I n d . E n g . Chem, Res. 1987,26, 1670-1672
1670
pg/dm2 migrated. Such levels approached the BHT detection limits. Registry No. B H T , 128-37-0;IPS, 108815-92-5;ethanol, 6417-5; acetic acid, 64-19-7; water, 7732-18-5. Literature Cited
Pfab, von W.; Mucke, G. Dtsch. Lebensm.-Rundsch. 1977, 73, 1. Till, D. E.; Ehntholt, D. J.; Reid, R. C.; Schwartz, P. S.; Schwope, A. D.; Sidman, K. R.: Whelan. R. H. Ind. Eng. - Chem. Fundam. 1982a, 21, 161. Till, D. E.; Ehntholt, D. J.; Reid, R. C.; Schwartz, P. S.; Schwope, A. D.: Sidman, K. R.: Whelan, R. H. Ind. Eng. Chem. Prod. Res. Dev. i982b, 21, 106. Till, D. E.; Schwope, A. D.; Ehntholt, D. J.; Sidman, K. R.; Whelan, R. H.; Schwartz, P. S.; Reid, R. C. Food Chem. Toxicol. 1986, in press. Uhde, W. J.; Woggin, H. Dtsch. Lebensm.-Rundsch. 1971,67, 257. I
Bergner, K. G.; Berg, H. Dtsch. Lebensm.-Rundsch. 1972,68,282. Gandek, T. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, 1986. Klahn, J.; Figge, K., "Application of a Mathematical Model on the Migration of Low Molecular Components out of Polystyrene into Fat", Third International Symposium on Migration, Hamburg, Oct 22-24, 1980.
Received f o r review October 27, 1986 Accepted May 7, 1987
Viscosity-Temperature Correlation for Glycerol-Water Solutions Yen-Ming C h e n and A r n e J. Pearlstein* Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona 85721
A four-parameter correlation of the temperature dependence of the viscosity of aqueous glycerol solutions is presented. For large temperature ranges, the errors associated with our correlation are 1-2 orders of magnitude less than those obtained with the exponential and Arrhenius fits and are considerably smaller than the errors due to using the form suggested by Litovitz (1952) for hydrogen-bonded liquids. For pure glycerol, our correlation is slightly better than the five-parameter fit of Stengel et al. (1982). Empirical relationships between fluid viscosity and temperature are required in a number of fluid mechanical situations, including convective stability problems (Palm, 1960; Stengel et al,, 1982; Busse and Frick, 1985) and multiplicity calculations for parallel shear flows in channels (Davis et al., 1983), as well as in numerous heat-transfer applications (Siebers et al., 1985; Carey and Mollendorf, 1980). In addition, the rates of diffusion-controlled chemical reactions depend linearly on temperature and inversely on solvent viscosity, through the Debye equation (Calvert and Pitts, 1966). Also, most methods for estimating self-diffusion and binary diffusion coefficients employ the viscosity of the fluid or solvent (Reid et al., 1977). Thus, it is important to be able to accurately correlate the temperature dependence of the viscosity of pure liquids and liquid solutions. A number of equations have been proposed for the correlation of viscosity-temperature data. Among these are the Arrhenius form p ( T ) = BebIT(Andrade, 1934),the exponential form p ( T ) = Ce-fT (Reynolds, 1886),the Litovitz form
p ( T ) = AeaIRT3
(1)
(Litovitz, 1952), and more complicated expressions (0'Donne11 and Zakarian, 1984). Glycerol and its aqueous solutions are important in many of the fluid mechanical (Stengel et al., 1982; Chen and Thangam, 1985),heat-transfer (Seki et ai., 1978), and chemical kinetics (Hasinoff, 1977) applications. Several correlations for glycerol (Litovitz, 1952; Stengel et al., 19821, as well as one for glycerol-water solutions (Litovitz, 1952), have been developed. Litovitz (1952) proposed a remarkably accurate twoparameter fit (eq 1) for the viscosity of glycerol-water solutions of fixed composition as a function of temperature. The basic idea is that there is an inverse square relation
between temperature and activation energy for the intermolecular hydrogen-bonding interactions in highly associated liquids. In (l), R is the universal gas constant, T is the absolute temperature, and A and a depend on composition. Equation 1 is much better than the exponential correlations used by Stengel et al. (1982) for pure glycerol and Chen and Thangam (1985) for aqueous glycerol solutions. For pure glycerol, Stengel et al. (1982) also proposed a five-parameter empirical fit for the kinematic viscosity (cm2/s)
+
v ( T ) = exp[4.5490 - 0.123092' 9.1129 X 10-4T24.7562 X 10?l"3 + 1.3296 X 10-8T4] (2)
in the range -16.5 OC IT I90 "C, where T is the temperature in "C,and the coefficient of the T 3term is corin Stengel et al. [Note that rected from -4.7562 X glycerol melts at 18.6 "C, but there is a nonequilibrium supercooled glassy liquid state which can persist to -50 "C (Stengel et al., 1982).] Four-Parameter Correlation In the present work, we propose a four-parameter correlation p ( T ) = DeE/T3+FT+GIT
(3)
for aqueous glycerol solutions, in which T is again the absolute temperature. The present correlation, based on the Litovitz, Arrhenius, and exponential forms, is superior to each of these at all of the compositions considered and for pure glycerol gives results which are slightly better than the five-parameter correlation of Stengel et al. (1982). We note that the Litovitz, Arrhenius, and exponential forms are special cases of (3), which is therefore, necessarily more accurate than any of them. In what follows, we compare (3) to the Litovitz, Arrhenius, and exponential correlations for various glycer-
0888-5885/87/2626-1670$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1671 Table I. Maximum and Average Relative Errors (peXpt- ploorr)/perpt, in Percent, for Four Correlations of p( T ), for Aqueous Glycerol Solutions w t %, glycerol le4plmar le4plav leelmar IeLlav leAlav IeLlmax IeAImaa leelav 17.05 0.14 0.61 1.48 8.61 3.80 7.94 0 0.30 1.34 1.64 12.06 23.00 0.46 0.80 5.62 11.24 40 0.33 0.60 6.69 12.88 1.37 14.14 25.97 1.02 50 0.33 16.41 1.15 8.30 0.90 15.30 2.03 29.39 60 0.53 1.94 9.90 17.99 4.95 21.01 35.11 1.36 70 0.41 2.55 12.53 22.22 24.39 40.30 1.11 5.55 80 29.47 47.25 0.52 3.56 1.34 15.90 27.86 7.26 90 1.04 2.19 17.36 32.18 1.83 3.89 30.18 53.70 99 1.02 4.24 42.59 71.60 1.88 20.59 40.08 2.76 100 Table 11. Coefficients in (3) for Various Glycerol-Water Solutions wt % glycerol D,CP E, K3 1.56569 X lo8 40 2.62498 X 10' 50 1.52362 X lo4 1.42058 X los 1.69756 X lo8 60 1.72310 X lo5 70 3.50774 X 10" 2.61076 X lo8 80 1.46345 X lo8 2.53866 X lo8 3.31214 X lo8 90 1.01266 X 10" 99 4.87171 X 2.25682 X lo8 2.21895 X lo8 100 1.00758 X 50
A
m
e
'
e
*
-20
-30
1
1
A
-4 0
A
-6 0 -10
0
10
20
30
40
60
60
70
BO
80
T 1%)
Figure 1. Relative error (pexpt- pCorr)/pexpt of pure glycerol for correlations of (A)Litovitz (1952), (+) Stengel et al. (1982), and (0) present work (eq 3).
01-water solutions. The data are taken from Segur and Oberstar (1951), corrected for a measured viscosity of pure water of 1.002 CPat 20 OC. The constants in the three correlations are determined, at each composition, by a least-squares fit to the data of Segur and Oberstar in the range 0 OC IT 5 100 O C . Table I shows the maximum and average errors associated with the use of the Litovitz, Arrhenius, and exponential correlations, as well as the four-parameter fit of (3). The maximum error is the relative error with largest magnitude in 0 "C IT I100 OC,and the average error is the mean (over T )of the magnitudes of the relative error. The relative error at a given temperature is defined as (Clexpt - P c o r r ) / k p t . Figure 1 shows the relative errors associated with the Litovitz correlation and (3) for pure glycerol. As might be inferred from Table I, the Arrhenius and exponential fits for pure glycerol give much larger errors than either the Litovitz form or (3) and for that reason are not shown in Figure 1. Figure 1 also includes the correlation by Stengel et al. for pure glycerol. [Note that Stengel et al. correlated the kinematic viscosity, v. We have multiplied v by the density, as computed from a linear least-squares fit of the data given by Newman (1968).] Table I1 shows the parameters in (3) for various glycerol concentrations obtained by the least-squares fit.
F, K-' -2.26380 X -1.31294 X -1.52632 X -3.40855 X -2.13821 X -3.08786 X 6.22448 X 7.99323 X
lo-' lo-' lo-*
lo-' lo-' lo-'
G, K -4.50002 X -2.82406 X -3.50154 X -7.66521 X -5.42997 X -7.72577 X 4.65536 X 8.80469 X
lo3 lo3 lo3 lo3
lo3 lo3 10' 10'
Discussion It is clear that (3) fits the original data much better than the Arrhenius or exponential forms, with the average and maximum errors for our four-parameter fit being 1-2 orders of magnitude smaller at all glycerol concentrations. As expected, (3) also gives smaller errors than the Litovitz correlation at every glycerol composition. At some glycerol concentrations, the average and maximum relative errors are 5 times (or more) larger for the Litovitz fit than for (3). We also note that, for pure glycerol, (3) is slightly better than the correlation of Stengel et al. (maximum and average errors of 2.91% and 1.07%) even though (3) has one less parameter. In conclusion, the present work shows that the Arrhenius and exponential forms are completely inadequate for representing the viscosity-temperature relationship for aqueous glycerol solutions over large ranges of temperature. Moreover, the Litovitz inverse cubic term is a sound building block for the development of empirical viscosity-temperature correlations for highly associated pure liquids and solutions. Acknowledgment This work was supported in part by Unocal and in part by NSF Grant MSM-8451157. Nomenclature A-D = preexponential factors a, b, f = constants in Litovitz, Arrhenius, and exponential fits e = relative error, b e , t - pcorr)/pexpt E-G = constants in (37 R = universal gas constant T = temperature, K or "C Greek Symbols p = dynamic viscosity v = kinematic viscosity Subscripts
av = average A = Arrhenius fit corr = correlation e = exponential fit expt = experiment L = Litovitz fit, (1) max = maximum 4p = four-parameter fit, (3)
Znd. Eng. Chem. Res. 1987, 26, 1672-1678
1672
Newman, A. A. Glycerol; CRC Press: Cleveland, 1968; p 6. O’Donnell, R. J.; Zakarian, J. A. Ind. Eng. Chem. Process Des. Deu.
Registry No. Giycerol, 56-81-5.
1984,23, 491-495.
Literature Cited
Palm, E. J . Fluid Mech. 1960,8, 183-192. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Reynolds, 0. Phil. Trans. SOC.1886, 177, 157-234. Segur, J. B.; Oberstar, H. E. Ind. Eng. Chem. 1951,43,2117-2120. Seki, N.; Fukusako, S.; Inaba, H. J. Fluid Mech. 1978,84,695-704. Siebers, D. L.; Moffat, R. J.; Schwind, R. G. J . Heat Transfer 1985,
Andrade, E. N. Phil. Mag. 1934,17(7) 698-732. Busse, F. H.; Frick, H. J.Fluid Mech. 1985, 150, 451-465. Calvert, J. G.; Pitts, J. N. Photochemistry; Wiley: New York, 1966; p 626.
Carey, V. P.; Mollendorf, J. C . Intl. J . Heat Mass Transfer 1980,23, 95-109.
107, 124-132.
Chen, C. F.; Thangam, S. J. Fluid Mech. 1985, 161, 161-173. Davis, S. H.; Kriegsmann, G. A.; Laurence, R. L.; Rosenblat, S. Phys. Fluids 1983, 26, 1177-1182. Hasinoff, B. B. Can. J . Chem. 1977, 55, 3955-3960. Litovitz, T. A. J . Chem. Phys. 1952,20, 1088-1089.
Stengel, K. C.; Oliver, D. S.; Booker, J. R. J . Fluid Mech. 1982,120, 4 11-43 1.
Received for review January 15, 1987 Accepted April 20, 1987
Toughness of Silicone Block Copolyimides Giuliana C. Tesoro and Gesindasamy P. Rajendran Department of Chemistry, Polytechnic University, Brooklyn, N e w York 11201
Donald R. Uhlmann* and Chan-E. Park Department of Material, Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Block copolyimides were synthesized from 3,3’,4,4’-benzophenonetetracarboxylic dianhydride by reaction with 4,4’-methylenedianiline and poly(dimethylsi1oxane) diamines. Aromatic diamineterminated poly(dimethylsi1oxanes) were prepared by the base-catalyzed equilibration of octamethylcyclotetrasiloxane with 1,3-bis(4-aminophenoxypropyl)tetramethyldisiloxane.The amount of poly(dimethylsi1oxane) incorporated in the copolyimides was 5-15 wt % The selected molecular weight of poly(dimethylsi1oxane) diamine was 4500. The fracture toughness of copolyimides was found to increase by a factor of 3 with only 5 w t % poly(dimethylsi1oxane)-modified copolyimide without sacrificing the mechanical strength or thermal stability. The glass transition temperatures of the copolyimides were very close to that of the homopolyimide over the range of poly(dimethylsiloxane) content from 5 to 15 wt % . The morphology and toughening mechanisms have been examined, as have the large-scale mechanical properties of copolyimide films.
.
The importance of polyimides in many applications including coatings, adhesives for bonding metals, thermally stable films and fibers, and advanced composites is well documented. The excellent thermal stability, chemical resistance, and dielectric properties of polyimides make them well suited for dielectric and passivant coatings in electronic devices. However, the brittleness of many polyimides can be a problem under various conditions of processing; and improvements in toughness are desirable for specific applications (Mittal, 1984). The toughening of polymers has been studied by many investigators, and extensive literature reports cover the relevant principles and experimental approaches (e.g., Bucknall, 1977). For example, McGany and his co-workers (Rabideau, 1981) have used Teflon microfibers in conjunction with specific addition polyimides and have reported an increase in the fracture energy of the resin by a factor of 17 at room temperature. However, the improved toughness was achieved at the cost of a reduction in tensile strength, to about 1/4 of its original value. The incorporation of an elastomer as a discrete second phase has been successful in several polymer systems, and this approach has also been investigated for polyimides. St. Clair and St. Clair (1980) evaluated the addition of varying amounts of poly(si1oxanes) elastomers such as Silastic (LS 420), Sylgard (1841, or experimental aromatic amine-terminated siloxane (ATS) and of aromatic amine-terminated butadiene-acrylonitrile copolymers to
a norbornene-based addition polyimide adhesive (LARC13) as an approach to improved toughness. These authors reported increases in peel strength by a factor of 6-7, with accompanying increases in fracture toughness by a factor of 3-5. More recently, the effects of aromatic amine-terminated butadiene-acrylonitrile elastomer (ATBN) on the mechanical properties of some condensation polyimides were investigated (Ezzbll and St. Clair, 1984). In this instance, the yield strength, tensile strength, and tensile modulus were found to decrease with an increasing amount of added elastomer. A carboxylic acid terminated butadieneacrylonitrile copolymer was evaluated in conjunction with a bis-maleimide polyimide by Kinloch and co-workers (1984). With the addition of large concentrations of the elastomer, a 20-fold increase in fracture energy was observed. To date, the study of elastomer additives in polyimides has been primarily focused on high-temperature adhesives, primarily based on addition polymers. In the case of condensation polyimides, where requirements for thermooxidative stability are more stringent, the addition of thermally stable elastomers has not been investigated extensively as an approach to improved toughness, and other concepts are being explored. Recent studies of the polymerization of cyclosiloxanes (McGrath et al., 1983a) coupled with developments in the synthesis of elastomeric poly(si1oxane)modifiers for epoxy resins (Riffle et al., 1983) have provided the impetus for
0888-5885/8~/2626-~672$01.50/0 0 1987 American Chemical Society
