THE JOURNAL OF CHEMICAL PHYSICS 140, 044508 (2014)
Does shear viscosity relaxation control the dynamics of critical fluctuations in polystyrene–cyclohexane? Sirojiddin Z. Mirzaev1,2 and Udo Kaatze1,a) 1
Drittes Physikalisches Institut, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 2 Institute of Ion-Plasma and Laser Technologies, Uzbek Academy of Sciences, Dormon Yuli Street 33, 100125 Tashkent, Uzbekistan
(Received 25 November 2013; accepted 7 January 2014; published online 28 January 2014) Between 20 and 90 MHz frequency-dependent shear viscosities of the polystyrene–cyclohexane mixture of critical composition have been measured at polymer molar weight Mw = 30 000. The viscosity data reveal dispersion, in conformity with relaxation characteristics in the non-critical background contributions to the ultrasonic attenuation, i.e., in the longitudinal viscosity of the critical system. The dispersion behavior is discussed with a view to its effect on the critical dynamics of the liquid near its consolute point. Attention is especially given to the relaxation rates of fluctuations of that system. The data as resulting from ultrasonic attenuation spectroscopy on the one hand and from quasi-elastic light scattering and viscosity measurements on the other hand differ near the critical temperature. It is concluded that likely an additional dispersion exists in the shear viscosity at frequencies below the presently available frequency range of measurement. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862825] I. INTRODUCTION
Phase separation of polymer solutions is a discerning phenomenon of critically demixing fluids. It is difficult to understand because of the competition between the correlation length ξ of critical fluctuations and the radius of gyration Rg of the polymer molecules as a second mesoscopic length scale. Naturally, even with polymer solutions, adequately close to the critical demixing point, ξ grows so large that the structure of the liquid gets insignificant. The system features universal behavior.1–7 When leaving the close proximity of the critical point, however, the radius of gyration of the macromolecules may play the role of a cut-off or screening length. On such condition a crossover from Ising-type to mean-field critical behavior occurs.8–11 As a consequence, the phase separation characteristics in mixtures of polymers with low-weight molecular solvents change substantially with Rg and thus with the degree Np of polymerization.12 Dynamic light scattering measurements have revealed valuable insights into the behavior of the mutual diffusion coefficient D of polystyrene–cyclohexane critical mixtures at different polymer length.13 At comparatively low Np (≈1881) the dependences of D upon temperature and scattering angle were found to be qualitatively similar to those of lowweight “simple” critical mixtures. Qualitatively, however, the D dependencies of the polymer systems were only compatible with mode-coupling theory for simple molecular mixtures if an apparent viscosity ηs,app instead of the macroscopically measured shear viscosity ηs was used in the relevant relations. This apparent viscosity appeared to be smaller than the shear viscosity ηs of the solution but larger than that of the solvent cyclohexane, ηso . Essentially ηs,app was identified with a a) Electronic mail: [email protected]
0021-9606/2014/140(4)/044508/6/$30.00
“mesoscopic” viscosity that characterizes the hydrodynamic environment of the polymer molecules.13 It was shown to scale with the polymer molar weight Mw as ηs,app = η*Mw 0.6 and with the reduced temperature ε = |T − Tc |/Tc as ηs,app = η0 ε−0.043 ηso . Here Tc is the critical temperature and in the above relations η* and η0 are coefficients. It is mentioned that the exponent 0.043 of the above power law is smaller than the universal viscosity exponent xη = 0.069.14, 15 The intermediate position of the mesoscopic viscosity ηs,app between the macroscopic viscosities ηs and ηso of the solution and the solvent, respectively, is demonstrated by the upper part of Fig. 1. That delineation brings the characteristics of extrapolated high-frequency shear viscosities ηs (∞) to mind which had been determined for some critical mixtures with water as one component, such as triethylamine–water,16 2,6-dimethylpyridine–water,17 and triethylene glycol monoheptyl ether–water18 mixtures. For the triethylamine–water mixture of critical composition ηs (∞) data are shown along with corresponding ηs and ηso data in the lower part of Fig. 1. The extrapolated high-frequency shear viscosities have been obtained from frequency-dependent shear impedance measurements which, for the liquid under consideration, yielded the complex shear viscosity spectra (i2 = −1), ηs (ν) = ηs (ν) − iηs (ν),
(1)
at frequencies between 20 and 140 MHz. As demonstrated by the spectrum displayed in Fig. 2 the real part ηs of the shear viscosity displays dispersion characteristics (dηs (ν)/dν < 0) in that frequency range and, consistently with the fluctuationdissipation theorem, the shear viscosity includes also a nonvanishing imaginary part (η (ν) > 0). Within their rather large uncertainty ranges, the frequency-dependent complex shear viscosity data can be well represented by a Debye-type
140, 044508-1
© 2014 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 220.225.230.107 On: Mon, 03 Feb 2014 05:16:03
044508-2
S. Z. Mirzaev and U. Kaatze
J. Chem. Phys. 140, 044508 (2014)
FIG. 2. Real part ηs (ν) and negative imaginary part ηs (ν) of the complex shear viscosity spectrum of the triethylamine–water mixture at critical composition and at 18 ◦ C16 (ε = 0.93 × 10−3 ). The lines are graphs of a Debyetype relaxation spectral function (Eq. (2)) with discrete relaxation time.
FIG. 1. Shear viscosities as a function of reduced temperature for two critically demixing binary liquids. The upper part shows results13 for a polystyrene–cyclohexane mixture of critical composition with polymer molecular weight Mw = 196 000 (Np ≈ 1881): the dashed line represents the experimental ηs data, the full line depicts the apparent mesoscopic viscosities as following from application of the dynamic mode coupling theory to the polymer solution, and the dotted line indicates the viscosity of the solvent (cyclohexane). The shaded area accentuates the gaps between the mesoscopic and the experimental viscosities. The lower part of the figure refers to the triethylamine–water mixture of critical composition:16 points (●) mark the “static” viscosities as measured with capillary viscometers, squares () identify high-frequency viscosities as extrapolated from viscosity spectra, and the dotted line shows again the viscosity of the solvent (water). Here the shaded area demonstrates the relaxation amplitudes.
relaxation spectral function, ηs (ν) = ηs (∞) +
ηs (0) − ηs (∞) . 1 + iωτη
(2)
Here ω = 2π ν is the angular frequency and τ η is a (discrete) relaxation time. The low-frequency shear viscosity ηs (0) of critical mixtures is somewhat smaller than the “static” viscosity ηs as determined with capillary or falling ball viscometers. This results reflects the fact that, due to the extrapolation from the MHz frequency range, ηs (0) does not contain contributions from the slowed fluctuations in the local compositions of the mixtures. The high-frequency parameter ηs (∞) represents the shear viscosity at frequencies well above the relaxation region (Fig. 2). The triethylamine–water system exhibits a lower critical point whereas the polystyrene–cyclohexane system possesses an upper one. For that reason, far from Tc the viscosity relations of the critical mixtures reveal opposite temperature dependencies (Fig. 1). Apart from that feature the ηs (∞) data of the aqueous solution behave similar as the apparent shear viscosities of the polymer solution. For that reason we became interested in the question whether polystyrene–cyclohexane mixtures display also relaxations in their background viscosity and whether such relaxations may be the reason for the need of apparent viscosities smaller than ηs when evaluat-
ing diffusion coefficients in terms of mode-coupling theory. In this paper we report results from shear impedance spectroscopy of polystyrene–cyclohexane mixture of critical composition with comparatively small polymer molecular weight. II. MATERIALS AND METHODS A. Critical mixture
Polystyrene (PS) with molar mass 30 000 g/mol (N = 288) and narrow mass distribution (Mw/MN < 1.03) has been received from Pressure Chemical (Pittsburg, PA). Cyclohexane (CH) was supplied by Sigma–Aldrich (Munich, Germany). Both chemicals were used as delivered. Solutions were prepared by weight using an analytical balance (BP 61 S, Sartorius, Göttingen, Germany). In order to avoid water uptake from the air the source materials and the samples were stored and handled in a dry nitrogen atmosphere. The polymer critical mass fraction Yc [=0.186 ± 0.001] and the critical temperature Tc [=(283.10 ± 0.02) K] were determined visually according to the equal-volume criterion. The same data as in a recent ultrasonic spectrometry study19 were found. Therefore, the density ρ(Tc ) [=(0.830139 ± 0.000001) g cm−3 ] and the sound velocity cs (Tc ) [=(1351.3 ± 7) ms−1 ] values at the critical temperature have been taken from the former study. The density had been measured with a vibrating tube densitometer (Physica DMA 5000, Anton Paar, Graz, Austria), the sound velocity had been derived from resonance frequency series of appropriate ultrasonic resonator cells.19
B. Shear impedance spectroscopy
Since liquids of low viscosity respond only marginally to shear stimuli we used a sensitive resonator method20 to investigate the suggested frequency dependence in the shear viscosity of the PS-CH solution of critical composition. The method is based on the determination of the shifts δν n in the
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 220.225.230.107 On: Mon, 03 Feb 2014 05:16:03
044508-3
S. Z. Mirzaev and U. Kaatze
J. Chem. Phys. 140, 044508 (2014)
FIG. 3. Delineation of a thickness shear vibrating AT-cut quartz disc (1, thickness dq ) loaded with a liquid (3). Low-resistance chromium-gold electrodes (2) are sufficiently thin (
Does shear viscosity relaxation control the dynamics of critical fluctuations in polystyrene–cyclohexane? Sirojiddin Z. Mirzaev1,2 and Udo Kaatze1,a) 1
Drittes Physikalisches Institut, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 2 Institute of Ion-Plasma and Laser Technologies, Uzbek Academy of Sciences, Dormon Yuli Street 33, 100125 Tashkent, Uzbekistan
(Received 25 November 2013; accepted 7 January 2014; published online 28 January 2014) Between 20 and 90 MHz frequency-dependent shear viscosities of the polystyrene–cyclohexane mixture of critical composition have been measured at polymer molar weight Mw = 30 000. The viscosity data reveal dispersion, in conformity with relaxation characteristics in the non-critical background contributions to the ultrasonic attenuation, i.e., in the longitudinal viscosity of the critical system. The dispersion behavior is discussed with a view to its effect on the critical dynamics of the liquid near its consolute point. Attention is especially given to the relaxation rates of fluctuations of that system. The data as resulting from ultrasonic attenuation spectroscopy on the one hand and from quasi-elastic light scattering and viscosity measurements on the other hand differ near the critical temperature. It is concluded that likely an additional dispersion exists in the shear viscosity at frequencies below the presently available frequency range of measurement. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862825] I. INTRODUCTION
Phase separation of polymer solutions is a discerning phenomenon of critically demixing fluids. It is difficult to understand because of the competition between the correlation length ξ of critical fluctuations and the radius of gyration Rg of the polymer molecules as a second mesoscopic length scale. Naturally, even with polymer solutions, adequately close to the critical demixing point, ξ grows so large that the structure of the liquid gets insignificant. The system features universal behavior.1–7 When leaving the close proximity of the critical point, however, the radius of gyration of the macromolecules may play the role of a cut-off or screening length. On such condition a crossover from Ising-type to mean-field critical behavior occurs.8–11 As a consequence, the phase separation characteristics in mixtures of polymers with low-weight molecular solvents change substantially with Rg and thus with the degree Np of polymerization.12 Dynamic light scattering measurements have revealed valuable insights into the behavior of the mutual diffusion coefficient D of polystyrene–cyclohexane critical mixtures at different polymer length.13 At comparatively low Np (≈1881) the dependences of D upon temperature and scattering angle were found to be qualitatively similar to those of lowweight “simple” critical mixtures. Qualitatively, however, the D dependencies of the polymer systems were only compatible with mode-coupling theory for simple molecular mixtures if an apparent viscosity ηs,app instead of the macroscopically measured shear viscosity ηs was used in the relevant relations. This apparent viscosity appeared to be smaller than the shear viscosity ηs of the solution but larger than that of the solvent cyclohexane, ηso . Essentially ηs,app was identified with a a) Electronic mail: [email protected]
0021-9606/2014/140(4)/044508/6/$30.00
“mesoscopic” viscosity that characterizes the hydrodynamic environment of the polymer molecules.13 It was shown to scale with the polymer molar weight Mw as ηs,app = η*Mw 0.6 and with the reduced temperature ε = |T − Tc |/Tc as ηs,app = η0 ε−0.043 ηso . Here Tc is the critical temperature and in the above relations η* and η0 are coefficients. It is mentioned that the exponent 0.043 of the above power law is smaller than the universal viscosity exponent xη = 0.069.14, 15 The intermediate position of the mesoscopic viscosity ηs,app between the macroscopic viscosities ηs and ηso of the solution and the solvent, respectively, is demonstrated by the upper part of Fig. 1. That delineation brings the characteristics of extrapolated high-frequency shear viscosities ηs (∞) to mind which had been determined for some critical mixtures with water as one component, such as triethylamine–water,16 2,6-dimethylpyridine–water,17 and triethylene glycol monoheptyl ether–water18 mixtures. For the triethylamine–water mixture of critical composition ηs (∞) data are shown along with corresponding ηs and ηso data in the lower part of Fig. 1. The extrapolated high-frequency shear viscosities have been obtained from frequency-dependent shear impedance measurements which, for the liquid under consideration, yielded the complex shear viscosity spectra (i2 = −1), ηs (ν) = ηs (ν) − iηs (ν),
(1)
at frequencies between 20 and 140 MHz. As demonstrated by the spectrum displayed in Fig. 2 the real part ηs of the shear viscosity displays dispersion characteristics (dηs (ν)/dν < 0) in that frequency range and, consistently with the fluctuationdissipation theorem, the shear viscosity includes also a nonvanishing imaginary part (η (ν) > 0). Within their rather large uncertainty ranges, the frequency-dependent complex shear viscosity data can be well represented by a Debye-type
140, 044508-1
© 2014 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 220.225.230.107 On: Mon, 03 Feb 2014 05:16:03
044508-2
S. Z. Mirzaev and U. Kaatze
J. Chem. Phys. 140, 044508 (2014)
FIG. 2. Real part ηs (ν) and negative imaginary part ηs (ν) of the complex shear viscosity spectrum of the triethylamine–water mixture at critical composition and at 18 ◦ C16 (ε = 0.93 × 10−3 ). The lines are graphs of a Debyetype relaxation spectral function (Eq. (2)) with discrete relaxation time.
FIG. 1. Shear viscosities as a function of reduced temperature for two critically demixing binary liquids. The upper part shows results13 for a polystyrene–cyclohexane mixture of critical composition with polymer molecular weight Mw = 196 000 (Np ≈ 1881): the dashed line represents the experimental ηs data, the full line depicts the apparent mesoscopic viscosities as following from application of the dynamic mode coupling theory to the polymer solution, and the dotted line indicates the viscosity of the solvent (cyclohexane). The shaded area accentuates the gaps between the mesoscopic and the experimental viscosities. The lower part of the figure refers to the triethylamine–water mixture of critical composition:16 points (●) mark the “static” viscosities as measured with capillary viscometers, squares () identify high-frequency viscosities as extrapolated from viscosity spectra, and the dotted line shows again the viscosity of the solvent (water). Here the shaded area demonstrates the relaxation amplitudes.
relaxation spectral function, ηs (ν) = ηs (∞) +
ηs (0) − ηs (∞) . 1 + iωτη
(2)
Here ω = 2π ν is the angular frequency and τ η is a (discrete) relaxation time. The low-frequency shear viscosity ηs (0) of critical mixtures is somewhat smaller than the “static” viscosity ηs as determined with capillary or falling ball viscometers. This results reflects the fact that, due to the extrapolation from the MHz frequency range, ηs (0) does not contain contributions from the slowed fluctuations in the local compositions of the mixtures. The high-frequency parameter ηs (∞) represents the shear viscosity at frequencies well above the relaxation region (Fig. 2). The triethylamine–water system exhibits a lower critical point whereas the polystyrene–cyclohexane system possesses an upper one. For that reason, far from Tc the viscosity relations of the critical mixtures reveal opposite temperature dependencies (Fig. 1). Apart from that feature the ηs (∞) data of the aqueous solution behave similar as the apparent shear viscosities of the polymer solution. For that reason we became interested in the question whether polystyrene–cyclohexane mixtures display also relaxations in their background viscosity and whether such relaxations may be the reason for the need of apparent viscosities smaller than ηs when evaluat-
ing diffusion coefficients in terms of mode-coupling theory. In this paper we report results from shear impedance spectroscopy of polystyrene–cyclohexane mixture of critical composition with comparatively small polymer molecular weight. II. MATERIALS AND METHODS A. Critical mixture
Polystyrene (PS) with molar mass 30 000 g/mol (N = 288) and narrow mass distribution (Mw/MN < 1.03) has been received from Pressure Chemical (Pittsburg, PA). Cyclohexane (CH) was supplied by Sigma–Aldrich (Munich, Germany). Both chemicals were used as delivered. Solutions were prepared by weight using an analytical balance (BP 61 S, Sartorius, Göttingen, Germany). In order to avoid water uptake from the air the source materials and the samples were stored and handled in a dry nitrogen atmosphere. The polymer critical mass fraction Yc [=0.186 ± 0.001] and the critical temperature Tc [=(283.10 ± 0.02) K] were determined visually according to the equal-volume criterion. The same data as in a recent ultrasonic spectrometry study19 were found. Therefore, the density ρ(Tc ) [=(0.830139 ± 0.000001) g cm−3 ] and the sound velocity cs (Tc ) [=(1351.3 ± 7) ms−1 ] values at the critical temperature have been taken from the former study. The density had been measured with a vibrating tube densitometer (Physica DMA 5000, Anton Paar, Graz, Austria), the sound velocity had been derived from resonance frequency series of appropriate ultrasonic resonator cells.19
B. Shear impedance spectroscopy
Since liquids of low viscosity respond only marginally to shear stimuli we used a sensitive resonator method20 to investigate the suggested frequency dependence in the shear viscosity of the PS-CH solution of critical composition. The method is based on the determination of the shifts δν n in the
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 220.225.230.107 On: Mon, 03 Feb 2014 05:16:03
044508-3
S. Z. Mirzaev and U. Kaatze
J. Chem. Phys. 140, 044508 (2014)
FIG. 3. Delineation of a thickness shear vibrating AT-cut quartz disc (1, thickness dq ) loaded with a liquid (3). Low-resistance chromium-gold electrodes (2) are sufficiently thin (
