Modeling particle inflation from poly(amic acid) powdered precursors. III. Experimental determination of kinetic parameters.
Although publications on polyimide foams and methods to produce them have been available in the patent literature for several decades [1-4], it has been only recently that archival publications have been published on the topic in scientific journals. NASA's development of a variant to the classical process, involving the use of a hydrogen bonding solvent that acts as a blowing agent has sparked renewed interest in the topic [5, 6]. In this novel process, termed solid-state powder foaming by the authors, poly(amic acid) (PAA) precursor powders are prepared by reacting aromatic diacid-diesters with aromatic diamines in the presence of a hydrogen bonding solvent (e.g., tetrahydrofuran, THF and methanol, MeOH). These powders are then heated and inflated to produce microspheres or neat foams with varying cell morphologies (Fig. 1).
More recently the authors have produced several articles where the different physicochemical phenomena involved in the inflation process have been studied and modeled [7-10]. They modeled the inflation process from first principles, considering the different concurrent phenomena involved . During this modeling effort, it was required to represent the changing properties of the polymer as the different phenomena advanced. As it will be discussed below, it was found that it was appropriate to follow the change in properties as dependent on the change of the glass transition temperature of the system. All the other transport properties (diffusion coefficients, thermal properties, density, etc.) were referred to this changing quantity. Although the published literature with respect to polyimide systems is expanding, there are very few sources for experimental data regarding the measurement of these properties or the effect of [T.sub.g] on them. Because of this, the authors had to develop experimental procedures to approximate the complex behavior of this changing system. This article presents the experimental and analytical approaches used to characterize the change in [T.sub.g] during the inflation process.
Change of Transport Properties During the Thermal Inflation of Precursor Particles
The physical properties that will determine the complex diffusive and Theological behavior undergone by the precursor particles as they transform from poly(amic acid) particles to polyimide bubbles will depend ultimately on the type of polymeric species involved. Most importantly, these properties may be directly related to the glass transition temperature of each species, which in turn will depend on molecular architecture, intermolecular attraction, and degree of plasticization by the solvents.
[FIGURE 1 OMITTED]
By following the inflation process in a thermogravimetric analyzer (TGA), three main desorption zones that exist during the foaming process have been previously identified (Fig. 2) . These zones are closely related to kinetic and thermodynamic processes, which themselves may be separated into three stages. Initially, a particle of solid-state oligomeric solution of PAA in THF/MeOH will show a depressed glass transition temperature due to the presence of the solvent. During the initial temperature rise, the rheology of the system will be affected due to solvent depletion providing a viscosity increase, while simultaneously the inherent thermal increase in Brownian motion affects the chain separation reducing the viscosity. These two processes act simultaneously and eventually give way to the much stronger temperature effect and an overall decrease in viscosity once the solution temperature equals its glass transition temperature. In a similar fashion, coefficients of mass diffusivity of the solvent will be affected as well by the changing concentration-temperature profiles. A decrease in concentration will reduce the effective solvent diffusion while a temperature increase allows higher intermolecular mobility of the diffusant.
At this point, the onset of higher mobility indicates the start of the second stage. When the process temperature exceeds the [T.sub.g] of the PAA matrix, unreacted amine and ester groups are able to react and increase the molecular weight. The byproduct of such reaction is methanol, which further plasticizes the polymer during its diffusion from the polymer chains to the particle surface. Additionally, the primary blowing agent still present in the particles becomes free upon decomplexation  and upon the onset of greater mobility due to the transition from a glassy to a rubbery state. This increases the rate at which volatile material desorbs from the particle.
The beginning of the third stage overlaps the end of the second stage and corresponds to the conversion from poly(amic acid) to polyimide by thermal cyclization of the amic-acid moieties. Because of the randomness of the imidization reaction on the chain, at this stage there is no single specie of macromolecule, but rather several chains with varying contents of amic acid or imide. The process of imidization of the poly(amic acid) chains into polyimide may be regarded as a random copolymerization process , in which different repeat units with amic-acid groups undergo thermal cyclation in no specific order. Therefore, the initially exclusive poly(amic acid) precursor transforms after a complete imidization process into a total polyimide system in the end. As it may be expected, the chemical transformations that take place during stages two and three directly affect all material and transport properties involved. The effect of such transformations is evidenced not only in the introduction of secondary volatile byproducts, but also in the change in glass transition temperature from that of the oligomeric poly(amic acid) ([T.sub.g] = 70-110[degrees]C depending on the degree of polymerization and plastization) to that of the corresponding polyimide ([T.sub.g] = 276[degrees]C).
In an effort to correlate the changing properties to the dynamic scenario that takes place during the bubble growth process, a single material property has been selected for common reference. This property is the glass transition temperature, [T.sub.g], of the polymeric chains. The [T.sub.g] may be regarded as the temperature at which long-range segmental molecular motions cease to exist during the transition from rubbery to glassy state . It is this change in molecular mobility that accounts for noticeable changes in material properties and serves as reference for their prediction. Although the glass transition temperature is a second-order transition which is kinetically controlled and depends on the experiment time-scale , its relationship to other material properties is well established. Significant research efforts by several different research groups addressed both fundamental and experimental relationships between the glass transition temperature of polymeric species and several properties such as viscosity, diffusion coefficients, and thermal properties among many others. These properties vary according to other parameters such as solvent content or molecular weight, but the plastization effect that solvents exert over polymers and the existing relation between chain length and chain stiffness allows for the inclusion of such effects within the variation of the glass transition temperature.
[FIGURE 2 OMITTED]
The Glass Transition of Poly(amic acid) Precursors and Polyimides as a Function of Conversion and Plastization
Chain length and chain architecture play significant roles in the segmental mobility of macromolecules. During the powder foaming process, the precursor particles experience a chemical transformation from poly(amic acid) oligomer to poly(amic acid) polymer, and finally to polyimide. It is considered that for poly(amic acids), molecular dimensions do not change substantially before and after thermal imidization . Any chain scission present due to anhydride regeneration is restored upon reaching higher temperatures and mobility. Therefore, the only change that takes place is the carboxylic and amidic moieties of the repeat units when they come into contact and taking place into what is known as dehydrocyclization. The cyclization reaction involves a change in chain architecture as two rings per repeat unit are added to the polymeric chains. These new rings plus the existing aromatic rings present from the aromatic dianhydride and diamine in the poly(amic acid) (i.e., six rings in total per repeat unit) hinder chain rotation and enhance the rigidity of the chain, consequently increasing the [T.sub.g] of the polymer (Fig. 3). Through this process, molecular architecture does play an important role in the glass transition temperature of the transient system.
Kotera et al.  performed curing experiments on several polyimide systems during curing reactions and found that the glass transition of the partially cured polyimide follows the so-called Fox equation, which is typical of the glass transition change expected for a random copolymer:
1/[T.sub.g] = [[w.sub.PAA]/[T.sub.g,PAA]] + [[w.sub.PI]/[T.sub.g,PI]] (1)
where [w.sub.PAA] and [w.sub.PI] correspond to the weight fractions of PAA and polyimide repeat units respectively.
Kim et al. [15-17] proposed an expression for the glass transition temperature of poly(amic acid) precursors plasticized with different solvents. Their comprehensive work deals with different types of polyimide systems and their expression is based on a modified Gordon-Taylor equation of the type:
[T.sub.g] = [[[DELTA][[alpha].sub.s][T.sub.g,s][w.sub.s] + [DELTA][[alpha].sub.P][T.sub.g,P][w.sub.P]]/[[DELTA][[alpha].sub.s][w.sub.s] + [DELTA][[alpha].sub.P][w.sub.P]]] + [[[DELTA]k [w.sub.s][w.sub.P]]/[[DELTA][[alpha].sub.s][w.sub.s] + [DELTA][[alpha].sub.P][w.sub.P]]] (2)
where [DELTA][[alpha].sub.i] is the difference between the volume expansion coefficients of the component i (S is solvent, P is polymer) in the rubbery (or liquid) and glassy states. The parameter [DELTA]k represents the excess volume change of mixing the two components.
[FIGURE 3 OMITTED]
Based on these two expressions, knowledge of the pure poly(amic acid), polyimide, and solvent glass transition temperatures, as well as their concentration and some other thermodynamic properties is sufficient to estimate the changes of the effective glass transition of the system with molecular architecture and solvent content during the inflation process.
Kinetics of Imidization for Aromatic Polyimide Systems
Thermal imidization of poly(amic acid) precursors into polyimide requires heating the poly(amic acid) to temperatures ranging from 250 to 400[degrees]C. Poly(amic acid) groups can suffer three types of transformations upon heating: the group can cyclize itself to produce either an isoimide group, an imide group, or to regenerate an anhydride molecule. Undesirable side reactions may occur during the imidization stage. Depolymerization of poly(amic acid) occurs to a certain degree during the thermal imidization. Similarly, hydrolysis in the anhydride formation can be compared to the temperature effect, and can be prevented if the dianhydride is reacted with a small alcohol (e.g. methanol, ethanol) before the reaction with the diamine to esterify the carboxyl groups in the dianhydride. This will prevent the action of water in carboxyl groups and cyclization due to temperature . Another transformation that can possibly occur is crosslinking. However, evidence of crosslinking has not been found below imidization temperatures of 300[degrees]C . As it was mentioned earlier, at normal imidization temperatures it has been shown that molecular weight does not change appreciably during the imidization reaction from poly(amic acid) precursor to polyimide .
Mechanism and Kinetics of Thermal Imidization. Significant research has been devoted to study the conversion of poly(amic acid) precursors into high-molecular-weight polyimides [12, 18, 19-28]. The attractive properties of polyimides develop during the imidization, and are a function of the degree of imidization (also called degree of cure even though for thermoplastic polyimides no crosslinking takes place). From the seminal study of imide cyclization by Kreuz et al. , an interesting phenomena has been observed. Under isothermal imidization conditions, two primary rates of imidization seem to exist. In the first one, the process is carried on at a constant rate of cyclization, while in a later stage, this rate decreases until an almost complete halt in imidization occurs even before the poly(amic acid) is consumed entirely. Laius et al.  presented a clear phenomenological explanation of the slowing of the reaction. He claimed that although the reaction of ring closure is an intramolecular one, which should be regarded as a first-order reaction, experimental evidence did not support this claim. Three primary explanations were proposed:
1. Amic acid groups in different, nonequivalent (i.e., activated and deactivated) kinetic states exist . These two different kinetic stages exist due to different conformations of the chains in the matrix, and as solvent is depleted during the thermal imidization, rigidity and lack of plastization hinder conformation of the nonactive groups into active ones, thereby slowing the reaction.
2. The reactivity of amic acid groups decreases during solvent depletion and later vitrification of the polymeric matrix. In the early stages, enough solvent is still present to allow the constant imidization. Because of the effect of heating, the amount of residual solvent will decrease and at the same time the rate of imidization decreases as well . One of the reasons why solvent helps the imidization process is the enhancement of molecular mobility in the presence of solvent that favors the previously mentioned optimum conformation between carboxyl and carboxamide groups in the same molecule. The rise in the glass transition temperature of the matrix as molecular weight builds up during the imidization reaches a point at which the [T.sub.g] of polyimide will be greater than the reaction temperature, thereby limiting the mobility of chains while it enters the glassy region. Eventually the imidization stops.
3. Important side reactions mentioned earlier which depolymerize the poly(amic acid) chain regenerating anhydride can reach important levels at higher temperatures and conversion levels.
Considering the different factors affecting imidization, it has been proposed by some researchers that poly(amic acid) imidization kinetics can be divided into three stages: a kinetics-controlled stage (fast early stage), a diffusion-controlled stage (slow final stage), and a transitional stage between the first two stages .
Imidization kinetics has been modeled as both first and second-order reactions. The mechanistic view of such process dictates that in an intramolecular reaction there is no logical reasoning to model it as second order [22, 28]. However, experimental data yield better agreement to the second-order kinetics than to the first order. Good agreement has been found as well in modeling it as two sequential, first-order reactions . The reason for this is the previously mentioned two-stage kinetics which to a certain degree mimics the diffusional limitations existent in second-order systems . Although complex models based on kinetically nonequivalent states  fit the data, lack of physical relation between the many parameters involved and the phenomena that take place in the reaction make them difficult to apply.
Fourier transform infrared spectroscopy, FT-IR, and more recently calorimetric analysis on curing systems of aromatic polyimides has brought insight into the interaction between poly(amic acid) chains and reaction solvents and the effect they have on imidization kinetics [12, 20, 21, 23. 26]. It is believed that commonly used solvents interact with the precursor molecules in the form of hydrogen bonds with the amidic and acid moieties. In particular, Brekner et al. [20, 21] mentioned molar complexes between the studied solvent, 1-methyl-2-pyrrolidinone (NMP) and pyromellitic dianhydride dianiline poly(amic acid)s where four molecules of NMP hydrogen bond to the repeat unit of the precursor creating populations of bound and free species of NMP. These bound solvent molecules are said to inhibit imidization while they remain attached to the polymer chain, but upon reaching a specific decomplexation temperature (dictated by the decomplexation activation energies for each specific bond) they are free to diffuse through the polymeric matrix plasticizing it and favoring imidization conformation.
To determine the kinetic parameters of both reactions involved in the chemical transformation, two primary steps were taken. The imidization parameters were determined first due to the ease of obtaining an isolated system of pure, high-molecular-weight poly(amic acid). This is necessary to exclude any interactions of possible amidation that may disturb the measurements from FT-IR spectra, TGA, and DSC (differential scanning calorimetry) thermograms. To obtain pure, high-molecular-weight PAA in the same solvent system as the low-molecular-weight oligomer, a direct amidation method was followed. Once the kinetic parameters had been obtained for the imidization reaction, they could be used to determine the kinetic parameters of the amidation reaction as both reactions overlap partially during the thermal inflation. Kinetic parameters from the amidation reaction were obtained from the thermal treatment of precursor powders obtained by the regular procedure (diacid-diester route) followed in this work.
Poly(amic acid) solutions were prepared by two different methods: the diacid-diester method which yields low-molecular-weight PAA oligomers (i.e., number average degrees of polymerization D[P.sub.n] between 1.2 and 3), and the direct amidation method which yields high-molecular-weight PAA (D[P.sub.n] ~ 29).
The diacid-diester PAA (DADEPAA) was prepared from BTDA and ODA in a two-step process consisting of an initial esterification step and a subsequent amidation step. An amount of 151.2 g of 3,3',4,4'-benzophenonete-tracarboxylic dianhydride (BTDA) was added to a solution of 96 g of THF and 56 g of methanol. Temperature in the reaction kettle was maintained at ~60-61[degrees]C. These conditions were maintained with constant stirring and total reflux, as well as a constant nitrogen blanket until the solution turned clear (dissolved diester, BTDE). Subsequently, an amount of 93.64 g of 4,4'-oxydianiline (ODA) was added to the BTDE solution and left to react at 70[degrees]C for 2 h (Fig. 3). The solution was then dried in the absence of moisture and subsequently ground into fine powder. The fraction of powder of particle size less than 75 [micro]m was used to increase surface area and enhance solvent desorption during testing.
The direct amidation PAA (DAPAA) was prepared according to the procedure described by Echigo et al. . In this method, the esterification step is omitted, higher reactivity is obtained and full conversion of the dianhydride and diamine into PAA is achieved. An amount of 2.65 g (0.0129 mol) of ODA was placed into a three-necked reaction flask fitted with a mechanical stirrer, thermometer, and nitrogen inlet. THF (31.47 g) and 7.87 g of MeOH were added to produce a solvent ratio of 8/2 by weight. Stirring was begun and once the ODA had dissolved, 4.29 g (0.0129 mol) of BTDA were added slowly during 30 min. The final solids content of the solution was 15% by weight. Temperature was maintained at room temperature (20-25[degrees]C) during the reaction. The solution was left with stirring under nitrogen for 24 h to achieve the most probable molecular-weight distribution for the polymer. After 24 h of reaction, a clean KBr plate was spin-coated with the poly(amic acid) solution. The plate was left inside a dessicator for 24 h to prevent hydrolysis of the PAA. Once dry, the sample was removed for FT-IR testing. Additionally, a sample of the precursor solution was poured into a Petri dish and dried in vacuum at 70[degrees]C for 72 h and at 110[degrees]C for 2 more hours to achieve complete elimination of reaction solvents. This sample was used for TGA and DSC analysis.
Infrared spectroscopy was performed on precursor samples during the imidization reaction from poly(amic acid) to polyimide to follow the imidization kinetics. The characteristic vibrations of certain bonds in the poly(amic acid) molecules, and their change with time and temperature were followed through the IR spectrum during the reaction. A Digilab Excalibur FTS-300 FT-IR equipped with a temperature-controlled cell (Omega CN 3251) was used in ramp mode under nitrogen atmosphere.
The kinetics of amidation was followed with a Q1000 TA Instruments DSC in the modulated DSC (MDSC) mode. This technique was especially useful as several transitions during the experiments occurred at similar temperature intervals (e.g. volatilization of solvents and glass transition of the precursor), and the use of MDSC permitted a separate identification and analysis. For all the experiments, hermetic aluminum pans were used. Because of the liberation of volatiles from the samples during the experiments, and to avoid pressure-related effects, most pans were modified by opening a pinhole on the lid to allow the volatiles to escape. The samples were subjected to an underlying linear heating rate of 5[degrees]C/min, amplitude of [+ or -] 0.663[degrees]C, and a period of 50 s.
Thermogravimetric analysis (TGA) was carried out under nonisothermal conditions. For this purpose, a TA Instruments Q500 TGA was employed. Through all the experiments a nitrogen purge was maintained in the furnace and the balance to minimize the effects of moisture or undesired oxidation. The heating rate was maintained at 5[degrees]C/min.
RESULTS AND DISCUSSION
Fourier Transform Infrared Spectroscopy (FT-IR) Analysis
The overall conversion of the imidization reaction may be examined by evaluating the relative change of the following absorption peaks during the imidization reaction (Fig. 4): C--N imide stretching band I at 1370-1375 [cm.sup.-1], C--N imide stretching band II at 720 [cm.sup.-1], carbonyl symmetric stretching band at 1780 [cm.sup.-1], and carbonyl coupled stretching band at 1726 [cm.sup.-1] . However, of these peaks only the one at 1375 [cm.sup.-1] does not appear to be affected by temperature, dichroism, or anhydride regeneration, making it the best choice . The internal standard chosen for quantification is the 1500 [cm.sup.-1] band, considered the "ring breathing mode" of the aromatic diamine which will remain constant during the reaction and eliminates any "film thickness" effect .
The nonisothermal conditions of the experiment with a ramp temperature increment of 5[degrees]C/min allow sufficient heat transfer between measurements to occur. The conversion was determined by the ratio of the height of the imide stretching band peak divided by the internal standard peak, where a conversion value of 1.0 is defined as such ratio when it stops changing.
[FIGURE 4 OMITTED]
[alpha] = [(D/D*).sub.T]/[(D/D*).sub.F] (3)
Eq. 3 relates the overall conversion, [alpha], to the ratio of peak heights (D for the imide stretching peak at 1375 [cm.sup.-1] and D* for the internal standard peak at 1500 [cm.sup.-1]) at any temperature T, to the same ratio when it ceases to change, denoted by F. The change in peak heights can be followed in Fig. 5 for different imidization temperatures during the ramping process. The conversion of the imidization reaction, as determined by following the 1375 [cm.sup.-1] peak, is presented in Fig. 6.
Calculation of Parameters and Governing Equations for the Kinetics of Imidization of Poly(amic acids)
Before calculating the kinetic parameters for the imidization reaction, it is necessary to determine the order of the reaction. This is accomplished following the procedure of Carroll et al.  for nonisothermal thermal analysis.
A generalized rate equation of the form:
d[alpha]/dt = k(1 - [alpha])[.sup.n] (4)
is assumed, where [alpha] is the conversion, k is the rate constant, and n is the order of the reaction. The heating rate, m, of the experiment can be expressed as:
m = dT/dt (5)
Substitution of Eq. 5 into Eq. 4 yields the nonisothermal generalized rate equation:
[FIGURE 5 OMITTED]
d[alpha]/dT = [k/m](1 - [alpha])[.sup.n] (6)
Additionally, the rate constant can be expressed as an Arrhenius function of temperature of the form:
k = A exp ([-[E.sub.a]]/RT) (7)
where A, the pre-exponential term is a constant characteristic of each reaction (referred to as the frequency factor), and [E.sub.a] is the activation energy for the reaction. Substitution of Eq. 7 in Eq. 6 gives the following expression of the rate of conversion with temperature:
[FIGURE 6 OMITTED]
d[alpha]/dT = [A/m]exp([-[E.sub.a]]/RT)(1 - [alpha])[.sup.n] (8)
By applying natural logarithms and on both sides of the equation, differentiating again with respect to temperature and some algebraic manipulation, the following linear expression is obtained:
[[d/dT](ln[d[alpha]/dT])]/[[d/dT](ln(1 - [alpha]))] = n + [E.sub.a](R[T.sup.2][d/dT](ln(1 - [alpha])))[.sup.-1] (9)
A plot of Eq. 9 will yield a straight line where the slope will correspond to the activation energy, [E.sub.a], and the intercept to the order of the reaction, n. From this analysis, and following the 1375 [cm.sup.-1] peak, the reaction appears to be of second order (i.e., n = 2.034) and the activation energy obtained is 148.5 kJ/mol. This result contradicts the expected single-order characteristic of an intramolecular reaction, but the diffusional and conformational limitations of the system, as discussed earlier, may account for this result.
For a second-order reaction the rate in Eq. 8 becomes :
d[alpha]/dT = [A/m]exp([-[E.sub.a]]/RT)(1 - [alpha])[.sup.2][C.sub.0] (10)
where [C.sub.0] is the initial concentration of carboxylic or amidic (i.e., same concentration of the PAA) functional groups. Manipulation of this equation yields:
ln(d[alpha]/dT) - ln([(1 - [alpha])[.sup.2][C.sub.0]]/m) = ln(A) - [[E.sub.a]/RT] (11)
The plot of this linear function (left hand side vs. 1/T) allows the determination of the pre-exponential factor and the activation energy from the intercept and slope, respectively. These two parameters provide the temperature dependence of the kinetic constants. The two parameters obtained for the imidization reaction from the 1375 [cm.sup.-1] peak were A = 3.21 X [10.sup.17] L [mol.sup.-1] [s.sup.-1] and [E.sub.a] = 148.5 kJ/mol. Experimental results of imidization reactions for this system from direct amidation in other solvents  present lower [E.sub.a] values of 93 kJ/mol, and have peak imidization rates occurring at temperatures close to 180[degrees]C compared to peak rates at temperatures of 210[degrees]C for the present experiment. A possible explanation for such behavior would be that solvents commonly used for poly (amic acid) synthesis (e.g. NMP [T.sub.b] = 202[degrees]C, DMF [T.sub.b] = 153[degrees]C, DMAc [T.sub.b] = 165[degrees]C) have boiling temperatures much greater than those of THF/methanol systems, which are around 60[degrees]C. The low volatility and affinity to complexation with the poly(amic acid) of these solvents, allow them to stay with the polymer for a much longer time while plasticizing the material and allowing faster conformation of the amic acid groups at lower temperatures.
[FIGURE 7 OMITTED]
Calculation of Parameters and Governing Equations for the Kinetics of a Midation of Poly(amic acids)
MDSC data provides information regarding the heat flow due to kinetic events occurring within the sample during the thermal treatment. A thermogram for the precursor sample from direct amidation is presented in Fig. 7. The plot presents two endotherms, a small one at 87[degrees]C (onset at 67[degrees]C) and a larger one at 164[degrees]C (onset at 152[degrees]C). The first endotherm is attributed to vaporization of residual solvents (THF and methanol), as well as absorbed water between the drying step and the DSC experiment. The second peak is clearly due to the highly endothermic imidization reaction. The onset for the imidization peak as calculated from the DSC correlates well with the value obtained from FT-IR.
The enthalpy or reaction for the imidization [DELTA][H.sub.RI], is obtained from the integration between the onset and endpoint of the imidization peak (123.8 J/g). To avoid pressurizing the reactive system and therefore resemble as much as possible the foaming scenario, a pinhole was perforated in each DSC lid to allow volatiles to escape. It is therefore necessary to correct the enthalpy measurement for the actual weight of material that underwent imidization. Such effective weight is obtained from TGA data for the same material (i.e., PAA from direct amidation) as the weight where the onset of volatile desorption occurs. The corrected value for [DELTA][H.sub.RI] is then 125.9 J/g, which is in accordance to reported values obtained for similar polyimides (i.e., 125 J/g) .
The heat of reaction from the amidation reaction can now be obtained from a MDSC thermogram of a dry (i.e., free of primary blowing agent) sample prepared by the diacid-diester method (Fig. 8). Three endothermic peaks are present: a small peak at 66[degrees]C (onset at 41[degrees]C), a second one at 148[degrees]C, and a third at 186[degrees]C. Onsets for these last two peaks were impossible to be determined due to overlapping. As in the previous sample, the first peak responds to the vaporization of remaining THF, methanol, and absorbed water. However, in this case, even though the samples were devolatilized for a long time, the remaining primary blowing agent was still significant. This was caused by the possible reaction of remaining monomers at the devolatilization temperatures. The second peak, which was not present in the first sample prepared by the direct amidation route, corresponds to methanol liberated during the continuation of the polycondensation between diacid-diester and amine groups. The end of the amidation peak overlaps with the onset of imidization and at 186[degrees]C peaks 22[degrees]C over the imidization from the first sample. The fact that not all the amic acid bonds are present initially (since this is the oligomeric PAA), but rather are produced simultaneously with part of the imidization could be responsible for such a shift. The heat absorbed by the second and third endotherms must correspond to the additive individual enthalpies of each individual reaction. Based on this assumption, and the previously obtained enthalpy for imidization, the heat of reaction for the amidation reaction can be calculated.
[FIGURE 8 OMITTED]
To determine the amidation enthalpy, another assumption must be made: the total weight of the sample after the imidization (i.e., when the thermal derivative in a TGA plot approaches zero approximately at temperature [T.sub.2] in Fig. 9) corresponds to polyimide at full conversion. Considering this assumption and with sufficient TGA data, the total expected release of water from the imidization can be calculated and compared to the measured amount, thereby obtaining the mass of methanol produced and the mass of PAA that undergoes imidization. The procedure can be summarized as follows: The total mass of water from imidization can be determined by knowing the remaining sample weight (at temperature [T.sub.2] in Fig. 9) and calculating the corresponding stoichiometric amount of water (assuming full conversion). The total mass that undergoes amidation ([m.sub.A]) corresponds to the weight measured by the TGA at [T.sub.1] in Fig. 9. The ratio of methanol produced during the TGA experiment to that expected for full conversion of the amidation reaction gives a measure of the existent amidation conversion prior to any inflation regime. To determine the amount of methanol produced during inflation, the total amount of water produced from the imidization is subtracted from the total weight change due to volatiles expelled during inflation (i.e., weight change experienced between [T.sub.1] and [T.sub.2]). The mass of PAA present before imidization ([m.sub.I]) is also easily obtainable by stoichiometric calculations upon knowing the final mass of polyimide. The existent amidation conversion prior to inflation can then be easily calculated by a ratio of the amount of methanol produced during inflation, to the total amount of methanol expected from full conversion of the amidation reaction (obtained as well by stoichiometric calculations assuming full conversion). Using this procedure and for different samples prepared via the diacid-diester process this conversion was calculated to vary between 17 and 36%, which with the aid of the following equation:
D[P.sub.n] = 1/[1 - [alpha]] (12)
yields an approximate degree of polymerization of 1.21-1.57 (M[W.sub.n] = 629-819). The calculated molecular weight is in close agreement with the measured value from GPC (M[W.sub.n] = 640)  and confirms the validity of the previous assumptions. Variation in the degree of polymerization of the different samples depends on several factors such as reaction time and temperature or drying time and temperature among others. By knowing the pre-TGA conversion and sample weight it is possible to calculate the effective weight to be used as a base for the total heat of amidation.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
The total heat used in the amidation and imidization reactions during the MDSC run reaction can be expressed as:
Q = [DELTA][H.sub.A] + [DELTA][H.sub.I] = [DELTA][H.sub.RA][m.sub.A] + [DELTA][H.sub.RI][m.sub.I] (13)
where [m.sub.A] and [m.sub.I] are the mass of material that undergo amidation and imidization, respectively (explained earlier). [DELTA][H.sub.RA] and [DELTA][H.sub.RI] are the respective enthalpies of reaction. From TGA data, the values of [m.sub.I] and [m.sub.A] are known and both Q and [DELTA][H.sub.RI] are calculated from the DSC data. Eq. 13 may be reorganized as:
[DELTA][H.sub.RA] = [Q - [DELTA][H.sub.RI][m.sub.I]]/[m.sub.A] (14)
The value for [DELTA][H.sub.RA] obtained by this procedure is 41.06 J/g.
An approximation to the kinetics of amidation may be obtained from the separation of the two overlapped endotherms by knowing a priori the behavior of the imidization kinetics (Fig. 10). From Eq. 14 and considering that the conversion can be expressed in terms of the heat flow as:
[alpha](T) = [[DELTA][H*.sub.I](T)]/[[DELTA][H.sub.I]] (15)
where [DELTA][H.sub.I] is the total heat for the imidization reaction (calculated from the mass and [DELTA][H.sub.RI]), the heat flow for the hypothetical case of only imidization occurring [DELTA][H*.sub.I](T) is obtained after algebraic manipulation. This hypothetical function is subtracted from the total nonreversible heat flow signal obtained from the MDSC and gives the resultant hypothetic endothermic heat flow for the amidation reaction [DELTA][H*.sub.A](T). After this procedure, two curves for each of the endotherms are separated from the overlapped total provided by the calorimeter (Fig. 10). The conversion of amic acid is straightforward by using an analogous equation to Eq. 15:
[alpha](T) = [[DELTA][H*.sub.A](T)]/[[DELTA][H.sub.A]] (16)
where [DELTA][H.sub.A] is the total heat of amidation.
The determination of the kinetic parameters for the amidation reaction was performed in a similar fashion as for the imidization. For the reaction of diesters of carboxylic dianhydrides and diamines, a first-order mechanism has been proposed in the literature . A fit of the conversion data to this model gives a good approximation for modeling purposes (Fig. 11). More complex models like the second-order autocatalytic reversible model  are difficult to compare to the DSC data in nonisothermal conditions owing to the need of six parameters for the data (i.e., frequency factor and activation energy for each of the three constants involved), therefore is not used in this work. The values for the kinetic parameters obtained from this procedure were: [E.sub.a] = 104.6 kJ/mol and A = 5.61 X [10.sup.14] [min.sup.-1]. A summary of the kinetic parameters for both reactions is presented in Table 1.
The successful representation of a polymer-solvent system for modeling purposes requires a complete understanding of the different material and transport properties. For the case of solid-state solutions of poly(amic acid) with THF and MeOH, there is no available information in the available literature (to the best of the authors' knowledge) that describes these relationships. With the goal of understanding the way [T.sub.g] was changing during the inflation process as a function of solvent loss, molecular weight increase and structural change, the authors have presented a methodology for kinetic analysis with the corresponding kinetic parameters for the specific system composed of a poly(amic acid) based on 3,3',4,4'-benzophenonetetracarboxylic dianhydride and 4,4'-oxydianiline in a solvent system composed of THF and MeOH. By using conventional graphical methodologies to determine the kinetic parameters of the easily isolatable imidization reaction, the authors were able to deconvolute both endothermic reactions (amidation and imidization) and approximate with good accuracy the more complex amidation reaction kinetics. Knowledge of the different kinetic parameters has enabled the authors to incorporate the rate of molecular weight change and of ring closure into the numerical model to have accurate representations of the rate of change of [T.sub.g] (and consequently other material and transport properties) in the system.
[FIGURE 11 OMITTED]
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Camilo I. Cano, (1) Meaghan L. Clark, (2) Thein Kyu, (3) R. Byron Pipes (1)
(1) School of Chemical Engineering, Purdue University, West Lafayette, IN 47907-2100
(2) Department of Chemistry, Purdue University, West Lafayette, IN 47907-1393
(3) Department of Polymer Engineering, The University of Akron, Akron, OH 44325-0301
Correspondence to: Camilo I. Cano; e-mail: firstname.lastname@example.org
Current address: Camilo I. Cano is currently at Pactiv Corporation, Materials Development, Canandaigua, NY 14424.
TABLE 1. Kinetic parameters for the chemical reactions in poly(amic acid) precursors. Amidation Imidization Apparent order n = 1 n = 2 of reaction Heat of reaction 41.06 J/g 125.9 J/g Frequency factor 5.61 x [10.sup.14] 3.21 x [10.sup.17] L [min.sup.-1] [mol.sup.-1] [min.sup.-1] (9.34 x [10.sup.12] (5.35 x [10.sup.15] L [s.sup.-1]) [mol.sup.-1] [s.sup.-1]) Activation energy 104.6 kJ/mol 148.5 kJ/mol
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|Author:||Cano, Camilo I.; Clark, Meaghan L.; Kyu, Thein; Pipes, R. Byron|
|Publication:||Polymer Engineering and Science|
|Article Type:||Author abstract|
|Date:||Mar 1, 2008|
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