Rheological Bases for Empirical Rules on Shear Viscosity of

Feb 25, 2013 - Tsuyoshi Yamaguchi,* Taiga Akatsuka, and Shinobu Koda. Department of Molecular Design and Engineering, Graduate School of Engineering ...
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Rheological Bases for Empirical Rules on Shear Viscosity of Lubrication Oils Tsuyoshi Yamaguchi,* Taiga Akatsuka, and Shinobu Koda Department of Molecular Design and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa, Nagoya, Aichi 464-8603, Japan ABSTRACT: The shear relaxation spectra of various lubrication oils were measured at 5−205 MHz and 10−70 °C, and the variation of the steady-state shear viscosity (η 0) was divided into the contributions of the high-frequency shear modulus (G∞) and relaxation time (τ). The temperature dependence of η0 was dominated by that of τ. The increase in molecular weight accompanies both increase in τ and decrease in G∞, and the increase in η0 results from the larger effect of the former. The flexibility of the chain reduced τ, while its effect on G∞ was rather small. The introduction of phenyl or cyclohexyl groups enhanced G∞. Oils of larger G∞ tend to exhibit a larger temperature dependence of η0.

shear viscosity, η(ν), is described as the Fourier transformation of the time correlation function. The spectrum η(ν) shows a relaxation around the relaxation frequency, 1/2πτ, so that the information on τ can be deduced from η(ν). The shear relaxation of ordinary liquids far from the glass transition cannot be measured with a conventional mechanical rheometer, because its frequency range is limited below the kHz region. However, the measurement in the MHz region is possible with ultrasonic methods, and the viscoelasticity of various viscous liquids has actually been studied.5 We have applied the shear impedance spectroscopy, which is based on the quartz crystal microbalance with a dissipation (QCM-D) method, to ionic liquids6,7 and organic electrolytes,8,9 and analyzed the mechanisms of the shear viscosity of these liquids. The shear relaxation of supercooled liquids in the MHz region was studied with ultrasonic spectroscopy in the middle of the last century, and several lubrication oils were employed as sample liquids.5 However, these studies were focused mainly on seeking a universal function that describes the shear relaxation, and they did not seem to be interested in extracting the factors that determine the variation of η0 from the relaxation measurements. We consider that it is probably due to the insufficient development of the nonequilibrium statistical mechanics on shear viscosity of liquids at that time. Owing to the Kubo−Green theory and its developments, we now know that the steady-state shear viscosity is determined by the correlation function, which is accessible experimentally with relaxation measurement. Since shear viscosity is calculated through the correlation function in current theories4,10 and

1. INTRODUCTION Shear viscosity is one of the properties of liquids which are important in various divisions of engineering science. Owing to the relative ease in its accurate measurement, shear viscosity has been studied intensively from the 19th century, and a huge amount of data has been accumulated. Understanding of shear viscosity in terms of molecular structure and intermolecular interaction is nonetheless yet to be established, and the shear viscosity sometimes appears in current topics of physical chemistry such as ionic liquids.1 Lubrication is a branch of mechanical engineering where shear viscosity of liquid is important, because lubrication properties are determined by the viscosity of lubrication oils in the hydrodynamic lubrication regime.2 In addition to the steady-state shear viscosity at ambient pressure, high-pressure viscosity and viscoelasticity of lubrication oils are involved in the elastohydrodynamic lubrication regime.2 Because of the necessity to provide lubrication oils required under given conditions, the relationship between chemical structures of oils and their viscosity was studied experimentally in the 20th century, and various empirical rules have been proposed.3 Shear viscosity is one of the transport coefficients of liquid. Transport coefficients are expressed as the time-integral of the corresponding currents of the conserved quantities according to the Kubo−Green theory.4 In particular, steady-state shear viscosity, η0, is related to the time integral of the time correlation function of the shear stress tensor. Therefore, the value of η0 is approximately given by the product of the relaxation time, τ, and the amplitude of the correlation function, G∞, which corresponds to the high-frequency shear modulus. The time correlation function of the shear stress tensor is accessible experimentally by measuring the shear viscosity as the function of frequency, ν, because the frequency-dependent © 2013 American Chemical Society

Received: January 31, 2013 Revised: February 24, 2013 Published: February 25, 2013 3232

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Figure 1. The chemical structures of the samples studied in this work.

simulations,11 experimental determination of the correlation function will serve as a test of these calculations. The purpose of this work is to relate the empirical rules on η0 of lubrication oils to the variations of G∞ and τ experimentally in order to give rheological bases to these rules. The target empirical rules are itemized as follows:3,12 (1) Shear viscosity is an increasing function of molecular weight within the same class of oils. (2) The flexibility of the chain reduces the shear viscosity. (3) The presence of the ring structure, i.e., phenyl or cyclohexyl groups, increases both the shear viscosity and its temperature dependence. In addition to the above three rules, we also examine the relationship between G∞ and the temperature dependence of η0. A theory on supercooled liquid relates the activation energy of the relaxation time to G∞, considering that the energy of the transition state results from the transient local shear distortion.13 Temperature dependence of the shear viscosity is an important property of lubrication oils, and its correlation with G∞, if present, would help in understanding the mechanism of the temperature dependence of η0.

In the Maxwell model of the viscoelasticity of liquids, the decay of the correlation function is exponential as

Cη(t ) = G∞e−t/ τ

and the real and imaginary parts of η(ν), defined as η(ν) ≡ η′(ν) − iη″(ν), are given by η′(ν) =

η″ (ν ) =

≡

V kBT

∫0

∞

∫0

∞

1 + (2πντ )2

(3)

2πG∞ντ 2 1 + (2πντ )2

η0 = G∞τ

(4)

(5)

Although the Maxwell model, eqs 2−4, cannot describe the properties of real liquids quantitatively, the correlation function Cη(t) is sometimes characterized with two parameters, G∞ and τ, as ⎛t ⎞ Cη(t ) = G∞ϕ⎜ ⎟ ⎝τ⎠

(6)

where ϕ(t′) is a master function in the time domain that does not depend on G∞ and τ. Substitution of eq 6 into eq 1 leads to

dt e−2πiνt ⟨σxz(0)σxz(t )⟩

dt e−2πiνt Cη(t )

G∞τ

The real part, η′(ν), decreases with frequency and the imaginary part, η″(ν), shows a maximum around the relaxation frequency, 1/2πτ. The steady state shear viscosity is described as

2. THEORETICAL BACKGROUND According to the Kubo−Green formula on shear viscosity, the frequency-dependent shear viscosity, η(ν), is related to the time correlation function of the shear stress tensor, σxz(t), as4 η (ν ) =

(2)

η(ν) = G∞τ

(1)

∫0

∞

dt ′e−2πiντt ′ϕ(t ′) ≡ G∞τϕ(̃ ντ )

(7)

In particular, the steady-state shear viscosity is proportional to the product of G∞ and τ as

where V, kB, and T stand for the volume of the system, the Boltzmann constant, and the absolute temperature, respectively. The steady-state shear viscosity, η0, is given by its zerofrequency value, η(0).

̃ η0 = G∞τϕ(0) 3233

(8)

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Therefore, the variation of η0 is ascribed to the changes of G∞ and τ. Hereafter we normalize ϕ(t) so that ϕ̃ (0) is equal to unity, in order to reduce eq 8 into eq 5. It is impossible to divide the contributions of G∞ and τ by measuring merely η0 with conventional viscometers. However, the division is possible with the analysis of η(ν). According to eqs 7 and 8, the normalized relaxation spectra, η(ν)/η0, reduce to a master curve when they are plotted against ντ. When τ is constant and the change in η0 is dominated by that of G∞, the plot of the normalized spectra against ν yields a master curve. On the other hand, the plot against 2πη0ν does a master curve in the case of constant G∞, because η0 is proportional to τ in such a case. In addition, the relaxation frequency on the 2πη0ν axis is a measure of G∞, because 2πν is equal to 1/τ there.

up to 205 MHz were performed using overtones up to the 41st mode. The resonance signal of the transducer was measured with a vector network analyzer (VNA) and an RF bridge. The output signal from VNA was divided into two. One was sent to the transducer, and the other was sent to the RF bridge, which was adjusted to cancel the nonresonant response of the transducer. The difference between the electric responses of the transducer and the RF bridge was sent to the input port of the VNA, and recorded as the complex transmission coefficient, S21(ν). The S21(ν) without sample liquid was fitted into the superposition of Lorentzian functions in order to take spurious modes into account, and the signal with sample was reproduced by shifting and broadening the Lorentzian functions. The temperature of the sample was controlled by flowing the thermostatted water through the sample cell. The temperature of the sample liquid was monitored with a thermistor directly immersed into the sample, and the fluctuation of the temperature was typically within 0.1 K.

3. EXPERIMENTAL SECTION 3.1. Samples. Squalane (Kishida Chemical Co., Ltd., >98%) and 2,4-dicyclohexyl-2-methylpentane (DCMP, Acros, 97%) were chosen as examples of paraffinic and naphthenic oils, respectively. Three diesters with different numbers of phenyl rings, di(2-ethylhexyl)sebacate (DEHS, Tokyo Chemical Industry Co., Ltd., 98%), di(2-ethylhexyl)phthalate (DEHP, Tokyo Chemical Industry Co., Ltd., >98%), and benzylbutylphthalate (BBP, Sigma-Aldrich, >97%), were measured to clarify the effects of phenyl rings. Silicone oils (polydimethylsiloxane, PDMS) of three different viscosity grades were also studied, whose catalog values of kinematic viscosity are 5 cSt (PDMS-5, Aldrich), 20 cSt (PDMS-20, ShinEtsu), and 100 cSt (PDMS-100, Shin-Etsu). Poly(methylphenylsiloxane) of 100 cSt grade (PMPS, Shin-Etsu) was also used for comparison. The chemical structures of the samples are summarized in Figure 1. All the samples were used without further purification. Measurements on all the samples were performed at 25 °C. In addition, the experiments on squalane, DCMP, DEHS, DEHP, and PDMS-20 were performed at temperatures from 10 to 70 °C with intervals of 15 °C. 3.2. Density and Steady-State Shear Viscosity. In shear impedance spectroscopy, the value of density, ρ, is required in order to convert shear impedance into η(ν). The values of η0 are also required in the analysis of η(ν). The values of ρ and η0 of squalane,14 DEHS,15 and DEHP14 were taken from the literature, and those of other samples were determined in this work. The density was measured with a vibrating tube densitometer (Anton Paar, DMA-60/602). The viscosity of PDMS-5 was determined with an Ubbelohde viscometer, while those of other oils were measured with a cone−plate viscometer (Blookfield, RVDV-IPCP equipped with a spindle CPE-40). Shear rate dependence of the shear viscosity was not observed. 3.3. Shear Impedance Spectroscopy. Shear impedance spectroscopy is an application of QCM-D technology to viscometry, which has been used to study liquid alkanes,16 alcohols,17 electrolyte solutions,8,9 ionic liquids,6,7 and so on. It determines the viscous friction on the surface of an AT-cut quartz crystal transducer in contact with the sample liquid through the measurement of the resonance signal of the crystal.18 The utilization of the overtone resonances makes it possible to determine η(ν) at frequencies equal to the fundamental resonance frequency of the transducer multiplied by odd integers. The detailed description of the equipment we used was given in the literature.6−9 Briefly, the fundamental frequency of the transducer was 5 MHz, and the measurements

4. RESULTS AND DISCUSSION 4.1. Temperature Dependence. Figure 2 shows η0 of five oils, squalane, DCMP, DEHP, DEHS, and PDMS-20, as a

Figure 2. The steady state shear viscosity of squalane (red circles), DEHP (blue squares), DCMP (green diamonds), PDMS-20 (black upward triangles), and DEHS (purple downward triangles) are plotted as a function of reciprocal temperature.

function of reciprocal temperature. Comparing those of oils with ring structures (DCMP with cyclohexyl groups and DEHP with a phenyl ring), those with linear structures depend more weakly on temperature. It has been well-known that the viscosity of naphthenic oils abundant in ring structures shows stronger temperature dependence (i.e., lower viscosity index) than that of paraffinic ones with linear structures, and the results in Figure 2 follow the traditional trend. The temperature dependence of η0 of silicone oil (PDMS-20) was the weakest among five. Figures 3, 4, and 5 demonstrate the normalized relaxation spectra, η(ν)/η 0 of squalane, DEHP, and PMDS-20, respectively, at temperatures from 10 to 70 °C as a function of 2πη0ν. We have already reported the spectrum of squalane at 25 °C,19 and the spectrum in Figure 3 agrees with that in the literature. The temperature alters the absolute values of η0 as is demonstrated in Figure 2, and the relaxation frequency also changes with temperature accordingly. However, the normalized spectra reduce to their respective master curves when plotted against 2πη0ν, as are demonstrated in Figures 3, 4, and 5, which indicates that the decrease in η0 with increasing temperature is dominated by the decrease in the relaxation time, τ. The similar reduction to a master curve is also observed in cases of DCMP and DEHS, although the results are not shown for brevity. 3234

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curve with the proposed temperature dependence of G∞. Since the estimated variation of G∞ is far smaller than that of η0, its consideration does not affect our conclusion that the temperature dependence of η0 is dominated by that of τ. We cannot state whether the variation of G∞ as was proposed is present or not based on our experiment due to the limitation of experimental errors. Figure 6 compares the normalized spectra of five liquids at 25 °C on the 2πη0ν axis. The relaxation frequency on this axis is a Figure 3. The normalized shear relaxation spectra, η(ν)/η0, of squalane at 10 °C (red circles), 25 °C (blue squares), 40 °C (green diamonds), 55 °C (black upward triangles), and 70 °C (purple downward triangles) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

Figure 6. The normalized shear relaxation spectra, η(ν)/η0, of squalane (red circles), DEHP (blue squares), DCMP (green diamonds), PDMS-20 (black upward triangles), and DEHS (purple downward triangles) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

Figure 4. The normalized shear relaxation spectra, η(ν)/η0, of DEHP at 10 °C (red circles), 25 °C (blue squares), 40 °C (green diamonds), 55 °C (black upward triangles), and 70 °C (purple downward triangles) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

measure of G∞, as was described in section 2. The five spectra are roughly divided into three groups, that is, DCMP and DEHP, squalane and DEHS, and PDMS-20. Therefore, the values of G∞ of the first two are larger than those of the second two, and G∞ of PDMS-20 is the smallest. Comparing the variation of G∞ with the temperature dependence of η0, we can notice that oils of the larger G∞ tend to show the larger temperature dependence of η0. The relationship between G∞ and the activation energy of η0 was proposed in the elastic model for glass transition.13 The activation energy was proportional to G∞ in the elastic model based on the idea that the activation energy required for the distortion of liquid structure in the transition state can be estimated with the continuum mechanics. The trend observed in this work that the oils with larger G∞ show larger temperature dependence of η0 is consistent with the elastic model at least qualitatively. 4.2. Ring Groups. It has been empirically known that the introduction of ring groups, either cyclohexyl or phenyl ones, increases both the absolute value and the temperature dependence of η0. In this subsection, relaxation spectra of oils with and without the ring groups are compared in order to clarify which of G∞ and τ is responsible for the larger η0 of oils with ring groups. The first comparison is that between squalane and DCMP. Their normalized spectra at 25 °C are plotted in Figure 6 as a function of 2πη0ν. The value of 2πη0ν of squalane at the relaxation frequency is smaller than that of DCMP. Since η0 of squalane (28.3 mPa s) is smaller than that of DCMP (36.5 mPa s), the relaxation time of squalane must be longer than that of DCMP. The larger η0 of DCMP is therefore ascribed to G∞ rather than τ. The second comparison is that among three esters, DEHS, DEHP, and BBP. Their values of η0 at 25 °C are 17.2 mPa s (DEHS), 57.5 mPa s (DEHP), and 44.3 mPa s (BBP), respectively. The viscosity of DEHS with no phenyl ring is

Figure 5. The normalized shear relaxation spectra, η(ν)/η0, of PDMS20 at 10 °C (red circles), 25 °C (blue squares), 40 °C (green diamonds), 55 °C (black upward triangles), and 70 °C (purple downward triangles) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

Our present result does not contradict with the analyses in previous works in which the temperature dependence of G∞ was taken into account. In these works, the values of G∞ were determined experimentally around the glass transition temperature where the relaxation frequency was much lower than the frequency used in the experiment, and the temperature dependence of G∞ was extrapolated to higher temperatures in order to estimate the temperature dependence of τ based on eq 5.5 Squalane was a target of these studies, and its temperature dependence of G∞ was proposed. The analysis of our present results on squalane with the proposed values of G∞ is described in the Appendix, together with the comparison of our results with raw data in the literature and a proposed model function. As is demonstrated in the Appendix, we can draw a master 3235

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lower than those of DEHP and BBP with phenyl rings, in harmony with the empirical rule. Although BBP with two rings has lower viscosity than DEHP with one ring, it may be probably due to the smaller size of the former. The normalized spectra of three esters are compared with each other in Figures 7 and 8 on ν and 2πη0ν axes, respectively.

Figure 9. The normalized shear relaxation spectra, η(ν)/η0, of PDMS100 (red circles) and PMPS (blue squares) are plotted against ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

liquids can naturally be understood as the consequence of the strong Coulombic interaction between ions. However, the same idea does not apply to the effects of ring groups. Although there may be weak additional interactions between phenyl rings, the interaction strength between cyclohexyl rings would be comparable to that between alkyl chains. One may consider that the higher rigidity of rings is associated with the higher shear modulus. However, the rigidity of chains is related to τ rather than G∞, as will be presented later. Microscopic studies such as molecular dynamics (MD) simulation will be required in future works in order to resolve the origin of the higher G∞ of oils with ring groups. 4.3. Molecular Size. The effects of molecular size on G∞ and τ are investigated by comparing the relaxation spectra of silicon oils with different viscosity grades. The values of η0 of the three silicon oils are 4.69 mPa s (PDMS-5), 18.4 mPa s (PDMS-20), and 94.8 mPa s (PDMS-100), respectively. According to the empirical relationship between the viscosity and molecular weight provided by a manufacturer,20 the molecular weights of PDMS-5, PDMS-20, and PDMS-100 are 670, 1900, and 6100, respectively. The normalized spectra of the three silicone oils are plotted in Figures 10 and 11 as the functions of ν and 2πη0ν,

Figure 7. The normalized shear relaxation spectra, η(ν)/η0, of DEHS (red circles), DEHP (blue squares), and BBP (green diamonds) are plotted against ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

Figure 8. The normalized shear relaxation spectra, η(ν)/η0, of DEHS (red circles), DEHP (blue squares), and BBP (green diamonds) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

The relaxation time is not correlated with η0 as is shown in Figure 7, and it rather seems that the relaxation becomes faster with increasing number of phenyl rings. Looking at the plot on the 2πη0ν axis, it is noticed that the increase in the number of phenyl rings leads to the increase in G∞. The third comparison is that between PDMS-100 and PMPS, silicon oils without and with phenyl rings. Their values of η0 are 94.8 mPa s (PDMS-100) and 98.0 mPa s (PMPS), respectively. The grades of the oils for comparison are chosen so that their values of η0 are close to each other. We guess that the molecular weight of PMPS is smaller than that of PDMS-100, although the molecular weight of the former is not available. The normalized spectra of these silicon oils are exhibited in Figure 9 as the functions of ν, which shows that the relaxation frequency of PMPS is higher than that of PDMS-100. The plot against 2πη0ν looks similar because η0 of these two oils are close to each other, although we do not show the plot for brevity. It can thus be concluded that G∞ of PMPS is higher than that of PDMS-100. The three examples above demonstrate that the introduction of ring groups increases G∞, which is responsible for the higher values of η0 of oils with ring groups. The higher value of G∞ may also explain the larger temperature dependence of η0 according to the relationship between G∞ and the activation energy of η0 suggested in section 4.1. We have shown in our previous work that G∞ of ionic liquids is higher than that of nonionic ones.19 The higher G∞ of ionic

Figure 10. The normalized shear relaxation spectra, η(ν)/η0, of PDMS-5 (red circles), PDMS-20 (blue squares), and PDMS-100 (green diamonds) are plotted against ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

respectively. The relaxation time increases with increasing chain length, as is shown in Figure 10. The increase in τ is quite natural because η0 increases. However, Figure 11 demonstrates that the increase in τ is far larger than that of η0. The increase in the chain length not only increases τ but also decreases G∞. The increase in η0 is the result of the larger effect of τ than that of G∞. We have already reported that η0, τ, and G∞ of a series of ionic liquids depend on the alkyl chain length of the cation in a similar way,6 although it is yet to be clarified whether the 3236

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Figure 12. The normalized shear relaxation spectra, η(ν)/η0, of PDMS-5 (red circles), PEO500 (blue squares), squalane (green diamonds), and DEHS (black upward triangles) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

Figure 11. The normalized shear relaxation spectra, η(ν)/η0, of PDMS-5 (red circles), PDMS-20 (blue squares), and PDMS-100 (green diamonds) are plotted against 2πη0ν. The filled and open symbols denote the real and imaginary parts of the spectra, respectively.

tendencies observed in the two different series of liquids are based on the same mechanism. It should be noted here that the shear relaxation spectra of poly-1-butene of various molecular weights were measured by Barlow and co-workers.21 They determined the high-frequency shear modulus from η(ν) in the MHz domain at temperatures near the glass transition temperature, and showed that G∞ increases with increasing chain length, which is opposite to our conclusion on silicon oils. Although the reason for the difference is not clear at present, our tentative idea is that the shape of the shear relaxation spectrum may depend on the molecular weight. What we measured in this work is the lowfrequency side of the shear relaxation spectra, while the highfrequency side was used by Barlow and co-workers in order to determine G∞. Therefore, the increase in the relaxation time distribution with increasing molecular weight may account for the apparent discrepancy between their conclusion and ours. 4.4. Chain Flexibility. It has been known that the molecular flexibility lowers η0 of oils. In particular, the low values of η0 of silicon oils has been ascribed to the high flexibility of Si−O−Si bonds. We have determined the shear relaxation spectra of three oils composed of linear molecules, squalane, DEHS, and PDMS-5, whose chain lengths are similar to each other. In this subsection, the relaxation spectra of these three liquids are compared in order to resolve how the chain flexibility of silicon oil reduces η0. The spectrum of poly(ethyleneglycol) dimethyl ether (PEO500), whose molecular weight is 500,9 is also taken from the literature for comparison. The comparison between the spectra of squalane and PEO500 was already reported in our previous paper,19 and their comparison with DEHS and PDMS-5 is regarded as its extension. Figure 12 demonstrates the normalized spectra of the four liquids as a function of 2πη0ν. All four spectra agree with one another in Figure 12, indicating that the difference in the values of η0 is determined by those of τ. In particular, high flexibility of the Si−O−Si chain reduces the shear viscosity of silicon oils through the promotion of structural relaxation. Although one may expect that a rigid chain increases G∞ through the enhancement of the rigidity of liquid structure against instantaneous shear deformation, it is the elongation of τ that is responsible for the increased η0. 4.5. Future Perspective. We have shown in this work how G∞ and τ affect the variation of η0, and the next question is the way these two parameters depend on the molecular structure, intermolecular interaction, and liquid structure. We consider that theoretical or computational studies are indispensable for

such a purpose. One of the present authors (T.Y.) extended the mode-coupling theory to shear viscosity of molecular liquids,10 which may be applied to present systems. MD simulation is a straightforward method to calculate the shear viscosity from the intermolecular interaction, and it was actually used to calculate the shear viscosity of squalane at two temperatures.22 The equilibrium MD simulation can yield η(ν) through the time correlation function, which is directly comparable with our experimental data. The high-frequency shear modulus in this work, G∞, is not an equilibrium quantity, because it corresponds to the amplitude of the time-correlation function associated with the α-relaxation.23 G∞ in this work is the shear modulus after the faster relaxations are completed, and it can be regarded as the rigidity of the vibrationally averaged structure, the so-called Vstructure.24 Yoshino recently proposed a theory to calculate the shear modulus of glass, which may be extended to evaluate G∞ of molecular liquids.25 Although the purpose of this work is to establish the bases of empirical rules on the shear viscosity of lubrication oils, we consider that our present result will also contribute to understanding the molecular mobility and reaction dynamics in viscous solvents. Shear viscosity of solvents has been used as a parameter to be correlated with dynamic properties of solutes such as diffusion coefficient, reorientational relaxation time, and reaction rate. As we have demonstrated in this work, however, the time scale of the microscopic dynamics of solvent is not characterized solely by η0. The difference in τ may affect the dynamics of solutes in solvents of similar values of η0. The effects of τ are expected to be large particularly in solute dynamics faster than τ. We believe that the liquids investigated in this work is suitable to examine the effect of the viscoelasticity of solvents on the fast dynamics of solutes, because specific solute−solvent interactions such as hydrogen bonding are expected to be weak, and the present work will provide fundamental data for such works in the future.

5. CONCLUSION The shear relaxation spectra of various oils are measured in this work, and the origins of the empirical rules on η0 are analyzed in terms of G∞ and τ. The increase in temperature decreases η0 through the decrease in τ. The role of G∞ is found to be marginal, although we cannot exclude its variation, as was proposed in the literature. The oils with higher values of G∞ tend to show a larger temperature dependence of η0, which is in harmony with 3237

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the elastic model proposed for the viscosity of supercooled liquids near the glass transition. The introduction of cyclohexyl or phenyl rings increases G∞, which explains the higher viscosity of oils with ring groups. Given the correlation between G∞ and the activation energy of η0, the higher value of G∞ can also be the reason for the larger temperature dependence of η0. We consider that the higher viscosity and lower viscosity index of naphthenic oils than paraffinic ones is ascribed to the larger G∞. According to the elastohydrodynamic lubrication theory, not only η0 but also the viscoelastic properties of oils are involved in the traction efficiency. DCMP studied in this work is a representative base oil used for traction oils, and the high G∞ and short τ of DCMP revealed in this work may explain the high traction efficiency of DCMP. The lengthening of the chain of a linear molecule both increases τ and decreases G∞. The increase in η0 is the result of the larger effect of the former than the latter. The flexibility of the Si−O−Si bond of silicon oils reduces its viscosity through shortening the relaxation time.

Figure 13. The normalized shear relaxation spectra, η(ν)/η0, of squalane are plotted against 2πτmν. The meanings of the symbols are the same as those in Figure 3 except for the orange triangles that indicate the values taken from ref 26. The solid and dashed curves show the real and imaginary parts of the model function, eq A2.

relaxation spectrum well, while the description of the higher frequency part is not so good. We consider it is because the model function and parameters were determined in order to reproduce the experiments below 100 MHz.

■

■

APPENDIX Analysis of the shear relaxation spectrum of squalane with temperature-dependent G∞ High-frequency shear viscosity of squalane was studied by Barlow and Erginsav with the refrectometry of transverse ultrasound. Extrapolating the temperature dependence of the shear modulus near the glass transition, they proposed that G∞ is a function of temperature given by26 1 1 = + C(T − T0) G∞(T ) G∞ ,0 (A1)

*E-mail: [email protected]. Phone: +81-52-789-3592. Fax: +81-52-789-3273. Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS This work was partly supported by JSPS KAKENHI Grant Number 24550019.

where G∞,0, C and T0 are constants. With this function, the Cole-Davidson function, given by5 J G∞ 1 =1+ + r (1 + 2πiντr)−β 2πiνη(ν) 2πiντm J∞

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REFERENCES

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(A2)

was suggested as a universal function, where τm = η0/G∞ is the Maxwell relaxation time, and Jr and τr are parameters that describe the relaxation part of the shear compliance. The values of Jr/Jm and τr/τm are assumed to be independent of temperature, so that η(ν)/η0 becomes a universal function of 2πτmν. In this Appendix, our experimental results on squalane is analyzed with G∞(T) proposed by Barlow and Erginsav in order to examine the consistency between these two experimental results. The normalized relaxation spectra of squalane are plotted in Figure 13 as the function of 2πτmν calculated with eq A1. Both experimental results at all the temperatures appear to fall on a single master curve, which indicates the consistency between two experiments. The plots against 2πη0ν (Figure 3) and 2πτmν (Figure 13) look similar to each other, because the variation of τm is dominated by that of η0. The value of η0 decreases by a decade from 10 to 70 °C, while the decrease in G∞ is predicted to be less than 50% by eq A1. Although we cannot determine within our experimental error which is the better reduced variable from the comparison between Figures 3 and 13, we can safely say that the consideration of the temperature dependence of G∞ proposed by Barlow and Erginsav does not affect our conclusion that the temperature dependence of shear viscosity is dominated by that of relaxation time. The model function, eq A2, describes the lower frequency part (2πτmν < 0.03) of the 3238

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The Journal of Physical Chemistry B

Article

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