Ind. Eng. Chem. Res. 1990,29,463-470 Greek Symbols t
= rate of dissipation of turbulent energy, m2/s3
viscosity, Pa s p = specific density, kg/m3 @ = (fuel-air ratio)/ (stoichiometric fuel-air ratio) = equivap =
lence ratio Subscripts 0 = initial (at flamefront) 1 = region ahead of flamefront 2 = zone of combustion 3 = region after flamefront
i = index LFR = laminar-flow reactor PFR = plug-flow reactor r = radial component R = reactant TSB = thermally stabilized burner z = axial component Superscript - = mixed-mean value
Registry No. NO, 10102-43-9; CO, 630-08-0; ethane, 74-84-0.
Literature Cited Aris, R. Introduction t o the Analysis of Reactors; Prentice-Hall: Englewood Cliffs, NJ, 1965. Bernstein, M. H.; Churchill, S.W. Multiple Stationary States and NO, Production for Turbulent Flames in Refractory Tubes. Sixteenth Symposium (International)on Combustion;The Combustion Institute: Pittsburgh, PA, 1977; pp 1737-1745. Chen, J. L.-P.; Churchill, W. W. A Theoretical Model for Stable Combustion inside a Refractory Tube. Combust. Flame 1972,18, 27-36. Churchill, S. W.; Pfefferle, L. D. The Refractory Tube Burner as an Ideal Stationary Chemical Reactor. Inst. Chem. Eng., Symp. Ser. 1985, NO.87, 279-285.
463
Collins, L. R. Prediction and Observation of Decay of Turbulence and Reaction Behind the Flame Front in a Thermally Stabilized Burner. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1987. Collins, L. R.; Churchill, S. W. The Decay of Turbulence in a Tube following a Combustion-Generated Step in Temperature. In review, 1990. Cutler, A. H.; Antal, M. J., Jr.; Jones, M., Jr. A Critical Evaluation of the Plug-flow Idealization of Tubular-Flow Reactor Data. Ind. Eng. Chem. Res. 1988,27,691-697. Hindmarsh, A. C. LSODE and LODI, Two Initial-value Ordinary Differential Equation Solvers. ACM SIGNUM Newsl. 1980, 15 (41, 10. Kee, R. J.; Miller, J. A.; Jefferson, T. H. CHEMKIN: A GeneralPurpose, Problem-Independent, Transportable, FORTRAN Chemical Kinetics Code Package. Technical Report SAND808003; Sandia National Laboratories: Livermore, CA, 1980. Pfefferle, L. D. Stability, Ignition and Pollutant formation in a Plug-flow Thermally Stabilized Burner. Ph.D. Thesis, University of Pennsylvania, Philadelphia, PA, 1984. Pfefferle, L. D.; Churchill, S.W. The Stability of Flames inside a Refractory Tube. Combust. Flame 1984, 56, 165-174. Pfefferle, L. D.; Churchill, S. W. The Kinetic Modelling of the Combustion of Ethane in a Refractory Tube Burner. h o c . World Congr. III Chem. Eng. Tokyo 1986a, 4, 68-71. Pfefferle, L. D.; Churchill, S.W. NO, Production from the Combustion of Ethane Doped with Ammonia in a Thermally Stabilized Plug Flow Burner. Combust. Sci. Technol. 198613, 49, 235-249. Tang, S. K.; Churchill, S. W. A Theoretical Model for Combustion Reactions Inside a Refractory Tube. Chem. Eng. Commun. 1980a, 9, 137-150. Tang, S. K.; Churchill, S. W. The Prediction of NO, Formation for the Combustion of Nitrogen-Doped Droplets of Hexane Inside a Refractory Tube. Chem. Eng. Commun. 1980b, 9, 151-157. Tang, S.K.; Churchill, S. W.; Lior, N. The Effect of Fuel-Sulfur on NO, Formation in a Refractory Burner. AIChE Symp. Ser. 1981, NO.211 , 77-86.
Receiued f o r review July 13, 1989 Accepted November 20, 1989
Development of a Reaction Injection Molding Encapsulant System. 2. Chemorheology of the Anionic Bulk Polymerization of Styrene Walter H. Christiansen, John G. Ekerdt, Isaac Trachtenberg, and Joel W. Barlow* Department of Chemical Engineering and Center f o r Polymer Research, The University of Texas at Austin, Austin, Texas 78712
An experimental protocol and model for predicting the viscosity rise with reaction time of a solution that contains anionically polymerizing styrene in styrene are presented. The solution simulates that to be used in a reaction injection molding (RIM) process for encapsulating microchips. T h e model prediction of viscosity is based on the use of material balances to relate the separately determined influences of polymer concentration, polymer molecular weight, temperature, shear rate, and chemical kinetics. T h e model is found t o predict the induction period for viscosity rise to within about 25% relative error. T h e primary source of error is found to be the rate constant for polymerization. Modification of the rate constant leads to predictions of induction times that agree t o within 10% relative error.
A reaction injection molding (RIM) process for encapsulating electronic components which will rapidly polymerize cross-linkable, substituted styrenics with anionic catalysts is being developed by this laboratory. The motivation for this approach has been discussed earlier in part 1of this series (Your et al., 1989). Because of the relatively high cost of monomers which lead to appropriate encapsulants, our initial kinetics of polymerization studies have used ordinary styrene. In this report, we continue to consider the formation of linear polystyrene by butyl0888-5885/90/ 2629-0463$02.50/0
lithium, because this permits us to use and verify the kinetic data generated in part 1. While the rheology of the linear system is quite different than that of the actual cross-linkable material, many of the general principles are nontheless similar, and the techniques developed to follow the viscosity of the linear polymerization can be applied to follow that of the nonlinear polymerization as well. Knowledge and control of the rate of viscosity increase of the polymerizing mixture is needed to ensure proper filling of the mold and to prevent such problems as in0 1990 American Chemical Society
464
Ind. Eng. Chem. Res. Vol. 29, No. 3, 1990
Figure 1 . Schematic diagram of torque rheometer
complete mold fill due to premature "gelation" and "wire sweep" where delicate wires are broken and detached from microchips by the viscous forces associated with the flowing encapsulant. There is a great wealth of information in the literature concerning the general effects of temperature, polymer molecular weight, polymer concentration, and shear rate on the viscosity of polymer-solvent solutions, yet these solutions continue to be studied because of the varied specific behavior that is observed. This study provides comprehensive data for polystyrenestyrene solutions over the ranges of parameters that are encompassed by RIM. This study also develops a fundamentally based model for predicting the viscosity vs time response for the bulk polymerization of butyllithium-initiated styrene. In contrast to empirically derived models (Roller, 1975, 1986), the more fundamental approach is able to account for changes in chemical formulation. Experimental Apparatus, Procedures, a n d Results T o r q u e Rheometer. A concentric cylinder torque rheometer with disposable cups and spindles was constructed to measure the viscosity rise of isothermally polymerizing styrene. In contrast to other viscometers (Perry et al., 1985; Broyer et al., 1978; Richter and Macosko, 1980), this instrument was designed with standard parts to operate in an isothermal mode. The instrument, shown in Figure 1,consists of a G. K. Heller Model HST20 torque motor stirrer/controller, equipped with an analog output signal which is proportional to the torque required to maintain the stirrer speed at the set value. This signal was recorded with a chart recorder. The spindles have an immersion length of approximately 4 in. and are formed from lengths of 5/,-in.-diameter aluminum rod. The disposable viscometer cups are 3/,-in.-inner diameter aluminum cigar humidors, available from M & N Cigar Company, Tampa, FL. Due to the method of their manufacture, form extrusion, the humidors have inner diameters that are precise to within *0.002 in. The viscometer cup was immersed in a Haake Model A80 refrigerated bath thermostated to hold the temperature to within *O.l "C. The 'Il6-in. annular gap provided a relatively short path for conduction of heat. Thermocouple measurements of temperatures in the gap during polymerizations verified the presence of' near-isothermal conditions, to within a few tenths of a degree centigrade at reaction temperatures below 25-35 "C, depending on the initiator concentration. Higher bath temperatures resulted in greater rates of heat generation by polymerization than could be removed by the bath. This resulted in runaway polymerizations. As determined by operation of the viscometer with 300-P and 1000-P silicone oils from Brookfield Engineering Laboratories, this instrument is capable of measuring viscosities to within 10% relative error. The chief source
of error appears to be the frictional drag in the stirrer gear box. Depending on the spindle speed, the apparatus can generate shear rates at the spindle surface between 6 and 380 s-l and can measure viscosities between roughly 10 and lo5 P. By use of the criterion for stable laminar flow developed by Taylor (1936), laminar flow without vortices is expected for the spindle and cup geometries used in this apparatus provided the fluid viscosity is greater than 0.08 P and the spindle speed is less than 650 rpm. Both criteria are easily met in our experiments. Materials a n d Procedures. All of the chemicals used in these studies were purchased from Aldrich Chemical Company. The styrene was received as 99% styrene monomer, which contained 10-15 ppm 4-tert-butylcatechol inhibitor. The n-butyllithium initiators were supplied as 1.6 and 10 M concentrations in isomers of hexane. As in the previous study (Your et al., 1989), moisture, oxygen, and carbon dioxide were removed to prevent preferential reaction of these materials with butyllithium. Samples to be used for studies of viscosity rise times were typically purified by placing 15 mL of monomer in a clean vial capped with a rubber septum. Oxygen-free argon was bubbled through the monomer for a minimum of 10 min. Larger samples for the purpose of preparing fixed molecular weight materials, used in the study of solution viscosities of polystyrene in styrene solutions, were purified by purging 240 mL of styrene for 45 min with dry, oxygen-free argon in a sealed flask. Regardless of its intended use, the purified styrene was then cooled to near 0 "C with an ice bath. Polymerization was then initiated by rapidly injecting the desired amount of butyllithium through the septum, followed by manual agitation of the container for 5 s to mix the ingredients. After initiation, the larger samples were then placed in an autoclave, constructed from a piece of 4-in. iron pipe, and quickly pressurized to 350 psig to prevent boiling of the monomer during the exothermic polymerization. The flask was removed from the pressure vessel after 45 min of reaction time to ensure that the reaction was complete and that the flask had cooled. The flasks were then broken to remove the polymerized materials. Prior to initiating the 15-mL monomer samples with butyllithium, the torque rheometer was placed in the bath to set it.9 temperature, and the baseline torque for a spindle speed of 100 rpm was determined. This spindle speed was selected because its use permitted a broad range of viscosity measurements, 10-104 P, to be measured at a reasonable shear rate, 60 s-'. Dry, oxygen-free argon was then flowed for several minutes into the viscometer cup to displace oxygen and water vapor. In less than 30 s, the sample vial was then injected with initiator and agitated, and the mixed reactants were dispensed into the viscometer cup with a syringe. The change in torque occurring with polymerization time was then recorded. The viscosity, 7, of the reacting mixture was computed from the torque, r, the spindle speed, a,and system geometry by the well-known Margules equation (Mooney, 1958) for Newtonian fluids:
where the symbols are defined in the Nomenclature section. As discussed below, very little dependency of viscosity on shear rate is observed for polystyrene in styrene solutions over a wide range of molecular weights and polymer concentrations and to shear rates of 30 s-l, and the use of the Margules equation seems justified for the conditions used. The temperature was ranged from 4 to
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 465 Table I. Summary of Polystyrene Molecular Weights sample M, M" MJM, 1
2 3 4 5 6
11500 27 000 33 000 53 000 175 000 215000
7 900 21 000 28 000 48 000 152 000 170 000
1.46 1.29 1.18 1.10 1.15 1.26
25 "C, initiator concentration from 0.015 to 0.050 M, and styrene/hexane volumetric ratios from 7.5 to 190 to investigate the effects of these variables and time on the viscosities of the polymerizing mixtures. The molecular weights of all samples were measured with a Waters Associates Model 244 gel permeation chromatograph (GPC), equipped with a Model R401 differential refractometer detector, using methods previously described (Your et al., 1989; Tung and Moore, 1977). Prior to measurement, each polymer sample was dissolved in toluene to form a 0.1 % solution. Atmospheric contaminants in the toluene terminated the living polymers, as evidenced by the disappearance of the red-brown color characteristic of the styryllithium complex. For each fraction, the elution time, ti,was related to polymer molecular weight, Mi, by calibration with known standards, and the height of the chromatogram from the baseline, hi, was assumed proportional to the weight fraction, wi. The molecular weight average were then computed from the chromatogram by evaluating 20-30 sets of molecular weight and peak height (Mi,hi) via 1/Mn = C(hi/Mi)/(Ch;) i
i
M , = Ch,Mi/(Chi) I
i
(2)
One reason for measuring the molecular weight distribution is to obtain an estimate of the effective initiator concentration, the initiator concentration that actually leads to polymerization of the monomer. This concentration, I , can be determined from M , by the assumption that ionic polymers do not terminate during the polymerization process (Morton and Fetters, 1975):
I = W/(M,V)
(3)
Theoretically, M w / M,, the polydispersity, of ionically polymerized vinyl monomers is near 1.0 so that I can be determined just as well by using the peak value on the chromatogram, which nearly corresponds to M,. This approximation is made for all of the initiator concentrations reported in this work. Roughly half of the butyllithium is found to be inactive by the GPC method when styrene is polymerized in the viscometer. This probably results from exposing it to the moisture adsorbed on the glass and metal surfaces of the viscometer during the preliminary stages of the experiment. The molecular weight distributions of viscometer polymerized materials are similar to those obtained for materials polymerized in more controlled environments; see below. For this reason, we believe that destruction of the ion by moisture occurs prior to propagation or polymerization. The molecular weights and polydispersities, Mw/Mn,are listed in Table I for the polystyrene samples that were prepared to study the effect of molecular weight on viscosity. The polydispersities were generally between 1.1 and 1.3, well above the theoretical value. However, this is not unusual (Morton et al., 1961) and can result from coupling termination reactions that occur when the reaction mixture is exposed to the atmosphere at the end of
-0-0-0-
.1 .1
10
1
Shear
Rate
100
(s-1)
Figure 2. Effect of polymer mass fraction and test shear rate on viscosity of M, = 175000 g/mol polystyrene in styrene at 24 "C. ( 0 ) w = 0.23, (0) w = 0.28, (+) w = 0.33, (0) w = 0.39, (m) w = 0.44, (0) w = 0.48, (A)w = 0.50, (A)w = 0.53.
the reaction (Morton and Fetters, 1975). Despite this broadening of the molecular weight distribution, the samples were judged to be suitable for investigating molecular weight effects on viscosity. Viscosities of Nonreacting Systems. The samples listed in Table I were dissolved in styrene monomer to prepare nonreacting solutions that could be studied to determine the effects of polymer concentration, polymer molecular weight, shear rate, and temperature on solution viscosity. Solutions, each containing 12% polystyrene with a different molecular weight, were initially prepared in sufficient quantity to fill a 600-mL beaker. The solutions were placed in a thermal bath and equilibrated, and their viscosities were measured with a Brookfield RTV rotating viscometer, following well-established procedures (Howard, 1985; Mitschka, 1982). Eight different speed settings with several different spindles were employed to examine the shear rate dependency of viscosity from 0.1 to 30 s-l. Viscosity measurements were made at three temperatures between 8 and 46 O C to explore the effect of temperature. Different polymer concentrations, ranging to 70% polystyrene, were prepared by a combination of solvent evaporation and polymer addition. Polymer concentrations were determined by measuring the loss of solvent from samples that were first dried at room temperature and then placed in vacuo at 100 OC. At least 95% of the solvent was removed at room temperature. While thermal polymerization of styrene at 100 "C is possible in the absence of inhibitor, it is relatively slow (about 2%/h) relative to the styrene stripping rate in the vacuum oven, and the error in polystyrene concentration caused by thermal polymerization is estimated to be less than 1% in most cases. As illustrated by Figure 2 for 175 000 molecular weight polystyrene in styrene, very little shear thinning was observed over the range of shear rates available to the Brookfield RTV viscometer. While the solution viscosity was found to be a function of polymer mass fraction, w, shear thinning of the solution was found to be independent of polymer concentration over the range studied. The change in viscosity with temperature normally follows an Arrhenius relationship at temperatures well above the glass transition temperature of the solution (Rosen, 1982; Nielsen, 1962): 70 = A e x p ( m , / R T ) (4) Figure 3 shows typical results from our studies for 175000 molecular weight polystyrene in styrene solutions that follow the behavior suggested by eq 4. This figure also clearly shows that the activation energy for viscous flow,
466 Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 1000
lo3
I
8
' " ' ' " I
I
)
'
*
'
I
"
'
/ 1
$ .-
100
0 Q
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.1' ' I " ' I ' I ' 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036
10
1iTemperature ('K) Figure 3. Effect of temperature on viscosity showing concentration-dependent Arrhenius behavior. ( 0 )w = 0.23, (0) w = 0.28, (+) o = 0.33, (0) o = 0.39; (M) w = 0.44, (0) o = 0.48, (A)w = 0.50, ( A ) (L' = 0.53,
-
.-
5
v)
0 0
.-
v)
>
0 2
0.4
0.3
0.6
0.5
Mass Fraction
0.7
PS
Figure 4. Effect of polymer mass fraction on the activation energy for viscous flow of polystyrene-styrene solutions. ( 8 )M = 11500, (+) M = 27000, ( X ) M = 33000, ( e )M = 53000, ( 0 )M = 175000, ( A ) M = 215000 g/mol.
AE,, is a function of the polymer weight fraction in the solution, a result that has been seen in other polymersolvent systems (Beisenberger and Sebastion, 1983; Mendelson, 1980). Figure 4 shows that LIE, is independent of polymer molecular weight over the experimental range from 11500 to 215000 g/mol. Lacking a theoretical model to explain this behavior, we fit these data to a third-order polynomial to obtain
LE,= -2564
+ 5.638 X 104w- 1.621 X
1 0 %+ ~~ 1.847 X 105w3f 63 cal/mol (5)
where the uncertainty is calculated from the average root-mean-square (RMS) deviation of the data from the curve. This corresponds to a relative percentage error of less than 4%. These results are similar to those obtained for styrene-ethylbenzene solutions (Mendelson, 1980), although there is some deviation from his results at polymer weight fractions greater than 0.5. This deviation could be due to the experimental temperature of the highly concentrated solutions being within 100 "C of the glass transition temperature, a point at which the Arrhenius equation should begin to show some error (Schwab and Murray, 1985). The viscosity of a polymer solution is generally expected to vary with molecular weight of the polymer according to 80
= KM"
(6)
where K is generally a function of temperature and concentration at low shear rate and where a varies from 1 to 3.4 as the molecular weight, M , is increased through the critical value. Figure 5 shows that the measured viscosities
'
'
'
1
*
c
5
.
'
'
" . " ' . I
lo4 105 106 (Mass Fraction)x(Molecular Weight)
Figure 5. Effect of molecular weight on the viscosity of polystyrene-styrene solutions at 25 O C . (0) w = 0.3, (A) w = 0.4, (0) w = 0.5, ( 0 )w = 0.6.
do vary with two different powers of molecular weight depending on whether the molecular weight is above or below the critical value. The critical value, M,, constructed from the intersection of the lines that describe the molecular weight dependency of viscosity in the two regions, was estimated to be about 45 000 g/mol for this system. This agrees well with 40 000 g/mol reported by Beisenberger and Sebastian (1983) for polystyrene-ethylbenzene solutions and with 46 000 g/mol reported by Graessley et al. (1967) for polystyrene in butylbenzene. There are too few data in Figure 5 to be able to obtain a conclusive estimate of the a values; however, the generally accepted values a = 1.0, uM < 45000 g/mol a = 3.4, wM
> 45000 g/mol
(7)
fit the data reasonably well, where w is the polymer mass fraction. These values are used in the model constructed below. The effect of polymer concentration on the viscosity of concentrated polymer solutions is not well characterized in the literature. Attempts have been made to use an empirical power dependency on concentration (Einaga et al., 1971; Graessley and Segal, 1969) as well as to employ shift factors to obtain a single composite curve of log viscosity vs concentration (Einaga et al., 1971). The former approach yields power coefficients that vary with polymer concentration from 3.7 to 26 for polystyrene concentrations between 0.2 and 0.7 g/mL in various organic solvents, and one suspects that the power coefficient may be a function of polymer molecular weight. The shift factors in the latter approach are found to be functions only of molecular weight at constant temperature; however, one could generally expect the shift factors to be functions of molecular weight, temperature, shear rate, and concentration. With some additional work, one could use these existing empiricisms to the concentration dependency of viscosity. We chose, instead, to determine the concentration function, F i w ) , from the viscosity and the separate variables discussed above via c
Equation 8 is purely empirical and has no profound basis for its form. Equation 8 simply states that the magnitude of the zero shear viscosity is determined by the product of three terms: an Arrhenius term that is found to be a function of temperature and composition, a term that is a function of the polymer molecular weight, and a term
-lo
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al
.-0VI
-
6000
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Ad B
n
.-c
4000
VI
0 0
.-VI
> 0.2
0.4
0.3
0.5
0.6
2000
0.7
n
Mass Fraction PS
0
Figure 6. Function F ( w )vs w in two different molecular weight regimes. Upper curve is for a = 1. Lower curve is for a = 3.4. (0) M = 11500, (+) M = 27000, ( X ) M = 33000, (+) M = 53000, (0) M = 175000, (A) M = 215000.
F(w) that is assumed to be a function only of composition. Figure 6 shows that a single characteristic curve of F ( u ) vs w does result from this empiricism for the low molecular weight materials where a = 1.0. A similar characteristic curve is obtained for solutions of high molecular weight materials where a = 3.4. These curves were best fit with third-order polynomials in polymer weight fraction, w , to give for a = 1,
F(u) = -9.939
-
+
8 2 . 9 7 ~ 2 5 4 . 4 ~- ~277.50~f 0.021 (9)
for a = 3.4, 7 0 . 3 5 ~+ 246.4~'- 2 8 0 . 9 ~f~0.018 (10) The uncertainties in eqs 9 and 10 are computed from the RMS average deviation. The relative percentage error in both curve fits is less than 0.1%. At low shear rates, eqs 5 and 8-10 are able to correlate the data for the entire range of temperatures, molecular weights, and polymer concentrations studied. Equation 8 can be rewritten to give
F(u) = -41.51
-
T~ =
Ma exp(AE,/RT
+ F(w))
10
20
30 40 time (min)
50
60
Figure 7. Viscosity rise as a function of time and temperature for bulk styrene polymerization with I = 0.048 M and CH = 0.10. (0) 20.7 "c, ( 0 ) 15.9 "c,(0) 12.6 "c, (A)8.6 "c,(8) 4.6 "c.
The first paper in this series (Your et al., 1989) established that the kinetics of polymerization followed the form dx/dt = k ( l - x)I'/' (12) where x is the fractional conversion of monomer to polymer and I is the butyllithium concentration. The rate constant, k, was found to be described by k = A exp(-AE/RT) = 4.5 X lo8 exp(-14400/RT) (L/(mol s'))"' (13) The uncertainty in A was found to be a factor of 2 (L/(mol s ~ ) ) "over ~ the range of uncertainty, f0.6 kcal/mol, in AE. An estimate of the viscosity rise with time for an isothermal polymerization can be made by integrating the rate expression to give x = 1 - exp(-kl%)
(14)
The mass fraction of polymer in the polymerizing mixture can be related to the conversion, x , by the relationship
(11)
from which the viscosity of polystyrene in styrene solutions could be predicted, as a further test of the correlation. Two studies from the literature of the polystyrene-styrene systems were examined, one with M , = 162000 and Mw/Mn= 1.74 (Nishimura, 19651, and the other with M , = 240000 and M,/Mn = 3.0 (Ide and White, 1974). Nishimura's viscosity values are a factor of 3 higher than predicted, and those by Ide and White are a factor of 1.5-4 times greater than predicted. The reasons for these differences are not known; however, it is interesting to note that the melt viscosity of polystyrene, prepared by anionic polymerization, has been previously observed (Penwell and Graessley, 1974) to also be a factor of 2 lower than other nearly monodisperse polystyrenes with similar molecular weights. Viscosities of Reacting Systems. Typical viscosity rise data for anionically polymerizing styrene, obtained with the torque rheometer described in Figure 1, are shown in Figure 7 for several temperatures between 4.6 and 20.7 "C. For these temperatures, the polymerizations are isothermal, as measured by implanted thermocouples, and the solution viscosity is observed to slowly build with time for an "induction period", after which a very rapid increase in viscosity occurs. Duplicate runs a t four of the five temperatures studied gave nearly identical results. This suggests that the torque rheometer is capable of measuring viscosity to within about 10% relative error, when operated according to the procedures described earlier.
cH
where = pHVH/ ( VMpM) is the mass ratio of hexane to monomer initially in the system. The hexane in these experiments is the carrier solvent for the butyllithium, and its presence in the reaction mixture, though small, is unavoidable. The apparent molecular weight of the growing polymer chains can be estimated by
M = 2MoS+/I (16) where the factor of 2 arises from the associaton of the growing polymer ends into dimers (Morton, 1983) when the solvent is not a Lewis base but is, instead, nonpolar and aromatic (Quirk and McFay, 1980). Equation 16 assumes that initiation is fast, relative to polymerization, that termination of the growing chain does not occur, and that each butyllithium initiator molecule initiates only one chain. Provided the initiator concentration, I , is determined from measurement of the molecular weight, as described above, these assumptions are all reasonable and verified by others (Morton and Fetters, 1975;Worsfold and Bywater, 1960). The prediction of isothermal viscosity rise with time requires the use of eqs 13-16 to compute M and w. From these values, the viscosity can be calculated from eq 11 with F ( o ) evaluated from either eq 9 or 10 and AE.,calculated from eq 5. A comparison of the calculated induction periods with those observed for isothermal polymerizations is shown in
Ind. Eng. Chem. Res., Vol. 29. No. 3, 1990
418
4000 3000 ).
.-
L
2
2000
0
.-v)
>
1000 .001
0 0
20
10
30
40
50
60
'
0.0033
time (min)
I
I
I
0.0034
0.0035
0.0036
0.0037
1iTemperature ( " K )
Figure 8. Comparison of viscosity rise data to predictions. (-) Prediction using eq 13; (- - -) prediction using eq 17; (0)20.7 "C, I = 0.035 M, {H = 0.01; (A) 14.7 "C, I = 0.035 M, {H = 0.07; (0) 4.6 "C, I = 0.048 M, [H = 0.1.
Figure 10. Comparison of rate constants found from viscosity rise with those obtained from integral analysis of chemical studies (0) kinetics, eq 17, upper line, and with those obtained from analysis of initial rates eq 13, lower line. 25
,
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I
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t
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0.0
0.2
0.4
0.6
0.8
1.0
Monomer Conversion
Figure 8. The calculated induction period for viscosity rise under isothermal conditions agrees well with observed values at the lowest temperature, 4.6 "C, but slightly overpredicts the induction time at higher temperatures. As discussed above, relative errors in the determination of molecular weight, less than 1070,and in the initiator concentration, less than 15%,can account for some of the discrepancy between experiment and prediction. This discrepancy may also be partly due to the uncertainty in the rate constant, eq 13. This rate constant was determined from analysis of the initial rate data, and its use does not accurately predict the isothermal conversion of styrene to polystyrene beyond the 60% conversion level (Your et al., 1989). As illustrated in Figure 9, the conversion range of interest to the prediction of induction periods is from 50% to 80%, where the depletion of monomer and increase in the average molecular weight cause the viscosity to rapidly build. The original conversion vs time data were reanalyzed to include the data to 80% conversion, using an integral method (Levenspiel, 1972) in which the rate constants were found from the slopes of lines formed by plotting -(ln (1 - x)) vs time. This approach gave a rate constant X
lo9 exp(-15300/RT)
(L/(mol s * ) ) ' / ~
4
8
12
16
20
24
time (min)
Figure 9. Viscosity rise as a function of conversion for bulk styrene 20.7 "C, (0) 15.9 polymerization with I = 0.048 M and fH = 0.10. (0) "c, (0)12.6 "C, ( A ) 8.6 "C, (8)4.6 " C .
k = 2.6
0
117)
where the uncertainty in the prefactor is f2.3 x lo9 (L/ (mol s~))'/~ and that in the activation energy is f0.6 kcal/mol. The predictions made with eq 17 are shown in Figure 8 to be quite accurate at the higher temperatures and to underpredict the induction time at 4.6 "C by no more than 10%. Rate constants were also found that gave the best fit of the model to the viscosity rise data at each temperature.
Figure 11. Comparison of nonisothermal viscosity rise data to predictions. (--) Prediction using eq 13; (- - -1 prediction using eq 17; (0)I = 0.040 M, (H = 0.0067.
These values are plotted in Figure 10, together with eqs 17 and 13. All of the best fit values fall within the range of the two rate expressions. This suggests that the discrepancy between experimental and predicted induction times is caused by uncertainties in the rate constant expressions. This also suggests that induction time data could be used to determine the chemical rate constant, provided that all of the other pertinent information is known. We intend to use the model, described above, to predict the changes in viscosity that occur during nonisothermal and high shear rate polymerizations such as the exist during the mold-filling step in the RIM process. Due to the highly exothermic nature of this polymerization and the low boiling point of styrene, it is not possible to run a polymerization in the torque rheometer used here under conditions that even begin to approach those expected in the RIM machine. However, some idea of the applicability of the model to nonisothermal reaction conditions can be made by ramping the setpoint of the refrigerated bath betwen 5 and 20 "C. Since thermal lag between the polymerizing medium and the bath could be important, the temperature of the reacting medium was measured with thermocouples embedded in the viscometer wall. Figure 11 shows a typical temperature profile and a comparison between observed and predicted induction periods. Observed and predicted periods are seen to agree to within an uncertainty of about lo%, a result that suggests that the model may be useful for describing nonisothermal polymerizations. No experimental work has been done in this project to measure the influence of high shear rate on the viscosity of the polymerizing mixture. It is nonetheless interesting to estimate the effect using information from the literature.
Ind. Eng. Chem. Res., Vol. 29, No. 3, 1990 469
h
Q
4000
.-VI0
P
Y
3000
r
.-VI
L
0
2000
0
.-
VI
>
1000
n 5
0
io
15
20
25
30
time (min)
Figure 12. Predicted effect of shear rate on viscosity rise at 12.6 "C, = 0.016.
I = 0.041 M, and
(,.,
A simple model has been developed (Graessley, 1974; Ree and Eyring, 1955) that relates the reduction in viscosity to the shear rate, y,and a characteristic time constant, 7, 7/70
=
( ~ / Y T )s i n k '
(77)
(18)
where tlo is the viscosity of the fluid at zero shear rate. The characteristic time constant has been shown (Graessley et al., 1967; Abdel-Amin et al., 1973; Penwall et al., 1974) to be of the order of the Rouse relaxation time, 7
= 6q&/ (a2cRT)
(19)
where c is the polymer concentration, grams/cubic centimeters. While the experimental and Rouse time constants are usually not identical, the Rouse time constant is sufficiently similar to provide a reasonable approximation of the shear thinning of viscosity when used with eq 18. This approximation is calculated at various shear rates, y, and illustrated in Figure 12 for an isothermal polymerizing bulk styrene system. The calculation follows the methods outlined above to establish qo, M, and c at a given time, from which 7 and 7 can be calculated by application of eqs 19 and 18, respectively. The effect of shear rate on the viscosity rise is predicted to be relatively small for shear rates less than lo3s-l, although substantial shear thinning appears possible for shear rates in the 104-s-' range. In the RIM process, the highest shear rates, typically in excess of io5 s-l for impingement mixers (Cross et al., 1985), occur in the mix head where conversions are typically low. As a result of these low conversions, shear thinning should not be a significant issue in the mix head. High shear rates will be generated during the mold-filling step, although these rates must be low enough to avoid turbulence. Although conversion could advance significantly during fill, the viscosities need to be relatively low, less than 1000 P, to ensure uniform, rapid fill. Consequently, shear thinning is not expected to be a significant issue during mold fill in a well-designed system.
Summary and Conclusions The torque rheometer, developed for this study, appears to be adequate for determining data of viscosity rise vs time for the anionic polymerization of styrene with butyllithium. Properly calibrated, this device can measure viscosity to within 10% relative error a t shear rates up to 30 s-l. Due to poor heat transfer, reaction temperatures and rates must be kept relatively low to ensure isothermal operation of the viscometer. Temperatures as high as 25 "C could be used when the reaction was initiated with 0.04 M butyllithium. While the viscosity rise is similar in appearance to the "gel point" seen in cross-linking systems, no three-di-
mensional structure is generated during polymerization. The reason for the rise in viscosity with reaction time is seen to be simply the high sensitivity of solution viscosity to polymer molecular weight and concentration. The model for predicting viscosity rise with reaction time, developed here, seems to predict the behavior of polymerizing styrene solutions quite well. Depending on the kinetic rate constant used, the model is able to predict the induction times prior to rapid rise in viscosity to within better than 25% relative error. More than half of the error is related to uncertainty in the rate constant, which describes the kinetics of polymerization. Because the viscosity only begins t o rise rapidly when the conversion exceeds 50%, rate constants determined from kinetic data a t high conversion seem to predict the viscosity rise more accurately than those determined from analysis of initial rates where the conversion is low. Over the limited range of temperatures tested, the model seems to accurately predict the induction period for nonisothermal polymerizations. This result will be important to the accurate modeling of polymerization in the RIM process. In contrast, calculations indicate that the shear thinning of viscosity may not be particularly important to the modeling of the RIM process.
Acknowledgment Support for this work by the State of Texas Advanced Technology Program is gratefully acknowledged.
Nomenclature A = preexponential factor, P a = power coefficient for the effect of molecular weight on viscosity c = polymer mass concentration, g/cm3 AI3 = polymerization activation energy, cal/mol AE, = activation energy for viscous flow, cal/mol F ( w ) = empirical function h = wetted height of viscometer spindle, cm hi = chromatogram height above baseline, cm I = initiator molar concentration, mol/L k = polymerization rate constant, &/(mol s ~ ) ) " ~ K = proportionality constant M = polymer molecular, weight, g/mol M , = critical molecular weight for entanglement, g/mol M i= molecular weight of species i, g/mol M , = number-average molecular weight, g/mol Mo = monomer molecular weight, g/mol M w = weight-average molecular weight, g/mol R = ideal gas constant, cal/mol/K or erg/mol/K ri = viscometer spindle radius, cm ro = viscometer cup radius, cm So = initial monomer concentration, mol/L T = temperature, K t = time, s , V = volume, cm3 VH = hexane volume in reaction mixture, cm3 VM = initial monomer volume in mixture, cm3 W = mass of polymer in sample, g x = fractional molar conversion Greek Symbols y = shear rate, s-l CH = initial mass ratio, hexane/monomer 7 = viscosity, P vo = zero shear viscosity, P pH = hexane density, g/cm3 pM = monomer density, g/cm3
r = Rouse relaxation time, s = mass fraction polymer
w
Ind. E n g ~Chem. Res 1990, 29,470-475
470 D = spindle speed, rad/s
Registry No. Butyllithium, 109-72-8; styrene, 100-42-5; polystyrene, 9003-53-6.
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