Langmuir 1988,4, 63-66
63
Viscosity Measurements Using Colloidal Particles Hani Saad, Y. C. Bae, and Esin Gulari* Department of Chemical and Metallurgical Engineering, Wayne State University, Detroit, Michigan 48202 Received March 18,1986. I n Final Form: June 15,1987 A method for determining viscosity at elevated pressures and temperatures is based on measuring diffusivities of colloidal particles by using photon correlation spectroscopy. Knowing the size of the probe, the Stokes-Einstein relation is employed to calculate the medium viscosity. The assessment of the method is made with respect to data along subcritical isotherms of the C02-heptane system.
Introduction Viscosity is a dynamic property, and its determination requires measurement of forces as a function of time. Liquid and gas viscosities have been measured by many techniques' based on various time-dependent phenomena such as descent of a sphere in a viscous fluid, flow in a capillary tube, torque transmitted from a rotating cylinder, and damping of an oscillating disk or vibrating ~ i r e . ~ P Under nonambient conditions, especially a t elevated pressures, sample containment is difficult. Only oscillating disk&' or vibrating wire3v8 type viscometers have been successfully used to measure viscosities of dense gases of near critical systems. We are proposing an alternate and nonintrusive method for determining viscosity a t elevated pressures and temperatures. It is based on measuring the diffusion coefficient of suspended colloidal particles by photon correlation spectroscopy (PCS).If the size of these probe particles is known, the Stokes-Einstein relation can then be employed to calculate the viscaity of the suspending medium. The method works very well with uniform-sized latex spheres suspended in a l i q ~ i d . Independent ~ measurements of diffusion coefficients and viscosities of the suspending media have established the validity of the Stokes-Einstein relation with stick boundary conditions over a wide range of viscosities.1° A major experimental difficulty in applying the technique to dense gases or liquids containing appreciable amounts of dissolved gases is finding a uniform-sized probe that can be suspended in such media. Aerosol OT (AOT, sodium bis(2-ethylhexy1)sulfosuccinate) forms micellar aggregates in nonpolar solvents. Zulauf and Eicke have used PCS to determine the size and shape of these aggregates.l' They found the radius of the inverted micelles to be 1.50 f 0.03 nm in isooctane. This value was independent of the concentration in the range 8X to 2 X 10-1 M and independent of the scattering angle. They also did not detect any changes in the radius when the temperature was varied between -20 and +95
also discuss the criticality of the assumptions made in determining the viscosity.
Experimental Section The light-scattering spectrometer used for the photon correlation experiments is similar to the one described in ref 12. The range of scattering angles of the spectrometer with the highpressure scattering cell was 6-11' for homodyne detection. The sample cell, machined out of a solid brass block, was fitted with flanged, quartz windows and tested up to 20 MPa. The temperature was measured by a thermistor embedded in the cell block. The pressure was monitored by a pressure transducer,the sensitivity and the reproducibility of which were *0.005 and 10.03 MPa; respectively. For each loading of the cell, both COz and n-heptane were introduced gravimetrically from high-pressure sample cylinders. The evacuated cell was first charged with n-heptane;then COz was injected into the cell containing n-heptane until the liquid reached the desired level and the cell was closed off. COz in the loading line was condensed back into the sample cylinder by emersing the cylinder into liquid nitrogen. The amounts were determined from the differences in the weights of the sample cylinders before and after loading. Spectrophotometricgrade n-heptane and bone-dry COz were used without further purification. Fluka purum grade AOT was purified by first being dissolved in methanol, filtered through active c h a r d , extracted with heptane, and dried under vacuum.
Methods of Data Analysis When colloidal-sizedparticles are suspended in a binary liquid mixture, there are two possible sources of optical fluctuations which contribute to the scattered intensity. One source is concentration fluctuations in the binary mixture, and the second contribution comes from the number fluctuations of the colloidal particles due to their Brownian motion. Both the concentration and the number fluctuations are exponentially decaying functions in time. The decay rate of the concentration fluctuations is proportional to DAB, the binary mutual diffusion ~0efficient.l~
"C. We measured the size of AOT micelles in n-heptane and confirmed the results of ref 11. We then seeded the liquid phase of the two-phase systems formed by dissolving C02 in n-heptane under pressure with AOT micelles. The PCS measurements were made with and without the AOT micelles. Two characteristic decay rates were detected in the presence of micellar aggregates. The faster component corresponded to binary mutual diffusion of C02 in nheptane, and the slower component was used to calculate viscosity from the Stokes-Einstein relational2 In the absence of other experimental viscosity data on this system, we compared our results to ideal mixture viscosities. We *Author to whom correspondence should be sent.
(1)Golubev, I. F. Viscosity of Gases and Gas Mixtures; Israel Program for Scientific Translations: Jerusalem, 1970. (2)Tough, J. T.;McCormick, W. D.; Dash, J. G. Rev. Sci. Instrum. 1964,35,1345. (3)Bruschi, L.;Santini, M. Rev. Sci. Instrum. 1975,46,1560. (4)Kestin, J.; Leidenfrost, W. Physica 1959,25,1033. ( 5 ) Haynes, W.M. Physica 1973,67,440. (6)Diller, D. E. Physica A (Amsterdam) 1978,104A,417. (7)Strumpf, H. J.; Collings, A. F.; Pings, C. J. J. Chem. Phys. 1974, 60,3109. (8)Carless, D. C.; Hall, H. E.; Hook, J. R. J. Low Temp. Phys. 1983, 50, 583. (9)Gulari, Es.; Gulari, Er.; Tsunashima, Y.; Chu, B. J. Chem. Phys. 1979,70, 3865. (10)Phillies, G.D. J. Phys. Chem. 1981,85,2838. (11)Zulauf,M.; Eicke, H. F. J. Phys. Chem. 1979,83,480. (12)Saad, H.; Gulari, Ea.J. Phys. Chem. 1984,88, 136.
0743-746318812404-0063$01.50/0 0 1988 American Chemical Society
64 Langmuir, Vol. 4, No. 1, 1988
Saad et al.
Table I. AOT Micelle Size in n -Heptane at 1 atm' T, O C D X lo8, cm2/s Tht nm 19.8 29.9 39.9 50.1 60.0 a VhepMe
3.31 f 0.08 3.93 f 0.04 4.59 f 0.04 5.29 f 0.04 6.02 f 0.06
Table 11. Pressure Dependence of AOT Micelle Size T," C P, MPa rhr nm
1.54 i 0.02 1.51 f 0.02 1.48 f 0.02 1.47 f 0.02 1.45 f 0.01
10.0 f 0.1 20.0 f 0.1
= 0.01492 exp(976/7') cP, with T in K.
30.0 f 0.1
The decay rate of the number fluctuations is proportional to D , a particle diffusion coefficient which can be related to t i e viscosity of the suspending medium by the Stokes-Einstein relation:
where rh is the hydrodynamic size of the particle. In photon correlation spectroscopy, the time dependence of the fluctuations is quantitatively determined from the measurements of the time autocorrelation function of the scattered light. Under the appropriate conditions, the correlation function of the scattered electric field is a superposition of the two exponentials and has the following form: g(')(q,t) = C1 exp[-DA~q2t] + C2 exp[-D,q2t]
(2)
where q = (4n7r/&,)sin (0/2), n is the refractive index, A,, is the wavelength of the incident light under vacuum, and 0 is the scattering angle. The coefficient C, is affected by the variation of the refractive index with the concentration of the binary mixture and the magnitude of the concentration fluctuations. C2 is related to the number of colloidal particles and their scattering power. Thus, a simultaneous determination of D m and D, become possible from analyses of correlation function profiles. A successful resolution of the two exponentials depends strongly on the magnitudes of the relative decay rates and the coefficients. The choice of the time window at which to sample the fluctuations is also important. The sample time can be changed to optimize the information about one phenomenon relative to the other.
Results and Discussion The reliability of the viscosities obtained from the measured diffusion coefficienb of micelles depends heavily on the assumption that the micelles have a known and uniform size. As a first step in testing this assumption, the hydrodynamic radii of AOT micelles in pure heptane were measured as a function of temperature. The measurements of a 0.1 M AOT solution in n-heptane are summarized in Table I. As the temperature increases, the viscosity of heptane decreases and the diffusion of micelles becomes faster. This effect accounts for most of the variation of D, with temperature while rh varies very slightly from 1.54 nm a t 20 "C to 1.45 nm a t 60 "C. The linear length of an AOT molecule is determined to be 1.2 nm from its chemical structure. Therefore, the minimum radius of the inverted AOT micelles is expected to be 1.2 nm. The measured value of 1.5 nm probably accounts for the presence of a few water molecules that facilitate the formation of micelles and the excluded volume in the association of the head groups. A possible explanation for the slight decrease of the radius of the micelles as the temperature is raised is the evaporation of some of the water molecules from the core of the micelles. These re(13)Mountain, R. D.; Deutch, J. M. J. Chen. Phys. 1969,50, 1103.
40.0 f 0.1 50.0 f 0.1
2.91 5.51 7.61 3.00 5.66 7.80 3.08 5.81 7.95 3.15 5.95 8.11 3.22 6.09
1.61 f 0.02 1.55 f 0.02 1.51 f 0.02 1.63 f 0.01 1.58 f 0.02 1.49 f 0.02 1.68 f 0.01 1.71 f 0.02 1.46 f 0.01 1.56 f 0.02 1.62 f 0.03 1.50 f 0.03 1.54 f 0.03 1.61 f 0.02
sults in n-heptane are in excellent agreement with those in isooctane reported in ref 11. An investigation was also done to find out whether AOT formed micelles in liquid C02. AOT did not dissolve in COz, and micelle formation was not detected by light scattering. A second test was run to determine the effect of pressure on the AOT micelle size. The solubility of N2 in liquid hydrocarbons is negligible compared to that of COP The volume above an AOT-n-heptane solution was pressurized with varying amounts of N2,and the hydrodynamic radii of micelles in the solution were measured as a function of pressure and temperature. The viscosity of pure heptane was used for computing rh. The results are listed in Table 11. rh values remain fairly constant over the temperature and pressure range of the data. The radius tends to decrease slightly with increasing presure. A similar decrease in radius with pressure has been observed by other investigator~'~ and has been attributed to the compression of the surfactant tails under pressure. Since the viscosity of heptane was not corrected for small amounts of dissolved N2,the rh values listed in Table I1 were only used to illustrate that the micelle sizes did not change with pressure. The rh values listed in Table I were taken as the correct size of AOT micelles for use in the calculations of mixture viscosities. Clearly, micelles are not as stable as latex particles. In this study, the requirements of chemical inertness and small buoyancies for the probes in a hydrocarbon environment could not be satisfied by latex spheres but could be satisfied by micelles. The insensitivity of the size of AOT micelles to changes of pressure and temperature could be established. The possibility that COP could infiltrate or swell the micelles could not be eliminated definitely. However, on the basis of the high solubility of C 0 2 in heptane, the insolubility of AOT in COz, and the nonpolar nature of COz, it was assumed that C 0 2 dissolved solely in heptane. It was less probable that nonpolar C 0 2 would be associated with the polar head groups of AOT. After accepting the viability of AOT micelles as uniform-sized probe particles, we made PCS measurements of two types of systems. For the first set of measurements, pure heptane was pressurized with COP, and the liquid phase which expanded due to dissolution of C 0 2 was probed. A similar set of measurements was repeated on systems containing -0.1 M AOT dissolved in heptane before the pressurization step with COP. Typical sections of correlation functions measured with and without AOT micelles a t the same temperature, pressure, and concentration are shown in Figure 1. In the case of the binary (14)Dawson, D.R.;Offen, H. W.; Nicoli, D. J. Colloid Interface Sci. 1981, 81,396.
Langmuir, Vol. 4, No. 1, 1988 65
Viscosity Measurements Using Colloidal Particles
y T=50'C
"1 ,/'
/X
65-
T=40'C T = 30 'C T = 20 'C
45-
T
0
0.4
0.2
0 1 0 2 0 3 0 4 0 5 0 6 0
Time(psec) Figure 1. Typical sections of measured autocorrelation functions of systems seeded with AOT micelles (open symbols) and binary mixtures (solid symbols). The two functions were normalized by their initial values. The parameters of the data were as follows. Seeded system: T = 10 "C, P = 3.54 MPa, DAB = 3.54 X cm2/s,D = 7.40 X lo4 cm2/s. Binary mixture: T = 9.9 "C, P = 3.56 d P a , Dm = 3.47 X cm2/s.
Figure 3. Plots of pressure versus mole fraction of C02 in the
liquid phase along isotherms. 0.5 1
R
1
L 2
3
4
P (MFb) Figure 2. Comparison of the binary mutual diffusion coefficients measured in two different ways at 10 "C. Open circles correspond to measurements of binary mixtures, and solid circles denote Dm determinations,which were simultaneous with D,. mixture of COP and heptane, the source of the optical fluctuations is predominantly concentration fluctuations. The decay rate of the autocorrelation function is related to the binary mutual diffusion coefficient. For the liquid mixtures seeded with AOT micelles, binary concentration fluctuations and Brownian motion of micelles contribute to the decay rate of the correlation function. As observed in Figure 1,the combined effect yields a slower decay rate, which must be sampled over a longer time range in order to determine both diffusion coefficients reliably. DABf D, varied from -5 to 10 over the range of the data. The correlation function measurements of a seeded system at a given temperature and pressure were repeated for delay times of 1,5, and 10 ps/channel. The diffusion coefficient of the micelles was obtained from the long-time behavior of the correlation function profiles. Then, DA* was determined from the initial decay rate when D, was known. Essentially, the correlation functions were analyzed in terms of eq 2 only in the overlapping short-time range of the data. D m determined from the two different types of measurements as a function of pressure a t 10 OC is shown in Figure 2. The agreement between the two measurements was very good over the entire range of the datal2 and
0.8
xcoz
0.01 0
3
0.6
= 10 'C
8
,
0.2
I
I
0.4
0.6
3
I
0.8
1
1
xcoz Figure 4. Plots of viscosity versus mole fraction of COz along
isotherms.
provided an indirect check for reliably extracting the two decay rates from the double-exponential analysis. The solubility of COPin heptane over the temperature and pressure range of the viscosity measurements was also measured. The overall composition for each loading of the constant volume cell was determined gravimetrically, and the volume increase of the liquid phase was monitored as a function of pressure at constant temperature. When the gas phase was assumed to be pure C 0 2 ,the mole fraction of COz in the liquid phase could be calculated. This assumption holds over the range of the reported data. From the values of vapor pressure of pure heptane, the worst case, at 50 OC and 4.7 MPa of total pressure, yields 0.003 for the mole fraction of heptane in the gas phase. The mole fraction of C 0 2in the liquid phase is plotted versus total pressure along isotherms in Figure 3. The viscosities of the liquid mixtures were calculated from the measured D, and known rh of the AOT micelles by using eq 1 and are plotted versus composition along isotherms in Figure 4. In the absence of other experimental viscosity data on this system, the measured viscosities were compared with ideal mixture viscosities calculated from pure component viscosities and their respective compositions. The results are listed in Table 111. For the viscosity of heptane, q = 0.0149 exp(976/T) CPand C 0 2viscosities reported by Michels et al.15 were used for (15) Michels, A.; Botzen, A,; Schuurman, W. Physica 1957, 23, 95.
66 Langmuir, Vol. 4, No. I, 1988
Saad et al.
Table 111. Comparison of Measured and Calculated Viscosities MPa
D, x lo6, Xcn. cm2/s vm, cP" v,, cPb T = 10.0 f 0.1 O C , rhd = 1.540 nm
devC
3.02 3.54 3.60 3.77 4.01
0.490 0.677 0.697 0.751 0.845
0.246 0.162 0.153 0.128 0.0855
7.9 -11 -15 -17 -41
3.48 4.23 4.35 4.63 4.91
0.432 0.649 0.684 0.765 0.846
= 1.525 nm 0.212 0.243 0.168 0.156 0.164 0.142 0.142 0.110 0.133 0.0781
15 -7.1 -13 -22 -41
P,
5.90 7.40 7.50 8.71 9.30
T = 20.0 f 0.1 "C, 6.64 8.40 8.56 9.91 10.6
0.228 0.182 0.179 0.155 0.145
%
rh
T = 30.0 i 0.1 " C , rh = 1.505 nm 3.92 4.92 5.11 5.57 5.88
0.431 0.634 0.673 0.766 0.829
4.33 5.58 5.83 6.52 6.92
0.418 0.616 0.655 0.765 0.828
7.20 9.30 9.67 11.7 11.9
0.205 0.159 0.152 0.126 0.124
0.219 0.147 0.133 0.101 0.0788
6.8 -7.5 -12 -20 -36
T = 40.0 f 0.1 " C , p rh = 1.490 nm 8.01 10.0 10.5 12.6 13.2
0.192 0.154 0.146 0.122 0.116
0.203 0.140 0.128 0.0936 0.0740
5.7 -9.1 -12 -23 -36
T = 50.0 f 0.1 "C, r, = 1.470 nm 4.68 6.17 6.53 7.45 7.78
0.413 0.593 0.637 0.748 0.789
8.10 10.7 11.6 13.8 13.6
0.199 0.150 0.139 0.117 0.118
0.187 0.135 0.123 0.0918 0.0808
-6.0 -10 -12 -22 -32
qrn is the measured viscosity. bvcis the calculated ideal mixture viscosity. c % dev = (vc - qrn)/vm X 100. dTemperature dependence of r,, was taken into account by fitting r h values in Table 1as a function of T.
calculating the ideal mixture viscosities. If XCO < 0.7, the agreement between the measured and calculated viscosities is within 15%. However, the deviations between the measured and calculated viscosities are systematic. As shown in Figure 5, ideal mixture viscosities show a sharper decrease with composition when compared to the measured viscosities. There are two competing effects which govern the behavior of viscosity a t constant temperature. Initial reduction of viscosity is due to "swelling" of the liquid phase by dissolved gas. As the pressure is increased and the saturation solubility of C02in heptane is approached, the viscosity levels off. Further increase of pressure may result in an increase of viscosity. In fact, similar viscosity behavior has been reported on crude oil-C02 mixtures.16
i 0.0 0
0.2
0.4
0.6
0.8
1
*co2 Figure 5. Comparison of measured (symbols and solid line) and ideal mixture viscaities (dotted line) along 10 and 40 "C isotherms.
In this study, the saturation solubilities were approached, but higher pressures, at which an increase of viscosity can be observed, were not accessible experimentally. The effect of pressure on viscosity can be more readily observed if the gas is somewhat less soluble in the liquid so that the saturation solubility is reached a t lower pressures.
Conclusions We have illustrated a method for measuring viscosities of liquids containing appreciable amounts of dissolved gases at elevated pressures and temperatures by using PCS to determine the diffusion coefficients of suspended probe particles. The crucial aspect of this method is the identification of a suitable probe that is chemically inert and uniform in size and that forms a stable suspension in nonpolar media of specific gravity -0.5. In this study, we first established the temperature and pressure insensitivity of the size of AOT micelles over the range of our data. The PCS experiments on the liquid heptane-COz mixtures seeded with AOT micelles yielded values of the viscosity and the binary mutual diffusion coefficient simultaneously. A t constant temperature, as the pressure is increased, the decrease of the viscosity of the liquid phase is due to dissolved COz.The viscosity levels off as saturation solubility is approached. Acknowledgment. Support of this work by National Science Foundation Grant CBT-8419755is gratefully acknowledged. Registry No. AOT, 577-11-7;COz,124-38-9; heptane, 142-82-5. (16) Killesreiter, H. Ber. Bunsenges. Phys. &em. 1984, 88, 838.
