Effect of pressure and temperature on nucleation.

INTRODUCTION

Microcellular plastic materials are usually defined as those having cell sizes less than or equal to 10 microns (1), and are used in applications such as separation media, adsorbants, controlled release devices, and catalyst supports (2, 3). Currently used methods for the formation of these materials make use of a temperature quench, or addition of a nonsolvent in order to induce phase separation in a homogeneous polymer solution and thus generate pores (4-6). The most commonly used thermally induced phase separation (TIPS) method involves dissolving the polymer in an appropriate hydrocarbon solvent--e.g. polystyrene in pentane (5)--raising the temperature high enough such that only one phase exists, then quenching the temperature to induce phase separation. This is followed by solvent removal, which is accomplished either by freeze-drying or by supercritical extraction. Removal of the solvent, if approached carefully, leaves behind micropores. Although interesting porous micromorphologies can be generated, temperature quench processes may lead to structure coarsening due to temperature gradients present during phase separation, and due to the surface forces encountered during solvent removal (7).

A novel recent development, use of C[O.sub.2] and [N.sub.2] as blowing agents for the generation of microcellular foams, has been reported by several authors (8-12). Typically, pellets of polymer are saturated with gas at moderate pressures (approximately 5 to 6 MPa), followed by heating to a temperature above the glass transition of the polymer in an appropriate high temperature solvent bath. It has been shown that this method can produce microcellular foams with cell densities up to [10.sup.10] per [cm.sup.3] and average cell sizes varying from less than 10 microns to 50 microns depending on the conditions.

In this paper, we explore an analogous, yet significantly different scheme by which to generate micro-pores in polymers using C[O.sub.2] as blowing agent. This is a constant-temperature variable-pressure method based on saturating the polymer with C[O.sub.2] at much higher pressures (25 to 35 MPa), in the supercritical region, followed by a rapid pressure quench. Growth of cells is allowed by the suppression of glass transition ([T.sub.g]) resulting from the diluent effect rather than heating the polymer to a temperature T above its normal glass transition temperature. Thus the driving force for growth of bubbles is still the temperature difference "T - [T.sub.g]," which, in our case, is controlled by manipulating [T.sub.g] via changes in C[O.sub.2] pressure, instead of changing T by directly heating the system. Nucleation is induced by supersaturation caused by a sudden pressure drop from the equilibrium solution state, and the nuclei grow until the polymer vitrifies at a lower pressure.

Supercritical fluids (SCFs), materials above their critical temperature and pressure conditions, exhibit interesting behavior by combining the properties of conventional liquids and gases (13). Specifically, their liquid-like densities allow for solvent power of orders of magnitude higher than gases, while gas-like viscosities lead to high rates of diffusion. These facts combine to ensure rapid swelling of polymers by SCFs to equilibrium values comparable to liquid solvents. In addition, SCFs will readily plasticize glassy polymers, just as liquid diluents will, A pressure quench from supercritical conditions at constant temperature ensures that no vapor-liquid boundary is encountered during the process of solvent removal. Thus some of the factors that could damage the delicate cellular structure are avoided.

In this study, an amorphous polymer sample is swollen by a supercritical fluid (C[O.sub.2]) over a sufficiently long period of time to ensure that an equilibrium amount of fluid is absorbed by the polymer. The amount of SCF absorbed is sufficient to reduce the [T.sub.g] of the polymer to below the ambient temperature, generating a liquid, albeit concentration, polymer solution. Quick reduction of the pressure at constant temperature both generates the pores and drives the system towards vitrification, freezing-in the micro-structure. We are using supercritical carbon dioxide, a nontoxic gas with a relatively low critical point ([T.sub.c] = 31 [degrees] C, [P.sub.c] = 7.376 MPa). Further, it acts as a very good plasticizer for poly(methyl methacrylate) (PMMA), resulting in a liquid state at gas pressures of 6 MPa and above (14). Consequently, as pressure is suddenly reduced from the equilibrium state, nuclei form as a result of supersaturation. These nuclei grow by diffusion of gas from the polymer matrix, as shown schematically in Fig. 1. The process continues until the pressure is reduced to a point where the polymer vitrifies, freezing-in the microcellular structure.

It has been shown that classical homogeneous nucleation theory is not able to fully describe the nucleation activity in the process employed by the research groups of Campbell (8), Suh (9), and Kumar (12). This inadequacy of a simple nucleation theory has been addressed by Kweeder, et al. (11), by hypothesizing the existence of a population of preformed microvoids in the system around which nucleation takes place. However, since in our process, we are operating at C[O.sub.2] pressures much higher than needed to plasticize the polymer at our operating temperatures, the system is in a homogenous liquid state. Therefore, classical nucleation theory, with appropriate accounting for the effect of diluent on the surface tension of the system and on the frequency factor of the gas molecules, should be expected to describe the nucleation activity under our conditions.

THEORY

It has been shown previously that C[O.sub.2] is an extremely good plasticizer for amorphous polymers like PMMA (15-17). Literature data (14, 16), plotted in Fig. 2, show that the glass transition temperature of PMMA drops to nearly room temperature at 12 to 15 wt% of C[O.sub.2] absorbed. These data can be transformed into a [T.sub.g] vs. pressure of C[O.sub.2] curve by combining them with the predictions of weight percent C[O.sub.2] absorbed by PMMA as a function of pressure and temperature, obtained by using a modified mean-field-lattice-gas (MFLG) model (17, 18). Figure 3 shows this transformation in the form of glass-liquid phase boundary for PMMA-C[O.sub.2], where it is clearly seen that the system is in a homogeneous liquid state at gas pressures of 6 MPa and above. Consequently, we have employed pressures greater than 6 MPa to generate microcellular foams, and have employed classical nucleation theory to describe trends vs. temperature and pressure.

The rate of homogeneous nucleation can be described by the following equation (19):

[Mathematical Expression Omitted]

where, [Mathematical Expression Omitted] is the number of nuclei generated per [cm.sup.3] per second, [C.sub.0] is the concentration of the gas (number of molecules per [cm.sup.3]), [f.sub.0] is the frequency factor of the gas molecules, k is the Boltzmann's constant, and T is absolute temperature. The term [Delta][G.sub.homo] is the energy barrier for homogeneous nucleation and is given by

[Delta][G.sub.homo] = 16[Pi][[Gamma].sup.3]/3[Delta] [P.sup.2] (2)

where [Delta] P is magnitude of the quench pressure and [Gamma] is the surface energy of the bubble interface, Because the bubble interface is a liquid mixture, rather than simply pure polymer, we have employed the correlation for the surface tension of mixtures given by Reid, et al. (20), to calculate [Gamma] as [[Gamma].sub.mix].

[Mathematical Expression Omitted]

where [[Chi].sub.i]'s are the mole fractions. [[Gamma].sub.i]'s are the surface tensions, and [Mathematical Expression Omitted] are the molar densities (moles/[cm.sup.3]) of the components.

Because the surface tension of an SCF is essentially zero, the expression reduces to:

[Mathematical Expression Omitted]

for the PMMA/supercritical C[O.sub.2] mixture.

Finally, by replacing the molar densities with mass densities (g/[cm.sup.3]), we obtain:

[[Gamma].sub.mix] = [[[Gamma].sub.polymer] [([[Rho].sub.mix]/[[Rho].sub.polymer]).sup.4] [(1 - [w.sub.gas]).sup.4] (5)

where [Rho]'s are the mass densities (g/[cm.sub.3]) and [w.sub.gas] is the weight fraction of C[O.sub.2] absorbed in the polymer. The surface tension for pure PMMA ([[Gamma].sub.polymer]) has been reported as 42 dynes/cm (21), and the density of PMMA ([[Rho].sub.polymer]) in the pressure range employed by us has been used as 1.25 g/[cm.sup.3] based on the MFLG predictions of the PVT behavior of PMMA.

Knowing the surface energy of the system as a function of pressure and temperature, we can also calculate the critical size of the nuclei generated at any conditions (19).

[r.sub.c] = 2[Gamma]/[Delta]P(6)

where [r.sub.c] is the radius of the critical nucleus.

The frequency factor of gas molecules, [f.sub.0] can be expressed as (23),

[f.sub.0] = Z[Beta]*(7)

where, Z, the Zeldovich factor, accounts for the fact that a large number of nuclei never grow, but rather dissolve. The rate at which the molecules are added to the critical nucleus, [Beta]*, can be calculated as the surface area of the critical nucleus times the rate of impingement of the gas molecules per unit area [a method suggested by Farkas (23)].

[Mathematical Expression Omitted]

Substituting Eq 8 into Eq 7,

[Mathematical Expression Omitted]

Equation 9 shows that the frequency factor of the gas molecules joining a nucleus to make it stable varies with the surface area of the nucleus. Equations 1, 2, 5, 6, and 9 form a complete set for the nucleation model of our system where, as a first approximation, ([ZR.sub.impingement]) can be used as a fitted parameter.

The quantities [C.sub.0], [[Rho].sub.mix], and [w.sub.gas] can be calculated as functions of temperature and gas pressure using MFLG model (17, 18). and the nucleation model can be solved to give [Delta][G.sub.homo], [Mathematical Expression Omitted] and [r.sub.c] as functions of temperature and gas pressure. In order to calculate the total number of nuclei generated in the system at given saturation conditions, the rate of nucleation needs to be integrated over the time period of nucleation. In our experiments, gas pressure falls as a function of time. Thus the starting saturation pressure ([P.sub.sat]) and the pressure at which the polymer vitrifies ([P.sub.g]) define the time scale over which the rate of nucleation should be integrated. Therefore,

[Mathematical Expression Omitted]

where [P.sub.g] can be calculated as a function of temperature using Fig. 3, and dP/dt is calculated as ([P.sub.sat] - 3.5)/60 based on the experimental observation that it takes approximately 1 min to reduce the pressure from the saturating pressure to approximately 500 psi (3.5 MPa). Integration of a particular isotherm on a [Mathematical Expression Omitted] vs. P graph from different saturation pressures down to [P.sub.g] gives the effect of pressure on nucleation, and different isotherms can be integrated from same saturation pressure down to [P.sub.g] to study the effect of temperature.

EXPERIMENTAL

Synthesis of Polymer Sample

Methyl methacrylate monomer (Aldrich) was washed with 5 wt% NaOH solution to remove the inhibitor and dried over molecular sieves. In a typical experiment, 1 ml of MMA and 1 mg of azo bis(isobutyronitrile) (AIBN, Polysciences) were added to a 20 mm x 70 mm glass vial. Polymerization was initiated under [N.sub.2] using 365 nm UV radiation (Spectronics Corp. model MB-100) at a distance of approximately 12 cm from the surface of the reaction mixture. After approximately 2 to 3 h, the samples were dried at 50 to 60 [degrees] C under vacuum for 24 h.

Generation of Foams

Our experimental setup is shown schematically in Fig. 4. A disc of PMMA (2 mm thick and 18 mm in diameter) is enclosed in a stainless steel high-pressure window cell. The cell is pressurized with C[O.sub.2] (Linde, bone dry, purified through 3A molecular sieves) using a single-acting/single-stage Haskel gas booster with a capacity of 3.1 cubic inch per cycle. Temperature of the cell is controlled to within [+ or -] 1/2 [degrees] C by a programmable PID controller (CN-2000; Omega Engineering). The pressure signal from the cell is converted to an electrical signal by a pressure transducer (280E; Setra) and indicated on a digital display. In a typical experiment, the cell was flushed with C[O.sub.2] for several minutes and then pressure was increased to the desired value (in the range 10 to 35 MPa) in three to four convenient steps while the temperature was held constant. The desired saturation pressure was maintained for 24 h to ensure equilibrium absorption of gas by the polymer. The pressure was then rapidly quenched (over 3 to 5 min) to atmospheric. The system was allowed to stabilize at zero pressure for approximately 1 hr and then the cell was cooled to room temperature.

CHARACTERIZATION

Foamed samples were fractured using liquid nitrogen, mounted on stubs with cross sections facing up and sputter coated with an approximately 100 [Angstrom] layer of gold. These were then studied under scanning electron microscope for a qualitative assessment of the microstructures. The number of cells seen in the micrographs were counted and the cell density of the foams was calculated using the method suggested by Kumar and Suh (10). The number of bubbles nucleated per [cm.sup.3] of the foam ([N.sub.f]) can be expressed by taking the cube of the linear bubble density as:

[N.sub.f] = [(n[M.sup.2]/A).sup.3/2] (11)

where n is the number of bubbles seen in the micrograph, A is the area of the micrograph ([cm.sup.2]), and M is the magnification factor.

In Kumar's system (10), nuclei grow by the thermal expansion of the entrapped gas, and the resulting volume increase of the system (which affects the cell density per unit volume of the final foam) is not accounted for in the nucleation rate calculations given by Eq 1. Therefore, [N.sub.f] calculated by Eq 11 needs to be converted to number of cells per unit volume of the unfoamed material (polymer plus absorbed gas at the saturating conditions) before it can be compared with [N.sub.total] given by Eq 10. However, owing to the basic difference in the method of generating cells, this correction is not required in our case. There is no thermal expansion of the absorbed gas at any point in our process, assuming that 1 to 2 [degrees] C changes associated with the pressure quench do not contribute significantly. Pressure reduction will lead to isothermal expansion of the gas; however, [([Delta]V/[Delta]P).sub.T] is negligibly small in the supercritical region. Isothermal expansion becomes large only at pressures below approximately 800 psi, which is close to the vitrification pressure ([P.sub.g]) for our system, and thus does not significantly affect the volume of the final foam. Consequently, it is only the volume of the absorbed gas at saturating conditions that contributes to the nucleation and growth, and thus to the final foam density. Therefore, the values of [N.sub.total] calculated by us should be directly comparable with [N.sub.f].

RESULTS AND DISCUSSION

Calculation of Nucleation Rate

As shown in Eqs 1 through 5, the rate of nucleation will explicitly depend on temperature and pressure as well as implicitly through their influence on the quantities [C.sub.0], [[Omega].sub.gas], and [[Rho].sub.mix]. Figure 5 shows the MFLG model predictions on the density of PMMA swollen by C[O.sub.2] as a function of pressure and temperature. With increases in pressure, the density decreases as more gas is absorbed in the polymer. Initially, the rate of decrease of the mixture density is relatively high (pressures up to 10.5 MPa), as this is the pressure regime where C[O.sub.2] density is rapidly increasing, and thus, the amount of C[O.sub.2] sorbed also increases rapidly (17). At higher pressures, the density of C[O.sub.2] reaches a plateau and the hydrostatic compression of the polymer leads to decrease in the mixture density. Density of the mixture also decreases with increasing temperature, probably because the decrease in the density of polymer with increasing temperature is higher compared to the effect of decrease in the amount of gas absorbed at higher temperatures. Density calculations, along with the weight fraction of gas absorbed, lead to the calculation of [C.sub.0], the number of gas molecules per [cm.sup.3] of the swollen network. Figure 6 shows the calculated values that exhibit the expected trends.

Figures 7 through 10 show the calculations based on the classical nucleation theory presented in Eqs 1 through 6. Increasing pressure has a dramatic effect on the energy barrier for nucleation, which decreases exponentially with increase in pressure up to approximately 21 MPa and then levels out. Not surprisingly, the opposite trend is shown by the rate of nucleation (9). Increasing temperature leads to an increase in the energy barrier and thus a corresponding decrease in the rate of nucleation. This can be explained by the lower gas concentration absorbed in polymer at higher temperatures, and further, by its effect on the surface tension of the system. The effect of temperature on nucleation, however, is more gradual, unlike the effect of pressure. The critical nucleus size decreases sharply with increasing pressure and levels out at higher pressures. The effect of temperature on nucleus size is very little.

Effect of Pressure and Temperature on Foam Structures

Figure 11 shows the typical foam structure generated in our constant-temperature, pressure-quench experiments. The samples exhibited an integral foam structure with a microporous core and an apparently nonporous skin.

The effect of saturation pressure on the foam structure has been studied at a constant temperature of 40 [degrees] C. As predicted by classical nucleation theory, there is marked difference between the foam structures generated by quenching from pressures of 21 MPa and below, compared to those generated at pressures of 27.5 MPa and above. To analyze these apparent differences, the total number of nuclei generated per [cm.sup.3] of the sample was calculated by integrating the 40 [degrees] C isotherm of Fig. 9 as per Eq 10 for different [P.sub.sat] values. These values are plotted against the observed number of cells per [cm.sup.3] of foam (calculated using Eq 11) in Fig. 13. Since our system is in a homogeneous liquid state at the temperature and pressure conditions employed by us, homogeneous nucleation theory should be expected to predict a pressure trend in nucleation similar to the pressure trend of experimentally observed cell density. The agreement between the data and the model calculations has, indeed, been found to be very good at higher pressures. However, we have observed some limited nucleation activity at pressures as low as 10.5 MPa, a phenomenon that is not predicted by classical nucleation theory. This could be due to heterogeneous nucleation at lower pressures, itself triggered by the presence of trace contaminants in the system. While important at low pressures, this contribution is negligible compared to homogenous nucleation at higher pressures. The reason for this could be that the rate of heterogeneous nucleation is dependent on the population of existing heterogeneous sites, which do not increase with pressure, whereas homogeneous nucleation activity increases quite dramatically with increasing pressure.

We have also studied the effect of changing the saturation temperature (40 [degrees] C and above to ensure supercritical nature of [CO.sub.2]) at constant saturation pressure (34.47 MPa). Results are presented in Figs. 14 and 15. Figure 14 shows a clear trend of increasing cell sizes, or equivalently decreasing cell density, with increasing temperature. In this case, the total number of nuclei were calculated by integrating the appropriate isotherm in Fig. 9 from 34.47 MPa down to [P.sub.g]. These are plotted in Fig. 15 against the observed cell density. While the predictions and data exhibit similar trends, the predicted values are slightly higher. Possible errors in the predictions of [C.sub.0] and [Gamma] (which itself has been calculated by using predictions on [[Rho].sub.mix] and [w.sub.gas]) may contribute to these differences. It has been shown previously that the MFLG model somewhat overpredicts the [w.sub.gas] (and therefore also [C.sub.0]) at higher pressures (18), leading to higher number of nuclei calculated by Eq 1.

Effect of Time of Saturation on Foam Structures

The effect of varying time of saturation of polymer with supercritical [CO.sub.2] has been studied at a fixed saturation pressure (34.47 MPa) and temperature (40 [degrees] C). It should be noted that the time of saturation represents the time for which the polymer sample is exposed to the high pressure [CO.sub.2] prior to the pressure quench, and has no relation to the rate of pressure quench, which controls the time period for nucleation and growth. Figure 16 shows foam structures obtained by stopping the saturation process short of equilibrium (approximately 22 h at these conditions, as seen in Fig. 17) exhibiting a clear trend that fewer cells per unit volume are generated at shorter times. This can be explained by reduced rate of nucleation resulting from a lower amount of fluid absorbed by the polymer at shorter times. There will likely be a concentration gradient across the sample thickness if the saturation process is stopped short of equilibrium, leading to a possible distribution of cell sizes across the thickness of the final foamed sample. For our present study, the micro-graphs were taken in the central region of the foamed sample, which will have the lowest concentration of the absorbed gas, and will exhibit the intended effect of time more clearly.

Theoretically, in order to be able to calculate the average rate of nucleation as a function of saturation time, one needs to study the kinetics of diffusion of [CO.sub.2] in PMMA. The relevant equations are given in the Appendix. Figure 17 shows the concentration of [CO.sub.2] absorbed at the center of PMMA disc as a function of time. Equilibrium absorption is reached at approximately 22 h of saturation. These values of concentration have been used in the nucleation rate equation (Eq 1) to calculate the number of nuclei generated in the system. Results are plotted in Figure 18 along with the observed cell densities as function of time of saturation. Both the experimental cell densities and calculated number of nuclei exhibit similar trends with time.

CONCLUSIONS

A process for synthesizing microcellular polymeric foams has been studied that makes use of a pressure quench at constant temperature to initiate nucleation in a homogeneous liquid solution of supercritical [CO.sub.2] in PMMA. It is the suppression of the glass transition of PMMA, due to the presence of the sorbed [CO.sub.2], which allows for the growth of cells in this process, rather than the heating of the polymer above its glass transition, which is commonly employed for the purpose. The foams generated in our experiments invariably have a microcellular core surrounded by a relatively nonporous skin, the characteristics of which can be manipulated by changing the process conditions.

We have shown that classical nucleation theory can be used to describe the effect of pressure and temperature on cell density. The rate of homogeneous nucleation increases very sharply with increasing fluid saturation pressure up to approximately 21 MPa, and then levels out at higher pressures. The effect of temperature on the rate of nucleation is by contrast rather small and gradual. The effect of saturation time on the cell density is entirely due to the kinetics of diffusion of supercritical [CO.sub.2] in PMMA. Cell density increases rapidly with increasing time of saturation, and then levels out at an equilibrium value. Experimental data on the cell density of foams show trends with pressure, temperature, and time similar to those suggested by our calculations.

Besides the cell densities discussed here, other characteristics of foams, such as average cell size and cell size distribution, skin thickness, and bulk density have also been studied. Cell sizes range from 0.4 to 20 microns, skin thickness from 10 to 600 microns, and bulk density from 0.4 to 0.9 g/[cm.sup.3], depending on the processing conditions. The phenomena of cell growth and skin formation will be discussed in a forthcoming paper.

APPENDIX: KINETICS OF DIFFUSION OF [CO.sub.2] IN PMMA

A schematic of the diffusion process is given in Fig. 19. For our sample geometry (2 mm thick x 18 mm diameter), the thickness is quite small in comparison with the diameter. Consequently, we have used a one-dimensional Laplace equation for mass transfer to describe the diffusion in our samples (24).

[Delta]c/[Delta]t = D [[Delta].sup.2]c/d[x.sup.2] (12)

where c is the gas concentration, x is linear distance from the center of the sample, and t is time. The diffusivity, D, is concentration and temperature dependent, and a value of 4.582 x [10.sup.-7] [cm.sup.2]/s has been used for PMMA/[CO.sub.2] at 34.47 MPa and 40 [degrees] C (25). Appropriate initial and boundary conditions for the problem are given by:

c = [c.sub.0], at x = [+ or -]L/2, for all t (13a)

[Delta]c/[Delta]x = 0, at x = 0, for all t (13b)

c = 0, at t = 0, for all x (13c)

where L is the thickness of the sample and [c.sub.0] is the equilibrium concentration at the operating conditions given by our MFLG model. In the limiting case:

c [approaches] [c.sub.0], as t [approaches] [infinity], for all x (14)

Equation 12 can be nondimensionalized and solved analytically using the initial and boundary conditions to yield the following exact series solution:

[c.sub.0] - c/[c.sub.0] = [summation of] 4[(-1).sup.n]/(2n+1)[Pi] cos((n+1/2)[Pi]2x/L)exp 

(-tD/[(L/2).sup.2][(n+1/2).sup.2][[Pi].sup.2]) (15)

ACKNOWLEDGMENTS

We thankfully acknowledge the financial support of the National Science Foundation (CTS-90005155) and the Petroleum Research Fund (ACS-PRF No. 22901-G7) for this work. We would also like to thank Mr. George McManus of the Material Science Department, University of Pittsburgh, for helping us with the SEM work, and Dr. Ross Gray of Calgon Corporation for his help with image analysis work.

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