An Equilibrium Model for Diffusion-Limited Solid-State Polycondensation

Ind. Eng. Chem. Res. 2000, 39, 2797-2806

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KINETICS, CATALYSIS, AND REACTION ENGINEERING An Equilibrium Model for Diffusion-Limited Solid-State Polycondensation Michael D. Goodner,†,‡,§ Joseph M. DeSimone,†,‡ Douglas J. Kiserow,*,†,‡,§ and George W. Roberts*,‡ Department of Chemistry, University of North Carolina, CB #3290, Chapel Hill, North Carolina 27599-3290, Department of Chemical Engineering, North Carolina State University, Box 7905, Raleigh, North Carolina 27695-7905, and United States Army Research Office, P.O. Box 12211, Research Triangle Park, North Carolina 27709-2211

A model for unsteady-state solid-state polycondensation (SSP) is developed and is applied to the polymerization of poly(bisphenol A carbonate) and poly(ethylene terephthalate) (PET). The model assumes that diffusion of the reaction condensate in the solid polymer is the rate-limiting step in the overall polymerization kinetics. Therefore, the reversible polycondensation reaction is at local equilibrium throughout the polymer particle at all times. The model is applicable to the three general types of step-growth polymerization: AB, A2, and A2 + B2 polycondensation. Through comparison with the predictions of a full kinetic model for polycarbonate synthesis, it is demonstrated that the equilibrium model provides an upper bound on molecular weight and its rate of increase. Model predictions are also compared to experimental data for PET SSP. These comparisons show that the equilibrium model provides a useful tool for understanding the effects of temperature and particle size as well as for establishing a lower bound on the diffusion coefficient of the reaction condensate in the solid polymer. Introduction Solid-state polymerization (SSP) is used to produce high molecular weight step-growth polymers for a wide range of applications. For example, the poly(ethylene terephthalate) (PET) that is used in soft drink bottles is produced exclusively through SSP. Solid-state polymerization also provides high molecular weight PET for several other applications and is used for the industrial synthesis of poly(butylene terephthalate)1,2 and highgrade, high molecular weight polyamides, including both Nylon-6 and Nylon-6,6.2,3 Recently, SSP has received attention as a possible technique for the formation of high molecular weight polycarbonate of bisphenol A4 using supercritical carbon dioxide as a processing aid.5,18 Solid-state polymerization can eliminate obstacles encountered in making high molecular weight polymers via other polymerization processes. For example, the standard synthetic route for polyesters such as PET is melt-phase transesterification. However, the melt viscosity increases dramatically with the molecular weight, giving prohibitively high viscosities at relatively modest molecular weights. This problem can be alleviated by using higher temperatures, at the expense of deleterious side reactions such as the formation of color bodies. Other polymerization processes (e.g., interfacial polym* To whom correspondence should be addressed. D.J.K.: tel, (919) 549-4213; fax, (919) 539-4310; e-mail, [email protected]. G.W.R.: tel, (919) 515-7328; fax, (919) 5153465; e-mail, [email protected]. † University of North Carolina. ‡ North Carolina State University. § United States Army Research Office.

erization) also have drawbacks, such as the generation of aqueous and/or organic waste streams which can be hazardous to the environment and expensive to recycle or remediate. Solid-state polymerization avoids these problems because no solvents are required and there is no need to handle a viscous melt. In SSP, particles of relatively low molecular weight polymer (referred to as prepolymer) are partially crystallized, either thermally or through the action of a penetrant or nucleating agent. The crystallization step is necessary to prevent the particles from flowing and sticking together above the glass transition temperature (Tg) of the amorphous polymer. These particles are then heated to between Tg and the melting point of the crystallites (Tm). The polymerization temperature must be below Tm to prevent particle agglomeration and above Tg to provide enough mobility for the polymer end groups to react. In practice, the process is usually carried out just below Tm to take advantage of the higher reaction rate afforded by the increased temperature. Because most step-growth polymerization reactions are reversible, a major challenge inherent to the SSP process is removal of the reaction byproduct, i.e., the condensate. Diffusion of the condensate through and out of the individual polymer particles can be rate-limiting. Removal of the condensate from the SSP reactor can also be important, and sweep fluids (primarily N2) are employed to help strip the byproduct from the packed reactor bed. Because of the industrial importance of SSP, mathematical modeling has been employed to gain a better understanding of the process and to investigate its

10.1021/ie990648o CCC: $19.00 © 2000 American Chemical Society Published on Web 06/24/2000

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limitations. The majority of these modeling studies have involved the solution of the full system of partial differential equations that describe the changes with time and position of all chemical species within the pellet. In these “comprehensive” or “full kinetic” models, each species balance contains the rates of all reactions in which the species participates, and the species balance for each condensate molecule contains an additional term for diffusion of the condensate within the particle. These models require the numerical solution of up to 12 coupled partial differential equations, and they contain a significant number of physicochemical parameters such as rate constants, diffusion coefficients, and their associated activation energies. Frequently, one or more of these parameters must be adjusted to fit the model to experimental data. Such models have been developed for SSP of a variety of polymers: PET,6-10 poly(butylene terephthalate),1 Nylon-6,3 Nylon-6,6,10 and poly(bisphenol A carbonate).11 In view of the mathematical complexity of these full kinetic models and in view of the fact that many of the required physicochemical parameters may be unknown or poorly defined, several simplified SSP models have been developed. These simplified models are based on the assumption that the rate-limiting step is either (1) the forward reaction kinetics, (2) the internal diffusion of the condensate within the polymer particle, or (3) the external transport of the condensate from the surface of the particle to the sweep gas. The full system of partial differential equations is then reduced to a more manageable system consisting of a single partial differential equation. This research focuses on SSP when the internal diffusion of the condensate molecule is the rate-limiting step. In this case, the polycondensation reaction is at equilibrium throughout the particle at all times. This assumption of instantaneous local equilibrium was first formulated by Chen et al.12 and subsequently reexamined by Chang.13 Although the model set forth in these works contains several errors in formulation, the paradigm of instantaneous local equilibrium set the stage for the development of more rigorous models. Early models for melt-phase condensation polymerization provided more accurate kinetic schemes and a more rigorous approach to condensate diffusion which established a basis for accurate SSP modeling.14,15 However, neither work employed the assumption of instantaneous local equilibrium. That concept was first rigorously applied by Pell and Davis in their analysis of the simultaneous reaction and diffusion of ethylene glycol in stagnant PET melts.16 However, the Pell and Davis model contains assumptions that limit its validity to low end-group concentrations, i.e., high degrees of polymerization. A model of SSP limited by internal condensate diffusion later was developed by Ravindranath and Mashelkar.7 Several mathematical errors in this model development were later corrected by Zhi-Lian et al.8 This model is not restricted to high degrees of polymerization but is otherwise very similar to the model of Pell and Davis. Both the Pell and Davis model and that of Ravindranath and Mashelkar are restricted to A2 type stepgrowth polymerizations in which all of the reacting end groups are chemically identical. These two models cannot be applied directly to AB or A2 + B2 polymerizations in which two different end groups react. More

importantly, neither set of authors explored the practical utility of the equilibrium limited model or its ability to elucidate the limits of polymerization behavior to any significant extent. Furthermore, while a variety of manuscripts have presented polycondensation models for AB and A2 + B2 reactions,3,14,17 the behavior of these reactions under diffusion-limited conditions has not been addressed. In this research, a model is developed for step-growth SSP in which internal condensate diffusion is the ratedetermining step in molecular weight buildup. The model applies to all three types of step-growth polymerizationsA2, AB, and A2 + B2sand employs the assumption of instantaneous local equilibrium. The predictions of this “equilibrium model” are then compared to predictions of a full kinetic model for polycarbonate SSP,11 to demonstrate that the equilibrium model provides an upper bound on the polymerization kinetics. Equilibrium model predictions are compared to experimental SSP data for polycarbonate synthesis, and the effects of a nonideal initial molecular weight distribution are demonstrated. Model predictions are also compared to PET SSP data. In this case, the model provides valuable insight into the SSP behavior as a function of temperature and particle size and into the value of the diffusion coefficient of ethylene glycol in PET. Model Development AB and A2 + B2 Polymerizations. In the SSP of step-growth polymers, a low molecular weight prepolymer is synthesized from the monomers using a traditional technique, such as melt-phase polycondensation. The prepolymer is then partially crystallized, and polymerization occurs in the amorphous regions of the polymer pellets as reactive end groups undergo further polycondensation. This reaction can be depicted as

(1)

where A and B are the reactive end groups, C is the condensate molecule, kf is the forward transesterification rate constant, and kb is the reverse rate constant, i.e., the rate constant for chain scission. For AB and A2 + B2 type polymerizations, A and B are different functionalities. For example, in the synthesis of polycarbonate (an A2 + B2 reaction), hydroxyl (A) and phenyl carbonate (B) end groups react to form carbonate linkages and phenol (C). Reaction 1 is reversible; the production of high molecular weight polymer depends on the removal of the condensate molecule to limit chain scission. An equilibrium constant (K) for the reaction can be defined in terms of species concentrations:

K ) CL/AB

(2)

In this expression, L is the concentration of polymeric repeats (e.g., carbonate groups in polycarbonate), C is the concentration of the condensate, and A and B are the concentrations of the two end groups. If A0 is the total concentration of A type end groups in the monomer mixture used for prepolymer synthesis, then the concentration of A at a point in the polymer particle can be written as A0(1 - x), where x is the

Ind. Eng. Chem. Res., Vol. 39, No. 8, 2000 2799

fractional conversion of the original end groups, provided that the volume of the particle does not change as the reaction proceeds. Assuming that the end groups are in a 1:1 molar ratio and that there are no side reactions, the concentration of the B type end groups is also A0(1 - x) and the concentration of polymer linkages is A0x. Using these relationships, eq 2 can be solved for the equilibrium condensate concentration that corresponds to a given conversion:

A0K(1 - x)2 C) x

only if the condensate concentration, C, decreases. The decrease in the condensate concentration is caused by diffusion of C through and out of the polymer particle. Taking the derivative of eq 3 with respect to time and substituting the result into the right-hand side of eq 10 yield

[

]

K(x2 - 1) ∂C ) D∇2C 2 2 ∂t K(x - 1) - x

(11)

∂C ) D∇2C + kfAB - kbLC ∂t

(7)

Equation 11 describes polymerization in a polymeric solid or stagnant melt in which the reactive end groups, reaction condensate, and polymer linkages are in instantaneous local equilibrium. This model provides an upper bound on the kinetic behavior of a real SSP. The assumption of instantaneous local equilibrium is equivalent to assuming that the rate constants, kf and kb, are so high that the polymerization reaction is immediate and that diffusion of the condensate out of the particle is the only kinetically significant step. Under such conditions, the rate of SSP can be increased by increasing D, the diffusion coefficient of the condensate. However, the rate cannot by increased by increasing the values of the kinetic constants. There are two limiting cases of the model given by eq 11. At low conversions (x ≈ 0), the term in brackets approaches unity, and eq 11 reduces to a simple Fickian diffusion equation. At low conversions, eq 3 shows that the condensate concentration must change considerably to produce a relatively small change in x; i.e., -dC/dx is large. Therefore, at low conversion, the first term on the left-hand side of eq 9 dominates the second. Ignoring this second term leads to the limiting form of eq 11 noted above. At high conversions (x ≈ 1), the right-hand side of eq 11 approaches zero and therefore dC/dt approaches zero. Now, from eq 3, dC/dx ≈ 0 so that very small changes in condensate concentration can affect significant changes in molecular weight. For this situation, generation of condensate by reaction essentially is balanced by diffusion of the condensate out of the pellet. Boundary and Initial Conditions. Two boundary conditions on the condensate concentration are required. For SSP in a spherical pellet of radius R, these are taken to be

∂A ) -kfAB + kbLC ∂t

(8)

∂C (r)0) ) 0 ∂t

(12)

C(r)R) ) 0

(13)

(3)

Conversely, the equilibrium conversion for a given phenol concentration is given by

x ) β - xβ2 - 1; β ≡ 1 +

C 2A0K

(4)

The negative root of the quadratic formula has been used to guarantee physically relevant values (i.e., x < 1). The conversion is related to the number-average molecular weight (Mn) through

1 M n ) Mr 1-x

(5)

If an ideal geometric distribution of polymer chain lengths is assumed, the weight-average molecular weight (Mw) is given by

1+x Mw ) Mr 1-x

(6)

In these equations, Mr is the average molecular weight of a repeat unit. The value of Mr is equal to the molecular weight of the structural repeat unit for A2 and AB polymerizations but is half the molecular weight of the structural repeat unit for an A2 + B2 polymerization. The species balances for the condensate and endgroup concentrations can be written as

In eq 7, D is the diffusion coefficient for the condensate molecule in the polymer particle, which is assumed to be constant. Adding these two equations gives

∂C ∂A + ) D∇2C ∂t ∂t

(9)

which can be rewritten in terms of the end-group conversion:

∂x ∂C ) D∇2C + A0 ∂t ∂t

(10)

If condensate diffusion in the polymer particle is the rate-limiting step, then reaction 1 will be at equilibrium at every point in the polymer pellet at any time. The fractional conversion, x, is determined by the phenol concentration at any point using eq 4, and x increases

The first boundary condition is required by symmetry. The second is necessary for the model to give the maximum possible rate of polymerization, consistent with the assumption that diffusion of the condensate inside the polymer particle is the rate-limiting step in the overall polymerization process. Physically, this condition requires that (1) the mass-transfer coefficient between the particle surface and the sweep fluid is very large (i.e., there is no additional mass-transfer resistance between the external surface and the surrounding fluid) and (2) the condensate concentration in the sweep fluid is essentially zero. A nonzero concentration in the sweep fluid would reduce the reaction rate because of decreased removal of the condensate. To the extent that these two conditions are not realized in actual practice and to the extent that the kinetic resistance is important, the actual rate will be lower than that predicted

2800


by the present model. Thus, this model provides an upper bound on the overall polymerization rate. An initial condition for the condensate concentration (C0) in the pellet is also needed. This value is considered constant throughout the pellet and is calculated in one of two ways. In the first case, C0 is determined using the equilibrium expression, eq 3. This method applies when the prepolymer and the condensate are in equilibrium; i.e., the prepolymer contains the equilibrium concentration of condensate corresponding to the initial end-group conversion, x0. The initial end-group conversion (x0) can be found from the molecular weight (Mn or Mw) of the prepolymer using either eq 5 or eq 6. In some scenarios, the prepolymer may not be in equilibrium with the condensate. This is the case for the SSP of polycarbonate when supercritical carbon dioxide is used as a crystallization agent. During the crystallization step, typically conducted at 70-100 °C,18 the condensate can diffuse out of the CO2-swollen prepolymer particle without further polymerization, and the condensate concentration in the prepolymer approaches zero. In this second case, the initial condition (C0) is obtained by assuming that the polymer comes to equilibrium instantaneously once the reaction begins. The concentration of the condensate produced will be A0(x0′ - x0), where x0′ is the new equilibrium conversion. Substituting this expression for C in the left-hand side of eq 3 allows x0′ to be computed for the case where the prepolymer contains no condensate initially. The value of C0 can then be calculated from x0′. Nondimensionalization. Five parameters appear in eqs 4 and 11-13: D, K, A0, R, and C0. This number can be reduced by introducing the following dimensionless variables:

r r ) r* R

(14)

t t ) t* R2/D

(15)

C C ) C* A xK/(xK + 1) 0

(16)

z) θ) P)

[

P 2(K + xK)

]

(17)

K(x2 - 1) ∂P ) ∇2P 2 2 ∂θ K(x - 1) - x

(18)

∂P (z)0) ) 0 ∂z

(19)

P(z)1) ) 0

(20)

with the initial condition for P given by

P(θ)0) )

K ) 4CL/AB

(2′)

A0K(1 - x)2 C) 2x

(3′)

1 ∂x ∂C ) D∇2C + ∂t 2 ∂t

(10′)

Upon taking the derivative of eq 3′ and substituting the result into eq 10′, the factors of 2 cancel, rendering eq 11 without modification. The calculation of the conversion given the condensate concentration is likewise modified. Solving eq 3′ for conversion gives

x ) β - xβ2 - 1; β ≡ 1 +

(K + xK)(1 - x0)2 x0

(21)

C A0 K

(4′)

The new scale factor for condensate concentration, C*, is found by recognizing that C* ) A0x/2 in a closed system. Substituting this into eq 3′ gives

C* )

The characteristic length and time scales in eqs 14 and 15 are common to diffusion problems. The characteristic condensate concentration, C*, is the equilibrium value in a closed system, found by setting C* ) A0x in eq 3 and solving for C*. These scale factors can then be used to nondimensionalize eqs 4 and 11-13, giving

x ) β - xβ2 - 1; β ≡ 1 +

where x0 is the initial conversion found using one of the two methods discussed above. In the system defined by eqs 17-21, only three parameters are required for solution: the equilibrium constant, K, the initial conversion, x0, and the time scale for diffusion, R2/D. The need to solve only one differential equation and the reduced number of parameters make this model considerably simpler than the majority of those which have been presented in the literature. A2 Polymerizations. Model development for an A2 polymerization, such as the formation of PET by transesterification of bis(2-hydroxyethyl) terephthalate, proceeds in an identical manner. The only differences are introduced by stoichiometry: two identical end groups disappear in the creation of one polymer linkage and one condensate molecule. These differences produce the following changes in the equilibrium expression, the equilibrium condensate concentration, and the overall species balance:

A0xK 2(xK + 1)

(16′)

When the nondimensionalization is performed, eqs 1721 are again recovered. Thus, the model is equally applicable to A2, AB, and A2 + B2 step-growth polymerizations. Diffusivity in SSP. It is necessary to know the crystalline volume fraction of the solid polymer particle in order to compute the diffusivity. In this work, it is assumed that the condensate diffusivity in the solid polymer is proportional to the condensate diffusivity in the amorphous phase and the amorphous volume fraction

D ) (1 - f)Da

(22)

where f is the crystalline volume fraction and Da is the diffusivity of the condensate in the amorphous (or melt) phase. This relationship has been shown empirically for ethylene glycol diffusivity in semicrystalline PET pellets19 and has been used previously in SSP modeling.1,8,10,11 Other relationships, such as free-volumebased approaches, have also been used in SSP modeling.17,20,21 However, these formulations add parameters


Figure 1. Average Mn values predicted by the equilibrium model and the SSP model11 for prepolymer having an ideal molecular weight distribution and K ) 30. The solid line represents the equilibrium model prediction, and the broken lines are SSP model predictions for different forward kinetic constants: kf ) 3 × 10-3 L/mol‚min (dashed), kf ) 3 × 10-2 L/mol‚min (dotted), and kf ) 3 × 10-1 L/mol‚min (dash-dotted). The parameters used in the equilibrium model are found in Table 1.

which are unknown for semicrystalline systems in many instances and thereby increase uncertainty rather than improve accuracy. Therefore, the simple relationship of eq 22 will be used. Factors such as tortuosity and blocking are not accounted for by such a simplified relation; however, at the crystallinities studied in this work, these phenomena should have a negligible qualitative effect on the overall polymerization behavior. Furthermore, the crystallinity is assumed to be constant over the course of the reaction, an assumption employed previously.7,13 Method of Solution. Equations 17-21 were solved using the method of lines. The spatial derivative was discretized using second-order centered differences on a 51-point computational grid. The points were equally spaced, with point 1 at the center of the particle and point 51 at the surface. Because of boundary condition (19), the resulting differential equations for the nondimensional condensate concentration were solved only for grid points 1-50. A commercial numerical integration package designed for solution of stiff differential equations was used to integrate temporally the resulting system of 50 differential equations (NAG Fortran Library, Mark 18, Numerical Algorithms Group, Oxford, U.K.). The nondimensional condensate concentration profiles were then converted into number- and weightaverage molecular weight profiles using eqs 17, 5, and 6. The calculation of the overall (particle) average molecular weights and polydispersities follows the method outlined in Goodner et al.11 The spherical pellet was divided into 51 shells corresponding to the grid points in the computational domain, and the volume fraction of each of these shells was computed. The volume fractions were then used to calculate the average molecular weights for the entire particle using existing formulas for mixtures of polymers having different molecular weight distributions.22 Results and Discussion Equilibrium Model-SSP Model Comparison for Polycarbonate SSP. Figures 1-3 compare the equilibrium model results to the predictions of a full kinetic model for SSP of polycarbonate (hereafter referred to as the “SSP model”) that is presented in Goodner et al.11

Figure 2. Average Mw values corresponding to the simulations shown in Figure 1. The solid line represents the equilibrium model prediction, and the broken lines are SSP model predictions for different forward kinetic constants: kf ) 3 × 10-3 L/mol‚min (dashed), kf ) 3 × 10-2 L/mol‚min (dotted), and kf ) 3 × 10-1 L/mol‚min (dash-dotted).

Figure 3. Overall polydispersities for the simulations shown in Figures 1 and 2. The solid line indicates the equilibrium model prediction, and the three broken lines are predictions of the SSP model for different forward kinetic constants: kf ) 3 × 10-3 L/mol‚ min (dashed), kf ) 3 × 10-2 L/mol‚min (dotted), and kf ) 3 × 10-1 L/mol‚min (dash-dotted). Table 1. Parameters Used in the Equilibrium Model Predictions for SSP of Polycarbonate As Shown in Figures 1-3a parameter

value

parameter

value

K x0 x0′ R (cm)

30 0.936 0.968 0.18

Da (cm2/s) f P(0) t* (min)

1 × 10-7 0.20 3.81 × 10-2 6750

a x is the initial conversion of the prepolymer simulated by the 0 SSP model of Goodner et al.,11 and x0′ is the initial conversion for the equilibrium model found using the process outlined in the Model Development section.

For this series of simulations, initial conditions for the SSP model were set to an ideal molecular weight distribution having an Mn of 2000 (giving x ) 0.936 and Mw ) 3870). From this value of x, the initial conditions for the equilibrium model were determined via the second method outlined in the Model Development section, which assumes no condensate in the prepolymer at the start of the reaction. The values used in the model are listed in Table 1. In Figures 1 and 2, three average molecular weight profiles predicted by the SSP model are shown for different values of the forward kinetic constant (kf) at a fixed value of the equilibrium constant (K ) 30). At the lowest value of kf (3 × 10-3 L/mol‚min), the molecular weight increases at a steady rate over the entire course of the polymerization. This behavior indicates that the

2802


reaction is kinetically limited; the time scale for phenol diffusion is much shorter than that for the forward transesterification reaction. The phenol produced quickly diffuses out of the polymer particle, and the condensate concentration is essentially zero throughout. Thus, chain scission is eliminated, and the growth of molecular weight is determined purely by the forward transesterification kinetics. As kf is increased to 3 × 10-2 L/mol‚min, a new behavior begins to manifest. Initially, the molecular weight increases at the constant rate determined by the forward polycondensation kinetics. However, the rate of phenol production by reaction is faster than its removal by diffusion. Phenol builds up in the particles, slowing the rate of molecular weight increase because the reverse reaction, chain scission, can now occur. The overall rate of polymerization is influenced by both the forward transesterification kinetics and the phenol diffusion rate. Increasing the forward kinetic constant further has three consequences. First, the rate of polycondensation in the initial, kinetic-limited region is increased. Second, the time at which the effects of diffusion begin to appear is shortened. Third and most important, the predictions of the SSP model closely approach the equilibrium model prediction. This behavior is demonstrated by the kf ) 3 × 10-1 L/mol‚min profiles in Figures 1 and 2. Simulations performed for kf values of 3 and 30 L/mol‚ min are not shown but have profiles asymptotically approaching the equilibrium model predictions. The equilibrium model provides the upper bound on both the molecular weight and its rate of increase at a given end-group concentration. In other words, the equilibrium model gives the maximum obtainable molecular weight for a given time, at fixed values of the equilibrium constant, condensate diffusion coefficient, particle size, and initial condition. This result is important from a practical standpoint. First, the equilibrium model provides a relatively simple means to estimate the fastest possible rate of molecular weight increase, without knowing the forward rate constant. Second, under some operating conditions encountered in SSP, internal diffusion of the condensate can be the rate-limiting step. If so, the overall apparent polymerization kinetics can be increased only by affecting an increase in the diffusion kinetics. Figure 3 shows the overall polydispersities predicted by the equilibrium and SSP models. Polydispersity is defined as Mw/Mn and is a measure of the breadth of the molecular weight distribution. The overall polydispersity is the average Mw for the entire polymer particle divided by the average Mn for the entire particle, both of which are calculated using the method discussed previously. For kf ) 3 × 10-3 L/mol‚min, the overall polydispersity barely increases. The polymerization rate is reaction controlled, and there are no condensate and molecular weight gradients in the particle that can lead to molecular weight distribution broadening.11 As the forward reaction kinetics are increased, diffusional limitations start to manifest, causing a radial phenol concentration gradient in the particle. This gradient causes a gradient in the local molecular weight, which in turn leads to a broadened molecular weight distribution over the particle as a whole, as evidenced by the drastic increase in the overall polydispersity. This effect is especially evident for the fastest reaction kinetics (kf ) 3 × 10-1 L/mol‚min); the polydispersity starts to rise

Table 2. Parameters Used in the Equilibrium Model Predictions for SSP of Polycarbonate As Shown in Figure 4a parameter

value based on Mw

value based on Mn

K x0 x0′ R (cm) Da (cm2/s) f P(0) t* (min)

30 0.903 0.958 0.18 1 × 10-7 0.20 6.49 × 10-2 6750

30 0.927 0.965 0.18 1 × 10-7 0.20 4.47 × 10-2 6750

a These values are based on the parameters used for the SSP model of Goodner et al.11 x0 is the initial conversion of the prepolymer, and x0′ is the initial conversion for the equilibrium model found using the process outlined in the Model Development section. The data in the second column are based on the conversion found using Mw ) 2500; the data in the third column are for Mn ) 1750.

appreciably within the first 20 min of polymerization. The equilibrium model once again predicts the upper limit. While the polydispersity predicted by the SSP model for kf ) 3 × 10-1 L/mol‚min approaches the equilibrium model prediction, it falls slightly below this upper bound, for two reasons. First, the SSP model takes 1015 min to approach an equilibrium condition, because of the initial condition of zero phenol concentration. Therefore, the increase in polydispersity predicted by the SSP model is delayed by the time required for equilibrium to be established. The second reason is that polymerization in a pellet can never be entirely diffusion-limited. At and near the surface of the pellet, diffusion is sufficiently fast (even for low diffusivity values) that the condensate concentration is near zero, and the forward reaction is rate-limiting. Molecular weight gradients therefore are small near the surface, leading to a reduction in the molecular weight distribution broadening calculated over the entire particle. Effect of Nonideal Molecular Weight Distributions. In the predictions displayed in Figures 1-3, a prepolymer having an ideal molecular weight distribution was assumed; i.e., the initial number- and weightaverage molecular weights were related to the initial conversion through eqs 5 and 6. However, an ideal distribution does not always exist in the prepolymer. For example, Goodner et al.11 discuss a poly(bisphenol A carbonate) with an Mn of 1750 and an Mw of 2500. These molecular weights yield different values of x when substituted into eqs 5 and 6, as shown in Table 2. Thus, it is useful to examine the discrepancies that may arise for nonideal initial distributions. This examination is performed by comparing the SSP model predictions and SSP experimental data presented by Goodner et al. to the predictions of the equilibrium model. In contrast to the equilibrium model, the “comprehensive” SSP model does not require that the molecular weight distribution be ideal at a given point in the polymer particle. With this model, the values of Mw and Mn are computed independently at each point in the polymer as a function of time. Figure 4 shows this comparison for an initial condition based on an initial Mn of 1750, which leads to a value of x0′ of 0.965, as shown in Table 2. The weight-average molecular weights predicted by the SSP model are in agreement with the experimental data and fall below the equilibrium model predictions, as expected. The difference between the weight-average molecular weight


Figure 4. Average molecular weights predicted by the equilibrium model compared to SSP model predictions and experimental Mw values (open circles) reported in Goodner et al.11 The initial condition used in the equilibrium model was found using an initial Mn of 1750. The values used in the equilibrium model are listed in Table 2.

predictions is significant. However, the number-average molecular weights predicted by the two models are comparable. Some of the difference in Mw values arises from the nonideal initial molecular weight distribution in the prepolymer. For an ideal distribution, the value of the polydispersity index is given by 1 + x, where x is the end-group conversion. With an initial end-group conversion of greater than 90%, the polydispersity index for an ideal distribution is no less than 1.9. However, for the prepolymer on which Figure 4 is based, the initial polydispersity is much lower, 1.4. As a result, the Mw predicted by the SSP model is much lower than the equilibrium model predictions, even though the numberaverage molecular weight values are similar. This leads to a smaller polydispersity for the SSP model than for the equilibrium model. It is important to note that this comparison was performed using the initial condition obtained from the initial number-average molecular weight. The initial condition based on Mn should be used when two conflicting conversions are found using eqs 5 and 6. Equation 5 is valid for all molecular weight distributions (neglecting end-group contributions to molecular weight), while eq 6 holds only for an ideal distribution. A simulation was performed using an initial condition based on the initial Mw value (Table 2). In that case, both the number- and weight-average molecular weights calculated using the SSP model exceeded the values predicted by the equilibrium model as the polymerization proceeded. Thus, the proper initial condition must be used to preserve the equilibrium model’s capability for predicting the upper bound on the polymerization process. Comparison with PET SSP Data. The SSP of PET has been studied by many investigators.8-10,12,13,23-25 The study of Wu et al.9 addressed the effects of temperature and particle size and also reported results in terms of average molecular weights. Therefore, that study is the basis for the following comparison between experimental data and equilibrium model predictions. The three parameters used in the equilibrium model, x0, K, and R2/D, must be known to compare experimental data with model predictions. The initial conversion can be found by applying eq 5 to the initial molecular weight values. The equilibrium constant for PET transesterification has been previously reported6 and is

Figure 5. Amorphous phase diffusion coefficients reported in the literature for the diffusion of ethylene glycol in PET. The data are tabulated in Table 3. Table 3. Values of the Diffusion Coefficient and Activation Energy for Diffusion of Ethylene Glycol in PET (Graphically Represented in Figure 5) source al.19

Yoon et Lee et al.27 Mallon and Ray10 Pell and Davis16 Rafler et al.26

D at 270 °C (cm2/s) 10-7

4.28 × 2.16 × 10-7 1.888 × 10-5 1.70 × 10-4 8.2 × 10-6

ED (kcal/mol) 28.0 38.4 29.67

generally accepted as having a value of 0.5, relatively independent of temperature. Wu et al. report the particle sizes used in their study.9 However, uncertainty exists concerning the value of the diffusion coefficient of ethylene glycol (the condensate in PET formation) in semicrystalline PET particles. Several studies report experimental values for ethylene glycol diffusion in PET melts,16,26,27 while others have reported diffusion coefficients in semicrystalline PET particles.10,19 Values for the ethylene glycol diffusion coefficient from five different studies are listed in Table 3 and are presented graphically in Figure 5. There are significant discrepancies between these values, with a range of almost 3 orders of magnitude at 270 °C. To a certain degree, these discrepancies are to be expected. Because the diffusion of ethylene glycol in PET cannot be decoupled experimentally from the reversible polycondensation reaction, the results are only as accurate as the analytical method (i.e., a model or fitting routine) employed in the analysis. The two studies that seem most appropriate for use in this investigation are the SSP model-based estimations of Mallon and Ray10 and the desorption-based measurements of Yoon et al.19 Both these studies involved ethylene glycol diffusion in semicrystalline PET pellets as opposed to PET melts. However, the diffusivity values at 270 °C from these studies still differ by more than 2 orders of magnitude. The crystalline volume fraction must be known to calculate the diffusivity in the solid pellets, using eq 22. Wu et al. concluded that the samples used in their study had an approximate crystallinity of 40%, so this value was used in the simulations. The parameters used in the equilibrium model predictions for PET SSP are found in Table 4. Figure 6 compares the predictions of the equilibrium model to Wu’s data over the temperature range from 215 to 245 °C using the ethylene glycol diffusion coefficient determined by Yoon et al. The equilibrium model predictions consistently fall below the experimental data, even though the model should provide an

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Table 4. Parameters Used in the Equilibrium Model Predictions for SSP of PET As Shown in Figures 6-8 parameter

value

parameter

value

K Da (cm2/s) at 270 °C (Figure 6) Da (cm2/s) at 270 °C (Figures 7 and 8)

0.5 4.28 × 10-7

ED (kcal/mol) f

28.0 0.40

2.14 × 10-6

Figure 7. Comparison of equilibrium model predictions (lines) and Wu’s experimental data (symbols) for various temperatures using the modified diffusion coefficient listed in Table 4. The particle radius is 0.25 mm.

Figure 6. Comparison of equilibrium model predictions (lines) and Wu’s experimental data (symbols) for various temperatures. The parameters used in the equilibrium model are listed in Table 4; the particle radius is 0.25 mm.

upper limit on the molecular weight. Moreover, the deviations appear to increase with increasing temperature. The most likely explanation for the deviation is that the ethylene glycol diffusion coefficients used in the model were too low. If the diffusion coefficients reported by Mallon and Ray (listed in Table 3) were used, the equilibrium model predictions do provide an upper limit on the data. However, the molecular weight is grossly overpredicted at all temperatures. While this overprediction is consistent with the properties of the equilibrium model, it may be that the value of the ethylene glycol diffusivity pertinent to Wu’s experiments lies somewhere between those of Yoon et al. and Mallon and Ray. To facilitate interpretation of Wu et al.’s data, an intermediate value of the diffusion coefficient was used in further comparisons between model predictions and experimental data. Specifically, the diffusivity reported by Yoon et al. was multiplied by an arbitrary factor of 5 at each temperature. This new diffusion coefficient was used in the remainder of this work and is given in Table 4. Figure 7 compares the model predictions using the new diffusion coefficient to the experimental data of Figure 6. With this diffusivity, the equilibrium model provides an upper bound on the molecular weight, as expected. At the lower temperatures, the experimental data approximately coincide with the model predictions, while at the higher temperatures, the deviation is greater. Although the close agreement at lower temperatures is due to the particular value of the diffusion coefficient used in this study, the general trend of greater deviation between the experimental data and the equilibrium model with increasing temperature would hold for any reasonable choice of diffusivity as pointed out in connection with Figure 6. This behavior can be understood within the framework of the equilibrium model. The model assumes that the reaction kinetics are extremely fast compared to the condensate diffusion process. While this assumption may be valid at lower temperatures, increasing the temperature may

Figure 8. Comparison of equilibrium model predictions (lines) and Wu’s experimental data (symbols) for various particle radii at 235 °C. The modified diffusion coefficient listed in Table 4 was used in the equilibrium model.

move the system away from the diffusion-limited regime by increasing the diffusivity more than the forward rate constant. At first glance, this behavior seems counterintuitive; both Chen et al.12 and Huang and Walsh25 report that the SSP process for PET is reaction-limited at low temperatures and diffusion-limited at high temperatures. However, the activation energy reported for PET transesterification (18.5 kcal/mol)6 is considerably smaller than the activation energies for the condensate diffusion coefficient reported in the literature and listed in Table 3. Thus, the deviation between the model predictions and the experimental data in Figure 7 may be due to the polycondensation reaction shifting from diffusionlimited at low temperatures to reaction-limited (or at least reaction influenced) at higher temperatures. The possibility of this behavior has been recognized previously.8 Figure 8 further reflects the ability of the model to clarify the interaction of diffusion and reaction processes on the overall SSP kinetics. This figure compares equilibrium model predictions to Wu’s data as the particle size is varied. For the largest particle sizes, the model predictions and the experimental data agree reasonably well, suggesting that the reaction is controlled by internal condensate diffusion at these conditions. However, there is considerable deviation for the smallest particle size, and the equilibrium model provides an upper bound on the experimental data. The


difference probably is due to a shift from diffusion control for the larger particles to reaction control (or influence) in the small particles. Conclusions A simple, equilibrium-based model was created to describe SSP kinetics in a single particle when the reaction is controlled by internal diffusion of the reaction condensate. The nondimensional form of the model requires the knowledge of three parameters: the initial conversion (which is obtained from the initial molecular weight), the equilibrium constant for the polycondensation reaction, and the time-scaling factor for diffusion, which depends on the particle size and condensate diffusivity. The model applies equally well to the three common types of step-growth polymerization (A2, AB, and A2 + B2). The predictions of the equilibrium model were compared to the predictions of a full kinetic model for polycarbonate SSP. The equilibrium model provided an upper bound on the obtainable molecular weight and polymerization rate for prepolymer with an ideal molecular weight distribution. As the forward rate constant was increased in the kinetic model (for a fixed value of the equilibrium constant), molecular weight versus time profiles approached those predicted by the equilibrium model. Thus, in polymerizations that are controlled by diffusion of the condensate within the particle, higher polymerization rates can only be achieved by increasing the condensate diffusivity or reducing the particle size. Equilibrium model predictions were also compared to kinetic model predictions and experimental data for prepolymer having a nonideal molecular weight distribution. The equilibrium model still provides an upper bound on molecular weight if the initial conversion is calculated from the initial Mn data. Equilibrium model calculations were compared to experimental PET SSP data from the literature. Large uncertainties in the true value of the ethylene glycol diffusion coefficient in PET led to discrepancies in the model predictions. Nevertheless, the equilibrium model provided insight into the limiting mechanism in SSP as the temperature and particle size were varied. As the particle size is decreased, the overall SSP progresses from diffusion control to reaction control/influence, as expected. The overall rate of molecular weight increase appeared to transition from diffusion-controlled at lower temperatures to reaction-controlled/influenced at higher temperatures, a counterintuitive result. This effect was attributed to the activation energy for condensate diffusion being higher than the activation energy for the transesterification reaction. Acknowledgment This work was performed while M.D.G. held a National Research Council-U.S. Army Research Office Research Associateship. The authors also thank the U.S. Army Research Office for funding and the Kenan Center for the Utilization of Carbon Dioxide for additional support. Literature Cited (1) Gostoli, C.; Pilati, F.; Sarti, G. C.; DiGiacomo, B. Chemical Kinetics and Diffusion in Poly(butylene Terephthalate) Solid-State

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Received for review August 30, 1999 Revised manuscript received February 16, 2000 Accepted May 10, 2000 IE990648O

An Equilibrium Model for Diffusion-Limited Solid-State Polycondensation

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