Dynamic Solvophobic Effect and Its Cooperativity in the Hydrogen

J. Phys. Chem. B 2008, 112, 16633–16641

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Dynamic Solvophobic Effect and Its Cooperativity in the Hydrogen-bonding Liquids Studied by Dielectric and Nuclear Magnetic Resonance Relaxation Tsuyoshi Yamaguchi,* Hiroki Furuhashi, Tatsuro Matsuoka, and Shinobu Koda Department of Molecular Design and Engineering, Graduate School of Engineering, Nagoya UniVersity, Furo-cho B2-3(611), Chikusa, Nagoya, Aichi 464-8603, Japan ReceiVed: August 20, 2008; ReVised Manuscript ReceiVed: October 20, 2008

The reorientational relaxation of solvent molecules in the mixture of nonpolar solutes and hydrogen-bonding liquids including water, alcohols, and amides are studied by dielectric and 2H-nuclear magnetic resonance (NMR) spin-lattice relaxations. The retardation of the reorientational motion of the solvent by weak solute-solvent interaction is observed in all the solvent systems. On the other hand, no clear correlation between the strength of the solute-solvent interaction and the slowing down of the solvent motion is found in N,N-dimethylacetamide, which suggests the importance of the hydrogen bonding in the dynamic solvophobic effect. The cooperativity of the reorientational relaxation is investigated by the comparison between the collective relaxation measured by the dielectric spectroscopy and the single-molecular reorientation determined by NMR. The modification of the dielectric relaxation time caused by the dissolution of the solute is larger than that of the single-molecular reorientational relaxation time in all the solvents studied here. The effect of the static correlation between the dipole moments of different molecules is calculated from the static dielectric constant, and the effect of the dynamic correlation is estimated. The difference in the effects of the solutes on the collective and single-molecular reorientational relaxation is mainly ascribed to the dynamic cooperativity in the cases of water and alcohols, which is consistent with the picture on the dynamic solvophobicity derived by our previous theoretical analysis (Yamaguchi, T.; Matsuoka, T.; Koda, S. J. Chem. Phys. 2004, 120, 7590). On the other hand, the static correlation plays the principal role in the case of N-methylformamide. 1. Introduction The dynamics of solvent in solutions is one of the important probes to investigate the solvation of a solute. The motion of solvent molecules in the solvation shell of the solute is influenced by the solute-solvent interaction. In addition, since the structure of the solvent around the solute is modified by the presence of the solute, the solvent-solvent interaction works differently in the solvation shell. The dynamics of solvent molecules in the vicinity of the solute thus reflects the solute-solvent interaction and the structure of the solvation shell. The traditional models on the hydration of ions owe much to the experiments on the dynamics of water in the aqueous electrolyte solutions.1 Samoilov proposed the concepts of the positive and negative hydrations based on the effects of ions on the activation energy of the self-diffusion of water.2 Frank and Wen referred to the viscosity B-coefficients of ions in water in the construction of their hydration shell model of ions.3 The hydrophobic hydration is another example for which the dynamics of solvent molecules in solution is intensively investigated.4-7 Experimentally, the dissolution of hydrophobic solutes into water is known to reduce the mobility of water, which appears to contradict the weak interaction between the solute and the solvent. The effect of the hydrophobic molecules on the dynamics of water has historically been ascribed to the formation of the ice-like structure called “iceberg”. The iceberg model on the hydrophobic hydration was originally proposed by Frank and Evans in order to explain the thermodynamic properties of hydrophobic hydration, that is, the * E-mail: [email protected]

dissolution of the hydrophobic molecules into water accompanies both enthalpic and entropic losses.8 Although the thermodynamics of the hydrophobic hydration does not necessarily require the formation of the iceberg structure,9 the iceberg model has been employed in the explanation of the dynamics of hydrophobic hydration.10 Because the iceberg model of the hydrophobic hydration is based on the tetrahedral network structure specific to water, its explanation is limited to aqueous systems. However, there are many experimental studies demonstrating that the retardation of the mobility of solvent molecules by solvophobic solutes is not limited to aqueous solutions.11-16 Recently, we measured the dielectric relaxation spectra of the mixtures of alcohols and nonpolar solute molecules.16 Our study showed that the solutes of weak solute-solvent interaction make the dielectric relaxation time larger and that the solutes possessing the hydrogen-bond accepting sites enhances the dielectric relaxation rate, just as is observed in aqueous systems. This work extends our previous one to liquid amides, which belong to another class of the hydrogen-bonding liquids. Because the hydrogen-bonding structure of liquid amides is different from those of water and alcohols, it is interesting to see whether the solvophobic slowing down is present in liquid amide systems. The dielectric relaxation spectroscopy and nuclear magnetic resonance (NMR) spin-lattice relaxation time (T1) measurements are two of the popular experimental methods to investigate the reorientational relaxation in solution. Both methods have been applied to study the reorientational relaxation of polar solvents in solution, and they have yielded qualitatively similar results. However, we consider that the study on the quantitative comparison between them is not satisfactory so far. The

10.1021/jp807473c CCC: $40.75  2008 American Chemical Society Published on Web 12/04/2008

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dielectric relaxation probes the relaxation of the collective dipole moment, whereas the NMR relaxation reflects the singlemolecule reorientation. Therefore, the difference between them can be related to the cooperativity of the reorientational relaxation. There are two sources of the cooperativity in the reorientational relaxation of liquids, as will be described in the next section.17,18 One is the static correlation between the orientations of different molecules, and the other is the dynamic correlation between the random forces acting on different molecules. In this work, we perform both the dielectric and NMR relaxation experiments on the same series of solutions, and the effects of solutes on the collective and single-molecular reorientational relaxations are compared. The contribution of the static correlation is determined from the concentration dependence of the static dielectric constant, and the effect of the dynamic correlation is estimated. It is found that the dynamic correlation plays a substantial role in the dynamic solvophobicity, which is discussed in terms of the theoretical analysis of the dynamics of the solvophobic solvation.19 2. Theory We shall present in this section the theoretical bases on the way how to compare the reorientational relaxation times obtained by the dielectric and NMR relaxation experiments. 2.1. Dielectric Relaxation. According to the linear response theory, the frequency-dependent complex dielectric spectrum, ε(ν), can be described theoretically in terms of the timecorrelation function of the collective dipole moment of the system. Because of the difference between the external and local electric field, however, the theoretical description is actually dependent on the electrostatic boundary condition, that is, the dielectric constant at the infinite distance. The boundary condition that has traditionally been used in the studies on the dielectric properties of liquids is that the dielectric constant at the infinite distance is the same as that of the bulk liquid. Although this boundary condition appears reasonable intuitively, the theoretical formulation is rather complicated due to the local field correction. In this work, we shall employ the conductive boundary condition, that is, the dielectric constant at the infinite distance is infinity. As we shall show later, the use of the conductive boundary condition makes the theoretical formulation simpler due to the lack of the local field correction, and the final relationship between the collective and single-molecular reorientational relaxation times does not depend on the boundary condition. In the case of the conductive boundary condition, ε(ν) is given by eq 1,20

ε(ν) - ε∞ )

F C (0) + 2πiν 3ε0kBT ε

[

∫0∞ dtCε(t)e2πiνt]

1 Cε(t) ≡ 〈P(0) · P(t)〉 N

(2)

(3)

i

where N stands for the total number of the polar molecules, i is the index for molecules, and µi denotes the dipole moment of the molecule i. The time-integrated relaxation time of Cε(t), denoted as τD, is defined by

τD ≡

1 Cε(0)

∫0∞ Cε(t) dt

(4)

According to eq 1, the static dielectric constant, ε(0), is related to the initial value of Cε(t) as

ε(0) - ε(∞) )

FCε(0) 3ε0 kBT

(5)

2.2. NMR Spin-lattice Relaxation. Because deuterium is the quadrupolar nucleus that possesses spin I ) 1, its spin-lattice relaxation is dominated by the quadrupolar coupling mechanism, that is, the interaction between the quadrupole moment of the nucleus and the electric field gradient at the nucleus produced by the electrons of the molecule. In the fast modulation limit where the reorientational relaxation of the molecule is sufficiently faster than the Larmor frequency of the nucleus, the quadrupolar coupling mechanism relates the single-molecular reorientational correlation time τ2R to the spin-lattice relaxation time T1 as13

1 3π 2 2 ) χ τ T1 2 D 2R

(6)

where χD stands for the quadrupolar coupling constant (QCC). We have assumed here that the electric field gradient at the deuterium nucleus has the uniaxial symmetry along the O-D or N-D bond. The rank-2 reorientational correlation time, τ2R, is given by

τ2R ≡ C2(t) ≡

1 N

∑ i

〈

∫0∞ C2(t) dt 3[ui(0) · ui(t)]2 - 1 2

(7)

〉

(8)

where ui denotes the unit vector parallel to the O-D or N-D bond of the molecule i. When the reorientational relaxation is diffusive, τ2R is related to the rank-1 reorientational relaxation time, τ1R, as follows:

(1)

where ν, F, ε0, kB, and T stand for the frequency, number density of the polar molecules, dielectric constant of the vacuum, Boltzmann constant, and absolute temperature, respectively. Cε(t) denotes the time-correlation function of the collective dipole moment of the system defined as

∑ µi(t)

P(t) ≡

τ1R ) 3τ2R τ1R ≡ C1(t) ≡

1 N

(9)

∫0∞ C1(t) dt

(10)

∑ 〈ui(0) · ui(t)〉

(11)

i

2.3. Reorientational Cooperativity Analyzed by the Generalized Langevin Formalism. Regarding the collective dipole moment and its time derivative as the slow variables, the time

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J. Phys. Chem. B, Vol. 112, No. 51, 2008 16635

dependence of Cε(t) is described by the generalized Langevin equation as C¨ε(t) -

C¨ε(0) 1 C (t) Cε(0) ε C¨ε(0)

∫

t

0

dτ

well-known Kirkwood g-factor, denoted as gK, is related to g∞ by eq 20.

〈P¨ · eiQLQτP¨〉 C˙ε(t - τ) ) 0 N

g∞ 3ε(0) ) gK 2ε(0) + 1

(20)

(12) Here, L stands for the Liouville operator, Q is defined as 1 P, and P denotes the projection operator onto P and P˙. The relaxation time τD is then given by eq 13

τD )

Cε(0) [C¨ε(0)]2

∫0

∞

g∞ )

〈P¨ · eiQLQτP¨〉 dτ N

1 N

∑ 〈µi(0) · µi(t)〉

(14)

∫0t dτ

∑ 〈µ¨ i · e

µ¨ i〉

iQLQτ

i

N

∫0∞ dτ

∑ 〈µ¨ i · eiQLQτµ¨ i〉 i

N

(16)

Comparing eqs 13 and 16, the relationship between the collective and single-molecular dipole moments is derived as

τD ) g∞ g
τµ

Cε(0) g∞ ≡ ) Cµ(0)

〈|

(17)

∑ µi|2〉 i

∑ 〈|µi|2〉

(18)

i




g ≡

(21)

∫0∞ dτ〈P¨ · eiQLQτP¨〉 ∞ ∑ ∫0 dτ〈µ¨ i · eiQLQτµ¨ i〉

3ε0 kBT Fµ

2

[ε(0) - ε(∞)]g
τµ

(22)

×

The rank-1 reorientational relaxation time of the singlemolecular dipole moment, τµ, is given by

[C¨µ(0)]2

[ε(0) - ε(∞)]

where µ ) |µ| stands for the dipole moment of the polar molecule. The dynamic contribution, g′, can be estimated from eqs 17 and 21. The substitution of eq 21 into 17 leads to the relationship between the collective and single-molecular reorientational relaxation time as

τD )

C˙µ(t - τ) ) 0 (15)

τµ )

Fµ2

i

C¨µ(0) 1 C¨µ(t) C (t) Cµ(0) µ ¨Cµ(0)

Cµ(0)

3ε0 kBT

(13)

The reorientation of the single-molecular dipole moment is treated in the similar way as

Cµ(t) ≡

The static factor, g∞, is then related to the static dielectric constant by eq 5 as

(19)

i

Here, C¨ε(0) is equated to C¨µ(0) because the linear and angular momenta of different molecules are not correlated at the same time. Equation 17 states that the difference between the collective and single-molecular reorientational relaxation times is determined by two factors, g∞ and g′. The former is the static correlation between the dipole moments of different molecules, and the latter is the dynamic correlation between the random accelerations of the dipole moments of different molecules. The

It is to be noted here that eq 22 is similar to that derived by Madden and Kivelson (eqs 6-21 of ref 17) particularly in the effect of the static dielectric constant. Our discussion above is based on the conductive electrostatic boundary condition, whereas the different boundary condition is employed in their derivation. The different electrostatic boundary condition leads to the different expression of the complex dielectric spectrum. The boundary condition also affects the static correlation between dipole moments of different molecules. However, the final relationship between τD and τµ is independent of the electrostatic boundary condition. In the present work, ε(0) and τD are obtained directly by the dielectric relaxation spectroscopy. Because τD is defined as the integration of the time correlation function in eq 4, it is regarded as the weighted average of the relaxation times in the case of multistep relaxations. The reorientational correlation time of the single-molecular dipole moment, τµ, is approximated by that of the O-D or N-D vector, τ1R, neglecting the anisotropy of the reorientational relaxation, and τ1R is then related to T1 of 2 H NMR by eqs 6 and 9. The high-frequency dielectric constant, ε(∞), has to include the contributions of all the processes faster than the reorientational degree of freedom. It therefore includes the electronic and vibrational polarizations, but not the librational modes. In this work, the refractive indices of the neat solvents are used to calculate ε(∞), neglecting the effect of the vibrational polarization and the concentration dependence of ε(∞). Since ε(0) is much larger than ε(∞) in the cases of all the mixtures studied here, we consider that the error in the estimation of ε(∞) does not affect the discussion. The relaxation times in the mixture are compared with those of the corresponding neat solvents in this work. According to eqs 9 and 17, the modifications of the reorientational relaxation times by the dissolution of the solutes are related to each other by eq 23.

τD τD,neat

)

g



g∞

τ2R

g neat g∞,neat τ2R,neat

(23)

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The ratio of g∞ in the second factor of eq 23 is evaluated by eq 21 as

g∞ Fneat [ε(0) - ε(∞)] ) · g∞,neat F [εneat(0) - ε(∞)]

(24)

Here, the dipole moment of a molecule is assumed to be independent of the concentration of the solute. The difference between the concentration dependence of τD/τD,neat and (g∞τ2R)/ (g∞τ2R)neat is then interpreted as the effect of the dynamic correlation on the dynamic solvophobicity. 3. Experimental Section 3.1. Samples. The solvents used for the dielectric spectroscopy are water, methanol (spectroscopic grade, Kishida), ethanol (spectroscopic grade, Nacalai), formamide (FA, 99.5%, Aldrich), N-methylformamide (NMF, 99%, TCI), and N,N-dimethylacetamide (DMA, anhydrous, 99.8%, Aldrich). FA is distilled once under the reduced pressure and dried by molecular sieves 3 Å (Wako). Methanol, ethanol, and NMF are dried by molecular sieves 3 Å prior to use. DMA is used as received. The solvents employed for the NMR T1 experiments are D2O, d4-methanol, d1-ethanol (C2H5OD), and d1-NMF (HCONDCH3). D2O (99%d, CEA) is used as received. d4-Methanol (99.8%d) and d1-ethanol (99%d) are purchased from Cambridge Isotope Laboratories and dried by molecular sieves 3 Å prior to use. The isotope substitution of the N-H proton of NMF is performed as follows: the mixture of NMF and D2O, whose molar ratio is 2:1, is first distilled under the reduced pressure and then dried by molecular sieves 3 Å. The yield of the deuterium substitution is about 50%, which is confirmed by 1H NMR spectrum. The solute molecules are cyclohexane (spectroscopic grade, Wako), carbon tetrachloride (∞pure, Wako), benzene (spectroscopic grade, Wako), 1,4-dioxane (∞pure, Wako), and pyrazine (99+%, Aldrich). The nondipolar solutes are chosen so that the reorientational relaxation of the solutes is not coupled to the dielectric response. The sizes of the solutes are similar to each other, and the accepter strength of the hydrogen bond is varied systematically in order to investigate the effect of the strength of the solute-solvent interaction on the reorientational relaxation of the solvents. According to the chemical intuition on the hydrogen-bonding acceptor strength, the strength of the solute-solvent interaction changes as cyclohexane < carbon tetrachloride < benzene < 1,4-dioxane < pyrazine. Although we could not find a measure to quantify the hydrogen-bonding accepter strength, the higher accepter strength of 1,4-dioxane is apparent as the higher value of the Gutmann’s donor number,21 for instance. All the solutes are used without further purification. The molar fraction of the solute is varied from 0 to 0.10. 3.2. Dielectric Relaxation. The dielectric relaxation spectra are measured with the frequency-domain microwave reflection method, using a vector network analyzer (HP8720D, HewlettPackard) and a dielectric probe kit (HP85070B, Hewlett-Packard).16,22 The measured frequency range is from 200 MHz to 20 GHz. The reference samples for mixtures are air, water, and neat solvent. In the cases of aqueous solutions, air, water, and methanol are used for the reference sample. The dielectric spectra of the reference liquids are taken from literatures.23,24 The temperature of the sample is controlled at 25.0 ( 0.1 °C by flowing the thermostatted water through the water jacket around the sample cell.

Figure 1. The dielectric spectrum of the NMF solution of carbon tetrachloride. The mole fraction of the solute is 0.06. Both the real and imaginary parts are plotted. The heavy lines show the experimental values (resolution of individual points is not possible at this scale), and the thin lines exhibit the fitting by the Debye function.

Figure 1 shows the typical example of the spectra. The spectra are then fitted into the sum of the Debye functions as N

ε(ν) )

∆ε

j + ε(νHF) ∑ 1 - 2πiντ j

(25)

j)1

and the dielectric relaxation time, τD, is determined as

τD ≡

∑ ∆εjτj j

∑ ∆εj

(26)

j

The number of the Debye relaxation, N, is 1 for water and amides and is 2 for alcohols. The fitting procedure works well as is demonstrated in Figure 1. The high-frequency value of the dielectric constant, ε(νHF), is treated as a fitting parameter. It is different from the dielectric constant at the optical frequency because of the libration and vibrational polarization. Although the estimate of the high-frequency dielectric constant is important in the detailed discussion on the high-frequency processes, it affects little the value of τD owing to the large relaxation amplitude of the low-frequency process. Some of the mixtures measured in this work have already studied in literatures, as will be introduced in Section 4. The agreement between them is satisfactory in most cases. However, small differences are found with our previous work on the alcohol mixtures.16 The vector network analyzer went out of order after the publication of the previous work, and some electric parts are repaired by the manufacturer. In the calibration procedure of the frequency-domain microwave reflection method, the impedance mismatch is approximated as the T-type filter.25 However, since the actual mismatch may not be described well by the T-type filter, we consider that the modification of the impedance mismatch within the vector network analyzer may lead to the systematic change in the calibrated spectra. We cannot judge which value is the better one at present. However, the difference in τD is within 5% at most, and it does not affect the discussions in this work and the previous one. 3.3. NMR Spectroscopy. The NMR spectrometer used in this work is JNM-A600 manufactured by JEOL. The resonance frequency of 2H nucleus is 92 MHz. The temperature is controlled at 25.0 ( 0.5 °C by flowing thermostatted air around the sample tube. The spin-lattice relaxation time, T1, is determined by the inversion recovery method. The measurements are performed with 39 different waiting times, and the

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Figure 2. Chemical shift of the OD-deuteron of d4-methanol are plotted against the mole fraction of the solute. The solutes are pyrazine (red circles), 1,4-dioxane (blue squares), benzene (green diamonds), carbon tetrachloride (black triangles), and cyclohexane (purple crosses).

value of T1 is deduced from the dependence of the peak intensity on the waiting time. The values of T1 of neat solvents are consistent with those reported in literatures.7,13,26,27 The 1H NMR spectra are also measured for the same samples in order to determine the chemical shifts of the O-H or N-H protons. The peaks of the methyl protons are used as the internal reference, assuming that the solvent effects on the chemical shifts of the methyl protons are smaller than those on the O-H or N-H protons. 3.4. Density Measurement. The density of the mixture is measured with a vibration tube density meter (Anton Paar DMA 602). The temperature is controlled at 25.00 ( 0.01 °C by thermostatted water. The references of air and distilled water are employed to determine the cell constants. 4. Results 4.1. Methanol. The experiments on the methanol solvent systems are performed with the solutes of cyclohexane, carbon tetrachloride, benzene, 1,4-dioxane, and pyrazine. The mole fraction of the cyclohexane is limited to no larger than 0.06 because stable results were not obtained at higher concentration, probably due to the wetting of one of the components on the surface of the dielectric probe. The chemical shift of the OD deuteron, denoted as δOD, is shown in Figure 2. The chemical shift of the CD3 deuterons of the neat solvent is used as the internal standard, assuming that it is not dependent on the concentration of the solute. The chemical shifts of the neat solvent are taken from the literature. According to Figure 2, the change in δOD in response to the dissolution of the solute is quite small, and no apparent correlation with the solute-solvent interaction is found. The chemical shift of the OH proton reflects the electron density around the proton, and it is known to be sensitive to the hydrogen bonding. The small chemical shift thus means that the environment of the OD group of the methanol is little affected by the species and concentration of the solute. In addition to chemical shift, the value of QCC also responds to the modification of the electronic density around the deuteron. On the basis of the ab initio MO calculation, Wendt and Farrar demonstrated that the linear correlation is found between the values of δD and χD, and they proposed that the chemical shift is utilized in order to estimate QCC.13 The relationship they obtained is given by eq 27,

χD /kHz ) 284 - 15.3δD /ppm

(27)

which is employed for the calculation of τ2R in the present work. The change in χ2D is no larger than 3% in all the mixtures studied here, which has only a marginal effect on the reorientational

Figure 3. The reorientational relaxation times of methanol solutions normalized to those of neat solvent are plotted against the mole fraction of the solute, Xs. The solutes are pyrazine (red circles), 1,4-dioxane (blue squares), benzene (green diamonds), carbon tetrachloride (black triangles), and cyclohexane (purple crosses). The dielectric relaxation time, τD, and the rank-2 single-molecular reorientational relaxation time, τ2R, are shown in panels a and c, respectively, and the former is divided by the static factor, g∞, in panel b.

relaxation time. Therefore, we neglect the concentration dependence of χD when no relationship as eq 27 is available. We have assumed in the derivation of eq 24 that the dipole moment of a solvent molecule is not altered by the dissolution of the solutes, although the dipole moment may be also modified in response to the electronic structure. The small change in the chemical shift can be utilized to justify the assumption of the constant dipole moment in the following way. Since both the chemical shift and dipole moment reflect the polarization of the O-H bond, there must be some correlation between them. In the case of supercritical water, Matubayasi et al. proposed the linear relationship between them.28 We suppose here the similar correlation in order to estimate the dipole moment of methanol in the mixtures. The chemical shift of OH proton of methanol is 4.81 ppm in neat liquid and is -0.11 ppm in the gas phase.29 The dipole moments are estimated to be 1.69 and 2.87 D in the gas and liquid phases, respectively.30 On the basis of these values and the linear relationship between them, the change in the dipole moment is no larger than 2% in all the mixtures, which is so small that we shall hereafter neglect the concentration dependence of the dipole moment. The collective and single-molecular reorientational relaxation times are exhibited in Figure 3, panels a and c, respectively. The dielectric relaxation spectra of carbon tetrachloride/methanol and 1,4-dioxane/methanol systems were previously reported by Buchner et al.11 and Mashimo et al.,12 respectively, and the 2H NMR T1 measurement on the carbon tetrachloride/methanol mixture was performed by Wendt and Farrar.13 Our present results are consistent with these previous works. As is described in Section 3.2, the present results of the dielectric relaxation times are slightly different from those of our previous works.16 However, the difference is small enough that it does not affect the discussions in the previous and present works. As is observed in both the dielectric and 2H NMR relaxations, the solute with hydrogen-bond accepting sites (1,4-dioxane and pyrazine) enhance the reorientational relaxation of the solvent, whereas those without such sites (cyclohexane, carbon tetrachloride, and benzene) reduce the rotational mobility of the solvent. However, the magnitude of the effects of the solutes is larger on the collective mode (dielectric relaxation) than on the single-molecular one (NMR). There are two sources in the difference in the concentration dependence of the collective and single-molecular reorientational relaxation times, that is, the static correlation of the dipole moments related to the static dielectric constant and the dynamic cooperativity, as is described in eq 23. In Figure 3b, the dielectric relaxation time is divided by the static factor, g∞, in order to

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Figure 4. The reorientational relaxation times of ethanol solutions normalized to those of neat solvent are plotted against the mole fraction of the solute, Xs. The meanings of the panels and symbols are the same as those of Figure 3.

Figure 5. The reorientational relaxation times of aqueous solutions normalized to those of neat water are plotted against the mole fraction of the solute, Xs. The meanings of the panels and symbols are the same as those of Figure 3.

remove the contribution of the static correlation. Therefore, the difference between panels a and b in Figure 3 reflects the static contribution, and that between panels b and c in Figure 3 reflects the dynamic one. It is clearly seen that the cooperativity in the effects of the solutes on the reorientational dynamics of the solvent methanol, that is, the difference between Figures 3, panels a and c, is mainly ascribed to the dynamic correlation. Although the static correlation reduces the slowing down and enhances the speed up of the solvent, as is observed in the difference between panels a and b in Figure 3, its role is a rather minor one. 4.2. Ethanol. The reorientational relaxation times of ethanol in various mixtures are demonstrated in Figure 4. Here, the value of QCC is estimated from the chemical shift as26

χD /kHz ) 297.60 - 15.286δD /ppm

(28)

and the dipole moment of a solvent molecule is assumed to be independent of the concentration of the solute. The estimated variation of QCC is as small as that of methanol mixtures. The results on the ethanol mixtures are qualitatively similar to those on the methanol mixtures shown in the previous subsection. The weaker solute-solvent interaction leads to the slower reorientational relaxation of the solvent. The effects of the solutes are stronger on the collective mode than on the single-molecular one, and the difference in the collective and single-molecular reorientational relaxation time is mainly attributed to the dynamic cooperativity. Comparing methanol and ethanol mixtures, the effects of the solvophobic solutes are smaller and those of the solvophilic ones are larger in the ethanol mixtures. 4.3. Water. Figure 5 exhibits the reorientational relaxation times of water in the aqueous solutions of 1,4-dioxane and pyrazine. The solute species are limited to the above two due to the limited solubility. In Figure 5, the values of QCC and

Figure 6. The reorientational relaxation times of NMF solutions normalized to those of neat solvent are plotted against the mole fraction of the solute, Xs. The meanings of the panels and symbols are the same as those of Figure 3.

dipole moment are assumed to be independent of the concentration of the solute. The mixture of water and 1,4-dioxane is a representative system to study the dynamic hydrophobicity, and both the dielectric relaxation12,31 and the 2H NMR T1 measurement7 have been reported by other researchers. The relaxation times obtained in the present work is confirmed to be consistent with the corresponding values in the previous studies. As is shown in panels a and c of Figure 5, the dissolution of 1,4-dioxane and pyrazine makes both the collective and the single-molecular reorientational relaxation slower. Comparing the effects of the two solutes, the hydrophobic slowing down is stronger for 1,4-dioxane, which appears to be consistent with the stronger proton acceptability and thus the hydrophilicity of pyrazine. From the comparison between panels a and b in Figure 5, it should be noticed that the effect of the solute on the reorientational relaxation of the solvent is larger on the collective mode than on the single-molecular one. The most part of the cooperativity in the hydrophobic slowing down of the reorientational relaxation is originated in the dynamic contribution, which is common to alcohol mixtures shown in the previous subsections, and the static correlation works so as to reduce the cooperativity. 4.4. N-Methylformamide. The experimental results on the relaxation times of NMF in various mixtures are summarized in Figure 6. It is assumed in the analysis that the values of QCC and dipole moment are not dependent on the concentration of the solutes. NMF has cis and trans isomers in association with the rotation around the C-N bond. The isomerization rate is so slow that these two isomers appear as the two different peaks in both 1 H- and 2H NMR spectra. According to the peak area of the NMR spectrum, the population of the cis isomer is about 12 times larger than that of the trans one, and the dependence of the population ratio on the solute concentration is rather small. Therefore, only the T1 of the cis peak is analyzed in the NMR experiment, and the existence of the trans species is neglected in the analysis and discussion. As is demonstrated in Figure 6, the weaker solute-solvent interaction tends to make the reorientation of NMF slower. The effects of the solutes are larger on the collective reorientation than on the single-molecular one. These tendencies are common to water and alcohols shown in the previous subsections. However, dividing the reorientational cooperativity into the static and dynamic correlations, the static one dominates the difference in the collective and single-molecular modes, in contrast with water and alcohols where dynamic contribution is important. Figure 7 exhibits the static dielectric constant of the NMF mixtures, which is the origin of the static factor that affects the

Dynamic Solvophobicity of Hydrogen-bonding Liquids

Figure 7. The static dielectric constant of NMF solutions are plotted against the concentration of the solvent NMF. Note that the largest concentration in the graph corresponds to the neat NMF, and the concentration decreases with the dissolution of the solutes. The meanings of the symbols are the same as those of Figure 3.

Figure 8. The dielectric relaxation times of FA solutions are plotted against the mole fraction of the solute, Xs. The meanings of the symbols are the same as those of Figure 3.

collective reorientational relaxation time. The concentration of the solvent is taken as the x-axis in Figure 7, because g∞ is proportional to the ratio of ε(0) - ε(∞) to the number density of the polar molecules. The static dielectric constant decreases with increasing concentration of the solute in all the mixtures, which can be attributed to the decrease in the concentration of the solvent. However, comparing the mixtures at the same concentration of the solvent, the dielectric constant still depends on the chemical structure of the solute. The solutes possessing the hydrogen-bond accepting site tends to reduce the dielectric constant, which can be understood qualitatively as the destruction of the network structure of the solvent. In our previous work, however, the same comparison of the static dielectric constant is performed on methanol mixtures (Figure 4 of ref 16), and the dependence on the solute species is not so large as NMF mixtures. It appears to suggest that the difference in the liquid structure is important in the effect of solutes on the static dielectric constant of the solution. 4.5. Formamide and N,N-Dimethylacetamide. The dielectric relaxation times of the mixture of formamide with 1,4dioxane or pyrazine are shown in Figure 8. The solutes are limited to the above two due to the low solubility of other solutes. These solutes make the dielectric relaxation slower, as is the case of aqueous solutions. The effect of 1,4-dioxane is larger than that of pyrazine, which is also the same as aqueous systems. The dielectric relaxation times of mixtures of DMA with various solutes are demonstrated in Figure 9. The solvent DMA is chosen as an example of amides without hydrogen-bonding. The dielectric relaxation time of neat DMA is smaller than those of FA (38 ps) and NMA (127 ps). The variation of τD with the addition of solutes is also smaller than those of FA and NMA. Comparing the solutions of different solutes, there appears to be no correlation with the hydrogen-bond acceptability of the solute.

J. Phys. Chem. B, Vol. 112, No. 51, 2008 16639

Figure 9. The dielectric relaxation times of N,N-dimethylacetamide solutions are plotted against the mole fraction of the solute, Xs. The meanings of the symbols are the same as those of Figure 3.

5. Discussion The retardation of the mobility of water induced by the dissolution of the hydrophobic solutes has long been ascribed to the strengthening of the hydrogen-bond between water molecules due to the hydrophobic hydration shell around the solute molecules, as is described in the Introduction. However, there have been some studies on alcohol solutions in which the mobility of the solvent alcohols are reduced by the solvophobic solute molecules, and our present work demonstrates that similar behavior is also found in hydrogen-bonding amide systems. The retardation of the reorientational relaxation of the solvents in our present work is regarded as being caused by the solvophobicity of the solutes, because the relaxation is made faster by the solvophilic solutes. Because the slowing down of the reorientational relaxation of the solvent molecules by solvophobic solute molecules is common to hydrogen-bonding liquids including water, alcohols, and amides, we need to provide an explanation that applies to these liquids in general, rather than resorting to the solvation structure specific to aqueous solutions. We recently performed a theoretical study on the mixture of water with model hydrophobic solutes using the mode-coupling theory for molecular liquids based on the interaction-site model. The slowing down of the mobility of solvent water was reproduced qualitatively well by the theoretical calculation, and its mechanism was analyzed according to the theoretical expression for the memory function.19,32 The analysis showed that the electrostatic friction on the collective dielectric mode plays a key role in the hydrophobic slowing down of the mobility of water. The collective dielectric mode couples to the product of the charge-density and the number-density modes of the polar molecules in the lowwavenumber region. In the presence of the fluctuation of the number-density of the polar molecules, the collective reorientation of the polar molecules leads to the heterogeneity of the electric polarization and, thus, the charge density. Because the generation of the charge fluctuation requires the excess electrostatic energy, the electrostatic force is exerted against the collective reorientation. The electrostatic force persists during the relaxation time of the charge-density fluctuation, and it works as the electrostatic friction on the collective dielectric mode. The dissolution of the hydrophobic solute accompanies the formation of the “cavity”, in which the solvent molecule is expelled. The cavity formation inevitably enlarges the fluctuation of the number-density of the solvent, so that the electrostatic friction on the collective dielectric mode is enhanced by the dissolution of the hydrophobic solutes. The large friction on the dielectric mode leads to the slow dielectric relaxation, which makes the relaxation of other modes slower through the dielectric friction. The above explanation applies not only to water but also to alcohols and amides, because it does not depend on the network

16640 J. Phys. Chem. B, Vol. 112, No. 51, 2008 structure specific to water. Rather, the same effect must be present in all the polar solvent systems irrespective of the hydrogen-bonding between solvent molecules. However, the reorientation of the solvent is influenced by intermolecular interactions other than the electrostatic one, such as the collision through the repulsive part of the intermolecular interaction. The dominance of the electrostatic friction in the reorientational relaxation dynamics is therefore required in order that the effect described above is experimentally observed as the concentration dependence of the relaxation time. The reorientational relaxation times of water, alcohols, and hydrogen-bonding amides are known to be larger than those expected from their viscosities and van der Waals volumes. It indicates the importance of the interaction through the hydrogen bond that involves the strong electrostatic interaction, so that the dynamic solvophobic effect is observed in these liquids. On the other hand, the dielectric relaxation time of DMA is much shorter than other amides in spite of its larger molecular size. The electrostatic friction on the dielectric relaxation of DMA is thus expected to be small, which we consider is the reason why the dynamic solvophobic effect is not evident. In our previous theoretical study, we employed a series of model solutes whose hydrophobicity is systematically changed. The mobility of solvent water is reduced in both hydrophobic and hydrophilic extremes, which corresponds to the dynamic aspects of the hydrophobic and positive hydration, respectively. On the other hand, weakly hydrophobic solutes tend to enhance the mobility of water, which is regarded as the negative hydration. The transition between the negative and hydrophobic hydration is interpreted as the competition between the effects of the cavity formation effect described in the previous paragraph and the reduction of the direct solute-solvent interaction. In the present work, it is shown that the solutes with hydrogen-bonding accepter site enhance the reorientational relaxation of methanol, ethanol, and NMF, which we consider is classified into the negative solvation. The comparison between the dielectric and NMR relaxation measurements in the present work has demonstrated that the dynamic solvophobic effect is larger on the collective dielectric mode than on the single-molecular reorientation, and the analysis has shown that the difference between the collective and singlemolecular modes is due to the dynamic correlation in the cases of water and alcohols. The larger dynamic solvophobicity due to the dynamic correlation is qualitatively consistent with the explanation described above. Since it is the dielectric mode that is influenced by the solvophobic solutes first, it must feel the largest effect of the dynamic solvophobicity, and singlemolecular reorientation experiences the smaller slowing down because the effect on it is secondary. Although the presence of the dynamic cooperativity is qualitatively in harmony with our explanation, there are some problems to be resolved. First, both the dielectric and singlemolecular reorientational relaxation times of water were calculated in our previous study as the function of the concentration of hydrophobic solute. Although the slowing down of the former was larger than that of the latter, their difference was much smaller than the experimental ones.19 Because the difference between the collective and single-particle dynamics is large in the case of the theoretical calculation on the systems of ionic liquids,33 we consider there are some deficiencies in the extension of the mode-coupling theory to molecular systems. For example, we have employed the modification of the modecoupling theory proposed by Yamaguchi and Hirata to include the coupling between the rotations around different axes.34 The

Yamaguchi et al. excess force due to the coupling was assumed to be the same for collective and single-molecular modes, because there was no appropriate approximation for the collective mode. The friction on the single-molecular mode is mixed into that of the collective mode through this assumption, which makes the difference between the collective and single molecular modes smaller. The second problem is the small dynamic cooperativity of NMF mixtures. Although the cooperativity is found in the dynamic solvophobic effect on NMF mixtures, the most part of it can be ascribed to the static correlation, g∞. The change in the static correlation between dipole moments of the solvent molecules indicates the modification of the liquid structure, which is also expected to affect the dynamic processes. However, our theory on molecular liquids at present is not able to treat the microscopic origin of g∞,35 and we believe further improvement of the theory is required at this point. There exists another model on the reorientational relaxation of hydrogen-bonding liquids, called the “wait-and-switch” model.36,37 The switching of the hydrogen-bonding is required for the reorientation of hydrogen-bonding liquid. When a molecule is going to change the partner of the hydrogen bond, it needs to wait for the hydrogen-bond accepting site of another molecule to be at the suitable position. Because the dissolution of solvophobic solutes reduces the hydrogen-bond accepting site through the decrease in the number density of the solvent, a molecule has to wait for a longer time in solution than in the neat liquid, which results in the slowing down of the reorientational relaxation. The wait-and-switch model has sometimes been used to explain the reorientational relaxation in alcohol mixtures,14,36,37 and it was also employed for aqueous solutions by Schro¨dle et al.31 The wait-and-switch model is, in a sense, interpreted as the real-space description of our explanation based on the modecoupling theory. The excess charge caused by the collective reorientation in the presence of the number-density heterogeneity corresponds to the free O-H (or N-H) group, and the relaxation time of the charge density is regarded as the time for the free O-H to make a new hydrogen bond, which can be read as the waiting time in the wait-and-switch model. However, there has been no explanation on the difference between the dynamic solvophobic effects on the collective and the single-molecular reorientational modes in the former, whereas the principal role of the collective mode is emphasized in the latter. The dynamic cooperativity in water and alcohol systems observed in the present work is therefore explained well by the latter, whereas the former can state nothing about it at present, although the translation of our explanation into the wait-and-switch model would be possible. 6. Summary The reorientational relaxation of hydrogen-bonding liquids including water, alcohols, and amides in their mixtures with nonpolar solute molecules are experimentally studied. The solute-solvent interaction is systematically varied by changing the hydrogen-bond acceptability of the solute. Both the collective and the single-molecular reorientational relaxation times are determined by the dielectric and NMR spectroscopies, respectively. The difference between the effects of the solutes on the collective and single-molecular modes is divided into the contributions of the static correlation between the dipole moments of different solvent molecules and the dynamic cooperativity. The static correlation is estimated from the static dielectric constant in order to extract the effect of dynamic cooperativity.

Dynamic Solvophobicity of Hydrogen-bonding Liquids The reorientational relaxation of all the hydrogen-bonding solvents is made slower by the solvophobic solutes, indicating that the solvophobic slowing down is the phenomenon common to various hydrogen-bonding liquids. Therefore, we need an explanation applicable to hydrogen-bonding liquid in general, rather than that resorting to the solvation structure model specific to aqueous systems. The solvophobic slowing down is not apparent in the non-hydrogen-bonding amide, DMA, which suggests the importance of the hydrogen bond. The effect of the solute on the collective dielectric relaxation time is larger than that on the single-molecular reorientational correlation time in all the solvents studied here. Dividing their difference into the static and dynamic contributions, it is found that the latter is dominant in water and alcohols, whereas the former is important in NMF. The results described above are discussed in terms of the theoretical explanation we proposed in our previous work based on the mode-coupling theory for molecular liquids. The generality of the dynamic solvophobic effect and the importance of the dynamic cooperativity are in harmony with our explanation, although there remain some problems to be resolved. Acknowledgment. The NMR measurements were performed at Chemical Instrumentation Facility, Research Center for Materials Science, Nagoya University. References and Notes (1) Hertz, H. G. In Water, Franks, F., Ed; Prenum Press: New York, 1972; Vol. 3, Ch. 7. (2) Samoilov, O. Y. Discuss. Faraday Soc. 1957, 24, 141. (3) Frank, H. S.; Wen, W.-Y. Discuss. Faraday Soc. 1957, 24, 133. (4) Zeidler, M. D. In Water, Franks, F., Ed; Prenum Press: New York, 1973; Vol. 2, Ch. 10. (5) Kaatze, U.; Pottel, R. J. Mol. Liq. 1992, 52, 181. (6) Laaksonen, A.; Stilbs, P. Mol. Phys. 1991, 74, 747. (7) Takamuku, T.; Yamaguchi, A.; Tabata, M.; Nishi, N.; Yoshida, K.; Wakita, H.; Yamaguchi, T. J. Mol. Liq. 1999, 83, 163. (8) Frank, H. S.; Evans, M. W. J. Chem. Phys. 1945, 13, 507. (9) Pierotti, R. A. Chem. ReV. 1976, 76, 717.

J. Phys. Chem. B, Vol. 112, No. 51, 2008 16641 (10) Hertz, H. G. Ber. Bunsenges. Phys. Chem. 1964, 68, 907. (11) Buchner, R.; Barthel, J. J. Mol. Liq. 1992, 52, 131. (12) Mashimo, S.; Miura, N.; Umehara, T.; Yagihara, S.; Higasi, K. J. Chem. Phys. 1992, 96, 6358. (13) Wendt, M. A.; Farrar, T. C. Mol. Phys. 1998, 95, 1077. (14) Schwerdtfeger, S.; Ko¨hler, F.; Pottel, R.; Kaatze, U. J. Chem. Phys. 2001, 115, 4186. (15) Smith, R. L., Jr.; Saito, C.; Suzuki, S.; Lee, S.-B.; Inomata, H.; Arai, K. Fluid Phase Equilib. 2002, 194-197, 869. (16) Nagao, A.; Yamaguchi, T.; Matsuoka, T.; Koda, S. J. Phys. Chem. A 2006, 110, 3377–12540. (17) Madden, P.; Kivelson, D. AdV. Chem. Phys. 1984, 56, 467. (18) Weinga¨rtner, H.; Nadolny, H.; Oleinikova, A.; Ludwig, R. J. Chem. Phys. 2004, 120, 11692. (19) Yamaguchi, T.; Matsuoka, T.; Koda, S. J. Chem. Phys. 2004, 120, 7590. (20) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987. (21) Gutmann, V. Coord. Chem. ReV. 1976, 18, 225. (22) Matsuoka, T.; Okada, T.; Murai, K.; Koda, S.; Nomura, H. J. Mol. Liq. 2002, 98-99, 319. (23) Ho¨lzl, C. Ph.D. Thesis, Regensburg University, Regensburg, Germany, 1998. (24) Barthel, J.; Buchner, R.; Wurm, B. J. Mol. Liq. 2002, 98-99, 51. (25) Wei, Y.-Z.; Sridhar, S. IEEE Trans. MicrowaVe Theo. Tech. 1991, 39, 526. (26) Ferris, T. D.; Farrar, T. C. Mol. Phys. 2002, 100, 303. (27) Weinga¨rtner, H.; Holz, M.; Hertz, H. G. J. Sol. Chem. 1978, 7, 689. (28) Matubayasi, N.; Wakai, C.; Nakahara, M. J. Chem. Phys. 1999, 110, 8000. (29) Hoffmann, M. M.; Conradi, M. S. J. Phys. Chem. B 1998, 102, 263. (30) McClellan, A. L. Tables of Experimental Dipole Moments; W. H. Freeman: San Francisco, 1963. (31) Schro¨dle, S.; Hefter, G.; Buchner, R. J. Phys. Chem. B 2007, 111, 5946. (32) Yamaguchi, T.; Matsuoka, T.; Koda, S. Phys. Chem. Chem. Phys. 2006, 8, 737. (33) Yamaguchi, T.; Nagao, A.; Matsuoka, T.; Koda, S. J. Chem. Phys. 2003, 119, 11306. (34) Yamaguchi, T.; Hirata, F. J. Chem. Phys. 2002, 117, 2216. (35) Perkins, J. S.; Pettitt, B. M. J. Chem. Phys. 1992, 97, 7656. (36) Petong, P.; Pottel, R.; Kaatze, U. J. Phys. Chem. A 1999, 103, 6114. (37) Petong, P.; Pottel, R.; Kaatze, U. J. Phys. Chem. A 2000, 104, 7420.

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