Kinetic Reaction Analysis of Gelation: Investigations of Random Step

1360

Ind. Eng. Chem. Res. 1997, 36, 1360-1372

Kinetic Reaction Analysis of Gelation: Investigations of Random Step-Growth and Chain-Initiated Resins D. J. Robbins, Q. Zhu, and D. C. Timm* Department of Chemical Engineering, University of NebraskasLincoln, Lincoln, Nebraska 68588-0126

Subject to intermolecular addition and intramolecular cross-linking reactions, deterministic models which describe chain infrastructure for a random A4 + B2 resin and a Poisson-type, chain-initiated epoxy anhydride thermoset were derived. Sol fractions are described in terms of molar concentrations of molecules of a given chemical composition and conversion and in terms of branch node distributions. Moments of the population density distribution form conditionally convergent series. Two limits were selected. Their difference is proportional to the substance in the gel. Purely deterministic derivations are made regarding the internal structure of the gel-structural parameters related to rubber elasticity, for example, which have been previously held to be obtainable only by statistical arguments. Numerical solutions are presented graphically to emphasize the distinct molecular structure within the two resins and the consequence with regards to mechanical performance. Deterministic solutions are equal to solutions derived by expectation theory.

Introduction Flory (1940, 1941a-c, 1942, 1944, 1953) and Stockmayer (1943, 1944, 1952) using probabilistic reasoning derived models for simulating step-growth, random polymerizations. In later research gelation was simulated by numerical methods (Bansil et al., 1984; Mikesˇ and Dusˇek, 1982) and mean-field (Gordon, 1962; Gordon and Ross-Murphy, 1975) and percolation theories (Stauffer, 1985). Advances based on expectation theory address step-growth (Dotson et al., 1996; Macosko and Miller, 1976a,b; Miller, 1987, 1988; Miller and Macosko, 1978, 1980; Sarmoria and Miller, 1991) and chaingrowth polymerizations (Tobita and Hamielec, 1988, 1989, 1992; Xie and Hamielec, 1993; Zhu and Hamielec, 1993, 1994). Stockmayer (1943, 1944, 1952), Ziff and Stell (1980), and Fukui and Yamabe (1964) illustrate deterministic methodology based on Smoluchowski (1917) type relationships, deriving expressions descriptive of molar concentrations of resin molecules as functions of degree of polymerization and conversion or population density distributions (PDD) (Flory, 1940, 1953; Nielsen et al., 1993; Noureddini and Timm, 1992; Stockmayer, 1952). When a chemical reaction forms the thermoset, stochastic methods have been used to simulate both polymerization reactions and network features associated with chain connectivity. With more complex competing reactions, hybrid models (Bokare and Ghandi, 1980; Dotson et al., 1996; Dusˇsek, 1985; Gupta and Macosko, 1990) have evolved that couple kinetic models for parts of molecules or superspecies and expectation theory to simulate effective network junctions and elastically active chain segments. Stochastic models are also capable of handling complex, competing reactions (Sarmoria and Miller, 1991). Authors (Dotson et al., 1996; Dusˇek and Sˇ omva´rsky, 1985; Williams et al., 1987) have stated that one must use stochastic methodology for the prediction of network topology. A kinetic approach is believed incapable of yielding descriptions of chain structure in the gel * Author to whom correspondence should be addressed. E-mail: [email protected]. Fax: 402-472-6989. S0888-5885(96)00461-7 CCC: $14.00

because of long-range connectivity dependencies. However, Ziff and Stell (1980) explicitly addressed reactions within the sol, between the sol and gel, and within the gel. Although the calculation of network structures required for the prediction of mechanical properties by the theory of rubber elasticity was not addressed, these principles can yield a deterministic procedure illustrated in this paper. Derivations are presented for two distinct resins. A paradigm of the pregel regime is the random A4-B2 polymer modeled by Stockmayer (1952) and a chain-initiated, Poisson-type resin modeled by Flory (1940, 1953) and Fukui and Yamabe (1964). Illustrations are based on thermosets comprised of a fourfunctional branch node plus a two-functional connecting link. Although the current paper does not address the case where the two polymerizations compete, Bokare and Gandhi (1980), Gupta and Macosko (1990), and Dusˇek and co-workers (Dusˇek, 1985; Dusˇek and Sˇ omva´rsky, 1985; Dusˇek et al., 1987) have, using hybrid models in the presence of first-shell substitution effects. Specifically, the reactivity of a primary amine was assumed to be distinct from that of a secondary amine. Riccardi and Williams (1986a,b) performed an experimental analysis using differential scanning calorimetry. Hybrid models are advanced and flexible. What advantage or disadvantage a deterministic theory of gelation has to offer can only be assessed after a theory is developed. Theory of Rubber Elasticity The theory of rubbery elasticity provides a structure/ property relationship (Mark and Erman, 1988). Idealized networks are described by affine (Flory, 1953) and phantom models (James and Guth, 1947). In the former, chain branch nodes are assumed to move in space linearly with respect to macroscopic deformation. No contribution results from the parts of a chain between junction points but only from the number of strands connecting elastically active junctions. In the phantom model, junction positions move affinely but fluctuate about their mean positions as fully mobile chain segments. Chain connectivity restricts motion © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1361

and, therefore, becomes a factor. In the constrained junction model (Flory, 1985a,b), the degree of interpenetration of a chain with neighboring junctions and strands is also emphasized. This model predicts that the elastic free energy of deformation is intermediate between the values for the phantom and affine networks (Erman and Flory, 1982; Mark and Erman, 1988). In continuing developments Ball et al. (1981) considered physical strand interactions in the form of entanglements. These interactions increase the elastic energy of deformation beyond that of the phantom model. Subsequently, Duering et al. (1993) addressed entanglement contributions relative to the affine model. Although entanglements were initially expressed by a “slip-link model”, several authors, including Macosko et al. (Macosko and Miller, 1976b; Macosko and Benjamin, 1981), Graessley and co-workers (Dossin and Graessley, 1979; Graessley, 1975; Pearson and Graessley, 1980), and Durand and Bruneau (1983), have envisioned physical entanglements as two intertwined strands. Dotson et al. (1996) and others have summarized molecular contributions to the equilibrium tensile modulus E in a single expression:

E ) (νc - hµc)RT + TeE0

(1)

The network is comprised of νc moles/volume of elastically active strands and µc elastically active junctions. A strand is a chain segment between two active junctions. An active junction has a minimum of three connections to the gel. The junction fluctuation parameter was represented as h. If h ) 0, the model is affine; if h ) 1, the model is phantom. The entanglement trapping parameter is Te, and the plateau modulus of an un-cross-linked material formed from the network strands is E0. The ideal gas constant is R, and the absolute temperature is T. An objective of our research was to use deterministic methods in the calculation of the several variables appearing in eq 1 as a function of conversion. Specific emphasis addresses chemical contributions to the number of strands and to the active junctions. Entanglement contributions use published stochastic theory. An A4 + B2 Step-Growth Polymerization Literature Review. (a) Population Density Distributions. The A4 + B2 polymerization is constrained by equal reactivity of functional groups, intermolecular second-order reactions, and an ideal, perfectly mixed batch reactor. The initial monomer has a functionality of four, and the second monomer has a functionality of two. Stockmayer (1943, 1944, 1952) and Ziff and coworkers (1980) are several who discussed a kinetic reaction approach based on a Smoluchowski expression:

dPi,j/dt ) -KAi, jPi, jB - KABi, jPi, j + K

∑i ∑j Ak,lPk,lBi-k, j-lPi-k, j-l

(2)

A discussion of the several variables and functions follows. The molar concentration of molecules that contain i A4 links and j B2 links is represented by the dependent variable Pi, j. An oligomer P9,13 appears in Figure 1 where A4 links are represented as crosses and the B2 segments as lines. Points a and b are unreacted A and B moieties. Initially, consider the chemical functionality of a molecule Pi, j. Since a minimum of i - 1 B2 links is required to connect i A4 segments, j - i

Figure 1. Cluster of A4 and B2 chain segments: Oligomer P9,13 if moieties at a and b are unreacted.

+ 1 bifunctional units have only one moiety reacted and Bi, j ) j - i + 1. Initially, the i A4 monomeric units contained 4i functional groups. Two moieties reacted with each of the i - 1 B2 connecting chain links and one group reacted with each pendent B2 chain link. Therefore, the unreacted moieties on the molecule are Ai, j ) 3i - j + 1. The maximum number of B2 units is 3i + 1 when Ai, j is zero. In eq 2 the first chemical reaction rate expression addresses reactions at A sites on the reactant Pi, j with all B moieties in the resin. The second rate expression correlates the reactions between the cumulative concentration of A moieties with B groups on the reactant. The last term represents permissible formation reactions Pk,l + Pi-k, j-l f Pi, j. The initial reactant supplies an A site and the second supplies a B moiety. For each degree of polymerization i, i - 1 e j e 3i + 1. The kinetic rate constant is K, B ∞ 3i+1 ∞ 3i+1 ) ∑i)1 ∑j)i-1 Bi, jPi, j + 2P0,1, and A ) ∑i)1 ∑j)i)1 Ai, jPi, j. In derivations time t is transformed to conversion. The extent of reaction of A moieties is F ) (A(0) - A)/ A(0) ) (B(0) - B)/(B(0) r). For B moieties, FB ) rF. The initial ratio of functional groups in a formulation r ) A(0)/B(0)) 2P1,0(0)/P0,1(0). A dimensionless time dτ ) KB dt is a convenient initial substitution. The disappearance of A groups is described by the relationship dA/dt ) -KAB or in dimensionless time dA/dτ ) -A. Differentiation plus rearrangement yields dF/(1 - F) ) dτ. Flory (1940), Fukui and Yamabe (1964), Ziff (1980), and Timm and co-workers (Noureddini and Timm, 1992; Noureddini et al., 1994) illustrate kinetic solutions for Pi,j. MAPLE V, a symbolic, numerical mathematical package, is also useful. Stockmayer noted that his solution satisfied the Smoluchowski equation, eq 2, subject to null initial conditions except for monomers P1,0(0) and P0,1(0). Solutions incorporate moments, including ∑i∑j pi, j, ∑i∑jipi, j, and ∑i∑j jpi, j plus integrating factors (Fulks, 1964). Derivations utilize a proof by induction in which equations are integrated sequentially, ultimately forming a general solution (Noureddini and Timm, 1992). For this resin population density distributions equal (Stockmayer, 1952)

pi, j )

Pi, j P1,0(0)

)

4(1 - F)(1 - rF)(3i)! × rF(3i - j + 1)!(j - i + 1)!i!

[

][

rF(1 - F)3 1 - rF

i

]

F(1 - rF) 1-F

j

(3)

The variable pi,j represents the normalized concentration of polymer molecules. Initially, p1,0(0) ) 1. Conversion is an explicit function of time:

1362 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

{

4KP1,0(0) t/(1 + 4KP1,0(0) t)

r)1

F ) 1 - exp(-(4P1,0(0) - 2P0,1(0))Kt) r - exp(-(4P1,0(0) - 2P0,1(0))Kt)

r*1

(4)

(b) Sol/Gel Reactions. Ziff and Stell (1980) unequivocally addressed chemical functionality in the sol and the gel. Reactions involving the set of molecules pi, j cause changes to occur in the concentrations of Ai, j chemical moieties:

d(Ai, j pi, j,sol)/dt ) -1KAi, j pi, j,solBsol Ai, jKAi, j pi, j,solBgel - Ai, jKBi, j pi, j,solAgel In the first rate expression, the stoichiometric coefficient 1 reflects the fact that each reaction in the sol consumes one A group. In the second and third rate expressions, the bond formed between the molecule and the gel results in all of its chemical moieties being transported to the gel. Thus, the stoichiometric coefficient is Ai, j; however, in the perspective of the gel, a net of Ai, j - 1 moieties is received since the required reaction consumed one. The notation pi, j,sol emphasizes that molecules addressed are in the soluble phase. Addition yields an expression descriptive of the total concentration of moieties Asol in the sol and Agel in the gel. For the soluble sol and insoluble gel

∑i ∑j {A2i, j pi, j,sol}Bgel K∑∑{Ai, jBi, j pi, j,sol}Agel i j

dAsol/dt ) -1KAsolBsol - K

dAgel/dt )

∑i ∑j {(Ai, j - 1)Ai, j pi, j,sol}Bgel + K∑∑{(Ai, j - 1)Bi, j pi, j,sol}Agel i j

-1KAgelBgel + K

Subject to the notation Asol + Agel ) A, the concentration of A moieties in the resin, addition yields

is a function of conversion and formulation as expressed by β ) rF2(1 - rF2)2 and the constant Ci ) (3i)!/((2i + 1)!i!). The subscript i continues to express the number of A4 links in a molecule. The function β is initially zero and at complete conversion if r ) 1, a condition which maximizes the extent of reaction. If a balanced stoichiometry is not employed, molecules become saturated with the excess reactant at complete conversion of the limiting reactant where β > 0. The first derivative of this double-valued function provides an alternate means for evaluating the critical conversion Fc. At the maximum

dβ dF

|

) 2rFc(1 - rFc2)(1 - 3rFc2) ) 0

Fc )

F′/F )

∞

ωA4 )

3i+1

∑ ∑ ipi, j i)1 j)i-1

(5)

In our work the convergence properties of this moment and other leading moments were determined by initially expressing equations as a single summation:

4(1 - rF2)2 ω A4 )

rF2

∞

i(3i)!

[rF (1 - rF ) ] ∑ i)1(2i + 2)!i! 2

2 2 i

Following Flory (1953) and Stockmayer (1943), the sum

x

3 F 1 - 2 4 2 rF

(8)

For the mass fraction of A4 nodes

{

1

0eFe1

2 2 rF′2 ωA4 ) 4(1 - rF ) rF2 4(1 - rF′2)2

0 e F′ e Fc e F e 1 (9)

The upper limit of 1 is also obtained when eq 2 is initially weighted by i, and algebra coupled with integration yields the value for the moment. The mass of the resin is invariant for conversions 0 e F e 1. If A is in excess, at complete conversion of B, rF ) 1 and F < 1. Beyond the gel point Fc the lower limit yields the mass fraction of the resin that is soluble. The difference in the upper and lower limits represents the fraction of A4 branch nodes in the gel. This equation was simplified using eq 8:

{( ) 1

(6)

x3r1 (7)

(b) Weight Fraction of the Sol. Ziff and Stell (1980) and Flory (1953) noted that by the critical conversion all population densities had passed their maxima. At higher conversions molecules in the sol are being lost to the gel. This phenomenon may also be discussed in terms of the convergence properties of the moments of the PDD. Prior to gelation, moments are absolutely convergent; therefore, their respective limits are unique (Fulks, 1964). At conversions greater than Fc they are conditionally convergent; therefore, multiple solutions exist (Fulks, 1964). Rearrangements that change the order of addition allow the moment to converge to any limit. Two limits illustrate. In the first rearrangement βi is expanded, using the binomial k theorem (a - b)k ) ∑m)0 (-1)mk!/(m!(k - m)!)ak-mbm before addition yields a polynomial. In the second rearrangement, the addition is in terms of the numerical values of weighted ipi, j’s. These rearrangements yield limits (Robbins, 1996; Zhu, 1995) which are expressed as functions of variables F and F′. In the interval Fc < F e 1 the variable F′ is the smaller root of β. Since β(F) ) β(F′) or rF2(1 - rF2)2 ) rF′2(1 - rF′2)2

dA/dt ) -KAB Since this expression is recovered when eq 2 is weighted by Ai,j and summed, Ziff and Stell (1980) concluded that Flory’s (Flory, 1953; Stockmayer, 1943) interpretation of gelation addressed reactions within the sol, between the sol and gel, and within the gel. These concepts are now incorporated into a kinetic analysis of gelation. Theoretical Derivations. (a) Critical Conversion. The conversion at the time of gelation is frequently derived from the weight-average molecular weight (Flory, 1953; Fukui and Yamabe, 1964; Macoska and Miller, 1976a). Our primary goal was to derive an independent variable for model development. The fraction of A4 branch nodes is represented by a first moment:

or

Fc

ω A4 )

4

F′ F

)

(x

1

rF2

-

)

3 1 4 2

0eFe1 4

0 e F′ e Fc e F e 1 (10)

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1363

The ratio F′/F is the fraction of A sites on a node that lead to finite chains, a prime variable for theoretical derivations. Chain segments that extend to the gel are considered infinite. (c) Cross-Link Concentration. The assumption of intermolecular reactions causes the population of molecules in the Smoluchowski formulation to be reduced by 1 for each bond formed. However, in the gel a molecule is considered to be infinite. As its molecular chains meander between the walls of the reactor, locations exist where two proximate chemical moieties that are attached to the same megamolecule react, forming a cross-link. Their locations are likely far apart in the sense of their respective mean end-to-end distances. This reaction does not diminish the number of molecules in the resin. The modeling error in the number moment is relatively inconsequential. If longrange segmental molecular motion provides sufficient collisions, intramolecular cross-linking reactions will be chemically controlled by a second-order reaction (Flory, 1953). The significance is the chemical moieties Ai, j, Bi, j, A, and B in eq 2 are unaffected by coupling intermolecular and intramolecular reactions. Solutions are correct. However, strategies must be developed to recover the fraction of sites cross-linked. Analysis of the cumulative molar concentration of molecules, subject to eq 3, yields ∞

3i+1

∑ ∑ pi, j ) i)1 j)i-1

{

1 - 2rF

0eFe1

(F′/F)4(1 - 2rF′)

0 e F′ e Fc e F e 1 (11)

When the conversion of B sites rF is 0.5, the number of molecules calculated by the upper limit is zero. The lower limit is greater than zero, as are the pi, j’s; see eq 3. At greater conversions, the upper limit becomes negative, but both the lower limit and the pi, j’s remain positive. At complete conversion pi,3i+1 ) 0 only if stoichiometry is balanced. A sol fraction exists if r * 1. The difference between the lower and upper limits actually represents the number of cross-links. Cross-linking should not be associated with chain cyclization as addressed by Jacobson and Stockmayer (1950) and Semlyen (1986). Chain cyclization involves the formation of small rings that contain a few atoms, such as five- and six-membered rings. These intramolecular reactions are first-order. In this work this competing reaction is neglected. Other moments of interest include

∑i ∑j jpi, j,sol ) 4F(1 - rF)(F′/F)

3

∑i ∑j ∑i ∑j

4(F′/F)3 ijpi, j,sol )

0 1 2 3 4

n0 n1 n2 n3 n4

0

1

2

3

4

X0,0 X1,0 X2,0 X3,0 X4,0 ∑4k)0Xk,0

X1,1 X2,1 X3,1 X4,1 ∑4k)1Xk,1

X2,2 X3,2 X4,2 ∑4k)3Xk,3

X3,3 X4,3

X4,4 X4,4

represented by nk. The reaction state of the four chemical moieties is k, 0 e k e 4. The dependent variables are normalized molar concentrations. Chain development is described by a series of second-order reactions nk + B f nk+1. Reaction rate expressions are of the form of KB(4 - k)nk. The rate constant is K; the cumulative molar concentration of chemical moieties is B; and the number of unreacted A sites on the node is 4 - k. Polymerization dynamics are described by the differential equations

dn0/dτ ) (1 - F) dn0/dF ) -4n0

n0(0) ) p1,0(0) ) 1

dn1/dτ ) (1 - F) dn1/dF ) -3n1 + 4n0

n1(0) ) 0

dn2/dτ ) (1 - F) dn2/dF ) -2n2 + 3n1

n2(0) ) 0

dn3/dτ ) (1 - F) dn3/dF ) -1n3 + 2n2

n3(0) ) 0

dn4/dτ ) (1 - F) dn4/dF ) 1n3

n4(0) ) 0

or in compact notation

(1 - F) dnk /dF ) -(4 - k)nk + (5 - k)nk-1(1 - δk,0) 0 e k e 4 (12) nk(0) ) δk,0 The Kronecker delta equals

δk,m )

{

1 0

k)m k*m

Solutions yield the molar concentration of branch nodes within the resin as a function of conversion:

n0 ) (1 - F)4

n1 ) 4F(1 - F)3

n2 ) 6F2(1 - F)2

n3 ) 4F3(1 - F)

n4 ) F4

nk ) 4!/(k!(4 - k)!)Fk(1 - F)4-k

(13)

+ 2rF (F′/F)

{rF2(F′/F)3 + F(1 - rF)}

1 - 3rF′2

2(F′/F)2

extent of cross-linking, m

reaction state k

6

i2pi, j,sol ) (F′/F)4(1 + rF′2)/(1 - 3rF′2)

∑i ∑j j pi, j,sol ) 2

2

Table 1. A4 Cross-Link Nodes Xk,m: k Reacted Sites, m Chain Segments Extending to Gel

{6(1 - rF)2F2 +

2

1 - 3rF′ (1 + 3rF′2)(F′/F)(2(1 - rF)F + rF2(F′/F)3)}

Recall that F′ may be set to F to calculate the second set of limits. MAPLE V was effective in determining limits. (d) Branch Node Dynamics. The resin is now expressed in terms of branch node distribution dynamics. A node is an A4 monomer or chain link and is

or

Solutions are binomial expansions of (a + b)4 where a ) 1 - F and b ) F; therefore, nodes are conserved 4 ∑k)0 nk ) 1. (e) Sol/Gel Node Dynamics. In order to address sol/gel reactions, nodes are partitioned between the sol (m ) 0) and gel (1 e m e 4) by the notation Xk,m; see Table 1. The subscript k represents the reaction states of the four chemical moieties on a node. The subscript m represents the number of chain extensions from a node to the gel. Each molecule pi, j transports i nodes to the gel with its first reaction with the gel. To illustrate, the site labeled a in Figure 1 is assumed to have reacted with the gel. Resultant dynamics may be expressed by

1364 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 ∞

4

d

∑

3i+1

∑ ∑ i)1 j)i-1

Xk,0/dt ) -K

k)0

iAi, j pi, j,solBgel ∞

K

3i+1

∑ ∑ iBi, j pi, j,solAgel i)1 j)i-1

(14)

The notation emphasizes that molecules pi, j,sol are in the soluble fraction. Sol/sol and gel/gel reactions effect conversion but have no immediate effect on the number 4 of nodes in the sol ∑k)0 Xk,0. Nodes in the sol have zero extensions to the gel but contain from zero to four reacted sites. The right-hand side of eq 14 was multiplied and 4 divided by 4∑k)0 Xk,0 ) 4∑i∑jipi, j,sol. Resultant ratios are average chemical functionalities 〈A〉 ) ∑i∑jiAi, jpi, j,sol/ 4∑i∑jipi, j,sol and 〈B〉 ) ∑i∑jiBi, jpi, j,sol/4∑i∑jipi, j,sol with respect to the four sites on branch nodes in the sol. The function # ) 〈A〉Bgel + 〈B〉Agel was incorporated into a dimensionless time dτ′ ) k# dt. Equation 14 became 4

d

4

∑ Xk,0/dτ′ ) -4k)0 ∑ Xk,0

k)0

4

∑ Xk,0(τ′c) ) p1,0(0) ) 1

k)0

The coefficient equals the four sites on A4 nodes in the sol, including reacted and unreacted chemical moieties. At the onset of gelation, the critical time associated with Fc is represented as τc′. At τc′ all nodes are in the sol; therefore, the initial condition is unity. Integration yields 4

∑ Xk,0 ) exp(-4(τ′ - τc′)) ) ωsol,A

Fx ) (1 - F′/F)/F

4

k)0

The substitution of eq 10 yields

exp(-(τ′ - τc′)) ) F′/F

and

dτ′ ) d(1 - F′/F)/(F′/F)

Therefore, eq 14 is equivalent to 4

d F′

∑ Xk,0

k)0

4

∑ Xk,0 k)0

) -4 F d(1 - F′/F)

4 4 sion ∑k)0 Xk,0 f ∑k)1 Xk,1. As it is not necessary for a site on the node to react with the gel, node connectivity is a major consideration. For a third time, our interpretation of the cluster of nodes appearing in Figure 1 is modified; this time we assume the site labeled b has also reacted with the gel. The fraction of the nodes pendent to the chain segment 4 a-b remain in the set ∑k)1 Xk,1, but the nodes connected by the strand a-b are now in the set Xk,2, 2 e k e 4. These nodes have reacted at least twice and have two paths to the gel. This event is represented 4 4 by the chemical transformation ∑k)1 Xk,1 f ∑k)2 Xk,2. If other chemical moieties appearing in Figure 1 had reacted with the gel, events causing some nodes in the sets to experience further first-order transformations, 4 4 4 ∑k)2 Xk,2 f ∑k)3 Xk,3 and ∑k)3 Xk,3 f X4,4, can be visualized. We use the past tense, since in the extent of crosslinking space conversion is invariant. Chemical moieties have either experienced intermolecular additions or intramolecular cross-linking reactions at F. This fact is analogous to a chemical moiety being reacted or unreacted at time t. Kinetic equations in cross-linking space can delineate the relative frequency of intermolecular and intramolecular reactions and their effects on molar concentrations of nodes Xk,m. 4 4 The series of events ∑k)m Xk,m f ∑k)m+1 Xk,m+1, 0 e m e 3, form a set of first-order differential equations when the extent of cross-linking Fx is the independent variable. The extent of cross-linking is the fraction of reacted sites on a node that extend to the gel. For the A4 + B2 resin

4

∑ Xk,0(0) ) p1,0(0) ) 1 k)0

(15)

The independent variable Fg ) 1 - F′/F represents the fraction of A4 sites connected to chains that extend to the gel. The ratio F′/F is the fraction of sites extending to chains of finite dimension, including reacted and unreacted chemical moieties. This variable is the extent of gelation Fg. (f) Concentrations of Cross-Link Nodes. It is of interest to note model constraints applied to eqs 2 and 14, respectively. In the former, intermolecular and intramolecular reactions are lumped into a single rate expression; in the latter, these reactions are explicitly distinguished. Our objective now is the calculation of the several Xk,m’s since they are the molecular features within the resin required for the calculation of the modulus. Before differential equations are developed that yield solutions, Figure 1 is again referenced. Initially, pendent moieties a and b were assumed unreacted. The site a was then assumed to have experienced a reaction with the gel. Since all nodes in this fragment now have a minimum of one reacted site and one path to the gel through a, they are in the set Xk,1, 1 e k e 4. The reaction at a with the gel transported nodes according to a first-order rate expres-

The numerator is the number of reacted sites extending to the gel. The denominator is the number of reacted sites. The distinction relative to eq 15 is that in eq 15 both reacted and unreacted sites are addressed. In cross-linking space only reacted sites are applicable. Initially, consider the monomer n0 ) p1,0. Since unreacted chemical moieties remain exclusively in the sol, its partial derivative is zero:

(1 - Fx) ∂X0,0/∂Fx ) 0

X0,0(0) ) n0

Integration recovers the known concentration X0,0 ) n0. The partial derivative indicates that conversion is being held constant. Integration is from 0 to Fx at F. Nodes with one reacted site n1 are distributed between X1,0, which has one finite chain extension, and X1,1, which has one chain extending to the gel. Since X1,0 f X1,1 only

(1 - Fx) ∂X1,0/∂Fx ) -1X1,0

X1,0(0) ) n1

(1 - Fx) ∂X1,1/∂Fx ) +1X1,0

X1,1(0) ) 0

Solutions are the fractions in the sol X1,0 ) n1(1 - Fx) and gel X1,1 ) n1Fx, respectively. Constants appearing in the normalized rate expressions equal the number of bond extensions to finite chains. The total number of nodes n1 at conversion F is expressed by the constant of integration which is from zero to the value of the independent variable Fx at F. In general, the distribution of nodes between the sol and the gel with k reacted sites is described by the series of first-order transformations Xk,0 f Xk,1 ... f Xk,k for which

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1365

(1 - Fx) ∂Xk,m/∂Fx ) (k - m)Xk,m + (k + 1 - m)Xk,m+1(1 - δ0,m) Xk,m(0) ) nkδ0,m

0eke4

0 e m e k (16)

and

Xk,m ) nkk!/(m!(k - m)!)(1 - Fx)k-m(Fx)m

(17)

probability that an A is reacted is F. In the A4 + B2 resin, the path out of a reacted A site leads into and out of single B sites. The probability that the chain segment is finite is P(FBout). The term 1 - rF addresses unreacted B moieties. A reacted B site leads into an A4 branch node which has three independent paths out. The probability that each of these chain segments is finite is P(FAout)3. The solution is

P(FAout) )

or

Xk,m ) nkk!/(m!(k - m)!)(1 - (1 - F′/F)/F)k-m × ((1 - F′/F)/F)m (18) Derivations were checked for model consistency. The 4 fraction of reacted sites that extend to the gel ∑k)1 ‚ k ∑m)1mXk,m equals the extent of cross-linking Fx ) (1 4 F′/F)/F. The fraction of nodes in the sol is ∑k)0 Xk,0 ) k (F′/F)4 as observed in eqs 10 and 15. Sums ∑m)0 Xk,m 4 conserve the population nk. The sum of nodes, ∑k)1 ‚ k 4 ∑m)1Xk,m ) 1 - (F′/F) , is the mass fraction of gel. In summary, the extent of cross-linking is a very effective variable. (g) Calculation of the Modulus. Discussion returns to an original objective, the calculation of the modulus from the theory of rubber elasticity. An elastically active junction µc contains three or more paths to the gel. Therefore, 4

µc )

∑ Xk,3 + X4,4 ) (1 - F′/F)3(1 + 3F′/F) k)3

Strands connect two elastically active junctions. The number of strands νc equals 4

νc ) (3

∑ Xk,3 + 4X4,4)/2 ) 2(1 - F′/F)3(1 + 2F′/F) k)3

4 4 ∑k)m mXk,m ) 4(1 - F′/ As expected the summation ∑m)1 F), the average number of sites on an A4 node that extend to the gel. If two strands entangle at bifunctional chain segments, the four chain segments entering adjoining branch nodes must lead to the gel for the entanglement to be permanent (Dotson et al., 1996; Macosko and Miller, 1976b). Each of the four branch nodes has three remaining sites that can lead to the gel. The probability that one chain segment exiting the entanglement leads to infinite chains is (1 - (F′/F)3). The probability that all four chain segments lead to the gel is

Te ) (1 - (F′/F)3)4 Deterministic/Stochastic Model Comparisons. Macosko and Miller (1976a,b) defined probability functions P(FAin) and P(FAout), the probabilities that paths out of reactive sites on a branch node lead to finite chains looking into and out of a node, respectively. Using the law of total probability and conditional probabilities, the following relationships were written:

P(FAout) ) (1 - F) + FP(FBout) P(FBout) ) (1 - rF) + rFP(FAout)3 The probability that the A moiety is unreacted is 1 - F. The probability that this site is finite is unity. The

x

3 1 1 - - ) F′/F 2 4 2 rF

The last equality is based on eq 10. Other functions in derivations can also be expressed in terms of probability functions used in expectation theory:

Fg ) 1 - F′/F ) 1 - P(FAout) Fx ) (1 - F′/F)/F ) 1 - P(FAout|A) The function P(FAout|A) is the conditional probability that a reacted A site leads to finite chains. Solutions derived by deterministic methodology equal those derived by expectation theory. A Chain-Initiated E2 + A Polymerization Literature Review. Attention is now directed to a chain-initiated thermoset. Our specific interest is a resin formulated from nadic methyl anhydride (NMA) and diglycidyl ether of Bisphenol A (DGEBA) and catalyzed by benzyldimethylamine (BDMA) in the presence of an initiator. The resin is used in advanced aerospace composites. Our emphasis is chain structure since this information is germane to eq 1. Therefore, overall reaction expressions are emphasized. Since kinetic rate expressions are mathematically equivalent to those used with polyethers, the analysis incorporates theory developed by Flory (1940), Fukui and Yamabe (1964), Dusˇek and co-workers (1985, 1987), Bokara and Gandhi (1980), Gupta and Macosko (1990), and Williams et al. (1987). (a) Anhydride/Epoxy Chemistry. Epoxy/anhydride chemical mechanisms are discussed by Mateˇjka et al. (1983, 1985) and Antoon and Koenig (1981), who used FT-IR spectroscopy to identify chemical intermediates during polymerizations of DGEBA/NMA/BDMA and updated previously reported reaction mechanisms. Trappe et al. (1991) relied on NMR analysis at the end of cures and reported consistent observations with a resin formulated with phenyl glycidyl ether (PGE), phthalic anhydride, and 1-methylimidazole. Nielsen et al. (1993) and Tadros and Timm (1995) polymerized PGE/NMA/BDMA and observed theoretical Poisson PDD’s and exponential decays in monomer concentration for extensive changes in conversion. At higher conversions diffusion effects caused reductions in the rates of polymerization. Number-average molecular weights were consistently less than expected. This was attributed to impurities, but Steinmann (1990) and Fedtke (1987) state that PGE forms a phenolate C6H5O+N-R3H and acrolein. Phenolate acts as a second initiator, causing time-dependent reductions in average molecular weights with 1-methylimidazole. In our work the number of polymeric molecules was invariant. Catalysts also affect PDD dynamics (Ayorinde et al., 1985). During a major portion of the cure, Antoon and Koenig (1981) observed that only carboxylic propagation sites were present, an indication of the presence of a rate-limiting step. Acid/epoxy reactions slowly form

1366 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

Figure 2. Representative chain configurations, DGEBA/NMA/ BDMA resin.

alcohols, whereas the alcohol/anhydride reaction immediately forms the acid. The thermosetting DGEBA/ NMA/BDMA resin is also void of competing reactions. Aromatic chains form rigid rods, eliminating chain cyclizing intramolecular reactions (Mateˇjka and Dusˇek, 1989; Rozenberg, 1986; Semlyen, 1986). The reaction state of one oxirane on DGEBA does not influence the reaction state of the second oxirane (Charlesworth, 1979, 1980; Dusˇek, 1986; Ilavsky´, 1984; Rozenberg, 1986). Polyether formation is also absent when both monomers are present (Antoon and Koenig, 1981; Nielsen et al., 1993; Steinmann, 1990; Tadros and Timm, 1995). The PGE/NMA/BDMA cure was accurately modeled with an initiation step having the same rate constant as the propagation reaction. However, the 1-methylimidazole resin experienced a distinct initiation rate constant (Fedtke, 1987; Hale and Macosko, 1989; Steinmann, 1990). The simplified analysis presented allows the emphasis to be placed on deterministic modeling, but we believe that constraints are representative of the DGEBA/NMA/ BDMA resin for substantial changes in conversion. We also emphasize that the model is not universally valid for chain-initiated polymers. Although published reaction mechanisms (Antoon and Koenig, 1981; Mateˇjka and Dusˇek, 1989; Mateˇjka et al., 1983, 1985, 1987; Steinmann, 1990) have variants in terms of chemical intermediates, they can be reduced to a common overall reaction sequence. Since chemical structure is paramount in the theory of rubber elasticity, this is emphasized. An abbreviated notation is used to describe molecular structure. A propagation site (I‚‚‚AH) reacts with an epoxy or oxirane (E), forming an ester and a terminal hydroxyl moiety (I‚‚‚AEH). The latter immediately reacts (Antoon and Koenig, 1981) with the anhydride (A), forming a second ester and re-forming the carboxylic moiety (I‚‚‚AEAH). Chain structure has the form -CH2CHREO‚‚‚OCRACOO- where the emphasis is the contribution from the oxirane and anhydride, respectively. The residue RE from PGE or DGEBA is pendent to the chain. With DGEBA, RE contains an unreacted oxirane or epoxy. The anhydride residue RA is integrated into the chain. Therefore, DGEBA has the potential of forming four esters, whereas the anhydride is associated with only two ester bonds. Reactions repeat by a step-growth mechanism, forming a linear polymer with PGE (I(AE)n-1AH). Nielsen and Timm (1993) and Tadros and Timm (1995) reduced the reaction mechanism proposed by Antoon and Koenig (1981) to an overall propagation reaction. Flory (1940, 1953) modeled propagation reactions with bifunctional monomers, deriving the Poisson PDD. Fukui and Yamabe’s (1964) derivation is applicable for cures with multifunctional epoxy monomers. Both deterministic derivations were constrained by initiation constants being equal to propagation constants. (b) Chain Structure. Chain structure is summarized in Figure 2. The difunctional epoxy contributed E-E branch nodes, and the anhydride contributed A connecting links. The paths I‚‚‚AH define chain branches and/or propagation sites on the P4,3 oligomer. In Fukui

and Yamabe’s notation (1964) the dependent variable Pi,j represents the molar concentration of molecules with i epoxy chain segments or branch nodes and j branches or propagation sites. In the figure, chain pendent oxiranes are shown as -E’s. In comparison with Figure 1 for the A4 + B2 resin, significant differences in chain structure are apparent. In the former resin, one to four chain segments exit a branch node, but with the chaininitiated resin only two or four chain segments exit a branch node. This is a direct result of the rate-limiting acid/epoxy reaction. At complete conversion initiator and propagation sites remain, causing to exist not only finite chain elements in the network even with a balanced stoichiometry between oxiranes and anhydrides but also a sol fraction comprised of molecules saturated with initiator sites. Similarities include fourth-order branch nodes connected by second-order chain links, but chain connectivity is distinct since nodes with one or three esters are absent. The polymerization is further distinguished from the A4 + B2 resin by an invariant number of propagation sites. Each reaction sequence reforms the propagation site. Only the number of oxiranes decays with time. If excess anhydride is formulated, the excess simply remains unreacted. If excess oxiranes are formulated, effects will be similar to those for the A4 + B2 resin. Molecules will become saturated with epoxy moieties, increasing both the sol fraction and the dangling ends within the network. Because of its distinct chemistry the chain-initiated resin was selected to further illustrate deterministic derivations. (c) Time-Conversion Transformation. Oxiranes of concentration O react with propagation sites of concentration P0,1(0). These reactions cause the monomer to decay according to dO/dt ) -KOP0,1(0). The rate constant is K. Dimensionless time τ ) KP0,1(0) t yields dO/dτ ) -O. Integration results in an exponential decay with respect to time:

O ) O(0) exp(-KP0,1(0) t) ) O(0) exp(-τ) The conversion of oxiranes F equals

F ) 1 - exp(-KP0,1(0) t)

and

dτ ) dF/(1 - F) (19)

Comparisons with eq 4 reveal distinct time dependencies. (d) Branch Node Dynamics. The epoxy branch node E-E has three reaction states, nk, 0 e k e 2. If oxiranes are unreacted, the branch node is equivalent to the monomer. Branch node dynamics are described by the differential equations

(1 - F) dn0/dF ) -2n0

n0(0) ) p1,0(0) ) 1

(1 - F) dn1/dF ) -1n1 + 2n0 (1 - F) dn2/dF ) 1n1

n1(0) ) 0 (20)

n2(0) ) 0

Coefficients represent the number of unreacted epoxides on a node. Solutions are

nk ) 2!/(k!(2 - k)!)(1 - F)2-kFk

0 e k e 2 (21)

In reference to (13) the major difference at a given conversion is the permissible values for the index k. (e) Population Density Distributions. Fukui and Yamabe (1964) selected the dependent variable Pi,j to describe the PDD. Stoichiometry considerations yield expressions for the number of chemical moieties on molecules in the set Pi,j, including oxiranes Oi, j ) i - j

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1367

+ 1, propagation sites Hi, j ) j and molecular weight MWi, j ) MWA(2j + i - 1) + MWIH j + MWE-Ei. The several molecular weights are that of the anhydride, initiator, and monomer, respectively. Bonding constraints are j g 1 and j - 1 e i e ∞. The initiator is a polymeric molecule; the diepoxy is a monomer. A polymer molecule contains a minimum of one propagation site. Molecular dynamics are described by an equation analogous to eq 2 when Oi, j and Hi, j are substituted for Ai, j and Bi, j, respectively. Moments are readily obtained by weighting the differential equation before addition followed by integration. Subject to a bifunctional monomer and an invariant number of propagation sites

pi, j ) Pi, j /P1,0(0) ) 2ji-1 (i - j + 1)!j!

Ri-1Fi+j-1(1 - F)i-j+1 exp(-R jF) (22)

∞

R

∞

∞

∑ ∑ j)1 i)j-1

jpi, j )

2 2 2

RF

∞

Cj β j ∑ j)1

(23)

In the single summation a molecule’s number of propagation sites remains j. The constant Cj ) j j-1/ j ! and the function β ) RF2 exp(-RF2). A balanced stoichiometry between oxiranes and anhydrides is assumed. In the interval 0 e F e 1 a maximum occurs in β at the critical conversion:

Fc ) (P0,1(0)/2P1,0(0))1/2 ) 1/R1/2

∑ ∑ jpi, j /2 ) j)1 i)j-1

F′

0eFe1

2

0 e F′ e Fc  F e 1

(25)

F

The constant R/2 is a result of normalization. The upper limit satisfies the Smoluchowski equation. The lower limit results from the addition of numerical values of jpi, j. Since F′ Fc. The difference in these limits is the number of propagation sites appearing in the insoluble gel phase. Diepoxy fourth-order branch nodes also become distributed between the sol and gel. Although moments do not normally include the monomer in this paper, the following relationships include monomer dynamics:

ωE2 )

∞

∑ ∑ ipi, j ) j)1 i)j-1 ∞

{1 - 1/j + RF(1 - F)}Cj β j + (1 - F)2 ∑ j)1

2/(R2F2)

(26)

and ∞

ω E2 )

{

∞

∑ ∑ ipi, j ) j)1 i)j-1

1

0eFe1 2

(F′/F)4F2 + (F′/F)22F(1 - F) + (1 - F)

0 e F′ e Fc e F e 1 (27)

The first through third terms represent nodes with two, one, and zero oxiranes reacted, respectively; see eq 21. In the sol fraction all chain segments are finite. The ratio F′/F is the fraction of esters connected to finite chain clusters. For conversions greater than the critical conversion, the sol’s mass fraction is less than unity. The difference between the two limits in (27) is the gel’s mass fraction based on DGEBA. Other moments of interest include

∑j ∑i

pi, j )

( )(

2 F′

1-

() () () 2 F′

∑j ∑i

∑j ∑i i pi, j ) 2F(1 - F)

2

2

F

1

F 1 - RF2(F′/F)2

F′

2

ijpi, j ) 2F

2

2

( ))

RF2 F′

2

R F

∑j ∑i j2pi, j ) R

(24)

Initially, β equals zero and at complete conversion β is essentially zero, depending on the formulation variable R. In commercial resins this variable equals approximately 10. The smaller root of β is represented by F′ and the larger root by F when conversion exceeds the critical conversion. The solution for F′ satisfies β(F′) ) β(F). An explicit solution was not obtained. A numerical algorithm based on interval halving converged. A numerical algorithm based on Newton’s method diverged. (b) Moment Analysis. Prior to gelation, absolutely convergent moments yield limits used in solving for the PDD. At conversions greater than Fc, moments are conditionally convergent. Two limits of interest for the distribution of propagation sites are

{( ) 1

∞

∞

Oxirane conversion is F. The formulation variable R ) 2P1,0(0)/P0,1(0) is the initial ratio of oxiranes to propagation sites. A comparison of eqs 3 and 22 reveals distinct PDD distributions. When a monofunctional epoxy is polymerized, the latter expression reduces to a Poisson molar distribution (Flory, 1940; Fukui and Yamabe, 1964). The most probable molar distribution for an A2 + B2 resin decays monotonically, but the Poisson distribution passes through a maximum near the resin’s average molecular weight. Theoretical Derivations. (a) Critical Conversion. The critical conversion was originally derived from the weight-average molecular weight (Fukui and Yamabe, 1964). We emphasize the derivation of the independent variable F′/F as a function of conversion. Initially, the convergence properties of the moments are addressed. The distribution of propagation sites I‚‚‚AH between the gel and sol was addressed using Maple V:

ωI‚‚‚AH )

ωI‚‚‚AH )

F

F′ F

2

1 - F + F(F′/F)2 1 - RF2(F′/F)2

1 + RF(1 - F + F(F′/F)2) 1 - RF2(F′/F)2

If F′ ) F, moments as functions of conversion only are recovered. This set of moments does not contain contributions from the monomer. (c) Cross-Link Dynamics. The distribution of crosslinked nodes Xk,m for the chain-initiated resin system is summarized in Table 2 as a function of the reaction state of the oxiranes appearing on the node, 0 e k e 2, and the bond extensions that lead to the gel, 0 e m e 2k.

1368 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 Table 2. E2 Cross-Link Nodes Xk,m: k Reacted Oxiranes, m Chain Segments Extending to Gel extent of cross-linking, m

reaction state k 0 1 2

n0 n1 n2

0

1

2

3

4

X0,0 X1,0 X2,0 ∑2k)0Xk,0

X1,1 X2,1 ∑2k)1Xk,1

X1,2 X2,2 ∑2k)1Xk,2

X2,3 X2,3

X2,4 X2,4

Moment solutions were explicitly incorporated into analysis, yielding concentrations for several cross-links Xk,m. The monomer p1,0 is the node X0,0 and, therefore, X0,0 ) (1 - F)2. Pendent oxiranes in the sol occur exclusively on the node X1,0 ) ∑j∑iOi, jpi, j,sol ) ∑j∑i(i - j + 1)pi, j. Algebraic manipulation of the moments revealed that X1,0 ) 2F(1 - F)(F′/F)2. Conservation laws also yield the nodes with two reacted oxiranes that are in the sol: X2,0 ) ∑j∑iipi, j,sol - X1,0 ) F2(F′/F)4. Since monomer is not normally included in moments, the difference is between the total nodes in the sol and nodes X1,0 only. Addition coupled with the binomial theorem reveals that the total concentration of nodes in the sol 2 is ∑k)0 Xk,0 ) (1 - F + F(F′/F)2)2. The concentration of nodes with one reacted oxirane that appear in the gel is now addressed in terms of all nodes that contain a single oxirane ∑j∑iOi, jpi, j ∑j∑iOi, jpi, j,sol ) 2F(1 - F)(1 - (F′/F)2). Factoring in terms of the independent variables yields (1 - F′/F)(1 + F′/F) ) (1 - F′/F)(2(F′/F) + (1 - F′/F)). Therefore, X1,1 ) 2F(1 - F)(2(F′/F)(1 - F′/F)) and X1,2 ) 2F(1 - F)((1 - F′/F)2). We now return to the extent of crosslinking Fx ) 1 F′/F for this resin. Recall that the extent of cross-linking is the fraction of chain segments exiting a node that extends to the gel. The variable F′/F is the fraction that is finite for the chain-initiated cure. Solutions derived from moments satisfy the set of differential equations

∂Xk,m ) -(2k - m)Xk,m + ∂Fx (2k + 1 - m)Xk,m-1(1 - δ0,m)

Chain entanglements are arbitrarily assumed to occur with two bifunctional anhydride links for consistency of model comparisons. A chain segment exiting the entanglement enters a branch node. The rate-limiting reaction causes the chain to continue via a second ester. The probability that this second ester is part of a finite cluster is F′/F. The pendent oxirane may or may not be reacted. The unreacted fraction 1 - F is of finite size. For the fraction reacted F, two chain segments exit. From the law of total probability, the expectation that the pendent oxirane is part of finite clusters is 1 - F + F(F′/F)2. Therefore, the probability that each of the four chain segments exiting the entanglement extends to the gel is

Te ) (1 - (F′/F)(1 - F + F(F′/F)2))4

Deterministic/Stochastic Model Comparisons. Dusˇek and co-workers (1975, 1986), Bokare and Gandhi (1980), and Gupta and Macosko (1990) addressed chaininitiated polymerizations using expectation theory. As applied to our resin system, their superspecies is equivalent to a chain branch, I-(AE‚‚‚)n-AH. In reference to Figure 2 the notation “‚‚‚” now represents the internal DGEBA structure that leads to the pendent oxirane. The branch contains n chain elements plus the initiator’s residual and the propagation site. Let the concentration of branches of degree of polymerization n be Bn. Since each chain grows by a propagation mechanism from time zero, a Poisson distribution describes Bn in dimensionless time:

Bn ) P0,1(0) (τ′′)n exp(-τ′′)/n!

Xk,m(0) ) nkδ0,m (28)

and

Xk,m ) nk(2k)!/(m!(2k - m)!)(1 - Fx)2k-m(Fx)m 0eke2 0 e m e 2k or

Xk,m ) nk(2k)!(m!(2k - m)!)(1 - F′/F)m(F′/F)2k-m (29) Equation 28 is analogous to eq 16. Differences in eqs 18 and 29 are directly attributable to distinct polymerization chemistries. Binomial expansions are appropriate for higher order branch nodes, but the variable F′ will have to be updated for Af or Ef nodes. (d) Calculation of the Modulus. Solutions are now directed at the calculation of the equilibrium modulus E. The number of elastically active strands νc is proportional to the number of chain extensions to the gel on elastically active junctions:

νc ) (3X2,3 + 4X2,4)/2 ) 2F2(1 - F′/F)3(1 + 2F′/F) (30) The concentration of elastically active junctions is

µc ) X2,3 + X2,4 ) F2(1 - F′/F)3(1 + 3F′/F)

(31)

where dτ′′ ) KO dt

The conversion of oxiranes may be expressed in terms of Bn:

F ) (O(0) - O)/O(0) )

(1 - Fx)

(32)

∑n nBn/O(0) ) τ′′/R

The last term incorporates a series expansion of an exponential plus the normalization factor R. Substitution yields

Bn ) P0,1(0) (RF)n exp(-RF)/n! Functions used in the literature are the probability that a finite chain cluster is present when looking into the node toward the chain pendent oxirane P(FEin) and the probability that a randomly selected oxirane leads to finite chain clusters P(FEout). The selected oxirane is either unreacted and finite or reacted and appearing in a branch. The probability that the reacted oxirane is in a chain of size n equals the fraction in these chains nBn/O(0). The branch exiting the reacted oxirane contains n - 1 additional pendent oxiranes. From expectation theory ∞

P(FEout) ) (1 - F) +

∑ nBnP(FEin)n-1/O(0) n)1

The probability of finding finite structures entering the pendent oxirane is the same as that of structures exiting the pendent oxirane along its branch P(FEin) ) P(FEout). Substitution and simplification yield

P(FEout) ) 1 - F + F exp(-RF) exp(RFP(FEout)) (33)

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1369

Figure 3. Double-valued function β vs conversion for different formulations illustrating Fc and the smaller root F′ for the epoxy resin.

Figure 4. Simulation of the soluble portion of branch nodes for the epoxy resin and A4 + B2 resin.

In comparing derivations with Bokare and Gandhi’s(1980) and Gupta and Macosko’s (1990) solution, the sol fraction was addressed: 2

Xk,0 ) (1 - F + F(F′/F)2)2 ) (P(FEout))2 ∑ k)0 Therefore,

P(FEout) ) 1 - F + F(F′/F)2 and the substitution for P(FEout) on both sides of eq 33 yields

F′2 exp(-RF′2) ) F2 exp(-RF2)

or

β(F′) ) β(F)

Since the function β is common, the deterministic and stochastic solutions are identical. Simulations To demonstrate distinct features of a Poisson-type polymerization, numerical simulations were performed for the two resin systems discussed. In Figure 3 the function β is correlated with conversion F and the relative initial concentration of epoxy groups compared to initiators for the DGEBA/NMA/BDMA resin. One will note that the maxima are independent of formulation R and that the critical conversion associated with the maxima is strongly dependent on R. When R ) 4, the roots of β are shown. At this level of initiator, substantial dangling ends exist in the network even at complete conversion, β > 0 and F′ > 0 when F ) 1. In commercial resins R = 10. In Figure 4 sol fractions, variables frequently used experimentally to measure the extent of the reaction, are addressed. Three cases are shown. Initially, consider the anhydride-epoxy resin. The formulation parameter is combined with the symbol E2 to represent solubility dynamics associated with the diepoxy monomer ωE2 (eq 27) and branch nodes within the developing network ωI‚‚‚H (eq 25). In the second case, the symbol IAH is used. For the third case the A4 + B2’s sol fraction ωA4 (eq 6) is shown as conversion advances. It is

Figure 5. Summation of third-order cross-links: X2,3 for the epoxy resin; ∑4k)3Xk,3 for the A4 + B2 system.

apparent that the dynamics for all three are distinct. For the Poisson-type system, the initiator is transported to the gel at a rate exceeding that for branch nodes in the resin near the gel point. The few propagation sites combine with many oxiranes to generate insoluble megamolecules prior to Fc for the most probable distribution resin. When a relatively low ratio of epoxy/ initiator ratio exists, R ) 4, a substantial sol fraction exists even at complete conversion. Oligomers pi,i+1 in the sol are saturated with propagation sites. Figures 5 and 6 express gelation in terms of the sums of third- and fourth-order cross-links as functions of conversion, resin type, and formulation. For the chaininitiated resin, the effect of increased initiator content is very apparent at complete conversion. For thirdorder cross-links, X2,3, substantial dangling ends exist when R is 4 or 9, but few are apparent when R is 16. Their presence and the presence of lower order branch nodes in the sol and in the gel contribute to the reduction of fourth-order cross-links X2,4 at high conver-

1370 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

Figure 6. Fourth-order cross-links: X2,4 for the epoxy resin, X4,4 for the A4 + B2 resin.

Figure 8. Chemical contribution to the modulus for phantom networks, νc - µc.

Figure 7. Chemical contribution to the modulus for affine networks, νc.

Figure 9. Entanglement contribution to the modulus, Te.

sion. If the ratio of oxiranes to initiators is increased excessively, cure times become prohibitive. The dynamics associated with the A4 + B2 system lag for substantial intervals in conversion and do not yield dominant X4,4 structures until high conversions are achieved. When condensation products form, every effort must be made to shift the ultimate equilibrium point to high conversions by stripping them from the resin. In Figures 7 and 8 molecular infrastructure is expressed in terms of chemical contributions to the modulus. Phantom (h ) 1) and affine (h ) 0) networks, respectively, are addressed. In the neighborhood of 90% conversion, the random and Poisson-type resins finally approach similar values. Junction fluctuations about their mean positions have appreciable effects on magnitudes but minor effects on the general shapes of the curves. In Figure 9 predicted effects on the modulus caused by chain entanglements are shown. With respect to ultimate values, effects of formulation are substantially less in this regard for the E2 + IAH epoxy.

Again, higher conversions are required to achieve similar values for the A4 + B2 system. Chain development dynamics are reflected in their rate of advancement. Discussion A deterministic procedure for modeling polymer networks with competing intermolecular and intramolecular cross-linking reactions has been illustrated for two resins, a random polymerization of comonomers A4 + B2 and a Poisson-type, chain-initiated polymerization of a diepoxy anhydride resin. Original contributions include exact solutions that relate PDD dynamics to cross-link concentrations. Derivations included deterministic solutions and algebraic manipulations of moments. Chemical reaction methodology yielded analytical solutions that were identical to published solutions derived by expectation theory. The problem solver MAPLE V proved to be a convenient means for achieving solutions. The equilibrium modulus at tempera-

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1371

tures in excess of the resin’s glass transition temperature was predicted to be very dependent on chain topology, a function of conversion. Optimal reaction strategies must be implemented to achieve high conversions to maximize the modulus. Acknowledgment The Anderson Fellowship is acknowledged. Assistance from the Center for Materials Research and Analysis, University of NebraskasLincoln, is appreciated. Nomenclature A ) total concentration of A moieties, M Agel ) concentration of A moieties in the gel, M Asol ) concentration of A moieties in the sol, M Ai, j ) A chemical functionality for molecules with i A4 and j B2 links B ) total concentration of B moieties, M Bgel ) concentration of B moieties in the gel, M Bn ) chain-initiated branches of degree of polymerization n Bsol ) concentration of B moieties in the sol, M Bi, j ) B chemical functionality for molecules with i A4 and j B2 links Ci ) constants in PDD E ) equilibrium tensile modulus, Pa E0 ) plateau modulus from un-cross-linked material, Pa h ) junction fluctuation parameter Hi, j ) acid functionality for molecules with i E2 links and j I‚‚‚AH K ) second-order rate constants, M-1 h-1 MWX ) molecular weight of species X, g/mol MWi, j ) molecular weight of molecule Pi, j, g/mol nk ) concentration of nodes with k reacted functional groups O ) cumulative concentration of oxiranes in the resin, M Oi,j ) oxirane functionality for molecules with i E2 links and j I‚‚‚AH sites Pi, j ) concentration of molecules with chemical composition i, j, M pi, j ) normalized concentration of polymer molecules relative to monomer pi, j,sol ) normalized polymer concentration stressing molecules in the sol P(Fin/out ) ) probability that paths in or out of reactive X sites X are finite r ) stoichiometric ratio, A/B R ) ideal gas constant, Pa L K-1 mol-1 t ) time, h T ) absolute temperature, K Te ) entanglement trapping parameter Xk,m ) concentration of nodes with k reacted moieties and m gel links Greek Symbols R ) ratio of initial epoxides to initiator β ) multivalued function in moment expression kernels δk,m ) Kronecker delta νc ) concentration of elastically active strands, M F ) chemical conversion F′ ) smaller root of β Fc ) critical conversion Fx ) extent of cross-linking µc ) concentration of elastically active junctions, M Fg ) extent of gelation τ ) dimensionless time τc ) critical dimensionless time at Fc

ωX ) soluble weight fraction of species X, X ) A4, E2, or I‚‚‚AH

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Received for review July 23, 1996 Revised manuscript received February 3, 1997 Accepted February 4, 1997X IE960461M X Abstract published in Advance ACS Abstracts, March 15, 1997.