# Bubble growth in reaction injection molded parts foamed by ultrasonic excitation.

INTRODUCTION

Reaction injection molding (RIM) is carried out by polymerizing very reactive monomers and oligomers in the mold. The monomers and oligomers, which have low viscosity at the processing temperature, are mixed in the mixing head by utilizing turbulent mixing prior to injection into the mold. HIM has some distinct advantages compared with injection molding because it is fast and requires less mold investment especially when a large complex part is to be produced. Polyurethane, polyurea, nylon, epoxy, and unsaturated polyesters may be processed by RIM but 95% of RIM products are made of polyurethane. For HIM of polyurethane, two components, polyol and isocynate, are mixed in the mixing head by impingement mixing, injected into the mold, and cured quickly as soon as the mold is filled. Polyurethane foams are usually produced by two different processing methods, high pressure mixing and low pressure mixing. Low pressure mixing method employs rotating mixing pins or beaters for mixing of the components. In high pressure mixing method impingement mixing is used as in HIM process. Blowing agents which have been added and mixed in the polyol yield gases by decomposition or phase change and the gas creates bubbles in the resin. Foamed plastics in general show inferior mechanical properties compared with unfoamed plastics. Microcellular polymers, which contain bubbles of around 10[[micro]meter] in diameter, have higher dimensional accuracy, higher impact strength, higher specific modulus and improved specific strength. To produce microcellular structure, high nucleation rate should be obtained and bubble growth should be controlled and stopped before coarsening of bubbles. For production of polyurethane foam by RIM, nitrogen gas is used as the blowing agent and ultrasonic excitation is applied to induced a high rate of nucleation. In this study bubble growth is modeled to understand the growth mechanism after the bubble is nucleated by ultrasonic excitation. Final size of bubbles is also predicted for different conditions and some experimental results are presented.

GROWTH MODEL

Second phase formation in a medium consists of the following three steps: nucleation of the new phase, growth of the nucleus, and coarsening. Bubble formation in polyurethane follows the three steps. Nucleation of bubbles in polyurethane resin has been modeled by nucleation theories (1). As soon as the bubbles are nucleated, the bubbles of critical size start growing. The growth of bubbles in a polymeric medium supersaturated with gas has been investigated theoretically by considering diffusion controlled growth (2-5). In the case of bubble growth in a thermosetting resin, reaction kinetics must be considered concurrently with diffusion and fluid flow. Gas pressure inside the bubble increases due to gas diffusion into the bubble from outside resin and the higher internal pressure overcomes the surface energy and pushes the matrix resin outward in the radial direction. As the reaction proceeds, the viscosity of the resin goes to infinity after passing the gelation point.

The growth of bubbles stops as the viscosity gets very large and solidification takes place. If the gas supply to the bubble from the gas-resin solution is limited, coarsening will not occur. For high pressure foaming of polyurethane by utilizing the RIM process, the polyol was saturated with nitrogen at a relatively low saturation pressure.

Reaction Kinetics

Viscosity influences bubble growth significantly and is a strong function of temperature and conversion. Variations in temperature and conversion of the polyurethane system can be predicted by an energy balance equation and a mass balance equation for the system.

Energy balance is expressed as

[K.sub.T][[Nabla].sup.2]T + ([Delta]H)(-[r.sub.OH]) = [Rho][C.sub.p] [Delta]T/[Delta]t (1)

where [K.sub.T] is the thermal conductivity of the resin, [Delta]H is the enthalpy of reaction, [[Gamma].sub.OH] is the reaction rate, and [C.sub.p] is the specific heat. If the system is adiabatic or the heat of reaction is much larger than the conduction heat transfer rate, the first term in Eq 1 is negligible. Mass balance is generally given by

D[[Nabla].sup.2]c - (-[r.sub.OH]) = [Delta]c / [Delta]t (2)

where c is the concentration of the reactive species. Since the diffusivity of the molecules, D, is very small, the first term in Eq 2 is neglected. From Eqs 1 and 2, temperature and concentration changes can be calculated by the Runge-Kutta method (6) with proper initial conditions. The reaction rate will be predicted by the following Arrhenius type equation.

- [r.sub.OH] = [k.sub.0] exp(- [E.sub.r]/RT) [c.sup.2] (3)

where [k.sub.0] is a constant, [E.sub.r] is the activation energy, and R is the universal gas constant. The activation energy is usually obtained by experiments and the conversion, [c.sup.*], is determined by

[c.sup.*] = [c.sub.0] - c / [c.sub.0] (4)

where [c.sub.0] is the initial concentration.

For determination of the pre-exponential reaction constant and the activation energy of the polyurethane system, different methods can be employed, e.g., infrared spectroscopic analysis (7), DSC (8), and adiabatic temperature rise method (9). Because reaction heat generation of the polyurethane is large and the thermal conductivity is small, the adiabatic temperature rise method was selected. For this study, a polyurethane system supplied by Dow Chemical, 35W, was used. The metered polyol and MDI were mixed in an insulated container and the temperature increase was measured. From the data, material constants, [r.sub.0], [E.sub.r], [Delta]H, [C.sub.p], were determined by comparing the measured temperature rise with temperature variation predicted by the numerical method as the material constants are varied. For comparison of the material constants, experimental values (10) for RIM2200 (Union Carbide) are chosen as an example. The measured adiabatic temperature increase is shown in Fig. 1 and the material constants for two polyurethane systems, 35W and RIM2200, are listed in Table 1.

[TABULAR DATA FOR TABLE 1 OMITTED]

Bubble Growth Analysis

Viscosity of the polyurethane resin changes largely during polymerization as a function of temperature, shear rate, and conversion (11). To conduct a bubble growth analysis in polyurethane resin processed by RIM, the viscosity variation must be predicted. By assuming a Newtonian fluid, Castro and Macosko (10) proposed a viscosity model,

[Eta] = [[Eta].sub.[infinity]] exp([E.sub.[Eta]]/RT)f([c.sup.*]) (5)

where [[Eta].sub.[infinity]] is a constant which indicates the viscosity when the temperature is infinitely high, [E.sub.[Eta]] is the activation energy, and f([c.sup.*]) is a function of conversion,

f([c.sup.*]) = [([c.sub.g]/[c.sub.g.sup.*] - [c.sup.*]).sup.A+[Bc.sup.*]] (6)

Table 2 lists constants for the viscosity model used for the bubble growth analysis. Figure 2 shows predicted data of conversion, temperature, and viscosity variations as a function of time and initial temperature. From the figure, the cure time is also identified when the viscosity becomes infinity.

Critical size bubbles are generated when the supersaturated resin is processed with ultrasonic excitation. The critical nucleus will grow by diffusion of gas into the bubble, pressure increase inside the bubble, and expansion of the interface by pushing the liquid outward, For analysis of bubble growth, the following assumptions are made.

1) The supersaturated resin is a non-Newtonian fluid.

2) The radius of the diffusion boundary, [r.sub.2], is determined by the number of bubbles nucleated, and the number of bubbles per unit mass of the resin remains constant during growth of bubbles.

3) Henry's law is applicable at the interface in equilibrium.

4) Diffusivity of the gas in the resin is constant.

5) Nitrogen in the bubble is an ideal gas, and viscosity, pressure gradient, concentration gradient, and inertia effects of the gas are neglected.

6) Latent heat of the gas dissolved in the resin is neglected.

7) Buoyancy of the bubble is small during bubble growth and bubble rise is negligible.

For modeling of bubble growth involving moving boundaries, mass transfer into the bubble, mass conservation of gas inside the diffusion boundary, momentum transfer by fluid flow, continuity of the fluid, and surface tension were considered to yield the following equation (2, 3, 12, 13), which is equivalent to the extended Rayleigh equation.

[Mathematical Expression Omitted] (7)

As bubbles grow in liquid resin, coalescence and coarsening may occur. But it is assumed that the bubbles will grow independently without any coalescence or breakage (5). Figure 3 shows a single growing bubble surrounded by the depleted zone, Diffusion boundary, [r.sub.2], is a spherical boundary through which gas diffusion does not occur during bubble growth. For prediction of bubble growth in a confined domain, quadratic gas concentration profile was assumed as the initial condition. Since the depleted zone outside the bubble is limited at the beginning of bubble growth, the size of the depleted zone will be smaller than the size of the diffusion boundary. As the radius of the depleted region increases with the bubble growth, a different gas concentration profile should be assumed because the depleted zone will be the same as the diffusion boundary. Gas concentration profile in the depleted zone is assumed as follows (14) when the boundary of the depleted zone is small.

X([Xi]) = [X.sub.1] - ([X.sub.1] - [P.sub.b]/K)[(1 - [Xi]/L).sup.2] (8)

where [Xi] is the distance from the surface of the bubble, [X.sub.1] is the gas concentration outside the depleted zone, and L is the size of the depleted zone. The distance between the bubble surface and boundary of the depleted zone, L, is given by

[P.sub.b][V.sub.b]/RT = [integral of] 4[Pi][([r.sub.b] + [Xi]).sup.] ([X.sub.1] - X)d[Xi] between limits L and 0 (9)

From Eq 9,

[L.sup.3] + 5[r.sub.b][L.sup.2] + 10[[r.sub.b].sup.2] L - 10[P.sub.b][[r.sub.b].sup.3]/RT([X.sub.1] - [P.sub.b]/K) = 0 (10)

The size of the depleted zone, L, is calculated by Eq 10 when the depleted zone does not reach the diffusion boundary.

[r.sub.b] + L [less than] [r.sub.2] (11)

After the depleted zone arrives at the diffusion boundary, the gas concentration profile is changed as

X([Xi]) = [X.sub.3] - ([X.sub.3] - [P.sub.b]/K) [(1 - [Xi]/L).sup.2] (12)

where the gas concentration at the diffusion boundary, [X.sub.3], is derived from mass conservation,

[Mathematical Expression Omitted] (13)

Bubble growth was predicted by the procedure shown by the flow chart in Fig. 4.

Nucleation rates and the diffusion boundaries were calculated based on the classical nucleation theory assuming that a negative pressure (1) was caused by the ultrasonic excitation. Table 3 lists nucleation rates and sizes of the diffusion boundaries. Figure 5 shows the results of bubble growth prediction using the data listed in Table 3. In the beginning of bubble growth after nucleation, the growth rate is larger when the saturation pressure is higher. However, if the saturation pressure is higher, the nucleation rate is larger and the size of diffusion boundaries is reduced, resulting in faster termination of gas supply to the bubble. The predicted final bubble sizes are reduced when the saturation pressure is decreased from 1.5 MPa to 2.0 MPa. To produce many cells smaller than 20[[micro]meter], a saturation pressure higher than 2.0 MPa is needed theoretically for ultrasonic foaming of polyurethane.

Mold Filling Analysis

The generalized Hele-Shaw flow was employed to simulate flow behavior during filling of the mold for reaction injection molding of polyurethane foam. In [TABULAR DATA FOR TABLE 3 OMITTED] order to estimate pressure distribution at the end of filling, Newtonian flow and isothermal conditions were assumed. A finite element program was written using triangular elements for analysis of the pressure field. The control volume method was utilized to advance the flow front. As a result of numerical analysis, 5 kPa was obtained at the entrance region in the mold when the mold was filled completely and the pressure was considered as the environmental pressure for prediction of the final bubble radius.

Prediction of Final Bubble Sizes

The final size of bubbles created during reaction injection molding can be predicted by solving the previous equations simultaneously with variation of the viscosity due to the polymerization reaction. The final bubble size is determined when bubble growth is terminated either by complete diffusion of gas into the bubble or by a rapid increase of viscosity to infinity according to progression of the curing reaction.

Variation in bubble radius was calculated at different saturation pressures and initial temperatures with consideration of the viscosity change due to reaction. Figure 6 shows that gelation occurs before the gas diffusion is completed for saturation pressures of 0.5 MPa and 1.0 MPa. In this case bubbles grow faster when the initial temperature is higher. However, final bubble sizes are smaller when the initial temperature is higher because gelation time becomes shorter and the growth is terminated faster. When the initial temperature was 50 [degrees] C, cure time was about 1.4 seconds. In the case of higher saturation pressures, i.e., 1.5 MPa and 2.0 MPa, the gas diffusion ends within 0.5 seconds before the resin is cured since the diffusion boundaries are small and supply of gas is limited. Especially when the saturation pressure is 2.0 MPa, the bubble growth stops after 0.1 seconds and the initial temperature does not influence final bubble size. The theoretical prediction states that the final radius of the bubble is about 18[[micro]meter] if the assumptions made for the prediction are valid.

The effects of environmental pressure on bubble growth are shown in Fig. 7. When the pressure outside the bubble is higher, the final bubble radius becomes smaller. Even in the same mold, bubble sizes will vary depending upon pressure distribution. If smaller diffusion boundaries are assumed for the same saturation pressure, the final bubble size is smaller as shown in Fig. 8. For production of microcellular foam by reaction injection molding, ultrasonic nucleation of bubbles with saturation pressure either below 0.5 MPa or above 2.0 MPa is desirable. If nucleation rate is increased by creating more cells per unit volume, the bubble radius will be minimized because the diffusion boundaries will be shrunk and supply of gas will be limited.

RIM EXPERIMENTS

A polyurethane system, Spectrum 35W, supplied by Dow Chemical Co. was used for the reaction injection molding. The polyurethane system consists of diphenyl methane diisocyanate and amine-modified polyol. Viscosity of the polyol was 107 Pa[center dot]s and that of the isocyanate was 162 Pa-s at 25 [degrees] C. At 50 [degrees] C, the viscosity was 28.3 Pa[center dot]s and 38.6 Pa[center dot]s respectively. Gelation time of the system was 1.35 seconds at the mold temperature of 70 [degrees] C and the cured part was ejected from the mold after 20 seconds. Pure nitrogen gas was used as the physical blowing agent for foaming of polyurethane.

An experimental set-up shown in Fig. 9 was designed and built. An ultrasonic horn produced of titanium was located opposite to the gate of the mold. An ultrasonic wave of 20 kHz was generated by the horn which has inside a piezo-electric actuator. Polyol and isocyanate were placed in the storage tank and evacuated by a vacuum pump to eliminate any gas and moisture contained in the resin. After removal of the dissolved gas and moisture, the resin was saturated with nitrogen at different pressure (0.5, 1.0, 1.5, 2.0 MPa) at 45 [degrees] C. The saturated resin was supplied to fill the material cylinder and the drive cylinder was operated to push the resin through a nozzle into the mixing head for impingement mixing. The mixed resin system entered the mold through the gate and was foamed by ultrasonic excitation. After filling and curing, the molded polyurethane was released from the mold and post-cured in an oven. Specimens were frozen in liquid nitrogen before fracturing and observation with a scanning electron microscope.

MEASUREMENT OF NEGATIVE PRESSURE DUE TO ULTRASONIC EXCITATION

A negative pressure field caused by an acoustic wave was studied and most experiments were done in water (15, 16). In this study, the amplitude of negative pressure caused by ultrasonic excitation in a polyol resin was measured by a hydrophone. In order to determine the sensitivity of the hydrophone, the sound pressure level was measured in air by both microphone and hydrophone for a sound generated by a speaker. The sound pressure level (SPL) is related to the effective pressure of sound wave, [P.sub.e], as follows.

SPL = 20log(s [multiplied by] [P.sub.e]/[P.sub.ref])[dB] (14)

where s is the sensitivity of the phone and [P.sub.ref] is effective reference pressure of 20[Mu]Pa. The hydrophone is placed in polyol and the effective pressure of the sound generated by the ultrasonic horn was determined by measuring SPL. The amplitude of the sound pressure at a point from the horn by the distance r is given by

P(r) = 2[[Rho].sub.0]c[U.sub.0][absolute value of sin{[Pi]fr/c[[-square root of 1 + [(a/r).sup.2] - 1]]}] (15)

where [[Rho].sub.0] is the density of the fluid, c is the velocity of the sound in the fluid, [U.sub.0] is the velocity of the horn surface, a is the radius of the horn, and f is the frequency of the sound. The maximum negative pressure generated by the horn was estimated from the measured SPL and listed in Table 4. The maximum negative pressure that can be generated by the horn employed for the experiment is about 19 MPa.

RESULTS AND DISCUSSION

Foamed structures created by the ultrasonic excitation of the polyurethane system supersaturated with nitrogen are shown in Figs. 10 to 14. Figure 10 shows the specimen foamed by ultrasonic excitation without any gas saturation. Since there is not any physical blowing agent in the resin, not even a void is observed. As shown in Figs. 11 to 14, as the saturation pressure increases, the number of bubbles per unit volume in the polyurethane also increases. The nucleation theory predicts that the nucleation rate, the number of bubbles created per unit volume per unit time, will increase with the increase in the saturation pressure because the critical free energy for bubble nucleation, the energy barrier, decreases with higher saturation pressure. According to the nucleation theory, the number of bubbles predicted per unit volume (1) should be in the order of [10.sup.14] to [10.sup.20]. However, the number of bubbles determined experimentally is in the order of [10.sup.13], which is smaller than the predicted value. More accurate nucleation theory is needed for exact prediction of the nucleation rate.

Effects of ultrasonic excitation on void nucleation can be identified in the micrographs. It is observed that more voids are created and a more uniform structure is obtained when ultrasonic wave is applied. When the saturation pressure are 1.5 MPa and 2.0 MPa, large size bubbles are observed since gas diffusion into the existing bubbles is faster and coalescence may occur because of the large amount of gas dissolved in the resin and short distance between bubbles. For production of a microcellular structure. lower saturation pressure should be used and the nucleation rate should be maximized by using ultrasonic excitation. The theoretical prediction of bubble growth does not match the experimental results because the theoretically calculated number of bubbles and the size of diffusion boundaries are different from the real values.

When the saturation pressure is 0.5 MPa, there are more bubbles at the melt front than in the middle of the specimen as shown in Fig. 11. Environmental pressure at the melt front is lower than that in the middle and lower environmental pressure will cause additional nucleation of bubbles. But if the saturation pressure is large enough to create a large number of bubbles as the resin enters the mold, the effect of environmental pressure on additional bubble nucleation is negligible as shown in Figs. 12 to 14.

CONCLUSIONS

The final sizes of bubbles were predicted numerically from a model by considering diffusion of gas into the bubble, fluid flow outside the bubble in radial direction, reaction kinetics for curing of polyurethane, mass conservation, energy conservation, continuity of fluid, and Henry's law for solubility. If the diffusion boundary is assumed to be infinite, bubbles will grow larger for higher saturation pressure. When the diffusion boundary is limited and the saturation pressure is lower than 1.0 MPa, growth of the bubble is terminated as the resin is cured. When the saturation pressure is higher than 1.0 MPa, growth is terminated by complete consumption of dissolved gas before the resin is cured. The final radius is determined by the size of diffusion boundary, level of gas concentration, and gelation time. For theoretical prediction of the gelation time, the change in viscosity was calculated by considering reaction kinetics.

The flow field in the mold was studied by a finite element analysis method and the pressure field was used as the environmental pressure for bubble growth modeling. Maximum pressure in the mold was calculated to be 5 kPa at the entrance region. Experimental results showed that the number of bubbles was increased when ultrasonic wave was applied and the saturation pressure was higher. It is believed that the negative pressure field around the ultrasonic horn accelerates nucleation rates. The negative pressure generated by the ultrasonic horn in the polyol resin was measured with a hydrophone placed in the liquid.

NOMENCLATURE

A = Surface area of bubble or Material constant for viscosity behavior.

a = Radius of ultrasonic horn.

B = Material constant for viscosity behavior.

[c.sup.*] = Conversion rate.

[c.sub.0] = Initial concentration.

c = Concentration.

[c.sub.g] = Conversion of gelation.

[C.sub.p] = Specific heat.

D = Diffusivity.

[E.sub.[Eta]] = Activation energy of viscous flow.

f = Frequency.

[Delta]H = Heat of reaction.

[[Kappa].sub.0] = Pre-exponential reaction constant.

K = Henry's law constant.

[K.sub.T] = Thermal conductivity.

L = Depleted zone.

[P.sub.0] = Environmental pressure.

[P.sub.b] = Pressure inside the bubble.

[P.sub.e] = Effective pressure of the sound wave.

[P.sub.ref] = Reference effective pressure.

r = Distance from ultrasonic horn.

[r.sub.2] = Diffusion boundary of bubble growth.

[r.sub.b] = Bubble radius.

R = Universal gas constant.

[Mathematical Expression Omitted] = Velocity.

[Mathematical Expression Omitted] = Acceleration.

[r.sub.OH] = Reaction rate.

s = Sensitivity of measurement.

SPL = Sound pressure level.

T = Absolute temperature.

t = Time.

[X.sub.1] = Initial gas concentration.

[X.sub.3] = Gas concentration at diffusion boundary.

X = Concentration of gas.

[Gamma] = Surface energy.

[Eta] = Viscosity.

[[Eta].sub.[infinity]] = Viscosity at infinite temperature.

[Xi] = Coordinate in r-direction.

[Rho] = Density.

REFERENCES

1. H. Park and J. R. Youn, Polym. Eng. Sci., 35, 1899 (1995).

2. C. D. Han and H. J. Yoo, Polym. Eng. Sci., 21, 518 (1981).

3. J. R. Street, A. L. Fricke, and L. P. Reiss, Ind. Eng. Fundam, 10, 54 (1971).

4. S. Y. Hobbs, Polym. Eng. Sci., 16, 270 (1976).

5. M. Amon and C. D. Denson, Polym. Eng. Sci., 24, 1026 (1984).

6. F. B. Hildebrand, Advanced Calculus for Applications, ch. 3, Prentice-Hall, Englewood Cliffs, N. J. (1976).

7. E. B. Richter and C. W. Macosco, Polym. Eng. Sci., 18, 1012 (1978).

8. R. B. Prime, Thermosets in Thermal Characterization of Polymeric Materials, p. 435, E. A. Turi, ed., Academic Press, New York (1981).

9. S. D. Lipshitz and C. W. Macosko, J. Appl. Polym. Sci., 21, 2029 (1977).

10. J. M. Castro and C. W. Macosko, AICHE J., 28, 250 (1982).

11. M. R. Kamal, Polym. Eng. Sci., 14, 231 (1974).

12. J. R. Youn and N. P. Suh, Polym. Compos., 6, 175 (1985).

13. L. E. Scriven, Chem. Engl Sci., 10, 1 (1959).

14. A. Arefmanesh, S. G. Advani, and E. E. Michaelides, Polym. Eng. Sci., 30, 1330 (1990).

15. M. Strasberg, J. Acoust. Soc. Am., 31, 163 (1959).

16. L. J. Briggs, J. Chem. Phys., 19, 970 (1951).

Reaction injection molding (RIM) is carried out by polymerizing very reactive monomers and oligomers in the mold. The monomers and oligomers, which have low viscosity at the processing temperature, are mixed in the mixing head by utilizing turbulent mixing prior to injection into the mold. HIM has some distinct advantages compared with injection molding because it is fast and requires less mold investment especially when a large complex part is to be produced. Polyurethane, polyurea, nylon, epoxy, and unsaturated polyesters may be processed by RIM but 95% of RIM products are made of polyurethane. For HIM of polyurethane, two components, polyol and isocynate, are mixed in the mixing head by impingement mixing, injected into the mold, and cured quickly as soon as the mold is filled. Polyurethane foams are usually produced by two different processing methods, high pressure mixing and low pressure mixing. Low pressure mixing method employs rotating mixing pins or beaters for mixing of the components. In high pressure mixing method impingement mixing is used as in HIM process. Blowing agents which have been added and mixed in the polyol yield gases by decomposition or phase change and the gas creates bubbles in the resin. Foamed plastics in general show inferior mechanical properties compared with unfoamed plastics. Microcellular polymers, which contain bubbles of around 10[[micro]meter] in diameter, have higher dimensional accuracy, higher impact strength, higher specific modulus and improved specific strength. To produce microcellular structure, high nucleation rate should be obtained and bubble growth should be controlled and stopped before coarsening of bubbles. For production of polyurethane foam by RIM, nitrogen gas is used as the blowing agent and ultrasonic excitation is applied to induced a high rate of nucleation. In this study bubble growth is modeled to understand the growth mechanism after the bubble is nucleated by ultrasonic excitation. Final size of bubbles is also predicted for different conditions and some experimental results are presented.

GROWTH MODEL

Second phase formation in a medium consists of the following three steps: nucleation of the new phase, growth of the nucleus, and coarsening. Bubble formation in polyurethane follows the three steps. Nucleation of bubbles in polyurethane resin has been modeled by nucleation theories (1). As soon as the bubbles are nucleated, the bubbles of critical size start growing. The growth of bubbles in a polymeric medium supersaturated with gas has been investigated theoretically by considering diffusion controlled growth (2-5). In the case of bubble growth in a thermosetting resin, reaction kinetics must be considered concurrently with diffusion and fluid flow. Gas pressure inside the bubble increases due to gas diffusion into the bubble from outside resin and the higher internal pressure overcomes the surface energy and pushes the matrix resin outward in the radial direction. As the reaction proceeds, the viscosity of the resin goes to infinity after passing the gelation point.

The growth of bubbles stops as the viscosity gets very large and solidification takes place. If the gas supply to the bubble from the gas-resin solution is limited, coarsening will not occur. For high pressure foaming of polyurethane by utilizing the RIM process, the polyol was saturated with nitrogen at a relatively low saturation pressure.

Reaction Kinetics

Viscosity influences bubble growth significantly and is a strong function of temperature and conversion. Variations in temperature and conversion of the polyurethane system can be predicted by an energy balance equation and a mass balance equation for the system.

Energy balance is expressed as

[K.sub.T][[Nabla].sup.2]T + ([Delta]H)(-[r.sub.OH]) = [Rho][C.sub.p] [Delta]T/[Delta]t (1)

where [K.sub.T] is the thermal conductivity of the resin, [Delta]H is the enthalpy of reaction, [[Gamma].sub.OH] is the reaction rate, and [C.sub.p] is the specific heat. If the system is adiabatic or the heat of reaction is much larger than the conduction heat transfer rate, the first term in Eq 1 is negligible. Mass balance is generally given by

D[[Nabla].sup.2]c - (-[r.sub.OH]) = [Delta]c / [Delta]t (2)

where c is the concentration of the reactive species. Since the diffusivity of the molecules, D, is very small, the first term in Eq 2 is neglected. From Eqs 1 and 2, temperature and concentration changes can be calculated by the Runge-Kutta method (6) with proper initial conditions. The reaction rate will be predicted by the following Arrhenius type equation.

- [r.sub.OH] = [k.sub.0] exp(- [E.sub.r]/RT) [c.sup.2] (3)

where [k.sub.0] is a constant, [E.sub.r] is the activation energy, and R is the universal gas constant. The activation energy is usually obtained by experiments and the conversion, [c.sup.*], is determined by

[c.sup.*] = [c.sub.0] - c / [c.sub.0] (4)

where [c.sub.0] is the initial concentration.

For determination of the pre-exponential reaction constant and the activation energy of the polyurethane system, different methods can be employed, e.g., infrared spectroscopic analysis (7), DSC (8), and adiabatic temperature rise method (9). Because reaction heat generation of the polyurethane is large and the thermal conductivity is small, the adiabatic temperature rise method was selected. For this study, a polyurethane system supplied by Dow Chemical, 35W, was used. The metered polyol and MDI were mixed in an insulated container and the temperature increase was measured. From the data, material constants, [r.sub.0], [E.sub.r], [Delta]H, [C.sub.p], were determined by comparing the measured temperature rise with temperature variation predicted by the numerical method as the material constants are varied. For comparison of the material constants, experimental values (10) for RIM2200 (Union Carbide) are chosen as an example. The measured adiabatic temperature increase is shown in Fig. 1 and the material constants for two polyurethane systems, 35W and RIM2200, are listed in Table 1.

[TABULAR DATA FOR TABLE 1 OMITTED]

Bubble Growth Analysis

Viscosity of the polyurethane resin changes largely during polymerization as a function of temperature, shear rate, and conversion (11). To conduct a bubble growth analysis in polyurethane resin processed by RIM, the viscosity variation must be predicted. By assuming a Newtonian fluid, Castro and Macosko (10) proposed a viscosity model,

[Eta] = [[Eta].sub.[infinity]] exp([E.sub.[Eta]]/RT)f([c.sup.*]) (5)

where [[Eta].sub.[infinity]] is a constant which indicates the viscosity when the temperature is infinitely high, [E.sub.[Eta]] is the activation energy, and f([c.sup.*]) is a function of conversion,

f([c.sup.*]) = [([c.sub.g]/[c.sub.g.sup.*] - [c.sup.*]).sup.A+[Bc.sup.*]] (6)

Table 2 lists constants for the viscosity model used for the bubble growth analysis. Figure 2 shows predicted data of conversion, temperature, and viscosity variations as a function of time and initial temperature. From the figure, the cure time is also identified when the viscosity becomes infinity.

Critical size bubbles are generated when the supersaturated resin is processed with ultrasonic excitation. The critical nucleus will grow by diffusion of gas into the bubble, pressure increase inside the bubble, and expansion of the interface by pushing the liquid outward, For analysis of bubble growth, the following assumptions are made.

1) The supersaturated resin is a non-Newtonian fluid.

2) The radius of the diffusion boundary, [r.sub.2], is determined by the number of bubbles nucleated, and the number of bubbles per unit mass of the resin remains constant during growth of bubbles.

3) Henry's law is applicable at the interface in equilibrium.

Table 2. Rheological Parameters of RIM2200. Parameter Value [[Eta].sub.[infinity]] (Pa[center dot]s) 10.3 x [10.sup.-8] [E.sub.[Eta]] (J/mol) 41.3 x [10.sup.3] [[c.sub.g].sup.*] 0.65 A 1.5 B 1.0

4) Diffusivity of the gas in the resin is constant.

5) Nitrogen in the bubble is an ideal gas, and viscosity, pressure gradient, concentration gradient, and inertia effects of the gas are neglected.

6) Latent heat of the gas dissolved in the resin is neglected.

7) Buoyancy of the bubble is small during bubble growth and bubble rise is negligible.

For modeling of bubble growth involving moving boundaries, mass transfer into the bubble, mass conservation of gas inside the diffusion boundary, momentum transfer by fluid flow, continuity of the fluid, and surface tension were considered to yield the following equation (2, 3, 12, 13), which is equivalent to the extended Rayleigh equation.

[Mathematical Expression Omitted] (7)

As bubbles grow in liquid resin, coalescence and coarsening may occur. But it is assumed that the bubbles will grow independently without any coalescence or breakage (5). Figure 3 shows a single growing bubble surrounded by the depleted zone, Diffusion boundary, [r.sub.2], is a spherical boundary through which gas diffusion does not occur during bubble growth. For prediction of bubble growth in a confined domain, quadratic gas concentration profile was assumed as the initial condition. Since the depleted zone outside the bubble is limited at the beginning of bubble growth, the size of the depleted zone will be smaller than the size of the diffusion boundary. As the radius of the depleted region increases with the bubble growth, a different gas concentration profile should be assumed because the depleted zone will be the same as the diffusion boundary. Gas concentration profile in the depleted zone is assumed as follows (14) when the boundary of the depleted zone is small.

X([Xi]) = [X.sub.1] - ([X.sub.1] - [P.sub.b]/K)[(1 - [Xi]/L).sup.2] (8)

where [Xi] is the distance from the surface of the bubble, [X.sub.1] is the gas concentration outside the depleted zone, and L is the size of the depleted zone. The distance between the bubble surface and boundary of the depleted zone, L, is given by

[P.sub.b][V.sub.b]/RT = [integral of] 4[Pi][([r.sub.b] + [Xi]).sup.] ([X.sub.1] - X)d[Xi] between limits L and 0 (9)

From Eq 9,

[L.sup.3] + 5[r.sub.b][L.sup.2] + 10[[r.sub.b].sup.2] L - 10[P.sub.b][[r.sub.b].sup.3]/RT([X.sub.1] - [P.sub.b]/K) = 0 (10)

The size of the depleted zone, L, is calculated by Eq 10 when the depleted zone does not reach the diffusion boundary.

[r.sub.b] + L [less than] [r.sub.2] (11)

After the depleted zone arrives at the diffusion boundary, the gas concentration profile is changed as

X([Xi]) = [X.sub.3] - ([X.sub.3] - [P.sub.b]/K) [(1 - [Xi]/L).sup.2] (12)

where the gas concentration at the diffusion boundary, [X.sub.3], is derived from mass conservation,

[Mathematical Expression Omitted] (13)

Bubble growth was predicted by the procedure shown by the flow chart in Fig. 4.

Nucleation rates and the diffusion boundaries were calculated based on the classical nucleation theory assuming that a negative pressure (1) was caused by the ultrasonic excitation. Table 3 lists nucleation rates and sizes of the diffusion boundaries. Figure 5 shows the results of bubble growth prediction using the data listed in Table 3. In the beginning of bubble growth after nucleation, the growth rate is larger when the saturation pressure is higher. However, if the saturation pressure is higher, the nucleation rate is larger and the size of diffusion boundaries is reduced, resulting in faster termination of gas supply to the bubble. The predicted final bubble sizes are reduced when the saturation pressure is decreased from 1.5 MPa to 2.0 MPa. To produce many cells smaller than 20[[micro]meter], a saturation pressure higher than 2.0 MPa is needed theoretically for ultrasonic foaming of polyurethane.

Mold Filling Analysis

The generalized Hele-Shaw flow was employed to simulate flow behavior during filling of the mold for reaction injection molding of polyurethane foam. In [TABULAR DATA FOR TABLE 3 OMITTED] order to estimate pressure distribution at the end of filling, Newtonian flow and isothermal conditions were assumed. A finite element program was written using triangular elements for analysis of the pressure field. The control volume method was utilized to advance the flow front. As a result of numerical analysis, 5 kPa was obtained at the entrance region in the mold when the mold was filled completely and the pressure was considered as the environmental pressure for prediction of the final bubble radius.

Prediction of Final Bubble Sizes

The final size of bubbles created during reaction injection molding can be predicted by solving the previous equations simultaneously with variation of the viscosity due to the polymerization reaction. The final bubble size is determined when bubble growth is terminated either by complete diffusion of gas into the bubble or by a rapid increase of viscosity to infinity according to progression of the curing reaction.

Variation in bubble radius was calculated at different saturation pressures and initial temperatures with consideration of the viscosity change due to reaction. Figure 6 shows that gelation occurs before the gas diffusion is completed for saturation pressures of 0.5 MPa and 1.0 MPa. In this case bubbles grow faster when the initial temperature is higher. However, final bubble sizes are smaller when the initial temperature is higher because gelation time becomes shorter and the growth is terminated faster. When the initial temperature was 50 [degrees] C, cure time was about 1.4 seconds. In the case of higher saturation pressures, i.e., 1.5 MPa and 2.0 MPa, the gas diffusion ends within 0.5 seconds before the resin is cured since the diffusion boundaries are small and supply of gas is limited. Especially when the saturation pressure is 2.0 MPa, the bubble growth stops after 0.1 seconds and the initial temperature does not influence final bubble size. The theoretical prediction states that the final radius of the bubble is about 18[[micro]meter] if the assumptions made for the prediction are valid.

The effects of environmental pressure on bubble growth are shown in Fig. 7. When the pressure outside the bubble is higher, the final bubble radius becomes smaller. Even in the same mold, bubble sizes will vary depending upon pressure distribution. If smaller diffusion boundaries are assumed for the same saturation pressure, the final bubble size is smaller as shown in Fig. 8. For production of microcellular foam by reaction injection molding, ultrasonic nucleation of bubbles with saturation pressure either below 0.5 MPa or above 2.0 MPa is desirable. If nucleation rate is increased by creating more cells per unit volume, the bubble radius will be minimized because the diffusion boundaries will be shrunk and supply of gas will be limited.

RIM EXPERIMENTS

A polyurethane system, Spectrum 35W, supplied by Dow Chemical Co. was used for the reaction injection molding. The polyurethane system consists of diphenyl methane diisocyanate and amine-modified polyol. Viscosity of the polyol was 107 Pa[center dot]s and that of the isocyanate was 162 Pa-s at 25 [degrees] C. At 50 [degrees] C, the viscosity was 28.3 Pa[center dot]s and 38.6 Pa[center dot]s respectively. Gelation time of the system was 1.35 seconds at the mold temperature of 70 [degrees] C and the cured part was ejected from the mold after 20 seconds. Pure nitrogen gas was used as the physical blowing agent for foaming of polyurethane.

An experimental set-up shown in Fig. 9 was designed and built. An ultrasonic horn produced of titanium was located opposite to the gate of the mold. An ultrasonic wave of 20 kHz was generated by the horn which has inside a piezo-electric actuator. Polyol and isocyanate were placed in the storage tank and evacuated by a vacuum pump to eliminate any gas and moisture contained in the resin. After removal of the dissolved gas and moisture, the resin was saturated with nitrogen at different pressure (0.5, 1.0, 1.5, 2.0 MPa) at 45 [degrees] C. The saturated resin was supplied to fill the material cylinder and the drive cylinder was operated to push the resin through a nozzle into the mixing head for impingement mixing. The mixed resin system entered the mold through the gate and was foamed by ultrasonic excitation. After filling and curing, the molded polyurethane was released from the mold and post-cured in an oven. Specimens were frozen in liquid nitrogen before fracturing and observation with a scanning electron microscope.

MEASUREMENT OF NEGATIVE PRESSURE DUE TO ULTRASONIC EXCITATION

A negative pressure field caused by an acoustic wave was studied and most experiments were done in water (15, 16). In this study, the amplitude of negative pressure caused by ultrasonic excitation in a polyol resin was measured by a hydrophone. In order to determine the sensitivity of the hydrophone, the sound pressure level was measured in air by both microphone and hydrophone for a sound generated by a speaker. The sound pressure level (SPL) is related to the effective pressure of sound wave, [P.sub.e], as follows.

Table 4. SPL, Pressure Amplitude, and Negative Pressure Produced by Different Ultrasonic Power. Ultrasonic SPL Pressure Negative Power (%) (dB) Amplitude Pressure (MPa) (MPa) 10 119 1.701 -1.600 20 130 6.035 -5.934 max [less than] 140 20.084 -18.983

SPL = 20log(s [multiplied by] [P.sub.e]/[P.sub.ref])[dB] (14)

where s is the sensitivity of the phone and [P.sub.ref] is effective reference pressure of 20[Mu]Pa. The hydrophone is placed in polyol and the effective pressure of the sound generated by the ultrasonic horn was determined by measuring SPL. The amplitude of the sound pressure at a point from the horn by the distance r is given by

P(r) = 2[[Rho].sub.0]c[U.sub.0][absolute value of sin{[Pi]fr/c[[-square root of 1 + [(a/r).sup.2] - 1]]}] (15)

where [[Rho].sub.0] is the density of the fluid, c is the velocity of the sound in the fluid, [U.sub.0] is the velocity of the horn surface, a is the radius of the horn, and f is the frequency of the sound. The maximum negative pressure generated by the horn was estimated from the measured SPL and listed in Table 4. The maximum negative pressure that can be generated by the horn employed for the experiment is about 19 MPa.

RESULTS AND DISCUSSION

Foamed structures created by the ultrasonic excitation of the polyurethane system supersaturated with nitrogen are shown in Figs. 10 to 14. Figure 10 shows the specimen foamed by ultrasonic excitation without any gas saturation. Since there is not any physical blowing agent in the resin, not even a void is observed. As shown in Figs. 11 to 14, as the saturation pressure increases, the number of bubbles per unit volume in the polyurethane also increases. The nucleation theory predicts that the nucleation rate, the number of bubbles created per unit volume per unit time, will increase with the increase in the saturation pressure because the critical free energy for bubble nucleation, the energy barrier, decreases with higher saturation pressure. According to the nucleation theory, the number of bubbles predicted per unit volume (1) should be in the order of [10.sup.14] to [10.sup.20]. However, the number of bubbles determined experimentally is in the order of [10.sup.13], which is smaller than the predicted value. More accurate nucleation theory is needed for exact prediction of the nucleation rate.

Effects of ultrasonic excitation on void nucleation can be identified in the micrographs. It is observed that more voids are created and a more uniform structure is obtained when ultrasonic wave is applied. When the saturation pressure are 1.5 MPa and 2.0 MPa, large size bubbles are observed since gas diffusion into the existing bubbles is faster and coalescence may occur because of the large amount of gas dissolved in the resin and short distance between bubbles. For production of a microcellular structure. lower saturation pressure should be used and the nucleation rate should be maximized by using ultrasonic excitation. The theoretical prediction of bubble growth does not match the experimental results because the theoretically calculated number of bubbles and the size of diffusion boundaries are different from the real values.

When the saturation pressure is 0.5 MPa, there are more bubbles at the melt front than in the middle of the specimen as shown in Fig. 11. Environmental pressure at the melt front is lower than that in the middle and lower environmental pressure will cause additional nucleation of bubbles. But if the saturation pressure is large enough to create a large number of bubbles as the resin enters the mold, the effect of environmental pressure on additional bubble nucleation is negligible as shown in Figs. 12 to 14.

CONCLUSIONS

The final sizes of bubbles were predicted numerically from a model by considering diffusion of gas into the bubble, fluid flow outside the bubble in radial direction, reaction kinetics for curing of polyurethane, mass conservation, energy conservation, continuity of fluid, and Henry's law for solubility. If the diffusion boundary is assumed to be infinite, bubbles will grow larger for higher saturation pressure. When the diffusion boundary is limited and the saturation pressure is lower than 1.0 MPa, growth of the bubble is terminated as the resin is cured. When the saturation pressure is higher than 1.0 MPa, growth is terminated by complete consumption of dissolved gas before the resin is cured. The final radius is determined by the size of diffusion boundary, level of gas concentration, and gelation time. For theoretical prediction of the gelation time, the change in viscosity was calculated by considering reaction kinetics.

The flow field in the mold was studied by a finite element analysis method and the pressure field was used as the environmental pressure for bubble growth modeling. Maximum pressure in the mold was calculated to be 5 kPa at the entrance region. Experimental results showed that the number of bubbles was increased when ultrasonic wave was applied and the saturation pressure was higher. It is believed that the negative pressure field around the ultrasonic horn accelerates nucleation rates. The negative pressure generated by the ultrasonic horn in the polyol resin was measured with a hydrophone placed in the liquid.

NOMENCLATURE

A = Surface area of bubble or Material constant for viscosity behavior.

a = Radius of ultrasonic horn.

B = Material constant for viscosity behavior.

[c.sup.*] = Conversion rate.

[c.sub.0] = Initial concentration.

c = Concentration.

[c.sub.g] = Conversion of gelation.

[C.sub.p] = Specific heat.

D = Diffusivity.

[E.sub.[Eta]] = Activation energy of viscous flow.

f = Frequency.

[Delta]H = Heat of reaction.

[[Kappa].sub.0] = Pre-exponential reaction constant.

K = Henry's law constant.

[K.sub.T] = Thermal conductivity.

L = Depleted zone.

[P.sub.0] = Environmental pressure.

[P.sub.b] = Pressure inside the bubble.

[P.sub.e] = Effective pressure of the sound wave.

[P.sub.ref] = Reference effective pressure.

r = Distance from ultrasonic horn.

[r.sub.2] = Diffusion boundary of bubble growth.

[r.sub.b] = Bubble radius.

R = Universal gas constant.

[Mathematical Expression Omitted] = Velocity.

[Mathematical Expression Omitted] = Acceleration.

[r.sub.OH] = Reaction rate.

s = Sensitivity of measurement.

SPL = Sound pressure level.

T = Absolute temperature.

t = Time.

[X.sub.1] = Initial gas concentration.

[X.sub.3] = Gas concentration at diffusion boundary.

X = Concentration of gas.

[Gamma] = Surface energy.

[Eta] = Viscosity.

[[Eta].sub.[infinity]] = Viscosity at infinite temperature.

[Xi] = Coordinate in r-direction.

[Rho] = Density.

REFERENCES

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Title Annotation: | 5th International Conference on Polymer Characterization |
---|---|

Author: | Youn, Jae R.; Park, Hyeok |

Publication: | Polymer Engineering and Science |

Date: | Mar 1, 1999 |

Words: | 4124 |

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