard free energy change between the initial state and the transition state, ΔG†, to write

† ΔGelec,dd =

i

j≠i k≠j

(14a)

1 ⟨∑ ∑ ∑ μ tr Tij[I + α T]jk −1 μk ⟩ 2 i j≠i k≠j i

(14b)

The exact calculation of eq 14b is difficult because each dipole polarizes its surroundings, inducing moments through the polarizabilities of the other molecules. Reasonable approximations provide a simpler approach, and calculations equivalent to the right-hand side of the equation have been carried out by many authors and discussed at length in several excellent reviews.31,51,52 The problem of evaluating eq 14b can be replaced by an equivalent problem that takes advantage of the long-range nature of a dipole field by assuming beyond a certain distance the interactions of a given molecule with its surroundings can be replaced by the interaction of the molecule with a continuum with dielectric constant εs. Consider the spherical region of volume V containing the N dipoles in the transition state. We first focus on a single molecule in the center of the spherical region; the molecule itself is assumed to be spherical and the dipole moment is located at the center of the molecule. Following Onsager,53 all short-range dipole− dipole interactions in the spherical region are neglected and the molecule polarizes the continuum outside of the spherical region, resulting in a reaction field at the molecule. Each of the N molecules polarizes the continuum in the same manner, resulting in a reaction field R at the molecule.

†

D = λ 2νt e−ΔG / RT

∑ ∑ ∑ μi tr Tij[I + α T]jk−1 μk

where μi is the electric dipole moment of molecule i (μitr is the transpose of μi), μj is the electric dipole moment of molecule j, α is the electronic polarizability of the target molecule, and Tij is the dipole field propagation tensor.50 This tensor relates the electric field at molecule i, Ei, due to a molecule at j with electric dipole moment μj, i.e., Ei = −Tij·μj. The molecular dipole moments are represented by three-dimensional vectors, while αij, Tij, and the unit tensor Iij are represented by 3 × 3 matrices. The sums run over all the molecules in the sample, NT, including the N molecules in the activated complex. Equation 14a describes the summation over all instantaneous dipolar interactions, which refers to a particular configuration of the NT molecules. However, what is required is the average configuration, and eq 14a is then more correctly written as

where k is the rate constant for diffusion. Again following Glasstone,20 we write the diffusion coefficient as D = λ 2k = λ 2νt K ′c

1 2

(13)

We note that this expression differs from the usual expression for D obtained from transition state theory20 by the absence of the factor kBT/h. 3.2. Polarization Energy and the Transition State. In the usual formalism for transition state theory, the quantity ΔG† is written as ΔH† − TΔS† and ΔH† is subsequently identified as the activation energy. However, as noted earlier, this procedure often leads to curved Arrhenius plots and the use of empirical equations to describe transport coefficients. 2426



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the exponential prefactor using the expression for the polarization energy on a per mole basis, given in Böttcher51

Here we will use the language and notation of Böttcher to write W ≈ ΔG†elec,dd, where the polarization energy Wi of a dipole with moment μi in its own reaction field R is Wi = −(1/ 2)μi·R.51 Therefore the polarization energy WN of N dipoles is simply WN = −(1/2)Nμi·R. In a polar liquid the molecules are in continuous random motion, which necessarily introduces an averaging process in the calculation of the reaction field. The formation of a highly localized transition state as discussed in the previous section and depicted in Figure 2 may be viewed as a slight decrease in the local dipole density of the transition state relative to the initial state that results from a thermal fluctuation; at any instant of time there are numerous regions in the macroscopic sample undergoing similar fluctuations. These regions, taken together, comprise the transition state of the liquid sample. The primary focus here is on the temperature dependence of the polarization energy, and it will be shown that the temperature dependence of W is due to the temperature dependence of the dipole density. The polarization energy of a target dipole changes with temperature because the spatial separation of the surrounding dipoles is temperaturedependent. The temperature dependence of this change in spatial separation is conceptually related to the free volume change with temperature inherent in the Vogel−Tammann− Fulcher equation.2−4 We assume here that the temperature dependence of the dipole density for a collection of dipoles in the transition state is similar to that in the initial state. In the view presented here, the dipoles comprising the transition state excluding the target dipole are “stationary” relative to the time required for the target dipole to polarize the surrounding medium and then interact with the resulting reaction field. The dipoles are considered “stationary” in the sense that they undergo only small changes in orientation and position during the transition, whereas the change in the position of the target molecule is relatively large. This relatively stationary configuration of dipoles allows application of Onsager’s reaction field model to the target dipole. The reaction field of the surrounding dipoles couples with the target dipole and therefore affects the transition state energy. A thermal fluctuation in the liquid produces a nonequilibrium transition state where the immediate surroundings of the target dipole are disrupted and subsequent translation of the target dipole occurs. Some workers have pointed out that the movement of the target dipole can be partially impeded if the surrounding dipoles do not instantaneously adjust to the motion of the target dipole.54,55 This lag in the reaction field produces a resistive force on the target dipole termed the dielectric friction.54,56,57 Here, we show that no assumptions regarding dielectric friction are required in order to describe the temperature dependence of self-diffusion in a pure polar liquid. The key point of our work is that the thermal activation model by itself does not sufficiently describe transport phenomena because it does not account for the effect of the reaction field on the target dipole, in particular the temperature dependence of that interaction. Although we have approximated ΔG†elec,dd by the polarization energy W calculated using the reaction field approximation described by Onsager, in section 3.5 we will demonstrate the critical result that the temperature dependence of both quantities is identical. Therefore we will continue to work with the polarization energy, which provides a convenient and compact notation for sections 3.3 and 3.4. 3.3. Temperature Dependence of the Exponential Prefactor. We now examine the temperature dependence of

W=−

4πNANdμ2 εs − 1 ε∞ + 2 3 2εs + ε∞ 3

(15)

Here NA is Avogadro’s number and Nd is the dipole density, which can be written as NAρ/M where ρ is the mass density and M is the molecular weight. The quantity ε∞ is often written as the square of the optical refractive index, nD2. The temperature dependence in W originates primarily in the dipole density, a point that will be more carefully developed in a later section. The temperature dependence of the ratio (εs − 1)/(2εs + ε∞) in the above equation is negligible across the range of our temperature measurements as illustrated in the Supporting Information. Therefore we approximate this ratio by 1/2 for values of εs that are relatively large compared with unity and write W≅−

2πNANdμ2 ε∞ + 2 3 3

(16)

We modify eq 13 to explicitly account for the intrinsic temperature dependence of ΔG† through its dependence on the polarization energy W. We write ΔG† = ΔG† + W − W and substitute into the expression for the diffusion coefficient, eq 13, regrouping the factors as D = λ 2νt exp( −W /RT ) exp(− (ΔG† − W )/RT ) †

(17)

†

With the definition ΔG′ ≡ ΔG − W, this expression becomes D = λ 2νt exp( −W /RT ) exp(−ΔG′† /RT )

(18)

†

The quantity ΔG′ may be more easily understood using Figure 3, which is the energy diagram of the transition depicted in Figure 2.

Figure 3. Free energy diagram of the transitions and states used in the CAF.

Here ΔG† is the free energy of activation of the polar liquid from the initial state (free energy GPi) to the transition state (free energy GP†). As noted earlier, in the initial state all molecules are in the average configuration of the bulk liquid, whereas in the transition state the N molecules depicted in Figure 2 have a more open environment due to the thermal fluctuations in the system. The quantity GnP† labels a 2427



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Figure 4. (Left) Exponential prefactors for the diffusion coefficients of several ketones measured from 5 to 80 °C and plotted as a function of Nd/T. (Right) Same data as the left-hand side of the figure now plotted as a function of 1/T. 2-Pentanone = cyan, 2-hexanone = black, 2-heptanone = red, 2-octanone = blue, 2-nonanone = green, 2-decanone = gray.

⎛ 2πN μ2 ε + 2 ⎞ ⎛ −ΔG′† ⎞ d ∞ ⎟ exp⎜ D = λ 2νt exp⎜ ⎟ 3 ⎠ ⎝ RT ⎠ ⎝ 3kBT

hypothetical state in which the N molecules have the configuration of the transition state, but all permanent electric dipole moments are set to zero. Consequently there are no dipole−dipole and dipole−induced dipole intermolecular interactions. This state can be viewed as a “non-polar transition state”. Formally, ΔG′† is the free energy of transition from the initial (bulk) state of the polar liquid to a hypothetical transition state in which all electric dipole moments have been “turned off”. As shown in the figure, this transition is thermodynamically equivalent to the transition from the bulk configuration in the polar liquid to the transition state configuration in the polar liquid, followed by a process with depolarization energy -W in which the electric dipoles are set to zero (recall from eq 15 that W is intrinsically negative). Increasing the temperature or alkyl chain length decreases |W| by reducing Nd. This follows because increasing the temperature causes thermal expansion of the liquid and leads to fewer dipoles per unit volume, while isothermally adding a longer alkyl chain also results in fewer dipoles per unit volume. Both cases produce an attenuation of the reaction field strength and a corresponding decrease in |W|. However much |W| decreases, the transition state of the polar liquid will be that much higher in energy. The resulting temperature dependence of ΔG† is experimentally observed as non-Arrhenius behavior in eq 13 in those systems where temperature-dependent changes in |W| are non-negligible relative to |ΔG†|. It is important to note that although W and ΔG† change with temperature or alkyl chain length (through the dipole density), the sum of ΔG† and −W (= ΔG′†) is independent of temperature and chain length. Substituting eq 16 into eq 18 results in ⎛ 2πN N μ2 ε + A d ∞ D = λ 2νt exp⎜ 3 ⎝ 3RT

⎛ −ΔG′† ⎞ 2⎞ ⎟ exp⎜ ⎟ ⎝ RT ⎠ ⎠

(20)

An analogous polarization calculation was used by Onsager to write the dielectric constant of a polar liquid with polarizable molecules.53 εs − ε∞ = 4π

3εs ⎛ ε∞ + 2 ⎞2 Ndμ2 ⎜ ⎟ 2εs + ε∞ ⎝ 3 ⎠ 3kBT

(21)

Onsager noted that for εs large relative to ε∞, this expression could be written εs ≈

2πNdμ2 ⎛ ε∞ + 2 ⎞2 ⎜ ⎟ kBT ⎝ 3 ⎠

(22)

Therefore the self-diffusion coefficient can be written ⎛ −ΔG′† ⎞ ⎛ εs ⎞ D = λ 2νt exp⎜ ⎟ ⎟ exp⎜ ⎝ ε∞ + 2 ⎠ ⎝ RT ⎠

(23)

It is important to recognize that the CAF compensates for the temperature dependence of the static dielectric constant, rather than the dielectric constant itself. The essential point here is that the temperature dependence of the first exponential factor in eq 20 is simply the intrinsic temperature dependence of εs. Therefore we rewrite eq 23 as ⎛ −ΔH′† ⎞ ⎛ ΔS′† ⎞ D = λ 2νt exp(Bεs) exp⎜ ⎟ exp⎜ ⎟ ⎝ RT ⎠ ⎝ R ⎠

(19)

(24)

where B is a temperature-independent constant. With ΔH′† ≃ Ea, the diffusion coefficient D can then be written as

We use R = NAkB and write 2428



Article

⎛ ΔS′† ⎞ ⎛ −E ⎞ D = λ 2νt exp(Bεs) exp⎜ ⎟ exp⎜ a ⎟ ⎝ RT ⎠ ⎝ R ⎠ ⎛ E ⎞ = D0 exp⎜ − a ⎟ ⎝ RT ⎠

exp(− (ΔG′† /RT )) D(T ) = Dr (Tr) exp(− (ΔGr′† /RTr)) †

⎛ D(T ) ⎞ E E ΔSr′† ΔS′† ln⎜ − ⎟=− a + a + RT RTr R R ⎝ Dr (Tr) ⎠ (26)

†

⎛ D(T ) ⎞ E E ln⎜ ⎟=− a + a RT RTr ⎝ Dr (Tr) ⎠

(27)

with A = λ2νt exp(ΔS′†/R). In the CAF procedure, B is treated as a constant by ignoring the weak temperature dependence of ΔS′†, which is negligible over the modest temperature range (ΔT = 80 K) of our data. The temperature dependence of D0(T) is given by the temperature dependence of the static dielectric constant εs(T) according to eq 27. We write the temperature dependence as εs(T) ∝ Nd/T, using the Onsager expression for εs given earlier (eq 22). The temperature dependence of the dipole density plays a critical role as evidenced by Figure 4, which compares plots of D0 against Nd/T and against 1/T for a number of ketones. The data on the left-hand side of the figure all fall on a single master curve, as illustrated earlier in Figure 1 that plots the prefactors against εs. However, when the same data are plotted as a function of 1/T, the data lie on six distinct curves. The failure to obtain a master curve when D0 is plotted against 1/T underscores the necessity of including the dipole density in order to account for the complete T dependence of εs. We have also previously demonstrated experimentally that the temperature dependence of the self-diffusion coefficient prefactor depends on Nd/T for families of nitriles, acetates and thiols.42 3.4. Thermodynamics of the Compensation Procedure. Previously we used the dependence of the diffusion coefficient on εs to allow cancellation of the exponential prefactors in the compensation procedure. We now reconsider the compensation procedure in terms of the role of the polarization energy, using the conventions of Figure 3. The selfdiffusion coefficient D(T) at a temperature of interest T is divided by the self-diffusion coefficient Dr(Tr) at a reference temperature Tr using the notation of eq 18. exp( −W /RT ) exp( −ΔG′† /RT ) D(T ) = exp( −Wr /RTr) exp( −ΔGr′† /RTr) Dr (Tr)

(30) †

The temperature dependence of ΔS′ and ΔSr′ is small; therefore the temperature dependence of the difference ΔS′† − ΔSr′† is negligible. This leads to

Finally, the temperature-dependent exponential prefactor D0 can be written in a more general form as D0(T ) = A exp(Bεs(T ))

†

We now substitute ΔG′ = ΔH′ − TΔS′ (with a similar expression for ΔGr′†) and take the natural logarithm, noting once again that ΔH′† ≅ ΔHr′† ≅ Ea.

(25)

where the exponential prefactor is ⎛ ΔS′† ⎞ D0 = λ 2νt exp(Bεs) exp⎜ ⎟ ⎝ R ⎠

(29) †

(31)

which can be recognized as the compensated Arrhenius equation, eq 5. 3.5. Generalization of the CAF. An important question is the generality of the resulting equations comprising the CAF, beginning with eq 13. In section 3.3, we wrote eq 18 in terms of the Onsager expression for W given by eq 16. A more general and formal notation would be to write the diffusion coefficient (eq 18) as ⎛ −ΔG† ⎞ †⎞ ⎛ elec,dd ⎟ exp⎜ −ΔG′ ⎟ D = λ 2νt exp⎜⎜ ⎟ RT ⎝ RT ⎠ ⎝ ⎠ †

†

ΔG†elec,dd

(32)

ΔG†elec,dd

where ΔG′ ≡ ΔG − and is given by eq 14. It is then necessary to ascertain the temperature dependence of ΔG†elec,dd. The dipole field propagation tensor Tij in eqs 14a and 14b can be conveniently represented as Tij =

3( rj⃗ − ri ⃗)( rj⃗ − ri ⃗) I − 3 | rj⃗ − ri |⃗ | rj⃗ − ri |⃗ 5

(33)

where ri⃗ and rj⃗ are the positions of the ith and jth molecules with respect to a set of laboratory fixed axes, and I is a threedimensional unit tensor.50 The units of Tij are 1/length3, and since the dimensions of the polarizability α are length3, the factor (I + αT)−1 has no units. Therefore from eq 14, ΔG†elec,dd has the (SI) units of μ2/(4πε0)length3, or μ2/(4πε0)V, where V is the volume and ε0 is the vacuum permittivity. Consequently the temperature dependence of the first exponential factor in eq 32 is proportional to 1/VT or equivalently, Nd/T for a closed system with a fixed number of molecules. This is exactly the result predicted by eq 27. Up to this point we have been concerned about the role of the temperature dependence of the dipolar energy W. However, it is interesting to examine eq 23 with attention to the magnitude of W. Equation 23 uses the Onsager approximation to write the prefactor as D0 ∝ exp(εs/(ε∞ + 2)). Figure 5 shows a plot of the D0 against εs for the ketones and nitriles, which is essentially the upper right portion of Figure 1; the dashed lines are best fits to D0 = A1 exp(A2εs), where A1 and A2 are fitting parameters. For both the ketones and nitriles, the curves in Figure 5 yield a value of about 0.20 for the quantity 1/(ε∞ + 2). Calculations of 1/(ε∞ + 2) using literature values58 for the refractive index nD and ε∞ = nD2 yield about 0.25, in surprisingly good agreement with the values obtained from Figure 5. However, the agreement between theory and experiment for the magnitude of the slope is not as good for diffusion data of low εs liquids such as the acetates and thiols.

(28)

Here Wr is the polarization energy at the reference temperature Tr and ΔGr′† is equal to ΔGr† − Wr, also at Tr. Recall that the reference curve is constructed from values of the diffusion coefficient measured at Tr for each member of a solvent family differing only in the length of the alkyl chain. Because the polarization energy depends on the dipole density Nd, which in turn depends on the alkyl chain length, changing the chain length changes the value of Wr. The key point to the compensation procedure is that W depends on Nd; therefore ΔG′† (=ΔG† − W) is the same regardless of whether we change Nd (and therefore W) by temperature or by the alkyl chain length. Cancellation of the exponential prefactors occurs when W/ RT = Wr/RTr; the ratio of the two diffusion coefficients is then 2429



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transport coefficient followed a simple Arrhenius equation. However, the CAF analysis offers a more interesting alternative: to increase the exponential prefactor by selecting or synthesizing organic solvents whose molecular characteristics lead to a larger exponential prefactor. Because the prefactor is proportional to exp(Bεs)), it would appear that increasing εs would increase the exponential prefactor. In particular, the molecular level view developed here provides guidance on how to accomplish the desired enhancement. We have explored this idea in electrolytes based on cyclic carbonates, which are of interest in lithium ion rechargeable batteries.59 Although derivation given here of a molecular level model has specifically considered self-diffusion, the similar successes of the CAF in describing conductivity, dielectric relaxation and fluidity lead us to suggest that this model may also be an appropriate description of mass and charge transport in those phenomena. The general theme common to these forms of transport appears to be some form of diffusion. Self-diffusion is translational diffusion of molecules in a local concentration gradient, fluidity is translational diffusion of molecules in the presence of applied stress, dielectric relaxation is rotational diffusion of dipoles in an electric potential gradient and conductivity is translational diffusion of charges in an electric potential gradient. Conductivity is a more complex phenomenon than diffusion, fluidity, and dielectric relaxation because it involves charge−charge and charge-dipole interactions in addition to the dipole−dipole interactions that predominate in pure polar organic liquids. However, we have successfully applied the CAF to describe ionic conductivity at reasonably high salt concentrations (0.3−0.8 m) in a variety of electrolyte solutions, obtaining activation energies similar to those found in self-diffusion measurements in polar liquid systems with εs > 5−6.12,41,42,59 In addition, the presence of prefactor master curves with the same qualitative dependence on εs as found in pure polar organic liquids by the other three kinds of measurements suggests that the molecular picture of ion transport closely resembles the picture emphasized in this paper for self-diffusion. Very recently, we have successfully applied the CAF to the conductivity of ionic liquids.60 The success of the CAF in its application to conductivity is not particularly surprising, given the picture of transport occurring as intrinsic to an ensemble of molecules, rather than a single molecule moving through a static matrix.

Figure 5. Plots of D0 as a function of εs for the nitriles and ketones used in this study. The red dashes are a best fit to D0 = A1 exp(A2εs), and the values of the fitting parameters A1 and A2 are shown on the plots.

4. CONCLUSIONS Conventional descriptions of mass and charge transport in liquids and liquid electrolytes are generally based on either a thermally activated or a hydrodynamic model; both fail to adequately describe experimental data as briefly discussed in the Introduction. The temperature dependence of a transport coefficient A(T) governed by a thermally activated process is given by the simple Arrhenius equation A(T) = A0 exp(−Ea/ RT) where Ea is the activation energy. Although this is an intuitively appealing picture, plots of ln(A(T)) against 1/T are often curved, suggesting that there is an additional temperature dependence not accounted for in the simple Arrhenius equation. This equation is insufficient because it assumes a static potential energy environment for a polar molecule over a particular temperature range. However, in a liquid the density changes with temperature, resulting from changes in the average intermolecular spatial separation in the system. This in turn leads to a temperature-dependent electrostatic interaction potential energy or dipole polarization energy. The energy of activation determined from the simple Arrhenius equation is temperature-dependent due to this dipole polarization energy contribution. If we write εs ∝ W/RT, the CAF separates out this contribution into the exponential prefactor so that the transport coefficient A(T) assumes an Arrhenius-like form consisting of two factors: a Boltzmann factor exp(−Ea/RT) that increases with increasing T, and an exponential prefactor A0(εs(T)) that decreases with increasing T due to the generally observed decrease in εs with increasing temperature.31 Successful application of this formalism to self-diffusion, conductivity, fluidity and dielectric relaxation in a wide variety of organic liquids and organic electrolyte solutions led to plots of the exponential prefactor against the static dielectric constant that appeared to have the common functional form A0(εs(T)) ∝ exp(Bεs)), where B is a constant, in all polar molecule systems studied to date. The integration of this molecular level view of transport into the CAF has potentially useful technological applications. An important goal in the development of advanced battery systems is to enhance the isothermal conductivity or diffusivity of an electrolyte. The most obvious method is to decrease the activation energy, a conclusion that would also be true if the

■

ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was supported by the Army Research Office, Grant No. W911NF-10-1-0437.

■ c 2430

†

GLOSSARY OF SYMBOLS steady state concentration of activated complexes dx.doi.org/10.1021/jp408899y | J. Phys. Chem. B 2014, 118, 2422−2432

The Journal of Physical Chemistry B c0 D D0 Dr Ea F†t F(i) t f ΔG†elec,dd ΔG† ΔG′† h ΔH† ΔH′† k kB kdr Kc K′c m N NT NA Nd nD R R ΔS† ΔS′† T T Tr W Wr α δ εs ε∞ ε0 η λ μ ν ρ σ σ0 σr

Article

τ

steady state concentration of molecules not in activated complexes self-diffusion coefficient self-diffusion coefficient exponential prefactor self-diffusion coefficient at reference temperature Tr energy of activation partition function of target molecule in transition state partition function of target molecule in initial state fluidity electrostatic dipole−dipole interaction potential energy free energy change between initial state and transition state free energy change between initial state and hypothetical nonpolar transition state Planck’s constant enthalpy change between initial state and transition state enthalpy change between initial state and hypothetical nonpolar transition state rate constant for diffusion Boltzmann constant dielectric relaxation rate constant equilibrium constant between molecules in activated complexes and remaining molecules in sample equilibrium constant between molecules in activated complexes and remaining molecules in sample with the contribution of the target molecule factored out mass of diffusing target molecule number of molecules in transition state containing target molecule t total number of molecules in the sample Avogadro’s number dipole (number) density optical refractive index gas constant reaction field entropy change between initial state and transition state entropy change between initial state and hypothetical nonpolar transition state temperature (Kelvin) dipole field propagation tensor reference temperature (Kelvin) dipolar polarization energy dipolar polarization energy at reference temperature Tr electronic polarizability length across top of transition barrier (transition state theory) static dielectric constant high frequency dielectric constant vacuum permittivity viscosity coefficient average distance traversed by target molecule during transition from initial to final state dipole moment translatory vibrational frequency of target molecule mass density ionic conductivity ionic conductivity exponential prefactor ionic conductivity at reference temperature Tr

■

average time to cross transition barrier (transition state theory)

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Molecular Model of Self Diffusion in Polar Organic Liquids

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