1210
Ind. Eng. Chem. Res. 1990,29, 1210-1218
Modeling Diffusion and Reaction in Cross-Linking Epoxy-Amine Cure Kinetics: A Dynamic Percolation Approach Donald F. Rohrt and Michael T. Klein* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Three models of reaction and diffusion in cross-linking addition polymerizations were developed and compared to experiments with the epoxy-amine system of the diglycidyl ether of Bisphenol A and bis(paminocyclohexy1)methane (DGEBA-PACM-20). All models were phrased in terms of a dynamic percolation grid whose sites were monomers and whose site connections were epoxy-amine bonds. The first model served as a control and was based on the standard equal reactivity approximation. In the second model, a single but average rate constant, which was a function of the weight-average molecular weight, was applied to all reacting pairs. Each reactive pair had a potentially unique rate constant in the third model. This was a function of the reaction environment, through the free volume, and the molecular weights of the reacting pair. Monte Carlo simulation provided reaction kinetics and information about the developing polymer that agreed well with the experimental data. The disguise of intrinsic kinetics by molecular weight related diffusion limitations during polymerization is well-known. The Trommsdorff effect (Trommsdorff et al., 1948) during chain polymerizations, where transport limitations on the rate of radical termination reactions lead to an increase in the steady-state concentration of free radicals and thus the rate of monomer consumption, is a classic example. A rigorous account of the influence of transport on this class of reactions exists (Tulig and Tirrell, 1982). Transport can also influence observable polycondensation kinetics (Horie et al., 1970; Huguenin and Klein, 1985; Lunak et al., 1978). The epoxy-amine reaction system provides a relevant example. The polymerization proceeds by substitution of an amine functionality at the less hindered epoxide carbon, which leads to formation of a carbon-nitrogen bond and a hydroxyl group on the more hindered carbon. Difunctional epoxies and amines lead to linear polymers, and higher functional reactants lead to cross-linked polymers. Transport limitations have been observed during polymerization of both (Sourour and Kamal, 1976;Huguenin and Klein, 1985; Rohr and Klein, 1988). Rohr and Klein (1988) recently applied the Rabinowitch (1937) analysis to link intrinsic kinetics and diffusion models into three separate models of linear epoxy-amine polymerization kinetics. This can be summarized in terms of an overall second-order reaction rate constant: ~~T~ABDABLUP exp(-E * / R T ) km = (1) (1/TAB) + u,(exp(-E * / R T 1) which can be phrased in terms of an observable intrinsic rate constant (kint) and an effectiveness factor (vAB):
where the Damkohler number Dam = kint/!4idAFLDm) is a measure of the influence of transport limitations on observable kinetics. The Rouse (1953) and de Gennes (1979) models provided estimates for Dm(M), whereas a free-volume model provided DAB(p). These workers showed that neither conversion nor instantaneous rate was a sensitive probe of transport limi-
* Corresponding author. ' Present address: General Electric Company, Corporate Research and Development, Schenectady, NY 12301. 0888-5885/90/2629-1210$02.50/0
tations. The models diverged most in their predictions of number- and weight-averaged molecular weights. The object of the present paper is to report on the development and predictions of three conceptually similar models for cross-linking epoxy-amine polymerizations. Our approach was to phrase polymerization in terms of a three-dimensional, dynamic percolation grid, where the grid points represented monomer units and the inter-point connections represented bonds. This approach rendered the structure of the developing oligomeric mixture explicit, which permitted exact calculation of M for each oligomer and also convenient estimation of p through a free-volume model. Thus, each potential pair of reacting sites on the percolation grid had a unique DAB,Da, and therefore T~ in eq 2. A Monte Carlo simulation computational technique, wherein the probability of reaction between A and B sites, Pm, was dependent upon k as deduced from eq 2, provided monomer conversion, M,, M,, the cluster size distribution, and cross-link types and densities as a function of reaction time. Model assessment is via comparison of each model to experimental results. The present model is an extension of vast literature concerning the analysis of nonlinear polymerizations. Flory (1953) and Stockmayer (1944) developed their theories of gelation in multifunctional polymers by invoking three assumptions: all functionally similar groups had equal reactivity; each group reacted independently; and no intramolecular reactions could occur. These assumptions allowed statistical analysis of nonlinear polymerizations and calculation of the gelation point, gel fraction, and weight- and number-averaged molecular weights as a function of monomer conversion. Gordon and Scantlebury (1964) employed the use of probability generating functions to account for first-shell substitution effects in polycondensation reactions. In their work, the equal reactivity assumption of Flory and Stockmayer was relaxed, and the effect of a neighboring site on the reactivity of a given functionality was examined. By varying the influence that a neighboring site might exhibit, Gordon and Scantlebury showed the gelation point could shift and the calculated molecular weights would differ when compared to the case where the equal reactivity assumption was invoked. More recently, Miller and Macosko (1976a,b) developed the recursive method for the analysis of nonlinear polymerization and postgelation network properties. Their approach does not require the use of probability generating 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1211 functions and is easily applied to many different reactive polymer systems. Unequal reactivity and first shell substitution effects can also be addressed through their method (Miller and Macosko, 1978). All of the foregoing statistical methods are extremely powerful. However, they fall into difficulty when applied to a situation where the rate constant for reaction exhibits polymer chain-length dependence. Stauffer (1985) has described the application of lattice percolation for modeling polymer reactions. Stauffer et al. (1982) discuss some of the differences between lattice percolation and the Flory-Stockmayer approach. These differences lie primarily in the treatment of intramolecular loops, spatial dimensionality, and excluded volume effects, which are subsequently reflected in the predicted behavior of both network and polymer properties close to the critical point of gelation. Lattice percolation is a more readily applicable method for examining the influence of diffusion on the apparent kinetics of nonlinear polymer reactions and the resulting network properties. This is the method of choice in the present work. Outlining the remainder of this report, we first describe the epoxy-amine chemistry in more detail. We then provide details of the diffusion model used to calculate vAB. This is followed by discussion of the Monte Carlo simulation, with special focus on the link of kAB to PAB.The overall polymerization kinetics and the development of the polymer structure for three models, which differ only in the calculation of qAB, are considered next. Finally, the model predictions are examined in the light of the experimental kinetics of the system of the diglycidyl ether of Bisphenol A-bis(p-aminocyclohexy1)methane (DGEBA-PACM-20).
drogen, Coh is the concentration of hydroxyl donor hydrogen formed by reaction, C,, is the epoxide group concentration, C, is the amine hydrogen concentration, and k, and k, are rate constants. Equation 3 therefore provides kht for evaluation of Da. Specification of DABprovides the final information necessary for evaluation of 9. Branched Polymer Diffusion. Two general situations were considered: pre- and postgelation diffusion. The diffusivity of an oligomer through an environment of finite species was affected by its molecular weight and the free volume of the surrounding environment. The diffusivity of an oligomer moving through an infinite network (gel fraction) was affected also by the molecular weight between cross-links. The effect of the glass transition temperature ( T ) was accounted for in the free volume. kree Volume. Equation 4 summarizes the basic approach, where the self-diffusion coefficient of an oligomer was dependent on its molecular weight and the free volume of the medium: (4)
The free volume, calculated as f = 0.025 + a(T - T,)
was related to the polymer structure through a model for TB' Di Benedetto's equation (eq 6) has been reported by Nielsen (1969) as a method for relating T, to monomer conversion. In eq 6, Ex/E, is the ratio of lattice energies
Model Development Epoxy-Amine Chemistry. The kinetics of the crosslinking polymerization of the diglycidyl ether of Bisphenol A and bis(p-aminocyclohexy1)methane monomers (structures I and 11, respectively, Chart I) provided a convenient vehicle for the development of the modeling approach, which should generalize easily. Smith (1961) has summarized the intrinsic chemistry of epoxide ring opening and subsequent formation of the carbon-nitrogen bond. The epoxide oxygen forms a weak bond with an available donor hydrogen, and the less substituted carbon is attacked by the nitrogen of an available amine. Subsequent rapid hydrogen shifts form products that result in a new carbon-nitrogen bond and a hydroxyl group. The hydroxyl groups formed by the reaction provide more donor hydrogen, which leads to autocatalytic reaction kinetics (Horie et al., 1970). The reaction rate of eq 3 is consistent with the Smith mechanism and provides the ?-
=
(&&p
+ k,Coh)CepCam
(5)
for cross-linked and uncross-linked polymers, F J F, is the ratio of segmental mobilities for cross-linked and uncross-linked polymers, TBois the glass transition temperature of the initial polymer, and p is the conversion of monomer. Clearly Tgincreases with increasing p . Thus, eq 6 provided the link between conversion and free volume as
(7) This in turn provided the effect of free volume on the self-diffusion of all oligomers. The more direct molecular weight dependence of the diffusivity was handled separately, as follows below. It should be noted, however, that a more recent form of the Di Benedetto equation (1987) would be more appropriate for models where T,was the direct property of interest. We use eq 6 as a simple correlation between Tgand p.
(3)
intrinsic basis for modeling epoxy-amine reactions. In eq 3, Cimpis the initial concentration of impurity donor hyChart I
CH,
DGEBA, I H
H PACM-PO. I1
1
1212 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
D = DoXM-2
Me
M
Figure 1. Regimes for the self-diffusion of a labeled chain, M. within a netwoik consisting of X monomers between cross-links (de Gennes, 1986).
Pregelation Molecular Weight Effects. The dependence of the self-diffusion coefficient on molecular weight prior to gelation followed the form used in the previous (Rohr and Klein, 1988) linear polymerization models: D = D&-l
(M < Me)
D = D&gw2
( M > Me)
(8)
(9)
Equation 8 pertains to free draining or Rouse diffusion, whereas eq 9 handles entangled situations. Pregelation effects of cross-linking were thus absorbed in the freevolume term. The use of Me = 30000 follows de Gennes' (1979) estimate that an average degree of polymerization of 100 marks the onset of entanglement effects. Postgelation Molecular Weight Effects. At the gel point, the polymer medium changes from a mixture of finite oligomers to an infinite network interspersed with finite oligomers. Although the network is of infinite molecular weight, the mobilities of sites attached to it are not necessarily zero. There were, therefore, two types of diffusing species after gelation: finite molecular weight oligomers and polymeric segments attached to the network. Sol-phase diffusivities were estimated by following the recent work by de Gennes (1986). He outlined three critical regimes of self-diffusion for a single chain of molecular weight M diffusing through a network with X monomers between cross-links. These are further illustrated in Figure 1,where the various regions are delineated on a plot of X versus M . The first regime is bounded by M < Me and X > M , where Me is the critical molecular weight where entanglements become important. In this regime, the self-diffusion coefficient follows an inverse dependence on molecular weight, exhibiting a Rouse-like (1953) diffusivity:
D = D&-'
(10)
In the second regime ( M > Me and X > Me),self-diffusion is by reptation. Summarized by eq 11, the diffusivity is
D = D&Jt2
(11)
modeled as chain reptation through spaces of the network, in a manner similar to the case of entangled linear polymer diffusion. The third region, where X is "small", accounts for the influence of the gel. Here the chain's movements are severely hampered by the small characteristic distances in network spacing. This situation arises for X < Me and X < M , and the resulting diffusion coefficient is both a function of the network spacing and the molecular weight. This is summarized by eq 12.
(12)
Gel-phase diffusivities were calculated for two types of lattice sites. The first type of sites was those connected to only one or two other sites along a linear chain segment of the network. The diffusivities of these lattice "linear" sites were estimated as that for an unattached linear oligomer of the same molecular weight (eq 8 or 9). The second collection of lattice sites was the cross-link points joining three or four sites. For this case, the chain segments extending from the cross-link were considered to be involved in its diffusive movement, and the characteristic molecular weight was the sum of all the linear chain segments directly attached to the cross-link. The work of Bartels et al. (1986) was then used to estimate the diffusivity of the starlike moiety consisting of the cross-link site and its attached polymer segments. These workers investigated the self-diffusioncoefficient (DB3)of a three-armed star polymer relative to the selfdiffusion of the linear polymer with the same molecular weight (DL). They showed, experimentally and computationally, that DB3/DLwas a function of the ratio of molecular weight divided by the entanglement molecular weight, MB3/Me. In their work, the ratio DB3/DLvaried from a value of lo-' at low values of MB3/Meto as low as loW5at higher ratios of MB3/Me. Herein the value of DB3(M)/DL(M) = was used for evaluation of the selfdiffusion of sites attached to the cross-link. The sum of the molecular weights of the linear chains attached to the cross-linked site was used in the evaluation of the linear self-diffusion coefficient, DL(M). To summarize the diffusion models, various scaling arguments were used to assign diffusion coefficients to lattice sites prior to and after gelation. Those lattice sites belonging to the gelled infinite network were assigned mobilities that depended on a characteristic molecular weight and whether the site was a cross-link or a linear chain segment. As a better understanding of the diffusional behavior of cross-linked polymers is developed, new concepts can be readily incorporated into this modeling approach. Dynamic Percolation as a Model of Epoxy-Amine Polymerization Kinetics. The cross-linking polymerization reaction model was phrased in terms of a structure-explicit, dynamic percolation grid. The reactions of oligomeric functional groups were chronicled in terms of individual reaction (transition) probabilities, which were in turn dependent on intrinsic chemistry and transport phenomena. This allowed relaxation of the classical Flory-Stockmayer equal reactivity percolation theory, which does not account for excluded volume effects, intramolecular reactions, or the variable reactivity of oligomers. The Monte Carlo computational technique allowed prediction of the temporal variation of the molecular weight distribution, reaction rate, reaction extent, and other system properties without the restriction of the equal reactivity assumption. Reaction Grid. A three-dimensional, pseudostatic reaction grid approximated the environment of epoxy-amine molecules distributed in an isothermal reaction element. Thus, the important aspects of randomness and spatial dimensionality were explicit, whereas molecular movement was simplified to a quasi-static situation. Although molecules were not allowed to translate within this reaction grid, a diffusion time constant was assigned to all species, based on the current state of the grid and their molecular weights. This ultimately influenced the reactivity of each. A regular cubic grid with L, sites per edge was generated by randomly distributing SI tetrafunctional amine sites
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1213
Y
/ 0 Arbitrary reactive site (X,Y,Z) within the reaction grid. Members of the coordination cube of the site (X,Y,Z). Figure 2. Representation of the coordination cube around an arbitrary point within the reaction grid.
and S2difunctional epoxy sites throughout the lattice. The initial (t = 0) ratio of S1/S2sites was constant for a given simulation and chosen such that the functionalities of each reactive site were in stoichiometric proportion. F1 X Si = F2 X S2 (13) In eq 13, Fiis the functionality of a site and Si is the number of type i sites distributed within the grid. At each site within the reaction grid, coordinated nearest-neighbors and rate constants for their possible reaction were determined as follows. At a given point (X,Y,Z) in the reaction grid, all possible combinations of (X&l,Yhl,Zhl)were considered to be immediate neighbors and thus composed a template coordination cube with three lattice points per edge and centerpoint (X,Y,Z). This is illustrated in Figure 2. If the point (X,Y,Z) was a member of the set of edge or face points on the L, X L, X L, reaction grid, a periodic boundary condition was applied by allowing reactions to occur between opposite faces of the reaction grid. The computation focused on the tetrafunctional amine sites as its basis. Rate constants (transition probabilities) were determined as follows. For each of the functionalities of the site, 8 events were randomly chosen from the 26 possibilities of encounter between (X,Y,Z) and the members of its coordination cube. The coordination number is usually between 6 and 12 for a condensed phase, and the chosen value of 8 was considered to be reasonable. Next, a separate pseudo-first-order rate constant for reaction was calculated for each of the 8 X Fireactive pairs. This rate constant was identically zero if the sites of the proposed reaction pair were of the same type: amineamine or epoxy-epoxy reactions were not allowed. The rate constant was also considered to be zero if a chemical bond previously existed between the chosen reaction pair. In addition, sites with Fipreviously existing bonds are not allowed to react further. For non-zero rate constants, the formalism of eq 2 was used. A pseudo-first-order intrinsic rate constant for the reaction of a site of type S1 (say amine) with epoxy sites (S2)was calculated by dividing eq 3 by the appropriate concentration of site Sl. This is shown in eq 14, where is the reactivity ratio of a primary amine hydrogen divided by that of a secondary amine hydrogen and S2is the concentration of Cs species averaged over the space of the percolation grid. h o t e that while the present form of eq 14 is based on the Smith (1961) reaction mechanism, the approach is general and can incorporate different or improved understandings of the controlling elementary steps.
Rk
Figure 3. (a) Percolation grid showing tetrafunctional amine sites. (b) Percolation grid showing tetrafunctional amine sites and the largest cluster at 33% conversion. (c) Percolation grid showing tetrafunctional amine sites and the largest cluster at percolation (58% conversion). (d) Percolation grid showing tetrafunctional amine sites and the largest cluster at 66% conversion.
The variable time step Monte Carlo computational method provided the connection between the developing cluster population and the kinetics of reaction. As above, a pseudo-first-order rate constant was assigned to each possible event. Then some random event was allowed to occur. The time step required for this single event was calculated via eq 15, In (1 - P ) A t = - i=N, (15) kf;;t-order
i=l
where P is a random number (0 5 P < 1) and the firstorder rate constants, ki, were summed over all possible, Ne, events. Random events were allowed to occur and temporal information about the percolation grid was stored until a predetermined final time was reached. At this point in the simulation, the percolation grid was reinitialized to time = 0. One scan through time defined one Markov chain and the subsequent results from multiple Markov chains were averaged to increase the precision of the simulation. The standard deviation from the averaged value of NM Markov chains was proportional to NM-1/2. In summary then, first-order rate constants were assigned to the 8 X Fl X S1 possible events within the reaction grid. This allowed construction of a cumulative probability distribution from within which a random event could be selected and the variable time increment to the next event calculated. A n Illustrative Simulation. The simple example where L, = 10, Fl = 4, F2 = 2, v = 1, Rk = 1, and SIFl = S2F2allows illustration. Figure 3 illustrates the state of a reaction grid at four successive degrees of epoxide conversion (p = 0, p < pc,p = pc, and p > pc). In this figure, the reactive sites are regularly distributed throughout the outlined cube and existing epoxy-amine bonds are represented by a line connecting the two reacted sites. Figure 3a depicts the reaction grid at p = 0. The collection of dots illustrates the random placement of tetrafunctional amine sites on the percolation grid. The ( 0 )
1214 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990
difunctional epoxides are at unmarked lattice points. Figure 3b illustrates the largest cluster at 33% epoxide conversion. Note that only the amine sites are shown. Reacted bonds are indicated by a line connecting the visible amine sites to the unmarked epoxide sites. The apparently unconnected bonds are manifestations of the periodic boundary condition applied to edge and side lattice points. Figure 3c shows a percolating (spanning) cluster at p = p c = 0.58 epoxide conversion. This cluster spans two opposing faces of the reaction grid and marks the onset of an infinite network. This is gelation. Note that many finite clusters remain distributed throughout the grid at this point. Figure 3d illustrates the close interconnection of lattice points at conversions higher than pc. The largest cluster (gel) in the grid at p = 0.66 contains most of the epoxyamine population at this point, but there are still some smaller finite clusters interspersed throughout the gel. In the later stages, where the weight fraction of gel is high, many of the reaction events are intramolecular. This is in stark contrast with the initial phase of the polymerization where primarily intermolecular reactions occur. Three Percolation Models of Epoxy-Amine Polymerization. Application of these concepts to the case of epoxy-amine polymerization was aimed at modeling the complex interplay of chemical kinetics, polymer structure, and material properties. The inputs to the epoxy-amine simulation included the intrinsic chemistry and kinetics, a model of the diffusion characteristics of the polymer, and the initial conditions of the grid. Three distinct models were studied. The first model was of intrinsic, molecular weight independent kinetics that were unaffected by the reactant mobilities during the cure. Thus, for model 1, the effectiveness factor was unity throughout the duration of the polymerization. The second model assigned to all species an average mobility that was a function of the free volume and T,,,. The glass transition temperature was calculated with the Di Benedetto relation and further used to calculate the free volume through eq 7. An average diffusivity for use in model 2 was determined from eq 10 using M,. A t Tg = T,, the effectiveness factor was set to zero for all species and the reaction was stopped. In model 3, a separate q was calculated for each epoxy-amine reactant pair. This was accomplished through the use of the various models of polymer diffusion summarized as eqs 4-12 coupled with the intrinsic kinetics of reaction. Several parameters were considered invariant in each model: cy, T (p=l),T (p=O), and f ( T ). The value of cy of 5.0 X and t i e relationship between f and T Tgwere identical with those used previously for the linear DGEBA-n-butylamine (NBA) system (Rohr and Klein, 1988). The primary difference in predicting the free volume for the linear and cross-linked system was in the model for TB‘ For the linear system, Tgwas linked to the polymer structure through M,,whereas for the crosslinking system, the Di Benedetto relation (eq 6) correlated Tgwith conversion @) given values of Tg@=O)and Tg@=l).The values of the initial Tg@=O)and final T @=1) were determined experimentally to be 251 and 433 The quantities E J E , and FJF, were regressed from the data as 0.57 and 0.33, respectively. Figure 4 illustrates that the Di Benedetto relation represents a quantitative fit of the experimental data. Optimized parameters included k,Chp and k2 for model 1and klCimp,k2, and &J2/A for models 2 and 3. The values
R.
450 0
Tcure-325 / /
4w / / /
-Y
..f ’
350
c”
0,
c -’
300
250
200 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Epoxide Converslon
Figure 4. Comparison of the Di Benedetto relation and data of Tg versus epoxide conversion at a cure temperature of 325 K. 1.o
0.8
.f E
0.6
s s $
:n:
0.4
0.2
I
1 0.0
I
100
I
200
I
300
I
400
3
Time (Minutes)
Figure 5. Model predictions and experimental data for the temporal variation of epoxide group conversion at 325 K and stoichiometric loadings.
of klCimpand k , were regressed from the initial data of epoxide conversion rate, using q = 1for all species, as 2.66 cm3/(mol min) and 8.03 X lo2 cm6/(mo12min), respectively. The term &6*/A in models 2 and 3 was taken as 1.26 X 10-l g cm2/(mol s), the value regressed from the linear DGEBA-NBA system (Rohr and Klein, 1988). The combination of all of the above parameters allowed prediction of the development of the cross-linked gel structure, sol-phase properties, extent of epoxide conversion, rate of epoxide group conversion,and the cluster size distribution. Results Figure 5 compares the predicted conversion from models 1-3 to experimental data for the DGEBA-PACM-20 system at stoichiometric initial conditions and an isothermal cure temperature of 325 K. All three models afforded similar predictions at the early stages of the polymerization. They diverged at later times. Model 1 eventually approached a limiting epoxide conversion of 96%. This is due to the finite dimensionality of the model and the trapping of reactive epoxide sites within cages of unreactive neighbors. This topological conversion limit has been
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1215
360
-
--0 --
Experiment Model-1 Model-2
_----------
__-r -
-.
c -
0.8
L 280
J 260
!’ 0.0
240
0
100
200
300
400
100
500
noted experimentally (Oleinik, 1986) to be approximately 93%. Model 2 followed the intrinsic kinetics initially, but at Tg = T,,,,, all reactions ceased and the polymer was “frozen”. The predictions of model 3 were initially similar to the intrinsic kinetics but at later times fell between those of models 1 and 2. Model 3 predicted a lower ultimate epoxide conversion than did model 1 because larger oligomers suffered diffusional limitations. Model 3 predicted a higher ultimate conversion than did model 2 because the latter froze all reactions at Tg= T. In this respect, model 2 imposes nonspecific diffusional limitations on all reacting oligomers. The prediction of each of the models for the rate of reaction throughout the cure also compared reasonably well with the experimental data. As was observed for the linear DGEBA-NBA system (Rohr and Klein, 1988),the rate of reaction and epoxide conversion did not appear to be sensitive indicators of the extent of diffusive control. This was better observed in examination of the resulting polymer properties and structure. The differences between the predictions of the three models were more pronounced when the epoxide conversion was used as input to the Di Benedetto equation to predict Tr The results are plotted in Figure 6 as TFversus time. In effect, this is a transformation of the ordinate of conversion and amplifies the differences between the three models. Model 1 predicted that Tg= 365 K at 500 min, which is higher than the predictions of models 2 and 3 of 325 and 335 K, respectively. The data are intermediate between the prediction of models 1 and 3. The predictions of the three models and the experimentally measured temporal variation of gel weight fraction are plotted in Figure 7. At long cure times, model 1predicted higher gel content than did models 2 or 3. This is because intrinsic rate constants for reaction were applied to all species and, as a result, higher molecular weight oligomen were incorporated into the gel more readily. The gel content determined from model 2 remained unchanged after the point where T = T,, because reactions were quenched. Model 3 predicted closer values to the experimentally measured weight fraction of gel and lies between the predictions of models 1 and 2. Gel formation occurred between 100 and 120 min at an epoxide conversion between 0.59 and 0.63. Models 1and 2 predicted lower values of ~ ~ ( 0 . and 5 5 0.56, respectively) than did model 3 ( p , = 0.59). The predicted effect that
I
300
I
400
Time (Mlnutes)
Tlme (Minutes)
Figure 6. Model predictions and experimental data for the temporal variation of the glass transition temperature at 325 K and stoichiometric loadings.
I
200
Figure 7. Model predictions and experimental data for the temporal variations of the gel fraction at 325 K and stoichiometric loading. 1.o
0.8
0.6
E
1 -
tCritical Polnt
0.4
E 0.2
b
O.’
I
0.01
0.w
, 0.05
I
I
I
0.10
0.20
0.25
0.30
1N-g)
Figure 8. Effect of percolation grid size on the location of the critical point.
diffusion has on the gel point is thus detectable but slight. Diffusional limitations delay the gel point to later times and somewhat higher conversions. This is primarily due to the preferential reaction of higher mobility, lower molecular weight species over reactions that involve higher molecular weight oligomers. The influence of the finite size of the reaction grid used to model the experiments was examined through replicate simulations. As the grid size increased, the percolation threshold approached the theoretical value (P,= 1/31/2 = 0.58) for model 1and, in the case of model 3, exceeded it. This is illustrated in Figure 8 where the critical point as determined with model 2 is plotted versus the inverse of the linear grid size, l / L r For l/Lg = 115 the critical point exhibits a higher standard deviation (f0.071) compared to 1/L, = 1/15, where the standard deviation is f0.015. The easily measured fraction of gel does not yield comprehensive structural information about the infinite network. The simulation provides more detail such as the quantity and type of cross-link points. Cross-linked sites in the DGEBA-PACM-20 system have either three or four polymeric arms. The cure time dependence of the probability of finding a tetrafunctional site with three attached
1216 Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 0.5
0.4
0.3
w
I-
5
td
0.2
-P(X3) Model-1 - - P(X3) Model-2
0.1
..- - - - _- - - - - - _ - -
P(X3) Model-3
0.0
0 100
200
300
400
3
0
I
100
T i m (Mlnutas)
I
200
I
300
I
400
Time (Minutes)
Figure 9. Model predictions of the temporal variation of P(X3) for a cure temperature of 325 K. Mn Experlment 0.6
0.5
Mn-3
P(X4) Model-1 P(X4) Model-2 P(X4) Model-3
0.4
.-r
8
0.3
P
t 0.2
0.1
0.0
p ‘ 100
200
300
400
Time (Minutes)
Figure 10. Model predictions of the temporal variation of P(X4) for a cure temperature of 325 K.
polymeric arms (P(X3))and the probability of finding a tetrafunctional site with four polymeric arms (P(X4)) is shown in Figures 9 and 10, respectively. Model 1exhibits a maximum in P(X3) at approximately 150 min, whereas models 2 and 3 approach asymptotic values. Figure 10 shows that model 1 predicts a higher fraction of X4 sites compared to model 2 or model 3. At the end of the cure, the network structure predicted by model 1is much higher in cross-links than the resulting structure from model 2 or model 3. Model 1 also indicates a higher rate of formation of X4 sites than the predictions of model 2 or model 3. Model 2 does not allow for further increases in the number of network cross-links at high conversion. Model 3 permits the conversion of X3 sites to X4 sites, but the probability is lower than in model 1due to diffusional limitations. The sol phase can be characterized by its molecular weight distribution as the reaction proceeds. Parts a and b of Figure 11 illustrate the weight- and number-average molecular weights, respectively, of the sol phase. Experimental gel permeation chromatography (GPC) measurements are added for comparison. Before the gel point, the predictions from each of the three models are similar. However, after gelation, model 1predicts the lowest values
0
100
200
300
400
5
Time (Minutes)
Figure 11. (a) Model predictions of the temporal variation of the sol-phaseM, compared to experimentaldata for a cure temperature of 325 K. (b) Model predictions of the temporal variation of the sol-phase M. compared to experimentaldata for a cure temperature of 325 K.
of weight- and number-average molecular weights. This is because all reactive species are assigned the same intrinsic rate constants for reaction. Thus, higher molecular weight species are allowed to add to the infinite gel as readily as lower molecular weight species. In model 2, all The molecular weight reactions stop when Tg= T,,,. distribution becomes set at this point. The postgelation weight- and number-average molecular weight predictions of model 3 are higher than those of model 1 or model 2. After the critical point in model 3, reactions between epoxy and amine sites with higher mobilities are more probable. The sol-phase molecular species are more likely to react with other sol-phase oligomers than with the gel. As a result, the averages of the molecular weight distribution are higher in model 3 than in model 1 or model 2. After the gel point, the molecular weights determined from the experimental GPC measurements are expected to be lower than the true molecular weight of a given sample. This is primarily due to the calibration of the GPC columns with linear polystyrenes. A sample containing cross-linked species will have a smaller hydrodynamic radius than its linear analogue and therefore yield a lower apparent molecular weight. Additionally, the higher mo-
Ind. Eng. Chem. Res., Vol. 29, No. 7, 1990 1217 lecular weight constituents of the sol phase may not have had sufficient time to disentangle from the gel phase and, as a result, could bias measurements of the sol molecular weight toward lower values. The experimental GPC data are considered to be low estimates of the true sol-phase molecular weights. In summary then, the three models of the effects of diffusion on cross-linking epoxy-amine reactions exhibit similar temporal epoxide conversion and rate of epoxide conversion predictions but different structural predictions. The cross-link density (P(X3)and P(X4)) is most sensitive to the extent of diffusional limitations. The sol-phase M , and M, predictions are similar prior to gelation but differ the most after the critical point. Finally, the temporal predictions of the models of the gel weight fraction are initially similar,but after gelation and at long times, model 1predicts the highest value of w gfollowed by models 3 and 2. The development of this stochastic percolation model and its subsequent application to modeling diffusion effects in epoxy-amine cross-linking polymerizations raises several questions that remain unanswered. First, since many properties depend on the Tgof the polymer, a structural basis would seem more appropriate for predicting TB‘ Thus, rather than correlate the material properties with conversion, the polymer structure could be used to predict Tg, f , viscosity, and other related quantities. The polymer structure also plays a part in the diffusion of reactive oligomers. The diffusion models for cross-linked polymers are in some respects extrapolations of the linear diffusion concepts. The effects of polymer branching on the self-diffusion coefficient were accounted for in the correlation of Tgwith conversion, but the present model did not provide a truly satisfying treatment of these ideas. Finally, the emphasis on the polymer structure suggests that another possible method for model discrimination lies in the nature of the critical point, the critical exponents that characterize gelation, and the fractal dimension of the polymer. Experimental measurements of quantities such M,, M,, wg,P(X4), and P(X3) close to the critical point would provide physical insight in this area. The stochastic percolation model could then be used to examine the effects that diffusion might have close to the phase transition of gelation.
Summary and Conclusions The effect of diffusion on the observable kinetics of a cross-linking epoxy-amine polymerization has been analyzed. The summarizing model is in terms of a three-dimensional, dynamic percolation grid, which was used to calculate the changes of the developing material structure and properties with respect to time and conversion. The three cross-linking models provided similar predictions of conversion. The predicted structures of the gel and sol, i.e., the probability for finding three- and fourarmed cross-links, P(X3) and P(X4), provided a more sensitive basis for model discrimination. Model 1 predicted a more cross-linked network structure and higher gel weight fractions than model 2 or model 3. The predictions of model 2 were invariant after Tg= Tm and represented general, nonspecific,diffusional limitations on all reactions. Model 3 allowed for differing reaction probabilities between epoxy-amine sites, and its predictions of P(X3), P(X4), and w gwere between the two extremes of models 1 and 2. The linear and cross-linking polymerizations can be viewed in another light. In each of these systems, two transitions are in competition: the critical point of the
polymerization and the glass transition. In the case of linear polymerizations, p c = 1and the glass transition is reached first, provided that T,,, < T,(=).However, in cross-linking systems, pc is usually less than unity and is dependent on the functionality of the specific monomers. There are thus two situations that can affect the way diffusion alters the global rate constant of reaction. One case arises when the glass transition is reached before the critical point, and the other situation occurs when gelation is reached first. If the glass transition is reached first, the system tends to “freeze”, and this results in the nonspecific hampering of molecular movement. As a result, general diffusional limitations on all reactions occur. This was observed in a linear epoxy-amine polymerization (Rohr and Klein, 1988), where parallels to both models 2 and 3 fit experimental data reasonably well. The dominant factor appeared to be the free-volume dependence of the self-diffusion constant. If the critical point occurs first, there is a change for specific diffusional limitations brought about by the range of species mobilities. Here deviations from the intrinsic kinetics for reaction can depend on the free volume and the reactant molecular weight. In cross-linking systems, these specific diffusional limitations can affect the crosslink density, sol-phase molecular weight, and gel fraction. If the system is linear, specific diffusional limitations can occur for very high molecular weight oligomers as p c = 1 is approached. Registry No. DGEBA, 25085-99-8;PACM-20, 1761-71-3.
Literature Cited Bartels, C. R.; Crist, B.; Fetters, L. J.; Graessly, W. W. Self diffusion in branched polymer melts. Macromolecules 1986,19, 785. de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornel1 University Press: Ithaca, NY, 1979. de Gennes, P. G. Conjectures on the transport of a melt through a gel. Macromolecules 1986, 19, 1245. Di Benedetto,A. T. Prediction of the Glass Transition Temperature of Polymers: A Model Based on the Principle of Corresponding States. J . Polym. Sci.: Part B Polym. Phys. 1987, 25, 1949. Flory, P. J. Principles of Polymer Chemistry; Cornel1 University Press: Ithaca, NY, 1953. Gordon, M.; Scantlebury, G. R. Non-Random Polycondensation: Statistical Theory of the Substitution Effect. Faraday SOC. Trans. 1964,60. Horie, K.; Hiura, H.; Sawada, M.; Mita, L.; Kambe, H. Calorimetric investigation of polymerization reactions 111. J. Polym. Sci., A-1 1970,8, 1357. Huguenin, F. G. A. E.; Klein, M. T. Intrinsic and transport-limited epoxy-amine cure kinetics. Ind. Eng.Chem. Prod. Res. Dev. 1986, 24, 166. Lunak, S.;Vladyka, J.; Dusek, K. Effect of diffusion in the glass transition region on critical conversion at the gel point during curing of epoxy resins. Polymer 1978,19,931. Macosko, C. W.; Miller, D. R. A new derivation of average molecular weights of nonlinear polymers. Macromolecules 1976a, 9, 199. Miller, D. R.; Macosko, C. W. A new derivation of post gel properties of network polymers. Macromolecules 1976b, 9,206. Miller, D. R.; Macosko, C. W. Average property relations for nonlinear polymerization with unequal reactivity. Macromolecules 1978, 11, 658. Nielsen, L. E. Crosslinkmg-effect on physical properties of polymers. J. Macromol. Sci. Rev. 1969, C3 (l), 69. Oleinik, E. F. Epoxy-Aromatic Amine Networks in the Glassy State: Structure and Properties. Adu. Polym. Sci. 1986,80, 49. Rabinowitch, E. Collision, coordination, diffusion, and reaction velocity in condensed systems. Trans. Faraday SOC.1937,33,1225. Rohr, D.F.; Klein, M. T. Modeling Diffusion and Reaction in Polymerization Kinetics. Ind. Eng. Chem. Res. 1988, 27, 1361. Rouse, P. E. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J . Chem. Phys. 1953,21 (7), 1272. Smith, I. T.The mechanism of the crosslinking of epoxide resins by amines. Polymer. 1961, 2, 95.
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Received for review September 12, 1989 Revised manuscript received March 12, 1990 Accepted March 26, 1990
PROCESS ENGINEERING AND DESIGN Design of Resilient Controllable Chemical Processes: An Autothermal Reactor Case Study Richard W. Chylla, Jr.,t and Ali Char* Department of Chemical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616
A technique for the analysis of statespace linear systems is applied to the problem of selection of resilient chemical process designs. Structural Dominance Analysis affords the evaluation of many process design and control configurations and assessment of the effects of potential manipulated variables and disturbances. After a brief presentation of the analysis method, a complex multibed tubular autothermal reactor system is examined. Resilient process configurations, ease of control, and effects of various inputs on reactor state variables and outputs are considered, and effective control configurations are selected, 1. Introduction An ever-changing world marketplace and stiff foreign competition have placed severe burdens on the chemical process engineer. Companies are more than ever being forced to squeeze profitability out of existing processing facilities. With the role of specialty chemicals and polymers coming of age, the problems have intensified. One manufacturing plant may be used to produce many different products, each requiring different process schedules and operating conditions. The control engineer must design control strategies that are flexible enough to regulate the process over a wide variety of conditions subject to both economic and physical constraints. One approach toward achieving resilient process operation is to couple the control system design with the chemical process design. Indeed, the most elegant process design is virtually worthless if the plant cannot be controlled about the nominal operating conditions with acceptable performance. Major contributions to control system performance and, hence, process operability often derive from perceptive and clever modifications of the process itself (Foss, 1973; MacGregor, 1985). The process design stage or process retrofit stage is the time to consider and resolve many important control issues along with the design decisions. Design and control decisions for a tubular packed-bed autothermal reactor system can illustrate the benefits of such an approach. Assuming that the plant capacity and chemical reaction characteristics dictate use of fixed-bed tubular reactors and energy management assessment Current address: Innochem Process Development, S. C. Johnson & Son, Inc., Racine, WI 53403. 0888-5885/90/2629-1218$02.50/0
dictates autothermal reactor operation, a major design decision that remains is the mechanism of heat exchange. In the tubular autothermal reactor shown in Figure 1, the exothermic heat of reaction is removed by using the cold feed gas as a heat-exchange medium, thus bringing the feed gas to reaction temperature, and the process becomes self-sustaining. This heat exchange may be done by internal heat exchange by using an annulus (Figure 1)and/or by a feed-effluent heat exchanger following an adiabatic reactor bed. When wall cooling is chosen, the coolant may be passed cocurrently or countercurrently. These reactor design decisions must be made when designing fixed-bed catalytic reactors for reactions that are highly exothermic such as ammonia synthesis (Baddour et al., 1965; Eschenbrenner and Wagner, 1972; Handman and Leblanc, 1982), methanol synthesis (Stephens, 1975), and others (Froment, 1980). The large variety of designs found in industry is due at least in part to the huge task of evaluating and optimizing all the possible reactor configurations. As will be presented later, these selections have a large influence on the stability of the reactor system and on the difficulty of controlling it. Once the reactor design is determined, the selection of controlled and manipulated variables remains. If on-line measurements of the controlled variables are not available, alternate measurements should be selected and the need for state estimation should be assessed. These are all design/control synthesis problems that must be addressed when designing an autothermal reactor system. Many of these control and design issues can be tackled simultaneously through a systematic analysis of detailed mathematical models of the process. Detailed process representations of large collections of algebraic and differential equations based on first principles avoid the issue 0 1990 American Chemical Society
