Modeling of imidization kinetics.
Polyimides are an important class of high-performance polymers used in the electronic industry as interlayer dielectrics in advanced packaging applications and photoresist materials (1-3). Since requirements in these high technology areas are stringent and properties depend on the degree of conversion of polyamic acid to polyimide, the optimization of processing conditions, especially the cure profile, is very important (4). Though the details of the imidization mechanism are still not completely understood, it has been generally accepted that in most cases the rate constant remains constant only during the initial step of the reaction, and then continuously decreases (5). The change in the rate constant on a logarithmic scale as a function of degree of imidization can be represented by two straight lines, indicating the existence of kinetically nonequivalent amic acid segments (6).
There have been numerous experimental studies on the kinetics of thermal cyclization of polyamic acids. However, a physical model of this process and correlations of theoretical calculations with experiments under any conditions have not yet been thoroughly developed (5, 7, 8). Though physically reasonable, fitting the kinetic data with two first-order models includes arbitrary separation of the entire process into two regions. Recently we introduced a new model that takes into account the evolution of amic acid groups with time (9). It is quite simple and describes the general behavior of the imidization process.
The purpose of this study was to assess the applicability of this model to thermal imidization, including films with residual solvents, and to the reformation of imidization by reactivation after a temporary stop. Results for imidization of solid solvent free film are reported in another study (9).
DESCRIPTION OF THEORETICAL MODEL
There are two fundamental reasons for the drop in the cyclization rate constant of solid polyamic acids: the change in amic acid group reactivity as a result of decreasing molecular mobility when the imide ring content increases, and the existence of various kinetically nonequivalent states of amic acid groups and the statistical distribution of these groups over these states (6, 7). In order to simulate the entire imidization process, two models of solid state cyclization of polyamic acids were developed (6). The first is the so-called "cage" model, based on the assumed existence of a spectrum distribution of kinetically non-equivalent states of amic acid groups. It considers two stages: 1) the diffusion of an amic acid group from the initial state to the present state with a rate constant that depends on the state parameter, and 2) the transition of the amic acid group from the present state to the imide state, with a rate constant in Arrhenius form. The second model, the so-called variable parameter model, includes a virtual transition state of the amic acid group that essentially changes the kinetics of the reactions to a second-order differential equation. The cyclization kinetics under any temperature and time conditions can be obtained by numerically solving the second-order differential equation. Though both these models describe the dependence of kinetics on the imidization degree, this dependence is obtained with an arbitrary initial distribution of the amic acid group states and/or by the numerical solution. Our model is similar to the variable parameter model in that it assumes that two states of polyamic acid exist, favored and unfavored, and that the transition from the latter to the former varies when the degree of imidization increases. However, this model differs from the variable parameter model in that the transitions between these two states are explicitly expressed as a function of time, so it does not require numerical solutions of the differential equation to obtain the imidization isotherms at any time and temperature.
In the imidization process, it is generally accepted that chemical glass transition serves to change the rate-determining stages (6, 10-13). In the softened state, chain segment mobility is unrestrained, and the transitions between kinetically nonequivalent states are very rapid. The rate-determining stage entails the chemical conversion of amic acid group to imide rings. The rate of this process is high, and its parameters depend only slightly on the degree of conversion and are close to those for the reaction in solution. When the polymer is solidified, the transitions between kinetically nonequivalent states are hindered, and they begin to limit the process rate. Now the chemical reactivity is no longer of significant value. The rate of the amic acid group transition from unfavorable to favorable states becomes the main factor. The effective rate of the chemical reaction begins to be controlled by the distribution of amic acid group states (6). According to this physical reasoning, the preexponential factor is a decreasing function of time; initially it is a constant or depends on the cyclization degree only very weakly; later, it exhibits the form of an exponential decaying function with cyclization degree (or time) as shown in Fig. 1. We assume that all the amic acid group states are combined into two: a favorable (activated) state and an unfavorable (deactivated) state. If both the amide and acid groups in polyamic acid are in the proper conformation for cyclization, then polylmide will be produced; if not, they may return to the inactive state. Hence, part of the activated state will be deactivated; and by the same token, part of the deactivated state can be transformed into the activated state. The state function can be expressed as exp(-at) where a is a rate constant. Then the deactivated state density will be on the order of exp(-2at), because the conversion requires simultaneous deactivation of both acid and amide group at the same time. Considering that part of the excited state reverts to the previous conformation and then moves again to reexcitation, the reacting state density can be expressed as
[a.sub.0][e.sup.-at](1 - [e.sup.-2at](1 - [e.sup.-2at](1 - [e.sup.-2at](...))))
= [a.sub.0][e.sup.-at](1 - [e.sup.-2at] + [e.sup.-4at] - [e.sup.-6at] + ...)
= [a.sub.0] [e.sup.-at]/1 + [e.sup.-2at] = [a.sub.0]/2 2/[e.sup.at] + [e.sup.-at] = [a.sub.0] sech (-at) (1)
where [a.sub.0] is the initial distribution factor for the reaction. Then the preexponential factor can be expressed as the following nonlinear function form,
f(t) = b sech(-at) (2)
where a and b are constants. The kinetic equation can be written as
- dx/dt = b sech(-at)x (3)
where b is a constant and x is the concentration of amic acid groups. Integration gives
-1n(1 - [Alpha]) = 2b/a [tan.sup.-1][e.sup.-at] + constant (4)
where [Alpha] is the conversion of amic acid groups.
This equation has recently been introduced to simulate the solid film imidization process (9). The sech(-at) function initially has a constant value and depends weakly on time. Later it becomes an exponentially decreasing function of time. This function's nonlinear form can simulate the variation of the amic acid group state with time represented by the sum of exponents, and each is used with its own weight (9). A schematic of the nonlinear function behavior of Eq 2 is shown in Fig. 2. A large a value induces a rapid decrease in reaction rate with time, so that the steady state is reached quickly, while the effect of b appears more slowly. The nonlinear function form of Eq 4 permits simulation of the general features of the imidization process. In the initial stage, this function is almost a constant, in accord with a process characterized by the participation of highly active amic acid groups and limited by the chemical conversion of amic acid groups into imide rings. In the final stage, it decreases exponentially because it is controlled by the evolution of active amic acid groups from the inactive state. Hence, it describes a change in the rate-determining stage and a sharp drop in reaction rate.
RESULTS AND DISCUSSION
To test the model, we compared the calculation with experimental data reported in the literature. Thermal conversion rates of polyamic acid derived from pyromellitic dianhydride and 4,4[prime]-diaminophenyl ether were reported by Kreuz et al. (13). Thermal ring closure rate studies were made with films cast from dimethylacetamide. The polyamic acid films contained about 28% dimethylacetamide. The conversion of polyamic acid is shown in Fig. 3. This conversion represents the typical character of the imidization process, i.e., initial rapid cyclization and later slow cyclization. In the initial stage, solvent diffusion control is unimportant. The solid lines are regression fits with the theoretical model. Good correlation can be observed between the calculation and the experimental data at all temperatures. The activation energy was obtained from the Arrhenius plot of rate constant a (1n(a) vs. 1/T) as shown in Fig. 4. The activation energy determined from the slope of this curve is [E.sub.a] = 22.6 Kcal/mole, which is in good agreement with the activation energy of fast cyclization of 26 [+ or -] 3 Kcal/mole and 23 [+ or -] 7 Kcal/mole obtained from the two first order fittings by Kreuz et al. (13).
The role of kinetic nonequivalence and chemical glass transition was investigated by Laius and Tsapovetsky, who compared the cyclization isotherms of pure polyamic acid and its solid solution in the amorphous polymeric matrix (6). In the first case, the reaction begins in the softened state and is completed in the glassy state while in the second case, the system remains in the glassy state from start to completion of imidization. Accordingly, the first isotherm has an extended initial high-rate section corresponding to the softened state and a slow-rate part corresponding to the glassy state of the polymer whereas the second isotherm has a very small high-rate section. Figure 5 contains kinetic plots of these two isotherms on a semilogarithmic scale. The model fits the experimental data very well. The parameter values for these two curves are 1) 0.1943 and 0.1586 for a, and 2) 0.2547 and 0.09341 for b, respectively. A small a value for the pure polyamic acid means its active state density decreases more slowly, indicating a high reaction rate at 180 [degrees] C. Small a values provide higher reaction rate values for a longer period as shown in Fig. 2. The values of the ratio (2b/a) are 1.17785 and 2.621; hence it is easily understood from Eq 4 that pure polyamic acid has higher conversion. According to Laius and Tsapovetsky, chemical glass transition serves to "switch on" or "switch off" the rate determining stages (6). In the softened state, chain segment mobility is unrestrained, and the transition between kinetically nonequivalent states is very rapid. The rate-determining stage entails the chemical conversion of amic acid group to imide rings. Hence, the rate of the process is high. Its parameters depend only slightly on the degree of conversion. On the other hand, when the polymer is solidified, the transitions between kinetically nonequivalent states are hindered, and they begin to limit the process rate. Hence the rate of amic acid group transition from unfavorable to favorable states becomes the main factor. Figure 5 and the parameter values are in accordance with this explanation.
This model also predicts the effect of cyclization activation in the stage of kinetic stop. The experiments on the activation of cyclization were carried out for the polyimide from 4,4[prime]-oxydiphthalic anhydride and p,p[prime]-diaminodiphenylmethylbenzene by Laius and Tsapovetsky (6). In one case the dissolution of a partially imidized sample was used to increase molecular mobility. The polymer was soluble in both polyamic acid and the imide forms. As seen in Fig. 6, when the sample is dissolved in the intermediate stage of imidization and then cast into a film again, the cyclization rate increases sharply. The kinetic curve was calculated for two steps. Before the reactivation, the curve was calculated using Eq 4. After the reactivation, one parameter was added in Eq 4 to take into account the previous imidization reaction time,
-1n(1 - [Alpha]) = 2b/a [tan.sup.-1][e.sup.-a(t- [t.sub.0]]) + constant (5)
where the previous degree of imidization is included in the value of the constant. As shown in Fig. 6, the calculated and experimental results coincide very well. The minimum points at the restarting times are numerical artifacts due to the function shape of Eq 5. The activation energy calculated from the a values using an Arrhenius relationship is shown in Fig. 7. This energy is 22.8 Kcal/mole. At the kinetic stop, the favorable states are almost consumed and the conversion from the deactivated state is almost halted. When the temperature is increased rapidly, the conversion from the deactivated state to the activated state is increased. This provides an increase in the cyclization rate at the first moment when the initial temperature of the experiment is reached again, which means a restarting of imidization after some time delay. The delay time to was obtained by fitting the data as 2490 sec, 2550 sec, 1200 sec, at 120 [degrees] C, 140 [degrees] C, and 160 [degrees] C, respectively. Considering experimental error, consistency with experimental findings is excellent. This result agrees very well with the cage model prediction (6), and confirms the validity of the theoretical model's prediction.
It has been demonstrated that the proposed model expressing the rate constant of the imidization kinetics as a function of time, sech(-at), can correctly describe the film imidization kinetics, including imidizations in the presence of solvent or after reactivation. In all cases, calculated kinetic values were in good agreement with experimental data. Activation energies calculated from the simulation were also in good agreement with those obtained from the first order model. The role of kinetic nonequivalence and chemical glass transition was well described by this model. Its applicability was assessed for simulating the reformed imidization process, which involved two separate stages. Simulation results revealed good quantitative and qualitative correlations with experimental data and also are in good agreement with the results of other, more complicated models.
The new model describes the general behavior of the imidization process very well because of its nonlinear form which correlates with the evolution of imidization kinetics. The model will be useful for analyzing the imidization process without arbitrary fitting of the process with two first-order kinetic models.
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|Publication:||Polymer Engineering and Science|
|Date:||May 1, 1997|
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