Ind. Eng. Chem. Res. 1988,27, 1361-1366
1361
Modeling Diffusion and Reaction in Epoxy-Amine Linear Polymerization Kinetics Donald F. Rohr and Michael T. Klein* Department of Chemical Engineering and Center for Composite Materials, University of Delaware, Newark, Delaware 19716 Three mathematical models of reaction and diffusion in linear addition polymerizations were developed and compared to experiments with n-butylamine (NBA) and the diglycidyl ether of bisphenol A (DGEBA). T h e first model served as a control and was based on the standard approximation that a single rate constant holds for reaction between all species a t all levels of polymerization. In the second model, a single but average rate constant was a function of the number- and weightaveraged molecular weights. Finally, the third model replaced this single, globally averaged rate constant with a matrix of rate constants accounting for the reactions of individual molecular species. Experimental kinetics data, obtained through FTIR, GPC, and DSC analyses, aided in model discrimination. Either model 2 or 3 provides a suitable account of the influence of diffusion limitations on linear polyaddition reactions. Although the intrinsic chemical activity of a reacting functional group may be essentially independent of chain length (Flory, 1953), overall global or observable polymerization kinetics can be a strong function of chain length or viscosity because of the intrusion of diffusional limitations (Trommsdorff et al., 1948; Sourour and Kamal, 1976; Tulig and Tirrell, 1981). Perhaps the most obvious of such situations is the increase in the rate of monomer consumption that is observed a t high conversions during free-radical polymerizations. This acceleration of rate known as the gel or Trommsdorff effect is actually due to diffusional limitations on the rate of radical termination reactions. More generally, such diffusional limitations might also be expected to suppress the apparent rate constant for reaction between any two oligomeric molecules, including the addition of an oligomeric epoxide and an oligomer with a reactive amine group. Diffusional influences on polymerization kinetics have been analyzed previously. Tulig and Tirrell (1981) used the Smoluchowski equation and a reputation-based diffusion model to describe the diffusion-controlled rate of termination of growing methyl methacrylate free radicals. It is noteworthy that a single, averaged rate constant that was dependent on the weight-averaged molecular weight represented the essential physics of the situation well. North and Reed (1963) had previously analyzed diffusion and reaction in the methyl methacrylate system experimentally through reaction in solvents of differing viscosity. Sourour and Kamal (1976) showed that an intrinsic, third-order, autocatalytic reaction model of epoxy-amine cure fit the early stages of polymerization quite well but failed at higher conversions because of diffusional limitations. Huguenin and Klein (1985) used the Rabinowitch analysis to link Kamal’s kinetic model with a free-volume-based diffusion model into an accurate correlation of epoxy-amine kinetics for the entire range of the cure. The most quantitative polymer reaction and diffusion models have concerned free-radical termination reactions under conditions of complete diffusive control. Also, these combinations of intrinsic kinetics and diffusion models have involved a single, averaged global rate constants that described the reaction between all diffusion-limited species. This motivated both the development of the present models and an analysis of the approaches for combining intrinsic kinetics and diffusion in polymerization systems. In particular, it seemed relevant both to tend the quantitative analyses to epoxy-amine systems and also to probe the appropriate level of model detail. It would be useful
to know not only when but also how to account for diffusion limitations: the appropriate balance of scientific detail and practical simplicity will in general vary with the goals of an analysis. This paper therefore describes the construction and predictions of three distinct models of diffusion and reaction in linear epoxy-amine reactions. A parallel representation of cross-linking systems will follow in a later paper. We first present background concerning the theory of reactions in liquid solution and the Rouse (1953), de Gennes (1971), and free-volume concepts of macromolecular diffusion. The intrinsic chemistry and rate equation used as the vehicle to probe diffusional restrictions are presented next. The three models feature (i) a single, constant rate coefficient; (ii) a single, average rate coefficient that is a function of the number- and weight-averaged molecular weights of the system; and (iii) a matrix of constant rate coefficients that describes the reactions between individual molecular species. Finally, examination of the model predictions in light of experimental kinetics of the polymerization of n-butylamine (NBA) with Shell Epon 828, the diglycidyl ether of bisphenol A (DGEBA), allows analysis of the modeling methods.
Reaction in Solution Rabinowitch (1937) considered a densely packed medium in which molecules were surrounded by others in coordination spheres. Bulk diffusion brought a molecule to within the first coordination sphere of its reacting partner to define a state of encounter. Comparison of the characteristic times for the duration of encounters and intrinsic chemical reaction allowed calculation of a global rate constant in terms of the diffusive and chemical parameters:
When the characteristic time for forming encounters, Tab = dab2/3D,is large relative to the characteristic time for reaction, l / ( v p exp(-E*/RT)), eq 1 reduces to the Smoluchowski equation, ke = 4ada&L (Moore and Pearson, 1981; Tulig and Tirrell, 1981), and represents control by diffusion. Conversely, when the characteristic time for reaction is large, eq 1reduces to the kinetic-controlled rate constant kht = (4/3)adab3Lvpexp(-E * / R T ) . A global rate constant for reaction can thus be written as in eq 2 as the product of an effectiveness factor, 7, and the intrinsic rate
0888-5885/88/2627-1361$01.50/0 0 1988 American Chemical Society
1362 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988
constant, kht, which would be measured in the absence of diffusional limitations: k = vkint (2) The effectiveness factor, 7, is thus given by 1
q=iTz
(3)
where the dimensionless Damkoehler number, Da, is given by
Da =
hint
4ada&D
(4)
For present purposes, eq 2 was used to calculate an apparent or global rate constant that could account for kinetic- and diffusion-controlled reactions as well as the transition between the two. In the modeling that follows, initial rates in the diffusion-unaffected regions are described by kint. Estimation of the diffusivity in terms of species' physical properties and the polymerization product spectrum allowed accounting for diffusion-limited reaction, as follows. The developments of Bueche (1962) and Marten and Hamielec (1979) allowed convenient estimation of the diffusion coefficient of a potential reacting pair in terms of their molecular weights and the viscosity of the surrounding environment. Marten and Hamielec's results are summarized in eq 5, essentially an extension of the
Stokes-Einstein relation with a free-volume model for viscosity and the polymer radius expressed in terms of the molecular weight. The value of the positive exponent, n, is in turn derived from the developments of Rouse (1953) and de Gennes (1971). Rouse (1953) described the diffusion of nonentangled polymer chains by using the concept of a submolecule, a portion of the polymer chain long enough to have a spatial Gaussian probability distribution. The complete chain consisted of N submolecules that formed junctions with the previous and next submolecule. The mobility of a submolecule, which depended on the viscous forces exerted by the surrounding fluid, was inversely proportional to the number of monomer units it comprised. The resulting scaling law for the diffusivity of each submolecule thus indicated an inverse dependence of the chain's diffusivity on its molecular weight: D M-l (6) Equation 6 fails to account for entanglements between polymer chains or other polymer-polymer interactions, which can be important above a critical length (de Gennes, 1971). The dynamics of molten polymer molecules under restriction by interchain entanglements were described by de Gennes (1971). In his development, a single polymer chain, trapped within a three-dimensional network, was confined to move only around and between fixed obstacles in a snakelike manner. The obstacles enclosing the chain's random path constituted a tube through which reptation occurred by the migration of point defects. The steadystate migration of these point defects caused a net forward motion of the polymer chain. Equation 7 summarizes the final result. Thus D is proportional to M - 2 in the entangled region. kBTUda,b2 D= (7) MZ
Equations 6 and 7 were used to account for the dependence of diffusivity on chain lengths below and above, respectively, a transition degree of polymerization of 200 (de Gennes, 1979). The dependence of the diffusivity on viscosity was deduced through free-volume theory. The free volume, vf, is the specific volume, u , less the hard-sphere close-packed volume, Uha. Normalization to the specific volume, as shown in eq 8, provides the fractional free volume, f . We follow the literature in the application of free-volume theory to polymer systems below. f = -
- uhs U
Bueche (1962) modeled the free volume as essentially a linear function of the departure of the temperature from the glass transition temperature, Tg.This is shown in eq 9, where the constant a is the thermal expansion coefficient and 0.025 is the fractional free volume at T , (f(T,)) (Williams et al., 1955; Bueche, 1962). Relating Tgto the polymer product distribution would thus provide the diffusivity as a function of molecular weight. f = 0.025 + CY(T - T,) (9) The dependence of Tgof a monodisperse polymer on its molecular weight, M , can be estimated following the analysis of Fox and Loshaek (1955), the conclusion of which is summarized as
where T,(m), Kl, and K2 are constants for a given polymer. This can be further applied to a distribution of polymers, as suggested by Marten and Hamielec (1979), by substitution of M , for the monodisperse molecular weight in eq 10. In the present analyses, eq 9 and 10 were combined in the calculation o f f as
Finally, then, substitution of eq 11 into eq 5 allowed calculation of a single, average diffusivity as
and, also, the mutual diffusivity, Dij,for the reacting pair i-j as
where n and m depended on the degree of polymerization, as before. With the appropriate parameter values, as specified below, these models for diffusion were used in
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1363 eq 1in the calculation of q for diffusion-limited reactions. Polymerization Model Development The linear polymerization of a difunctional epoxy with a difunctional amine was used as a vehicle reaction. The amine group substitutes a t the less-hindered epoxide carbon, forming a nitrogen-carbon bond; a hydroxyl group results at the other epoxide carbon. The intrinsic reaction is first order in each of the amine and epoxy concentrations and is catalyzed by hydroxyl groups on the surrounding oligomers and present as impurities in the resin. The intrinsic reaction rate expression (Sourour and Kamal, 1976) thus comprises an impurity-catalyzed and an autocatalytic hydroxyl-catalyzed term as illustrated in eq 14-16, which represent material balances for oligomeric moieties with amine-amine, amine-epoxy, and epoxy-epoxy reactive end groups, respectively. In eq 14-16, the global rate constant k’ = qlklCbp + v2kzCoh,where q1 and q2 were calculated from eq 3. j=Nt
-dAA(i)/dt =
2k’(AA(i)AE(j)) j=O
+
Figure 1. Apparent rate constant surface for diffusion-affected additional reaction.
j=i-’
4k’(AA(i)EE(j ) ) j=Nt
-dAE(i)/dt =
J=o
(14)
2k’(AE(i)AA(j)) +
2k’(AE(i)EE(j ) ) 1j=i-1
C 2k’(AA(i-j-l)AE(j))
j=O
+ 2k’(AE(i)AE(j ) ) j=z
- C 2k’(AE(i-j-l)AE(j)) 2 j=o
- C4k’(AA(i-j)EE(j)) (15) j=O
J=Nt
-dEE(i)/dt =
C 4k’(EE(i)AA(j)) +
J=o
2k’(EE(i)AE(j)) -
jzi-1
C 2kt(AE(i-j-1)EE( j ) )
(16)
1-0
The three models differed only in the calculation of the effectiveness factors qI and qz. The first model is of intrinsic kinetics and sets q1 and tza t temporally invariant values of unity. In the second model, M , and M , were used to calculate an average diffusivity and thus average but conversion-dependent values of t1 and q2, applicable to all reacting species. The third model used a discrete pair of effectiveness factors that depended on the molecular weights of the members of the reactant pair. A matrix of rate constants was thus established to account for the reactions of individual molecules, and its elements were dependent on the conversion only through the dependence of viscosity (e.g., via eq 13) on number-average molecular weight. The differences among the models are illustrated in Figure 1, a three-dimensional plot showing the dependence of the t i j on the molecular weight of the reactant pair (i and j ) in an addition reaction at a given system viscosity. Figure 1 is reminiscent of the effectiveness factor-Thiele modulus relationship in catalytic reaction engineering. Diffusion limitations occur when both reactants are of high molecular weight, whereas, at low viscosity, reaction of a monomer with a long polymer chain would essentially occur with intrinsic chemical kinetics. In summary, the intrinsic model uses q = 1 for all species. For the average model, the dependence of the effectiveness factor on molecular weight can be viewed as the q-Ma or q-Mb planes in Figure 1, with the monodisperse molecular weights replaced by M,. For the matrix model, the effectiveness factor of a reacting pair of molecules is given by the complete surface illustrated in Figure 1. Finally, M,, affects both model 2 and 3 through its influence on viscosity.
These three models were solved for identical initial monomer concentrations and identical intrinsic kinetics (measured from the initial rate data, as discussed below) and diffusion parameters (estimated from experimental molecular weight averages, also discussed below). Note, however, that the conclusions to follow are independent of the absolute values of these parameters. In solution, then, initial reaction rates were calculated based on the initial monomer concentrations. These rates were then used to predict the molar concentration of the species and their molecular weight averages a t an incremental time step. Iteration allowed simulation of the entire duration of the cure. One simplification eased the computation: rates of less than Rmin= mol/(cm3 min) were set to zero. This allowed truncation of the computer summations in the finite-difference-based algorithm to include only those species that produced a significant rate. Material balance calculations showed an accounting of 99% of all oligomers. Trial calculations showed that a 6 order of magnitude reduction in Rminto a value of mol/(cm3 min) did not alter the material balance significantly. Experimental Section n-Butylamine (NBA) was obtained from Aldrich (Gold Label) and the difunctional epoxide resin, Epon 828, diglycidyl ether of bisphenol A (DGEBA), was obtained from Shell Chemical Company. NBA reacts with the epoxide ring to form a linear adduct with butyl chains extending from the DGEBA backbone. A stoichiometric mixture was prepared to begin the 298 K polymerization whose kinetics were monitored by FTIR, DSC, and GPC. FTIR samples were prepared on NaCl salt disks in a standard manner (Enns and Gillham, 1983). Samples were scanned at 5-min intervals with a resolution of 4 cm-’. Monitoring the area of the epoxide absorptance peak a t 915 cm-I relative to that for the nonreactive DGEBA bridge carbon between the two p-phenylene groups (1184 cm-’) gave a direct measure of the temporal variations of the conversion of epoxide groups. Numerical differentiation provided the reaction rate. Replicate experiments allowed calculation of a standard deviation for the measurement of rate as (r = 1.5 X min-’. In parallel with the FTIR experiments, the quench of replicate samples, in an acetone-ice mixture, a t discrete time intervals for GPC and DSC analyses gave molecular weight and Tginformation, respectively, as a function of
1364 Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988
c
Tp Nesrurcd Predicted @q IO)
'7 - Tg 300
h
280
4
1
t
'\
b..,
P
o.8
0
0.4
1
8" __
Slnoe Conslant Yodel
o experlmmni
?3C
1 3030
1 :co1
0002
0003
'
1
'
I
I 0
0004
2W
400
'
i MD
'
I
'
M O
I
'
1OW
I 1100
-
' 1400
Time (Minutes)
1 /Mn
Figure 2. Comparison of model predictions and experimental data for the DGEBA-NBA system glass transition.
Figure 3. Model predictions and experimental data for the integral disappearance of epoxide groups.
reaction time. GPC sample vials were prepared by mixing the quenched reaction products with HPLC-grade tetrahydrofuran (THF) spiked with HPLC-grade benzene as an internal reference. This mixture was analyzed with a Hewlett-Packard 1090 GPC/HPLC using Waters Associates Ultrastyragel linear and Ultrastyragel 500-A columns in series to provide Mn (f15%) and M, (f16%) as a function of time. Calibration was accomplished using polystyrene standards. The DSC samples, obtained at the same time intervals as the GPC samples, were prepared in Du Pont sample pans and analyzed for Tg( f 2 K). The discontinuity in the slope of the calorimetric signal versus temperature provided an estimate of Tg. The glass transition temperature of the infinite polymer, Tg(m), was the converged value of T gobtained by repeated analysis of a DSC sample held a t 400 K for 4 h.
of the initial rate data compared to the model predictions for 7 = 1. The thus-determined intrinsic values of klCimp and k, used in all three models were 1.474 x lo-' cm3/(mol min) and 7.848 X lo2 cm6/(mo12min),respectively. The third-order rate constant is approximately 1.8 times the value published by Horie et al. (1970) for the reaction of n-butylamine with phenyl glycidyl ether (4.38 X 10, cm6/(mo12min) at 298 K). This comparison is quite reasonable. The high-conversion data for Mn and M, were used to obtain separate estimates of 4062/A for models 2 and 3. This was accomplished by using the simplex routine to minimize the objective function F defined as
Results and Discussion The analyses were aimed a t resolution of the effects of species' molecular weight on the kinetics of DGEBA-NBA polymerization. The experiments can be organized according to their intent. The first set comprised the DSC and GPC experiments and was directed at estimation of the model parameters. A second set comprised the FTIR and GPC analyses and provided kinetics data, against which the predictions of the three models could be compared. The models are quantitatively represented by eq 1-16, wherein K1, K2, a , f(T,), and T g ( m )are considered material properties and therefore constant. Model parameters were therefore klC,, and k 2 for model 1, and klCimp,k 2 , and do6,/A for mo8els 2 and 3. Their estimation allowed prediction of the temporal variation of the polymer product distribution, degree of epoxide group conversion, and rate of epoxide group conversion. We consider the material properties first. The value for CY in eq 13 was obtained from Huguenin and Klein (1985) the fractional and was invariant at 5.0 X lo-* K-', At Tg, free volume was taken as 0.025, following Bueche (1962). The GPC and DSC experiments provided Mn(t)and Tg(t), respectively. On a time-free basis, these data are represented as Tgversus l/Mn in Figure 2 and were therefore used to estimate the best fit values K1 = 4.785 X lo5 and K2 = 2.273 X 1G2 for eq 10; T g ( m ) was estimated from repeated DSC analysis at 400 K where a converged value of 324 K was obtained. Turning next to model parameters, the intrinsic kinetic constants (klCimpand k,) for use in eq 14-16 were determined from a simplex (Nelder and Mead, 1965) analysis
N
F = C ( Mni.Expt - Mni.Model)2 +
(MwiExpt
- MWiModel)2
(17)
r=l
where N = 3 is the number of high-conversion experimental data points that were used. The separate minimization of F for models 2 and 3 provided values of 4062/A as 2.741 X lo-' and 1.261 X lo-' (g cm2)/(mol s), respectively. The foregoing estimates of a , f(T,), K1, K2, T,(m), klCimp,k,, and 4062/Apermitted solution of the model equations and further comparison against predictions. FTIR data and the predictions of the three models for the temporal variation of the conversion of epoxide during the polymerization of NBA with DGEBA at 298 K are shown in Figure 3. At low time and conversion, where diffusional influences were minor, all three models followed the data closely. The models and data diverged at higher conversion, where the predictions of the single rate constant model were higher than not only the experimental values but also the predictions of the other two models. This was because the single rate constant model did not address diffusional limitations. The second (average rate constant) model predicted the lowest epoxide conversion because its rate constant was impeded by both viscosity (M,) and an overall diffusivity applied to all species, including smaller, more oligomers whose rate constants might be expected to depend on viscosity but otherwise be at intrinsic values. The predictions of model 3 (the matrix model) were intermediate between those of models 1 and 2 because the mobilities and therefore reactivities of molecules were based on their molecular weight and the available free volume. The rate constant for monomeric epoxy and amine groups was always higher in model 3 than the corresponding value in the average rate constant model and was lower than the value in model 1only because of the low free volume. This was true during the initial stages
Ind. Eng. Chem. Res., Vol. 27, No. 8, 1988 1365 Table I. Comparison of Model Discrimination Criteria for the Single, Average, and Matrix Rate Constant Models model single rate constant av rate constant matrix criterion X2 Q x2 Q Y2 Q rate of reaction 4.47 x 101 1.20 x 10-3 3.58 X 10' 1.63 X 3.64 X 10' 1.96 X 3.77 x 10' 0.00 2.03 X 10' 6.20 X 2.06 X 10' 5.58 X MW Mn 8.32 X 10' 0.00 3.04 X 10' 2.41 X 3.25 X 10' 1.15 X rate of reaction and M, 0.00 1.00 x 10-2 7.87 x 10-3 rate of reaction and M, 0.00 7.72 x 10-4 3.65 x 10-4 Mn and M, 0.00 1.15 x 10-3 5.51 X lo4 rate of reaction, Mn, and M, 0.00 2.81 x 10-4 1.26 x 10-4 ~
5000
~~
'
Y a l r u Yodel Avarag. Yodel Siryle Constant Yodel
Matrix Yodel Average Yodel Smile Constant Model
Exprlment
0 0 003
'" h
\
3000
$ 1
0.002
t
2000
rr
0 001
0000
,
0
-----_____
,
l 200
400
'
l 600
'
i
'
M)(I
l IO30
'
i 1200
'
l 1400
0
200
400
T i m e (Minutes)
Figure 4. Model predictions and experimental data for the rate of disappearance of epoxide groups.
800
1000
1200
1400
I600
Figure 5. Model predictions and experimental data for the temporal variation of M,.
of reaction as well as a t higher conversions while comparing the three models at the same free volume. Experimental and model rates are plotted versus time in Figure 4. As for the integral results, the predictions and data coincided initially. Model 1 overpredicted the rate at times greater than 400 min, whereas both models 2 and 3 agreed more closely over the duration of cure. More quantitative model discrimination was achieved through a chi-square ( x 2, analysis. For each model, the value of x 2 is given by
where N is the total number of experimental data points, Cyi(x)- yi(x,a))is the difference between the jth data point and model prediction using parameter vector a and at experimental conditions x , and ui is the standard deviation associated with each experimental measurement. Equation 18 allows calculation of Q(x2/uf), the probability that the given x 2 (i.e., the deviations between the experimental data and model predictions) is due to chance. Ideally, a good model should yield a high Q, but a model providing Q > 0.001 will frequently be accepted (Press et al., 1986). Numerical values for the three polymerization models are summarized in Table I, which shows that although the average and matrix models have higher values of Q (1.63 X and 1.96 X respectively) than the single rate constant model (Q = 1.19 X the fit is reasonable for all three models. Thus, the divergence based on predictions of data for rate alone does not allow decisive model discrimination, although models 2 and 3 appear preferred over model 1. The molecular weight averages are more sensitive to the modeling approach. This is illustrated in Figures 5 and 6, where the single rate constant (e.g., Flory) model greatly overpredicts both M,, and M,, respectively. The average and matrix rate constant models follow the data more
600
T i m e (Minutes)
Matrix Yodel Averege Yodel
Single Conatant Model
0
200
400
600
800
1000
1200
1400
1600
T i m e (Minutes)
Figure 6. Model predictions and experimental data for the temporal variation of M,.
closely, and the latter predicts a more polydisperse molecular weight distribution than the average rate constant model. Possible x analyses on the three models are shown in Table I. For all analyses, the values of Q = 0.00 for the single rate constant model confirm the qualitative conclusion that model 1is inadequate for predicting molecular weight averages and distributions. Models 2 and 3 have reasonable values of Q and are therefore considered equally likely and useful for modeling linear polymerization kinetics. Thus, for these systems, the level of detail in accounting for the influence of species size on global reactivity is rather unimportant. However, it is important to account for diffusion limitations.
Conclusions The fact that the three models predicted different molecular weight distributions is especially interesting in light
1366
Ind. Eng. Chem. Res. 1988,27, 1366-1369
of the similar reaction rates shown in Figure 4. Global kinetics of epoxy-amine polymerization are a much less sensitive probe of species mobility than their size distributions and their moments. Sensitivity was highest at conversions greater than 80%, where diffusivities were influenced strongly by both viscosity and species' chain lengths. Whereas the single rate constant model seems unacceptable in this case, either model 2 or 3 appears capable of reasonable prediction of the development of the molecular weight distribution. Since the average rate constant model is somewhat simpler, it would seem to be the approach of choice. Nomenclature A = diffusion proportionality constant, mol/g AA(i) = oligomer with amine-amine end groups and i repeat groups, mol/cm3 AE(i) = oligomer with amine-epoxy end groups and i repeat groups, mol/cm3 Cimp= concentration of impurities, mol/cm3 Coh = concentration of hydroxyl groups, mol/cm3 D = diffusion coefficient of the reactant pair, cmz/s Da = Damkoehler number, dimensionless E * = activation energy for the intrinsic reaction, kcal/mol EE(1') = oligomer with epoxy-epoxy end groups and i repeat groups, mol/cm3 K1 = glass transition equation parameter, (g K)/mol K2 = glass transition equation parameter, g/mol L = Avogadro's number M i= molecular weight of species i, g/mol M , = number-average molecular weight, g/mol M , = weight-average molecular weight, g/mol N = number of repeat units in a polymer chain, dimensionless Nt = number of repeat units in the largest chain, dimensionless Q = measure of the model goodness of fit, dimensionless R = gas constant T = absolute temperature, K T&n) = glass transition temperature of polymer with n repeat units, K T a b = time constant for diffusion, s U = defect mobility, s/g a = length of a repeat unit, cm b = free volume coefficient, dimensionless bl = length stored in one defect, cm d a b = collision diameter, cm d,, = average defect density per chain length, cm-'
k = global rate constant, cm3/(mol s) kint = intrinsic rate constant, cm3/(mol s) kg = Boltzmann constant, kcal/K f = fractional free volume, dimensionless p = steric factor, dimensionless u = specific volume of polymer, cm3/g uhs = close-packed hard-sphere volume, cm3/g Greek Symbols a = thermal expansion coefficient, K-' 6 = jump distance, cm
& = jump frequency, 7 = effectiveness factor, dimensionless u = frequency of vibration for two reacting molecules, s-l uf = degrees of freedom, dimensionless Registry No. NBA, 109-73-9; Epon 828, 25068-38-6. Literature Cited Bueche, F. Physical Properties of Polymers; Interscience: New York, 1962. Enns, J. B.; Gillham, J. K. J . Appl. Polym. Sci. 1983, 28, 2567. Fox, T. G.; Loshaek, S. J . Polym. Sci. 1955, 15, 371. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. de Gennes, P. G. J. Chem. Phys. 1971, 55, 572. de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornel1 University Press: Ithaca, 1979. Horie, K.; Hiura, H.; Sawada, M.; Mita, I.; Kambe, H. J . Polym. Sci. 1970, 8, 1357. Huguenin, F. G. A.; Klein, M. T. Ind. Eng. Chem. Prod. Res. Des. 1985, 24, 166. Marten, F. L.; Hamielec, A. E. ACS Symp. Ser. 1979, 104, 43. Moore, 3. W.; Pearson, R. G. Kinetics and Mechanism; Wiley: New York, 1951. Nelder, J. A.; Mead, R. Comput. J. 1965, 7, 308. North, A. M.; Reed, G. A. J. Polym. Sci. 1963, I , 1311. Press, W.; Flannery, B.; Teukolsky, S.; Vetterling, W. Numerical Recipes'; Cambridge University Press: Cambridge, MA, 1986. Rabinowitch, E. Trans. Faraday SOC. 1937, 33, 1225. Rouse, P. E. J . Chem. Phys. 1953,21(7), 1272. Sourour, S.; Kamal, M. R. Thermochem. Acta 1976, 14, 41. Trommsdorff, E.; Kohle, H.; Lagally, P. Makromol. Chem. 1948, I, 169. Tulig, T. J.; Tirrell, M. Macromolecules 1981, 14, 1501. Williams, M. L.; Landel, R. F.; Ferry, J. D. J. Am. Chem. SOC. 1955, 77, 3701. Received for reuiew September 24, 1987 Accepted April 20, 1988
Thermal Chemistry of a Utah Tar Sand: An Electron Spin Resonance (ESR) Characterization Leslie R. Rudnick* and Costi A. Audeh Mobil Research and Development Corporation, Central Research Laboratory, P.O. Box 1025, Princeton, New Jersey 08540
A study of the thermal chemistry of a Utah tar sand is described. T h e coking behavior of the intact Utah tar sand is compared with t h a t of t h e bitumen after extraction from the sand. T h e yield structure and oil properties are compared. Study of the free radicals present in the samples was made by using electron spin resonance (ESR) spectroscopy. Radical generation and termination reactions during coking at two heating rates (12 and 120 OC/min) of the intact tar sand and the extracted bitumen were studied directly in the ESR spectrometer. At present, crude oil is the main source of liquid hydrocarbon fuels. However, the known reserves of crude oil are finite, and the search for alternative resources from which liquid fuels could be derived has not stopped. Recently, shale and tar sand deposits have again attracted
the attention of many planners as potential sources of liquid fuels. Tar sands are a well-established resource base and include over 4 x 10l2barrels of hydrocarbon in place (Canada-Venezuela Oil Sands Symposium, 1977). Like shale,
0888-5885/88/2627-1366$01.50/0 0 1988 American Chemical Society
