High-Pressure Viscosity and Density of Polymer Solutions at the

Jun 19, 2002 - ane) (PDMS) + CO2,2 and polyethylene (PE) + n- pentane.3 In a ..... exp[B/(1 - VoG)] for the Density Dependence of Viscosity. T, °C. A...
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High-Pressure Viscosity and Density of Polymer Solutions at the Critical Polymer Concentration in Near-Critical and Supercritical Fluids Cigdem Dindar and Erdogan Kiran* Department of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061

This paper reports on the density and viscosity of solutions of poly(dimethylsiloxane) (PDMS; Mw ) 93 700) in supercritical carbon dioxide and of polyethylene (PE; Mw ) 121 000) in n-pentane. The measurements have been carried out at critical polymer concentrations, which are 5.5 wt % for a solution of PDMS in CO2 and 5.75 wt % for a solution of PE in n-pentane. For the PDMS + CO2 system, the measurements were conducted at 55, 70, 85, and 100 °C in the pressure range from 30 to 50 MPa. For the PE + n-pentane system, the measurements were conducted at 140 and 150 °C in the pressure range from 10 to 50 MPa using a falling-cylinder-type viscometer. At these temperatures and pressures, the viscosities were observed to be in the range from 0.14 to 0.22 mPa‚s for the PDMS + CO2 system and from 2.3 to 4.6 mPa‚s for the PE + n-pentane system. The flow activation energies and activation volumes in the PDMS + CO2 system were about 7 kJ/mol and 55 cm3/mol, respectively, compared to about 18 kJ/mol and 70 cm3/mol for the PE + n-pentane system. The close-packed volumes suggested from density correlations were found to be around 0.33 cm3/g for the PDMS system and 0.48 cm3/g for the PE system. Comparison of the viscosity data with the data on the kinetics of pressure-induced phase separation confirms that the slower kinetics in the PE + n-pentane system stems from the higher viscosity in this solution compared to the PDMS + CO2 system, despite the similarity in the molecular weight of the polymer and the critical polymer concentrations. 1. Introduction The motivation for the determination of the viscosity of polymer solutions in dense fluids at the critical polymer concentration stems from the need to understand the factors that influence the time scale of phase separation in systems that undergo spinodal decomposition upon a pressure quench. Pressure-induced phase separation is a relatively new technique that provides an alternative path to enter the unstable regions in polymer solutions. We have investigated the kinetics of pressure-induced phase separation in a number of polymer-fluid systems such as polystyrene + methylcyclohexane,1 poly(dimethylsiloxane) (PDMS) + CO2,2 and polyethylene (PE) + npentane.3 In a recent investigation of PDMS + CO2 and PE + n-pentane where molecular weights of the polymers and the critical polymer concentrations were comparable, significant differences were observed in the time evolution of new phase growth as reflected by the time evolution of the angular variation of scattered light intensities.2,3 This is schematically shown in Figure 1. In Figure 1a,b, the solid lines show the binodal envelopes and the dotted lines show the spinodal envelopes. The concentrations where the binodal and spinodal merge are the critical polymer concentrations. The region between the binodal and spinodal curves represents the metastable region. The region below the spinodal envelope is the thermodynamically unstable region. In polymer systems that are not monodisperse, the location of the critical polymer concentration is * To whom correspondence should be addressed. E-mail: [email protected]. Tel: (540) 231 4213. Fax: (540) 231 5022.

shifted to concentrations higher than the apex of the binodal. The schematic is drawn to demonstrate the actual system as close as possible. When a solution is subjected to a quench at the critical polymer concentration, the system enters its unstable region and phase separation proceeds spontaneously by a mechanism known as spinodal decomposition. The characteristic fingerprint of spinodal decomposition is the formation of a spinodal ring that in time becomes intense, but eventually the ring becomes smaller and collapses as time progresses. This is shown in Figure 1c in terms of the change in the angular variation of scattered light intensity and its evolution in time. In the systems depicted in Figure 1a,b, even though the molecular weights of PDMS and PE were similar and the critical polymer concentrations were comparable, the time evolution of phase growth was found to be much faster in the PDMS + CO2 system. For example, upon a 0.25 MPa quench into the region of instability, the ring collapse in PDMS + CO2 takes about 14 s, whereas in PE + n-pentane, even with a deeper quench of 1.1 MPa, the ring collapse takes about 24 s. Among the reasons that contribute to the difference in phase separation kinetics is the viscosity of the solutions. We therefore wanted to generate viscosity data for these solutions. For these systems, we have the most extensive kinetic information on phase separation near the critical polymer concentrations. Several techniques are available to measure the viscosity of polymer solutions at high pressures. Among these are the vibrating quartz-crystal viscometer,4 the oscillating-disk viscometer,5 the vibrating-wire viscometer,6 the high-pressure capillary viscometer,7 and the magnetoviscometer.8 However, the most widely used

10.1021/ie0108999 CCC: $22.00 © 2002 American Chemical Society Published on Web 06/19/2002

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Figure 1. (Top) Representation of the binodal and spinodal envelopes and experimentally determined critical polymer concentration for PDMS + CO2 (a) and PE + n-pentane (b). For actual data, see refs 2 and 3. (Bottom) Angular variation of the time evolution of new phase growth at the critical polymer concentration upon a pressure quench. In PDMS + CO2, the new phase growth proceeds much faster than in PE +n-pentane, and the spinodal ring collapses faster (tPDMS+CO2 , tPE+n-pentane).

viscometers are based on the principle of a falling (or rolling) body, which are based on the measurement of the fall time of a sinker in the fluid under pressure.9 Indeed, the falling-body-type viscometers have been used to measure the high-pressure viscosity of dilute polymer solutions in good solvents10 and polymer solutions in supercritical fluids.11 In these systems, the viscosity is determined at a constant temperature and pressure by measuring the fall time of the sinker (which is converted to terminal velocity Vt), the density difference between the falling body and the solution (Fs - Ff), and a calibration constant for the instrument (K) according to

η)

1 (F - Ff)K Vt s

(1)

Depending on the density of the sinker used, the measurable viscosity range can be varied from that of gaseous fluids to that of polymer solutions at relatively high concentrations. In our laboratory we use a fallingcylinder-type viscometer and have in the past determined the viscosity of n-alkanes,12 polystyrene solutions in n-butane,13 PE in n-pentane,14 PDMS in supercritical CO2,15 and polystyrene in n-hexane.16 This viscometer has been designed not only to measure the viscosity but also to provide information on the phase state and the density of the solutions. This is achieved by combining a view cell with a fall tube and a variable-volume attachment as shown in Figure 2.

Figure 2. Schematic diagram of the viscometer system: VC ) view cell; SW ) sapphire window; FT ) fall tube; VVP ) variable volume part; FI ) fluid inlet; SI ) solid inlet; CP ) circulation pump; PM ) pull-up magnet; PS ) position sensor; PGN ) pressure generator; LVDT ) linear variable differential transformer; P, T ) pressure and temperature sensor.

The view cell with sapphire windows permits observation of phase boundaries. The variable-volume part has a movable piston and helps change the solution pressure for a given loading. Monitoring the position of the piston with a built-in sensor (PS) permits the calculation of the internal volume from which the density of the solution can be determined at any given temperature and pressure from the initial mass of the solution charged to the viscometer. The fall tube is fitted

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with three linear variable differential transformer (LVDT) sensors to detect the passage of the falling sinker. Upon demagnetization of a pull-up magnet (PM) used to position the sinker, the cylindrical sinker falls through the tube and, as it passes through each LVDT coil, generates a characteristic voltage readout, as shown in the figure inset. The zero-voltage readings correspond to the time the sinker passes through the center position of each LVDT coil. Knowing the distance between the coils and the time for the fall provides a method to calculate the velocity for the sinker. Until the present study, we had used the distance and time corresponding to the passage between the center positions of the three coils. Comparison of the velocities based on the distance between coils 1 and 2 and between coils 2 and 3 was used to assess whether terminal velocities were reached. This method, even though reliable in most cases, left some degree of ambiguity. It also added some degree of inconvenience when working with solutions of high viscosities in that long times would be required for the sinker to travel the full distance of the fall tube and pass through all of the center points of the three sensors. In this study we have adopted a new methodology by developing a procedure by which we convert all of the voltage readings to the actual distance traveled through each coil. This then permits the generation of a complete distance versus time plot for all sections of the fall tube from which terminal velocities are determined without ambiguity. The details are described in the Experimental Section. 2. Experimental Section 2.1. Materials. The measurements were performed using previously characterized PE samples with Mw ) 121 000 and Mw/Mn ) 4.3 and PDMS with Mw ) 93 700, Mw/Mn ) 2.99. n-Pentane (Sigma-Aldrich) with a stated minimum purity of 99+% and CO2 (Air Products) with a minimum purity of 99.99% were used without further purification. 2.2. Viscometer and Operational Aspects. Full design and basic operational details of the viscometer have already been published.9,12,14 However, some significant features are briefly described below. The fall tube is made of nonferromagnetic stainless steel. The sinker, 0.7781 cm in diameter and 2.094 cm in length, is made of an aluminum core and a ferromagnetic stainless steel shell. This combination permits the adjustment of the sinker density and thus the fall time for effective measurement of a reasonable range of both low and high viscosities. The sinker used in this study has a density of 4 g/cm3 and is magnetically permeable, thus permitting the detection while it falls in the tube. The ratio of sinker-to-tube radii is 0.9799. For concentric falls, a value of 0.93 is required for this ratio. The temperature and the pressure in the viscometer are monitored with an accuracy of (0.5 K (read with a resolution of 0.1 K) and (0.06 MPa (read with a resolution of 0.007 MPa), respectively. The LVDT coils that are used as sensing elements were designed specifically for this viscometer. The three coils are all wound on a single brass washer with a known separation distance between the coils. The signal from each coil is monitored with a dedicated computer as voltage versus time data as the sinker passes through.

Figure 3. Normalized plot of “distance versus voltage” data from manual calibration.

The total internal volume of the viscometer including the view cell, fall tube, and circulation loop is 42.6 cm3. This volume is variable with a movable piston. The displacement of the piston is measured with an accuracy of 0.013 mm with a special position sensor (PS, in Figure 2) based on again LVDT-type detection of the location of a ferromagnetic slug on an extension rod attached to the piston (Figure 2). In loading the viscometer with a solvent or fluid of interest, a specially designed transfer vessel is utilized. Basically, this is a high-pressure vessel that is preloaded with the fluid and then placed on a sensitive balance. The vessel is then connected to a high-pressure pumping unit to pump the fluid from the vessel to the viscometer inlet (FI in Figure 2). The amount charged is monitored from the balance reading. When working with solid polymers, a previously weighted amount of polymer is first loaded to the viscometer through the inlet port for solids (SI in Figure 2) or by removing the fall tube attachment and loading the polymer from the top of the view cell. Liquid polymers such as PDMS are loaded through the solid inlet port using a syringe which is weighted before and after loading to determine the amount charged. Once the viscometer is charged with the polymer and/ or fluid, then the system temperature and pressure are adjusted with the aid of the pressure generator and movement of the piston. In this study we used a gear pump for circulation of the viscometer content instead of a magnetic circulation pump used earlier. Once the homogeneous conditions are verified by observation through the sapphire windows, the fall-time measurements are carried out. 2.3. Calibration. The calibration of the viscometer involves the determination of an instrument constant K, according to eq 1. This is achieved by determining the terminal velocities for fluids of known viscosity and density. In this study such calibrations were conducted using literature viscosity data for n-pentane and for CO2. A new procedure was adapted for the unambiguous determination of the terminal velocities. In this procedure we first generate complete signal versus position history for the sinker as it falls and passes through the coils in the absence of any fluid (simply in air at ambient conditions) by manually moving the sinker along the viscometer fall tube. These voltage signal versus position data are generated at 1 mm intervals for the full fall length of 14 cm. Figure 3 shows this position versus

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Figure 4. Normalized “voltage versus time” data obtained from calibration with n-pentane at 100 °C and 21 MPa.

Figure 6. Variation of viscosity, calculated using Kavg, with pressure. Literature viscosity data at 50 °C are taken from Audonnet and Pa´dua,17 and data at 100 °C are taken from Estrada-Baltazar et al.18

this low-viscosity fluid also. For example, at 97 °C and 40 MPa, the total time for the sinker to travel the full length of the fall tube is about 18 s compared to about 40 s in n-pentane at 100 °C and 21 MPa. Using the terminal velocity and densities determined experimentally, the known density of the sinker, and the literature value of viscosities, the value of the instrument constant was evaluated based on each viscosity data point and an average value of 0.0199 ( 0.0005 was assigned. Using this value, viscosities at different T/P conditions for these solutions could be determined with reliability. Figure 6 shows such a comparison for n-pentane viscosities, where literature data17,18 are different from the data used in generating the instrument calibration constant. Figure 5. “Distance versus time” data obtained from calibration with n-pentane at 100 °C and 21 MPa (where slope ) Vt).

3. Results and Discussion

signal output which has been normalized with maximum and minimum voltage readings to +1 and -1 V. In the second step of the calibration, LVDT signal versus time data are generated for the calibration fluid, i.e., n-pentane. Figure 4 shows such an output for n-pentane at 100 °C and 21 MPa. The voltage readings in this figure are also normalized with respect to maximum and minimum values. The data sets in Figures 3 and 4 then permit the generation of sinker travel distance versus time plots as shown in Figure 5. The slope gives a direct reading of the velocity of the sinker. As shown in Figure 5, the sinker for this system at this temperature and pressure clearly reaches its terminal velocity by the time it enters the third coil. The terminal velocities are typically assigned using an L-t portion corresponding to the third coil even though there is high degree of linearity throughout the three coils. This procedure eliminates the ambiguity that may arise if the times corresponding to only the zero-voltage crossing points corresponding to the center positions of the coils were to be used. For calibration we measured the viscosity of n-pentane at 50, 75, 100, and 125 °C up to 50 MPa pressures at each temperature. The densities of the fluid were directly measured at each T/P condition from the changes in the internal volume of the viscometer. Similar measurements were made with CO2 at 37, 52, 77, and 97 °C up to 50 MPa. Even though the fall times were much shorter in CO2, the plots similar to Figure 5 showed that the sinker reaches its terminal velocity in

We have measured the densities and viscosities of solutions of PDMS in CO2 and PE in n-pentane. The concentrations of the solutions were chosen to correspond to the critical polymer concentration, which is 5.5 wt % for the solution of PDMS in CO2 and 5.75 wt % for the solution of PE in n-pentane. These concentrations for these systems were determined earlier.2,3,19 For the PDMS + CO2 system, the measurements were conducted at 55, 70, 85, and 100 °C and at pressures up to 50 MPa for the 5.5 wt % solutions. Figure 7 is an example of the “voltage versus time” and “distance versus time” plots generated at 70 °C and 48.5 MPa. The density and terminal velocity data at different temperatures and pressures along with the calculated viscosity based on eq 1 and a K value of 0.0199 are presented in Table 1. Figure 8 shows the variation of viscosity with pressure at different temperatures. At these conditions the viscosities are low, being less than 0.22 mPa‚s. For the PE + n-pentane system, the measurements were conducted at 140 and 150 °C. Figure 9 shows the “voltage versus time” and “distance versus time” plots generated at 150 °C and 30.2 MPa for this system. The data at different temperatures and pressures are presented in Table 2. Figure 10 shows the variation of viscosity with pressure at these temperatures. In contrast to the PDMS + CO2 system, the viscosities for these solutions are much higher (nearly 10 times), ranging from about 2 to 4.5 mPa‚s.

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Figure 8. Variation of the viscosity with pressure for a 5.5 wt % solution of PDMS (Mw ) 93 700) in CO2 at different temperatures.

Figure 7. 5.5 wt % PDMS (Mw ) 93 700; Mw/Mn ) 2.99) in CO2 at 70 °C and 48.5 MPa. Table 1. Density and Viscosity of 5.5 wt % PDMS (Mw ) 93 700) in CO2 T, °C

P, MPa

F, g/cm3

Vt, cm/s

η, mPa‚s

55

48.70 44.93 41.74 39.29 34.72 33.19 33.19 30.78 28.04 48.45 46.38 44.90 43.38 41.54 40.07 38.17 35.86 35.59 35.16 48.28 44.83 41.59 39.41 37.75 48.21 46.55 44.83 43.79

0.9801 0.9663 0.9534 0.9420 0.9189 0.9090 0.9090 0.8959 0.8765 0.9349 0.9349 0.9222 0.9203 0.9064 0.9030 0.8887 0.8759 0.8761 0.8723 0.9054 0.8901 0.8734 0.8548 0.8476 0.8610 0.8503 0.8396 0.8325

0.2785 0.3008 0.3182 0.3282 0.3508 0.3603 0.3708 0.3834 0.3922 0.3161 0.3284 0.3334 0.3414 0.3578 0.4021 0.3858 0.4030 0.3964 0.4169 0.3531 0.3707 0.4039 0.4341 0.4388 0.4035 0.4158 0.4311 0.4475

0.2158 0.2007 0.1905 0.1854 0.1749 0.1707 0.1659 0.1611 0.1585 0.1930 0.1857 0.1837 0.1795 0.1721 0.1533 0.1605 0.1543 0.1568 0.1493 0.1744 0.1670 0.1541 0.1442 0.1430 0.1548 0.1507 0.1459 0.1409

70

85

100

3.1. Temperature Dependence of Viscosity. Temperature dependence of viscosity for polymer solutions

Figure 9. 5.74 wt % PE (Mw ) 121 000; Mw/Mn ) 4.3) in n-pentane at 150 °C and 30.2 MPa.

is often described by an exponential Arrhenius-type relationship11,20,21

η ) A exp(E#/RT)

(2)

ln η ) ln A + E#/RT

(3)

or

where η is the viscosity, E# is the activation energy of

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Figure 10. Variation of the viscosity with pressure for a 5.74 wt % solution of PE (Mw ) 121 000) in n-pentane at different temperatures.

Figure 11. Variation of ln η with reciprocal temperature, 1/T, for a 5.5 wt % solution of PDMS (Mw ) 93 700) in CO2 at different pressures. Table 2. Density and Viscosity of 5.74 wt % PE (Mw ) 121 000) in n-Pentane T, °C

P, MPa

F, g/cm3

Vt, cm/s

η, mPa‚s

140

48.51 41.81 34.40 24.26 21.55 18.62 17.59 48.29 44.86 41.65 34.88 31.34 27.76 20.83 15.52 12.97

0.5751 0.5623 0.5511 0.5346 0.5296 0.5296 0.5255 0.5819 0.5780 0.5737 0.5648 0.5597 0.5538 0.5411 0.5310 0.5255

0.0162 0.0170 0.0192 0.0234 0.0251 0.0273 0.0299 0.0147 0.0149 0.0154 0.0165 0.0178 0.0186 0.0221 0.0246 0.0261

4.1986 4.0180 3.5771 2.9509 2.7548 2.5288 2.3113 4.6239 4.5605 4.4368 4.1549 3.8521 3.6805 3.1193 2.8007 2.6496

150

viscous flow, R is the ideal gas constant, and T is the absolute temperature. The present data have been analyzed using this relationship by plotting ln η vs 1/T and assigning values for flow activation energy based on the slopes. Figure 11 shows the variation of the logarithmic viscosity with inverse temperature at different pressures (42, 45, and 48 MPa) for 5.5 wt % PDMS in CO2. The slope of the fitted equation, E#/R, is used to

Figure 12. Variation of ln η with reciprocal temperature, 1/T, for a 5.74 wt % solution of PE (Mw ) 121 000) in n-pentane at different pressures.

calculate flow activation energies. The activation energy is around 7 kJ/mol. Flow activation energies for 1, 2, and 5 wt % solutions of a PDMS sample of lower molecular weight (with Mw ) 38 900) in CO2 have been reported to be in the range 7-9 kJ/mol.15 Figure 12 shows the variation of ln η with 1/T for the PE + n-pentane system at 21, 34.5, and 48.4 MPa. The flow activation energy for this system is around 18 kJ/ mol. This should be viewed with some reservation because of the limited data over a very narrow range of temperatures. The activation energies for 1 wt % solutions of PE (with Mw ) 2150, 15 520, and 108 000) in n-pentane that were reported previously are lower, being in the range from 8 to 12 kJ/mol.14 The difference may stem from the much higher concentration (5.74%) and the higher molecular weight (121 000) of the PE samples in the present study. The flow activation energies obtained for PDMS + CO2 are observed to be lower than those for the PE + n-pentane system. This may be related to the greater backbone flexibility with PDMS. 3.2. Pressure Dependence of Viscosity. As shown in Figures 8 and 10, viscosities increase with pressure at a given temperature. The pressure dependence of viscosity for polymer solutions is often discussed in terms of the volume of activation21,23,24 according to

η ) A exp[(V#/RT)/P]

(4)

(∂ ln η/∂P)T,conc ) V#/RT

(5)

or

where R is the gas constant, T is the temperature in K, and V# is the apparent volume of activation. For low molecular weight liquids, it is reported that V# amounts to 1/4-1/3 of the molar volume; in the case of high molecular weight liquids, V# has been considered as a similar portion of the volume of the flow unit, i.e., of the independently moving part of a macromolecule. For polymer solutions, V# is a composite quantity that lies close to the corresponding value of the solvent.20 Activation volumes of the solution of 5.5 wt % PDMS in CO2 were calculated from the slopes of ln η versus P plots at 55, 70, 85, and 100 °C as shown in Figure 13. The activation volumes were around 55 cm3/mol. The activation volumes of 40-60 cm3/mol have been reported

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Figure 13. Variation of ln η with pressure for a 5.5 wt % solution of PDMS (Mw ) 93 700) in CO2 at different temperatures.

Figure 15. Variation of the density with pressure for a 5.5 wt % solution of PDMS (Mw ) 93 700) in CO2 at different temperatures.

Figure 14. Variation of ln η with pressure for a 5.74 wt % solution of PE (Mw ) 121 000) in n-pentane at different temperatures.

Figure 16. Variation of the viscosity with density for a 5.5 wt % solution of PDMS in CO2 and a fitted Doolittle type of equation for this system. Fitted equation: Y ) 0.000454 exp[4.156/(1 0.3297X)].

for 1, 2, and 5 wt % solutions of a lower molecular weight PDMS (Mw ) 38 900) in CO2.15 Figure 14 shows the variation of ln η with pressure at 140 and 150 °C for the PE + n-pentane system. The flow activation volumes were about 70 cm3/mol in the pressure range up to about 42 MPa and about 22 cm3/ mol for pressures above 42 MPa. Activation volumes reported for 1 wt % solutions of different molecular weight PE (Mw ) 108 000) samples with a narrower molecular weight distribution in n-pentane14 were in the range of 31-44 cm3/mol. 3.3. Density Dependence of Viscosity. A convenient way to interpret the effect of temperature or pressure on the viscosity of a polymer solution is to relate it to a change in the free volume or density. This is achieved by a Doolittle-type equation14,15

η ) A exp[B/(Vf /Vo)]

(6)

variation of the viscosity for PDMS + CO2 as a function of density of the solutions in Figure 16. The data correspond to a temperature range of 55-100 °C and a pressure range of 25-50 MPa. The viscosity data corresponding to different pressures and temperatures tend to collapse to a single curve when plotted as a function of density. The experimental data are used to calculate the optimal values for the three parameters in eq 7 by a nonlinear regression method using a Levenberg-Marquart type of iteration method. The obtained parameters are summarized in Table 3. A significant temperature dependence has not been observed, and all data were represented by an overall equation with a standard error of 4.334 × 10-3 in viscosity (Table 3). The correlative equation given by

η ) 0.000454 exp[4.156/(1 - 0.3297F)]

(8)

or

η ) A exp[B/(1 - VoF)]

(7)

where A and B are constants, F is the density, Vf is the free volume, and Vo is the close-packed volume. The experimentally determined densities for the solution of 5.5 wt % PDMS in CO2 as a function of pressure at four different temperatures are shown in Figure 15. These density values are used to show the

is shown as a solid curve in Figure 16. The estimated close-packed volume is 0.33 cm3/g, which is a little higher than the values reported for 1, 2, and 5 wt % PDMS (Mw ) 38 900) in CO2.15 The close-packed volumes for the solutions of PDMS in CO2 do not appear to change significantly with the polymer concentration and with the molecular weight in the range studied. Figure 17 shows the experimentally determined densities for the solution of 5.74 wt % PE in n-pentane as

Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002 6361 Table 3. Coefficients for the Exponential Equation η ) A exp[B/(1 - VoG)] for the Density Dependence of Viscosity T, °C

A

B

Vo

SEa

55 70 85 100 overall

5.5 wt % PDMS in CO2 4.51 × 10-4 4.16 0.3312 4.53 × 10-4 4.15 0.3300 4.45 × 10-4 4.17 0.3319 4.55 × 10-4 4.15 0.3347 -4 4.54 × 10 4.16 0.3297

4.33 × 10-3

overall

5.74 wt % PE in n-Pentane 6.37 × 10-7 11.44 0.4772

3.799 × 10-3

a Standard error of estimating η values using the given coefficients. It is calculated according to the equation SE ) (∑[yi ycal,i]2/n)1/2, where yi are the experimental values of a given property, ycal,i are the calculated values after regression, and n is the number of data points.

volume is found as 0.48 cm3/g for this solution, which is comparable with that of the previous work done for 1 wt % solutions of PE but of different molecular weight also in n-pentane.14 Doolittle-type relationships have also been found to be effective by other researchers in describing the effect of pressure on viscosity.14 Values for parameter B have been reported to be in the range from 1 to 10 for various fluids, which are similar to present observations. For relatively dilute polymer solutions, Vo values should be of similar magnitude and representative of the closepacked volume of the solvent.14 A value of 0.42 cm3/g is suggested for n-pentane,14 where 0.48 cm3/g was found for the solution of PE + n-pentane. A similar analysis with a viscosity of CO2 results in a Vo value of about 0.30 cm3/g,15 where 0.33 cm3/g was determined for the solution of PDMS + CO2 in the present study. 4. Conclusions

Figure 17. Variation of the density with pressure for a 5.74 wt % solution of PE (Mw ) 121 000) in n-pentane at different temperatures.

This investigation has shown that the viscosities of polymer solutions at the critical polymer concentrations in the pressure/temperature conditions corresponding to a one-phase region can be described by Arrheniustype relationships. The flow activation energies and the volumes are observed to be generally greater in the PE + n-pentane system compared to those in the PDMS + CO2 system. In either solution the density is observed to be an effective scaling parameter. The absolute value of the viscosity for those systems are consistent with the phase separation kinetics in these solutions. The system showing the faster phase separation kinetics, i.e., PDMS + CO2, displays much lower viscosities compared to PE + n-pentane. Literature Cited

Figure 18. Variation of the viscosity with density for a 5.74 wt % solution of PE in n-pentane and a fitted Doolittle type of equation for this system. Fitted equation: Y ) 6.369 × 10-7 exp[11.44/(1 - 0.4772X)].

a function of pressure. Figure 18 shows the variation of viscosity as a function of density for this system. The data correspond to temperatures of 140 and 150 °C and pressures up to about 50 MPa. The Doolittle-type correlation parameters are presented in Table 3, and the corresponding curve given by the equation

η ) 6.369 × 10-7 exp[11.44/(1 - 0.4772F)]

(9)

is displayed in Figure 18. The estimated close-packed

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Received for review November 1, 2001 Revised manuscript received May 7, 2002 Accepted May 8, 2002 IE0108999