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Describing Temperature-Dependent Self-Diffusion Coefficients and Fluidity of 1- and 3‑Alcohols with the Compensated Arrhenius Formalism Allison M. Fleshman,*,† Grant E. Forsythe,† Matt Petrowsky,‡ and Roger Frech§ †

Department of Chemistry, Lawrence University, Appleton, Wisconsin 54911, United States Maxwell Technologies, San Diego, California 92123, United States § Department of Chemistry and Biochemistry, University of Oklahoma, Norman, Oklahoma 73071, United States ‡

S Supporting Information *

ABSTRACT: The location of the hydroxyl group in monohydroxy alcohols greatly affects the temperature dependence of the liquid structure due to hydrogen bonding. Temperature-dependent self-diffusion coefficients, fluidity (the inverse of viscosity), dielectric constant, and density have been measured for several 1-alcohols and 3-alcohols with varying alkyl chain lengths. The data are modeled using the compensated Arrhenius formalism (CAF). The CAF follows a modified transition state theory using an Arrhenius-like expression to describe the transport property, which consists of a Boltzmann factor containing an energy of activation, Ea, and an exponential prefactor containing the temperature-dependent solution dielectric constant, εs(T). Both 1- and 3-alcohols show the Ea of diffusion coefficients (approximately 43 kJ mol−1) is higher than the Ea of fluidity (approximately 35 kJ mol−1). The temperature dependence of the exponential prefactor in these associated liquids is explained using the dielectric constant and the Kirkwood−Frölich correlation factor, gk. It is argued that the dielectric constant must be used to account for the additional temperature dependence due to variations in the liquid structure (e.g., hydrogen bonding) for the CAF to accurately model the transport property.

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INTRODUCTION

The CAF models transport properties using an adapted transition state theory11−13 with an Arrhenius-like form for selfdiffusion coefficients, D, and fluidity, F (inverse of viscosity η):

A complete model for molecular motion in liquids is difficult to attain given the varying degrees of intermolecular interactions that arise in the liquid state. Several models describe temperature-dependent transport properties of polar liquids rather well but require empirical fitting parameters, an example being the Vogel−Tamman−Fulcher equation.1−3 Monohydroxy alcohols are an excellent system for testing a new transport model, as these liquids exhibit degrees of hydrogen bonding, and can be acquired with high levels of purity. Many studies have measured transport properties and dielectric behavior of the monohydroxy alcohols and their isomers as a function of temperature.4−9 A recent review by Böhmer et al.10 demonstrates the complex dynamics and the numerous avenues of physical exploration these alcohol systems provide through dielectric relaxation studies over the past 100 years. Here, we present a study of transport properties of two monohydroxy alcohol systems1-alcohols and 3-alcoholsusing the compensated Arrhenius formalism (CAF). The goal of the work is to demonstrate how the model incorporates the effects of hydrogen-bonding and liquid structure on transport properties through the inclusion of the temperature dependent static dielectric constant. © XXXX American Chemical Society

⎡ −E ⎤ D(T ) = D0(T ) exp⎢ a ⎥ ⎣ RT ⎦

(1)

⎡ −E ⎤ 1 = F(T ) = F0(T ) exp⎢ a ⎥ ⎣ RT ⎦ η(T )

(2)

where D0 and F0 are the exponential prefactors for diffusion and fluidity, respectively, Ea is the energy of activation, R is the gas constant, and T is temperature. Unlike a simple Arrhenius expression, D0 and F0 are temperature-dependent. A key postulate of the CAF states that temperature dependence of the exponential prefactor is due to the temperature dependence of the solution static dielectric constant, εs, i.e., D0(εs(T)) and F0(εs(T)) for self-diffusion and fluidity, respectively.12,14 The CAF uses the solution dielectric constant in a scaling procedure that compensates for the additional temperature dependence in the exponential prefactor providing a means to determine the Ea. The scaling procedure is given in detail in the Results Received: April 7, 2016 Revised: August 30, 2016

A

DOI: 10.1021/acs.jpcb.6b03573 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B section. The CAF has been applied to mass transport and charge transport for a variety of pure solvents and concentrated electrolyte systems,12,14−17 and recently Dubois et al.18 used the CAF to determine Ea values for the conductivity of poly(propylene glycol) diacrylate systems. CAF results for self-diffusion coefficients divide solvents into two broad classes: aprotic liquids with an Ea of roughly 25 kJ mol−1 15 and protic liquids whose Ea values are in the range of 40−50 kJ mol−1.13,14 These results indicate that a molecule undergoing hydrogen bonding requires more energy to reach the transition state for diffusion compared to it is aprotic analogue. Fluidity studies employing the CAF yield similar results, with Ea values for aprotic liquids at least 10 kJ mol−1 lower than the hydrogen-bonded n-alcohols.19 To investigate the range of Ea for hydrogen-bonded liquids, and to further develop the effect liquid association has on the exponential prefactor, the CAF is applied to two solvent families: 1-alcohols and 3-alcohols. A solvent family is defined as a group of solvent molecules with the same functional group at the same location on the alkyl chain. Family members differ from each other only by the number of methylene groups in the alkyl chain. The 1alcohol family members studied here consist of 1-hexanol, 1heptanol, 1-octanol, 1-nonanol, and 1-decanol. The 3-alcohol family consists of 3-hexanol, 3-heptanol, 3-octanol, 3-nonanol, and 3-decanol. Many studies of liquid structure have been performed on the various isomers of several alcohols,4,8,9,20−26 but few compare multiple members of a solvent family.23 It is well-known that 1alcohols can hydrogen bond to form extended linear chains.4,27−29 However, relocating the hydroxyl group from the terminal to an interior carbon hinders the formation of polymer-like linear networks due to increased shielding of the hydroxyl group,4,20,21,30,31 which affects the relaxation dynamics.32,33 Using 2D N-IR correlation spectroscopy, Czarnecki et al. found that pure 1-octanol favors extended linear polymeric chains, whereas 3-octanol tends toward forming cyclic species. Diluting the pure octanol solvent with CCl4, however, reduces the associated species to monomers via different pathways that depend on both the location of the hydroxyl group and temperature.21,34 A recent study by Wikarek et al.33 used broadband dielectric spectroscopy to study temperature- and pressure-dependent relaxations of several monohydroxy alcohols. They concluded that the relocation of the hydroxyl group from a terminal carbon to an interior carbon greatly reduced the amplitude of the Debye relaxation peak, which is associated with transient motion of the H-bonded chains.32 Wikarek et al.33 also found that additional methyl groups along the chain or extensions of the alkyl chain of the monohydroxy alcohol did not significantly affect the dynamics. These works support using collective solvent family members in the CAF to determine Ea values because they undergo similar relaxation precesses. The CAF also successfully models temperature-dependent rate constants for dielectric relaxation, k = 1/τ,35 supporting that the underlying mechanism of these transport processes is the same. The CAF requires temperature-dependent transport data and dielectric constants for each member of the solvent family. Varying the alkyl chain length changes the static dielectric constant at constant temperature,13 providing the foundation for the CAF scaling procedure. Onsager’s equation, which relates the dielectric constant to the dipole density (polarization), is

(εs − ε∞)(2εs + ε∞) εs − (ε∞ + 2)2

=

4πN (T )μ2 9kBT

(3)

where N(T) is the temperature-dependent dipole density, μ is the permanent dipole moment of the molecule, ε∞ is the high frequency value of the real part of the dielectric constant, kB is Boltzmann’s constant, and T is temperature.36 Equation 4 simplifies to the following: εs ≈

2πN (T )μ2 (ε∞ + 2)2 9kBT

(4)

if εs is large compared to ε∞, a fair assumption for our systems. We intentionally write N(T) to emphasize the temperature dependence in the dipole density as it plays a primary role in the CAF. Assuming Onsager’s model is valid, either the dipole density or εs can be used in the CAF scaling procedure13 and will be demonstrated in the Results section. The dipole density is calculated by dividing the solution density by the molecular weight. In the case of monohydroxy alcohols, N(T) is also the number density since each molecule is contributing one dipole. The relationship between dipole density and dielectric constant is an important aspect of the CAF. Through Onsager’s model, the dielectric constant provides a link to molecular-level properties, i.e., the dipole density and the magnitude of the dipole moment of the polar molecule. For associating liquids, however, Onsager’s model becomes inadequate, and the Kirkwood−Frölich36 model must be used to interpret the dielectric constant behavior with temperature. Our work will show that fortunately the CAF incorporates these subtleties in the temperature dependence of εs, contained within the exponential prefactor by using both Onsager’s and Kirkwood−Frölich’s models and comparing the results. The complicated nature of associated liquids makes modeling mass and momentum transport difficult. Numerous studies attribute increases in viscosity or decreases in diffusion coefficients to increased association due to hydrogen bonding.6,37−40 However, it is well-known that the dielectric constant is also greatly affected by hydrogen bonding.36,39,41 The CAF takes into account the temperature dependence of the liquid structure through use of the solvent dielectric constant, thereby incorporating the relationship of transport to the hydrogen-bonding network. The goal of this work is to demonstrate the usefulness of the CAF as a model for transport properties of 1- and 3-alcohol systems and explore the importance of the temperature dependence of the exponential prefactor in describing these hydrogen-bonded liquids. 36

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EXPERIMENTAL SECTION All solvents (99% pure) were obtained from Sigma-Aldrich or Alfa-Aesar and used as received. All chemicals were stored and samples prepared in a glovebox (≤1 ppm of H2O) under a nitrogen atmosphere at ambient glovebox temperature (approximately 27 °C). Diffusion coefficients were measured using pulsed field gradient nuclear magnetic resonance (PFG NMR) with a VarianVNMRS-400 MHz NMR and an Auto-X-Dual broadband 5 mm probe tuned to the 1H frequency 399.87 MHz. The pure solvents were measured in a glass NMR tube with a 5 mm outer diameter and a sample height of 0.8 cm. The temperature was regulated from 5 to 85 °C in 10 °C increments using a FTS XR401 air-jet regulator. The sample was allowed to equilibrate B


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The Journal of Physical Chemistry B

temperature from 5 to 85 °C for 3-hexanol and 1-hexanol and from 15 to 85 °C for 3-decanol and 1-decanol. Similar temperature-dependent behavior is observed for fluidity data shown in the bottom plot of Figure 1. It is known that 3alcohols exhibit less extensive hydrogen bonding networks than 1-alcohols, which might explain the behavior of D(T) and F(T) observed at higher temperatures. However, quantitatively modeling these transport properties requires a better understanding of the origin of each temperature-dependent aspect. The CAF separates a given transport property’s temperature dependence into two contributing factors: the exponential prefactor (D0(εs)) and the Boltzmann factor (exp[−Ea/RT]). A scaling procedure eliminates the exponential prefactor such that the Ea in the Boltzmann factor can be determined. The scaling procedure is described below and also given in previous works.12,14,16,19 A reference diffusion, Dr, is determined from a plot of isothermal diffusion coefficients versus εs composed of each member of the solvent family at a reference temperature, Tr. Figure 2 displays the reference diffusion curve for Tr = 45 °C

at each temperature for 10 min. At each temperature, a standard Stejskal−Tanner42 pulsed field gradient spin-echo sequence was used. The gradient field strength was varied in 0.023 T m−1 intervals from 0.05 to 0.63 T m−1. At each interval the signal was integrated and the resulting intensity values were plotted as the natural logarithm versus the square of the gradient field strength. The capacitance (C) and phase angle (θ) were measured using an HP 4192 A impedance analyzer and an Agilent 16452A liquid test fixture. The measurements were taken with a logarithmic sweep over a frequency range of 1 kHz−13 MHz. The instrument was set to parallel circuit and averaging (slow) mode. The static dielectric constant εs was calculated from the measured capacitance C through the equation εs = α × C × C0−1, where α is a variable to account for stray capacitance and C0 is the atmospheric capacitance.43 Temperature-dependent viscosity data were collected using a VISCOlab 4000 small sample viscometer with thermal jacket housed in a nitrogen glovebox, paired with an external Huber ministat recirculating temperature bath. Measurements of a 2 mL sample were taken in increments of 10 °C, from 5 to 85 °C. Samples were allowed to thermally equilibrate for 15 min. Density measurements of pure solvents were made using an Anton-Paar DMA 4500 M density meter with internal temperature regulation. Control samples were checked against literature data and found to be within 0.1%.44

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RESULTS AND DISCUSSION Applying Compensated Arrhenius Formalism to SelfDiffusion Coefficients and Fluidity. The top plot of Figure 1 shows the self-diffusion coefficients as a function of

Figure 2. Temperature-dependent diffusion coefficient versus dielectric constant for 1-octanol (symbols given in legend). Black diamonds create the diffusion reference curve, Dr, for the 45 °C reference temperature using members of the 1-alcohol family. The empirical fit of the reference curve is given in the figure.

intersecting the temperature-dependent diffusion data for 1octanol. The reference curve originates from Onsager’s model of the dielectric constant given in eq 4, which links εs to the number of dipoles per unit volume, N(T).36 Thermal expansion of the liquid gives rise to the temperature dependence in N(T). Increasing temperature will decrease the dipole density, but so will increasing the alkyl chain length of the molecule as demonstrated with the black diamonds in Figure 2. This method of isothermally decreasing the dipole density by increasing the alkyl chain length of the molecule is the key to creating an isothermal reference diffusion to which the temperature-dependent diffusion can be scaled.13 The temperature-dependent diffusion, D(T), is divided by the corresponding Dr for a given value of εs using the empirical fit for a particular reference temperature: −E D0(εs(T )) exp⎡⎣ RTa ⎤⎦ D(T , εs) = ⎡ −E ⎤ Dr (Tr , εs) D0(εs(Tr)) exp⎣⎢ RTa ⎦⎥

Figure 1. (top) Diffusion coefficient vs temperature and (bottom) fluidity (1/η) versus temperature for pure 1-hexanol (circles), 1decanol (diamonds), 3-hexanol (triangles), and 3-decanol (plusses) from 5 to 85 °C.

r

C

(5) DOI: 10.1021/acs.jpcb.6b03573 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B The exponential prefactors, D0(εs), are the same for D and Dr since they share the same value of εs. The prefactors cancel, resulting in a scaled diffusion value. The natural logarithm of the scaled diffusion coefficients yields the compensated Arrhenius equation: ⎛ D(T , εs) ⎞ −Ea 1 E + a ln⎜ ⎟= ε ( , ) D T R T RT ⎝ r r s ⎠ r

(6)

where the Ea is determined from the slope or intercept. The temperature-dependent diffusion data for 1-octanol shown in Figure 2 are scaled to the 45 °C reference curve using the empirical fit shown in the figure. For example, D(25 °C) for 1octanol has εs of 10.1 and is divided by the value Dr(10.1) = 3.64 × 10−10 determined from the equation given in the figure. The natural logarithm of the scaled diffusion data for 1-octanol is presented in Figure 3 (open diamonds). A similar scaling

Figure 4. Compensated Arrhenius plot of fluidity for 1-octanol (open diamonds) with a Tr of 45 °C and 3-octanol (filled circles) with a Tr of 65 °C.

Table 1. CAF Energies of Activation for Pure 1-Alcohols and 3-Alcohols Calculated from Self-Diffusion Coefficient and Fluidity Data and Corresponding Reference Temperature, Tr, Used in the CAF Scaling Procedure 1-alcohol

diffusion

fluidity

Figure 3. Compensated Arrhenius plot of diffusion coefficients for 1octanol (open diamonds) with a Tr of 45 °C and 3-octanol (filled circles) with a Tr of 65 °C.

family member

CAF Ea (kJ mol−1)

Tr (°C)

CAF Ea (kJ mol−1)

Tr (°C)

hexanol heptanol octanol nonanol decanol average hexanol heptanol octanol nonanol decanol average

42.3 ± 0.9 41.6 ± 0.5 42.2 ± 0.5 42.5 ± 0.3 43.0 ± 0.2 42.3 ± 0.5 37.0 ± 0.3 36.6 ± 0.2 36.0 ± 0.4 35.9 ± 0.4 36.9 ± 0.8 36.5 ± 0.5

25 35 45 65 75

41.8 ± 0.2 43.6 ± 0.5 42.7 ± 0.6 44.6 ± 0.8 44.3 ± 0.9 43.4 ± 0.9 34 ± 1 35 ± 1 34 ± 1 33.6 ± 0.8 34 ± 1 34 ± 1

15 35 65 75 85

15 25 45 65 85

15 35 65 75 85

in Figure 2, with the range of εs for Dr approximately spanning the range of D(T) for 1-octanol. Our previous work determined the average Ea for selfdiffusion of 1-alcohols to be 37 ± 1 kJ mol−1.14 The smaller Ea previously determined results from selecting different 1-alcohol family members to comprise the family used in the scaling procedure, which is a trend also observed for conductivity data in acetate electrolyte solutions.15 The earlier alcohol study14 used ethanol, propanol, 1-butanol, 1-hexanol, and 1-octanol, whereas this study uses longer alkyl chain 1-alcohol family members. The average Ea values for self-diffusion given in Table 1 are the same to within the experimental error for both 1- and 3alcohols: 42.3 ± 0.5 and 43.4 ± 0.9 kJ mol−1, respectively. Previous work found the average Ea for self-diffusion of aprotic liquids (e.g., ketones, acetates, and nitriles) to be approximately 25 kJ mol−1.13 The increase in the Ea for the alcohol systems is primarily due to the increased intermolecular interactions brought about by hydrogen bonding. Average Ea values for 1and 3-alcohol fluidity data are 36.5 ± 0.5 and 34 ± 1 kJ mol−1, respectively. Ea values for aprotic systems were only slightly lower for fluidity data compared to self-diffusion data, approximately 22 kJ mol−1 compared to 25 kJ mol−1, respectively.19 A greater difference between the Ea for fluidity and self-diffusion data is observed for systems with hydrogen bonding present and will be discussed in greater detail in the next section.

procedure is followed for 3-octanol, using a reference curve created from isothermal diffusion coefficients of 3-alcohol family members. The resulting scaled diffusion for 3-octanol is also plotted in Figure 3 (open circles). The scaled diffusion coefficient data for 1- and 3-octanol have similar slopes, yielding similar Ea values. The reference temperature used in the scaling procedure for each respective alcohol is given in the figure caption. The same scaling procedure using a reference fluidity, Fr, is applied to temperature-dependent fluidity data that cancels out an exponential prefactor, F0, and results in the compensated Arrhenius equation for fluidity: ⎛ F(T , εs) ⎞ −Ea E ln⎜ + a ⎟= RT RTr ⎝ Fr(Tr , εs) ⎠

3-alcohol

(7)

The CAF as applied to fluidity has been discussed previously.19 An Ea is determined from the slope or the intercept of the compensated Arrhenius plot, shown in Figure 4 for 1- and 3octanol, just as with compensated diffusion plots. All Ea values determined for 1- and 3-alcohol data for selfdiffusion and fluidity are given in Table 1. It is important to note that the choice of reference temperature does not greatly affect Ea.16 To avoid extrapolation, it is best to select a reference curve that spans a εs range matching the data being scaled.16 Here, the values of Tr increase with increasing chain length to avoid extrapolation in the scaling procedure, which can be seen D


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The Journal of Physical Chemistry B The CAF takes into account the additional temperature dependence due to the dielectric constant contained in the exponential prefactor. A simple Arrhenius equation (SAE) assumes the exponential prefactor is temperature independent and determines an Ea from the slope of either ln D(T) or ln F(T) versus T−1. For comparison, Ea values were determined from SAE plots for each alcohol studied and are given in Table 2. The SAE Ea values increase with increasing alkyl chain for the Table 2. Energies of Activation from a Simple Arrhenius Equation for Pure 1-Alcohols and 3-Alcohols Calculated from Self-Diffusion Coefficient and Fluidity Data SAE Ea (kJ mol−1) family member diffusion

fluidity

hexanol heptanol octanol nonanol decanol hexanol heptanol octanol nonanol decanol

1-alcohol 27.7 27.9 29.7 30.3 30.5 22.2 23.6 25.2 26.2 27.5

± ± ± ± ± ± ± ± ± ±

0.4 0.1 0.2 0.3 0.4 0.3 0.5 0.5 0.5 0.6

3-alcohol 35.2 36.0 34.3 36.8 37.3 30 30 29 28.6 30

± ± ± ± ± ± ± ± ± ±

0.5 0.8 0.3 0.8 0.7 1 1 1 0.9 1

Figure 5. (top) Diffusion coefficients vs dielectric constants for 1alcohols (A) and 3-alcohols (B). The temperature-dependent curves are labeled as (6) hexanol, (7) heptanol, (8) octanol, (9) nonanol, and (10) decanol for both 1- and 3- alcohols. (bottom) Exponential prefactors, D0, vs dielectric constant for 1-alcohols (C) and 3-alcohols (D). The symbols correspond to the temperatures as shown. Average Ea from the CAF values used to create the master curves are given.

1-alcohol family for both diffusion and fluidity. Shinomiya determined similar Ea values using Eyring’s equation for viscosity on a family of 1-alcohols; however, the Ea values reported for the 3-aclohols for viscosity are approximately 10 kJ mol−1 higher than those presented in Table 2.23 The 3-alcohol Ea values do not show a distinct increase with increasing alkyl chain length as compared to the 1-alcohols for both diffusion and fluidity. If the intermolecular interactions are the same for members of the solvent family, the Ea values should be approximately the same as well. The SAE Ea values are also closer in magnitude to the CAF Ea values for the 3-alcohols as compared to the 1-alcohols. To further discuss the reasons for this similarity, and the differences that arise between diffusion and fluidity data for these systems, the exponential prefactor must be determined. After the compensation (scaling) procedure is performed, the exponential prefactors are calculated by dividing D(T) or F(T) by the Boltzmann factor, exp[−Ea/RT], using the average Ea obtained from the compensated Arrhenius plots. Figure 5 illustrates the dielectric constant dependence of D(T) and D0(T) for 1-alcohols (left) and 3-alcohols (right). The top plots of Figure 5 show the individual reference temperature curves that separate into the temperature-dependent diffusion coefficients for each family member. The curves are numbered according to the number of carbon atoms in the alkyl chain for a particular family member (e.g., (6) corresponds to either 1hexanol or 3-hexanol). Upon dividing the diffusion coefficient data in (A) and (B) by the Boltzmann factor with the appropriate average Ea value (given in the figure), the data collapse to form a master curve, D0(T) versus εs, for each alcohol family as shown in plots (C) and (D). The formation of a master curve supports one of the primary postulates of the CAF, that the temperature dependence of the exponential prefactor originates in the inherent temperature dependence of the dielectric constant, i.e., D0(εs(T)). Note plots (A) and (B) are shown on the same scale for comparison, as are plots (C) and (D).

A major difference between the D0(εs(T)) for 1-alcohols, plot (C), and 3-alcohols, plot (D), of Figure 5 is the shape of the master curves. The curvature of the master curve for the 1alcohols follows a similar exponential growth trend as seen for aprotic solvents studied to date.15 The curve formed from the 3-alcohol data separates at lower temperatures (higher dielectric constant) and does not follow an exponential growth function previously seen using the CAF. The values of D0 are higher for the 3-alcohol data than the 1-alcohol data; this difference is the dominant contribution in Figure 1. The difference in the curvature of the two master curves originates in the different temperature dependences of εs for the two alcohol families, and will be discussed in the next section. In a format similar to Figure 5, Figure 6 displays the fluidity data (top) and fluidity exponential prefactors (bottom) for 1alcohols (left) and 3-alcohols (right) versus εs. The fluidity data for the 1- and 3-alcohols resemble the trends observed in the top plots of Figure 5. Plots (C) and (D) show the exponential prefactors, F0(T), on the same scale for comparison. The 1alcohol F0 data follow a similar exponential growth trend with εs, as the exponential prefactor (D0) data for self-diffusion seen in Figure 5C. However, the 3-alcohol F0 data also show deviations at higher εs, similar to D0 for diffusion shown in Figure 5D. It is clear that an analysis based on the CAF reveals multiple temperature-dependent factors in the transport property. To further discuss the CAF results, the next section will explore the temperature dependence contained within εs. Temperature Dependence of the Dielectric Constant. Figure 7 shows temperature-dependent dielectric constants for three members of the 1-alcohol family (top) and the 3-alcohol family (bottom) from 5 to 85 °C. Over this temperature range, ∂εs/∂T for 1-alcohol data is relatively constant and negative. The dashed lines show a linear best fit, with R2 equal to 0.997, 0.993, and 0.988 for 1-hexanol, 1-octanol, and 1-decanol, E


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longer alkyl chain 3-alcohol family members causes the concave downward shape observed in the master curve in Figures 5 and 6, particularly at higher temperatures (lower εs). The limited temperature range of our εs data was extended using literature values and plotted in Figure 8 for 1-octanol25

Figure 6. (top) Fluidity vs dielectric constants for 1-alcohols (A) and 3-alcohols (B). The temperature-dependent curves are labeled as (6) hexanol, (7) heptanol, (8) octanol, (9) nonanol, and (10) decanol for both 1- and 3- alcohols. (bottom) Exponential prefactors, F0, vs dielectric constant for 1-alcohols (C) and 3-alcohols (D).

Figure 8. Dielectric constant versus temperature for 1-octanol (circles) and 3-octanol (crosses). Our 1-octanol data are fitted to a linear trend line, while our 3-octanol data are connected as a guide to the eye, with the vertical lines marking our measured temperature range.

(blue circles) and 3-octanol45 (green crosses). Data from this work are also shown for 1-octanol (open circles, with linear best fit line) and 3-octanol (open crosses, connected by line as a guide to the eye), with vertical dotted lines representing the measured temperature range. Our εs 1-octanol data are linear over the measured temperature range, but when extended to higher temperatures a change in slope is observed at approximately 100 °C. Our 3-octanol data show marked curvature (open crosses), while the εs literature data (green crosses) show a linear trend with decreasing temperature below 0 °C. Within our measured temperature range, the ∂εs/∂T for 3-octanol is not constant. The CAF utilizes the inherent temperature dependence of εs in the exponential prefactor to model fluidity and diffusion. It is therefore important to discuss the CAF results in terms of the variation in ∂εs/∂T for these monohydroxy alcohols. Onsager’s equation for εs (eq 4) does not account for extended association, as observed for monohydroxy alcohols.36 The Kirkwood−Frölich model for εs (eq 8) is similar to Onsager’s model but includes a factor, gk, to account for extended molecular association.36,46−48

Figure 7. Dielectric constant vs temperature for (top) 1-hexanol, 1octanol, and 1-decanol and (bottom) 3-hexanol, 3-octanol, and 3decanol. The dashed lines are linear best fit lines.

3εs(ε∞ + 2) εs − 1 ε −1 4πN gμ2 − ∞ = (2εs + ε∞)(εs + 2) 9kBT k 0 εs + 2 ε∞ + 2

respectively. As the alkyl chain length increases, the dependence of the dielectric constant on temperature deviates only slightly from linearity for the 1-alcohols. The dielectric constants for 1-heptanol and 1-nonanol also show a linear decrease with temperature (data not shown). The dielectric constants for the 3-alcohols, however, exhibit a nonlinear ∂εs/∂T compared to the 1-alcohol data shown in the lower plot of Figure 7. The shortest alkyl chain member (3hexanol) has marked curvature from 5 to 85 °C with an R2 of 0.955. As the alkyl chain becomes longer, ∂εs/∂T decreases in magnitude. The R2 values are 0.915 for 3-octanol and 0.911 for 3-decanol. The decreased temperature dependence of εs for the

(8)

Here the variables are the same as in eq 4. For nonassociating liquids, gk is unity.36 It is accepted that gk greater than unity corresponds to molecules with dipoles oriented parallel to neighboring dipoles, while gk less than unity molecules favor antiparallel orientation.25,36,47 Studies show that gk varies with both temperature and alkyl chain length.4,8,9,26,49 Figure 9 shows gkμ02 values determined from eq 8 for several members of the 1- and 3-alcohol family versus temperature. gkμ02, termed the apparent dipole moment by Campbell et al.,24 is used to provide a better comparison of the short and long chain alcohol, since μ0 can vary depending on the solvent in which it is determined.24 The refractive index values used are F


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N/T as shown in eq 4, and the resulting Ea values are the same.13 This modified CAF also elucidates any additional temperature dependence not due to the N/T factor that contributes to the exponential prefactor in associated liquids, namely, that due to gk. The CAF described in eqs 9a−9c was applied to 1- and 3alcohol diffusion coefficients and fluidity data. Average CAF Ea values are given in Table 3. The Ea values for each member and the reference temperature used in the scaling procedure are included in Table S1 of the Supporting Information. Table 3. Average Ea Values of Diffusion Coefficients and Fluidity for 1-Alcohols and 3-Alcohols Values Determined Using the CAF Scaling Procedure with N(T)/T as Shown in Eq 9c Ea (kJ mol−1) diffusion fluidity

1-alcohol

3-alcohol

35.9 ± 0.5 32.9 ± 0.2

43.5 ± 0.4 35 ± 1

Figure 10 shows the exponential prefactors D0 determined using the modified CAF for diffusion of the 1-alcohol family

Figure 9. Kirkwood−Frö lich correlation factor, gk, times the permanent dipole moment μ02 versus temperature for (top) 1-alcohols as labeled and (bottom) 3-alcohols as labeled. The solid lines are given as guides to the eye.

1.43 for 1-alcohols44 and 1.42 for 3-alcohols.44 The similarity of ∂gkμ02/∂T in Figure 9 and ∂εs/∂T in Figure 7 suggests that the nonlinear temperature dependence of εs for the 3-alcohols originates in the structure factor, gk. A plot of N/T versus T for the 3-alcohol family shows a linear dependence (data not shown), which is the only other term in eq 8 that could contain a temperature dependence. Compensated Arrhenius Formalism Using N/T. Previous work has shown a modified CAF can be used to model diffusion, conductivity, and fluidity data for aprotic liquids and electrolyte solutions, if N/T is used to represent the temperature dependence in the exponential prefactor.11,13,19 The modified CAF scaling procedure using N/T for diffusion is ⎛ N (T ) ⎞ ⎛ N (T ) ⎞ ⎡ −E ⎤ D ⎜T , ⎟ = D0⎜ ⎟ exp⎢ a ⎥ ⎣ RT ⎦ ⎝ ⎝ T ⎠ T ⎠

(

D T,

N (T ) T

(

N Tr

Dr Tr ,

) = D ( )e 0

) = D ( )e N 0 T r

⎛ N (T ) ⎜ D T, T ln⎜ ⎜ Dr Tr , N Tr ⎝

(

(

N (T ) T

−Ea / RT

−Ea / RTr

(9b)

Figure 10. Diffusion exponential prefactors, D0, versus dipole density divided by temperature, N/T, for 1-alcohols (top) and 3-alcohols (bottom). Average Ea values correspond to those given in Table 3. D0 was determined using modified CAF scaling procedure described in the text.

⎞

) ⎟ = −E

)

(9a)

⎟ ⎟ ⎠

a

RT

+

Ea RTr (9c)

(top) and 3-alcohol family (bottom). Scaling to εs takes into account all of the temperature dependence included in both N/ T and gk for associated liquids. The master curves formed in Figure 10 demonstrate the contribution of the temperature dependence of N/T to the exponential prefactor excluding any T dependence due to gk. D0 values are only reduced in magnitude for the 1-alcohol family compared to Figure 5, but a master curve still forms. This corresponds to N/T having the same temperature dependence as εs, i.e., a linear dependence with temperature. In addition, the value of the Ea is

Values for N are calculated by dividing the density by the molecular weight. Equation 9a is the rewritten Arrhenius-like form of the diffusion coefficient replacing εs with N(T)/T. Equation 9b demonstrates the scaling step such that D0

( ) = D ( ) and therefore cancel. Equation 9c is the N (T ) T

N 0 T r

resulting CAF equation using N(T)/T in the scaling procedure. An Ea is determined using eq 9c, and the exponential prefactor is calulated using the average Ea in eq 9a. Transport data can be scaled using eq 9b if the temperature dependence of εs is due to G


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The Journal of Physical Chemistry B approximately 10 kJ mol−1 less that the Ea determined using the εs because not all of the temperature dependence is compensated when scaling to N/T for 1-alcohols. This value is closer to the values of Ea determined for diffusion from the simple Arrhenius equation (Table 2) which assumes no temperature dependence in D0. Figures 7 and 9 show εs(T) is nonlinear because gk(T) is nonlinear. When D(T) for 3-alcohols is scaled with N/T which does not contain the gk(T) dependence, a master curve forms (bottom plot of Figure 10). A gk closer to unity represents molecules in which specific directional correlations of the dipoles are negligible;47 thus, N/T becomes the dominant contributor of the temperature dependence contained in D0(εs). Longer alkyl chain members of the 3-alcohol family show gk approaching unity assuming a μ0 ≈ 1.68 D.36 A decreased temperature dependence in gk (or almost negligible for 3-decanol) will reduce the contribution of the dipole orientation to the functional dependence of D0. Therefore, any remaining temperature dependence must originate from N/T. The formation of a master curve in the lower plot of Figure 10 supports this claim. It is known that the extent of hydrogen bonding is reduced upon relocating the hydroxyl group to an interior carbon.21,31,50 The reduced temperature dependence of gk, as well as the reduced magnitude, could be a result of a reduction in hydrogen bonding or another change in the liquid structure due to the steric hindrance introduced from the hydroxyl relocation that is not greatly affected by temperature. The Ea determined for the 3-alcohols using N/T in the scaling procedure is similar to the average Ea found by compensating for ε(T). Previously, the CAF was applied to temperature-dependent diffusion and conductivity for aprotic liquids and electrolytes13 scaling the data with both εs and N/T. Both scaling procedures yielded the same Ea and exponential prefactor values because aprotic molecules do not have strong preferential alignment of dipoles (the gk values are close to unity), and Onsager’s model applies.13 In associating liquids, εs(T) has three contributing factors: N(T), gk(T), and T. For the 3-alcohols to have a similar Ea from both scaling procedures (using εs or N/T), the contribution of gk(T) to the prefactor must be small, as observed for aprotic liquids. The magnitudes of both diffusion master curves are also similar (Figures 5D and 10), which further supports that the gk(T) contribution is small in D0 but non-negligible. Figure 11 shows F0 for 1-alcohols (top) and 3-alcohols (bottom) versus N/T. A master curve is formed with slight deviations for the 1-alcohol data, and the magnitude is lower than the master curve shown in Figure 6C. F0 versus N/T for the 3-alcohols displays a significant deviation from a master curve. Note the scales are different for the top and bottom plot. The deviation from a master curve displayed in Figure 6D suggests that the temperature dependence of F0 is not the same for each member of the 3-alcohol family. The magnitude of F0 in Figure 11 is also an order of magnitude smaller than F0 determined from εs (Figure 6D), suggesting that gk(T) contributes a significant amount to the functional dependence of F0(T). The variation in gk(T) for each member of the 3alcohol family could also explain the deviations observed in both fluidity master curves. These results indicate that although the temperature dependences of self-diffusion and fluidity presented in Figure 1 appear similar, the contributing temperature-dependent factors are quite different.

Figure 11. Fluidity exponential prefactors, F0, versus dipole density divided by temperature, N/T, for 1-alcohols (top) and 3-alcohols (bottom). Average Ea values correspond to those given in Table 3. F0 was determined using modified CAF scaling procedure described in the text.

■

SUMMARY AND CONCLUSION Temperature-dependent properties (self-diffusion coefficients, fluidity, dielectric constants, and densities) are compared for two monohydroxy alcohol families: 1-alcohols and 3-alcohols. The compensated Arrhenius formalism is applied to the transport properties of both systems using the dielectric constant, εs. Self-diffusion coefficient Ea values for both 1and 3-alcohols (42.3 ± 0.5 and 43.4 ± 0.9 kJ mol−1, respectively) and fluidity Ea values (36.5 ± 0.5 and 34 ± 1 kJ mol−1, respectively) are presented. A lower fluidity Ea compared to diffusion Ea values could be due to the molecules involved in viscous flow moving as a cluster as opposed to a single molecule. A smaller Ea is then expected because less energy is required to break the hydrogen bonds. Both 1- and 3-alcohol Ea values are higher than Ea values of the aprotic systems studied (≈25 kJ mol−1).15 The increased association between molecules in the alcohol systems increases the energy needed to disrupt the intermolecular interactions so that transport can occur, whereas the aprotic systems are considered nonassociated; thereby a lower Ea is determined for transport. The CAF assumes a temperature dependence in the exponential prefactor due to the temperature dependence contained in εs. The temperature dependence of εs originates from the dipole density factor, N(T)/T, and the Kirkwood− Frölich correlation factor, gk(T), for associating liquids,47 which is affected by the location of the hydroxyl group along the alkyl chain.4,8,26 The CAF exponential prefactors scaling with εs and N/T were determined for both diffusion and fluidity of 1- and 3-alcohols. Scaling with N/T does not include the temperature dependence of gk, allowing for further analysis of the origin of the temperature dependence contained in the exponential prefactor. The dependence of the resulting prefactors D0 and F0 on N/T revealed that the temperature dependence of D0 and F0 due to gk is nontrivial. It is unclear why the temperature dependence of the exponential prefactors D0 and F0 for 3H


Article


mol−1 K−1.23 Vij et al.9 measured dielectric relaxation times and showed a similar increase for the ΔS†τ from 31.8 to 129.7J mol−1 K−1 for 1-heptanol and 3-heptanol, respectively. A recent study by Mesele et al.52 used an extended jump model of hydroxyl reorientation using molecular dynamics simulations on isomers of butanol and found ΔS†τ to be larger for sterically hindered butanols. As mentioned in the Introduction, the CAF successfully models dielectric relaxation times via the relaxation rate constant. CAF Ea values for dielectric relaxation rate constants of 1-alcohols were found to be 38.6 kJ mol−1,35 which are approximately the same Ea determined for fluidity in this work. This work has shown that the CAF offers a means to uncouple the differences observed in these transport properties, in particular the nontrivial temperature dependencies in the exponential prefactors, that would otherwise remain unnoticed.

alcohols varies dramatically, but the variation in εs(T) for each member of the 3-alcohol family could play a role. So what information is gained from the nature of the exponential prefactor’s temperature dependence? Frech and Petrowsky11 developed the theoretical basis for the CAF in terms of diffusion for aprotic liquids. The CAF adapts Eyring’s transition state theory by assuming a temperature dependence in the exponential prefactor based on thermal expansion of the liquid. They determined that the expression for the exponential prefactor is ⎛ ΔS′ † ⎞ D0 = λ 2νt exp[Bεs] exp⎜ ⎟ ⎝ R ⎠

(10)

where λ is the average distance traversed by molecule t during a transition, νt is the translatory vibration frequency of the t molecule in the initial state, ΔS′† is the entropy of activation, B is a constant, and R is the gas constant.11 The Gibbs free energy of activation is given by ΔG′†. The CAF claims that ΔG′† = ΔG† − W, where the polarization energy W represents the energy of polarizing the reaction field as described by Böttcher.47 W has both a temperature and alkyl chain length dependence.11 The CAF incorporates the energy necessary to polarize the local environment of the target molecule in the activation energy, Ea.11 It follows that ΔG′† = ΔH′† − TΔS′†, where ΔH′† is the Ea determined via the CAF. For aprotic liquids, Frech assumed that the constant, B, incorporated the negligible temperature dependence of ΔS′† over the measured temperature rangea satisfactory approximation for nonassociating liquids. Johari51 used Frölich’s model for the entropy of a dielectric material, S,46 as ΔSconfig

ε ∂ε = S − S0(T ) = 0 s E2 2 ∂T

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b03573. Temperature-dependent self-diffusion coefficients, fluidity, density, and dielectric constants for all systems studied; a table of activation energies determined using the modified CAF scaling procedure with N/T (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: allison.m.fl[email protected] (A.M.F.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the Army Research Office for support of this work through Grant No. W911NF-10-1-0437 at OU. We thank the National Science Foundation for funding the NMR equipment (Grant CHE #0639199) and Susan Nimmo for her assistance with NMR diffusion experiments. We thank Matthew Johnson, Jeremy D. Jernigen, R. S. P. Bokalawela, and Chris Crowe in the OU Physics Department. We thank Lawrence University Excellence in Science Fund for funding of G.F.

(11)

where ΔSconfig is the change in the configurational entropy, S0(T) is the entropy in the absence of the field, ε0 is the permittivity of free space, and E is the electric field. We write S0 as S0(T) to emphasize the temperature-dependent component introduced by Frö lich.46 Johari51 does not include the temperature dependence of S0, as his work involved isothermal ∂ε studies. The sign and magnitude of s contributes to the ∂T quantity ΔSconfig. It is possible that ΔSconfig and ΔS′† proposed by Frech and Petrowsky11 are related. If we combine eqs 10 and ∂ε ∂ε 11, the s factor appears in the exponential. The s behavior ∂T ∂T varies with alkyl chain length within the 3-alcohol solvent family and could manifest in the temperature dependence of the configurational entropy. Deviations in the exponential prefactor as a function of ε would result, as demonstrated in Figures 5D and 6D for the 3-alcohols. Relocating the hydroxyl group on the alkyl chain does affect the temperature dependence in εs. Further studies into the affects on ΔSconfig are necessary to validate the claims made in the previous paragraph. Thermodynamic activation energies for viscous flow (η) and dielectric relaxation times (τ) were found by Shinomiya23 using Eyring’s equation for several linearalcohol families with different locations of the hydroxyl group. The entropy of activation increased for both relaxation and viscosity upon relocating the hydroxyl group from a terminal to an interior carbon. The entropy of activation for relaxation, ΔS†τ , increased from 41.2 to 117 J mol−1 K−1 for 1-octanol and 3-octanol, respectively, while ΔS†η increased from 18.4 to 49.4 J

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REFERENCES

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