Theoretical and visual study of bubble dynamics in foam injection molding.
Thermoplastic foams are widely used in a wide variety of applications due to their many advantageous characteristics, such as low thermal conductivity, light weight, and good impact strength. Because of their wide usage, numerous experimental and theoretical research works have been conducted on foaming process including bubble dynamics. Foaming process includes the following three major stages: (1) dissolution of a blowing agent into the polymeric phase under an elevated pressure and temperature, (2) nucleation of bubbles via a pressure drop, and (3) growth of bubbles.
Available research works on the foaming process have enhanced our knowledge of this subject (1-13). Considerable studies have been performed on the bubble growth phenomenon. Among them, the early work conducted by Barlow and Langlois (2) presented a model for stationary single bubble in a purely viscous (Newtonian) fluid considering isothermal condition. They also studied both the diffusion and momentum transfer processes in their theoretical investigation. Han and Villamizar (3) carried out an experimental investigation on flow behavior of gas charged molten polymer in a foam extrusion process. Using a rectangular slit die with a glass window, dynamic behavior of gas bubbles was studied where the melt was flowing between the two parallel planes. Patel (4) developed a Newtonian model to study the diffusion-induced growth of a stationary bubble in an isothermal condition. The assumptions considered the bubble consisting only of the diffusing species, spherical, and also completely homogeneous and constant physical properties for the gas charged molten polymer. Han and Yoo (5) performed a theoretical and experimental study on bubble growth mechanism during mold filling in a structural foaming process. Amon and Denson (6), (7) offered a mathematical analysis of bubble growth in an expanding foam based on the cell model in a viscoelastic medium and isothermal condition, using it as the microscopic building block of an analysis of low pressure structural foam molding. Shah et al. (8) suggested a model for the free expansion foam processing which simultaneously encompassed both nucleation and growth stages. Arefmanesh et al. (9) presented a numerical method to predict bubble size in a structural foam molding process assuming an isothermal condition. Ramesh (10) extended the model given by Patel (4) to predict diffusion controlled bubble growth in microcellular foaming of a non-Newtonian fluid in an isothermal condition. He modeled the fluid as a power law fluid and also considered that bubbles grow with spherical symmetry in the system, assuming constant physical properties for the polymer during growth stage.
Favelukis (11) developed a mathematical model to predict the growth behavior of a spherical gas bubble in the limited amount of a viscous liquid, assuming both momentum and mass transfer phenomena. Leung et al. (12) used the in situ visualization data from a batch foaming system and developed a mathematical model and simulation algorithm to study bubble growth dynamics. Everitt et al. (13) carried out a theoretical study on the effect of viscoelasticity on the expansion of gas bubbles using finite element method. Behravesh and Rajabpour (14) performed an experimental study on filling stage of microcellular foam injection molding process and showed that formation and growth of bubbles strongly depend on the shot size. They mentioned that too large a shot size could lead to the formation of a single-phase solution of gas-polymer. Also, their research presented a foaming characteristics diagram for foaming process.
The mathematical formulation presented in this research work is an enhanced state of the model given by Patel (4) and Ramesh (10). The main scope is to develop a power law model to predict growth and collapse of bubbles in the foam injection molding (FIM) process in a nonisothermal condition. The main emphasis is to observe and predict the cell collapse phenomenon that has been found to have a significant influence on the final foam structure as encountered in practice (14).
Equipment and Materials
To investigate the bubble dynamics in foam injection molding, a special setup was designed and prepared with the schematic shown in Fig. 1. The setup was found to be a very effective visualization system for injection molding process (15-18). The system consists of four major units: (i) a 70-ton injection-molding machine, (ii) a visual mold equipped with a transparent glass window, (iii) a [CO.sub.2] gas injection apparatus, and (iv) a visualization set up including a high-speed camera, light sources, and a data acquisition system. A mold with a rectangular cavity (62 x 50 x 5.5 mm) was designed and manufactured to produce plate shaped specimens with the specification given in Fig. 2. Polystyrene grade TATTAREX GPPS 861N, manufactured by Taita Chemical, Co. Ltd., was used as the experimental material.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
To remove moisture, prior to injection, polystyrene granules were dried in a dryer at 80[degrees]C for 3 hr. The injection-molding machine was set and prepared to produce plate-shaped parts. Melt and mold temperatures were set at 290[degrees]C and 15[degrees]C, respectively. [CO.sub.2] gas was injected into the melt using a calibrated dosing system. As shown in Fig. 1, a pneumatic-actuated nozzle was used to preserve and pressurize the solution before injecting into the mold cavity. As the melt entered the cavity, the highspeed camera system was activated to record the images of the flowing melt and foaming behavior. The prepared apparatus could detect bubbles larger than 400 [micro]m in size. A higher magnification, for detecting the smaller size, could limit the visualizing area, which in turn, could make tracking of a moving object (here a nucleated cell), in an unpredictable path, highly difficult or even impossible. Hence, a 3X magnification lens was used to view entire area of the cavity. In this study, the cavity was filled by a full shot injection. Injection pressure was also set at 15 MPa for all the experiments. To study the collapse behavior of the bubbles, holding pressure was exerted on the polymer--gas system. The exerted holding pressure was applied for 25 sec after various delay times of 10, 15, and 25 sec, during initial growth stage. The exerted pressure was also set at various values of 10, 15, and 18 MPa. Thus, the effects of delay time and holding pressure on bubble growth (and collapse) were studied. Figure 3 depicts the schematic pressure-time history of the process. Note that the shown pressure designates the pressure at the injection nozzle.
[FIGURE 3 OMITTED]
The theoretical study here considers the isolated bubble growing spherically in a nonisothermal condition in a gas charged molten polymer system. As the melt cools down, melt properties such as density, Henry's law constant, surface tension, and diffusion constant change.
The mathematical model to investigate the effect of the processing parameters on bubble dynamics considers the following assumptions:
1. Polymer is considered to be a non-Newtonian and incompressible liquid.
2. Melt temperature continuously decreases in the cavity.
3. Bubble grows in an infinite sea of polymeric solution.
4. Thermodynamic equilibrium exists at the blowing agent-polymer interface which follows the Henry's law, [P.sub.g] = [k.sub.h][c.sub.w].
5. Bubble grows spherically in system.
6. Bubble growth is due to the gas diffusion from the solution into the bubble.
Figure 4 shows a schematic view of the growth phenomenon due to the gas diffusion into the bubble.
[FIGURE 4 OMITTED]
The power law model used to model the polymer can be given by the following equation (19):
[tau] = [eta] [gamma] (1)
where [eta] = K[[gamma].sup.[n-1]].
The governing equations for the growth of a cell are as follows (4):
RR + [3/2][R.sup.2] + [[4K[(2[square root of 3]).sup.[n-1]]]/n[[rho].sub.l]][([R/R]).sup.n] = [1/[[rho].sub.l]]([P.sub.g] - [P.sub.[infinity]] - [2[SIGMA]/R]), (2)
[[partial derivative]c/[partial derivative]t] + [[upsilon].sub.r][[partial derivative]c/[partial derivative]r] = [D/[r.sup.2]][[partial derivative]/[partial derivative]r]([r.sup.2][[partial derivative]c/[partial derivative]r]);r [grater than or equal to] R (3)
where [[upsilon].sub.r] is the radial velocity component in the melt and is given by the following equation (4),
[[upsilon].sub.r] = [R[R.sup.2]/[r.sup.2]],
[d/dt]([[rho].sub.g][R.sup.3]) = 3[R.sup.2][[rho].sub.l]D[([[partial derivative]c/[partial derivative]r]).sub.[r=R]]. (4)
Equations 2-4 represent the continuity equation, the diffusion equation, and the mass conservation equation, respectively. Initial and boundary conditions can be presented as below,
R(0) = [R.sub.0], c(r, 0) = c([infinity], 0) = [c.sub.[infinity]], c(R, t) = [c.sub.w](t).
Initial bubble radius is usually considered between 0.1 and 1 [micro]m depending on the various foaming conditions (10). In this study, initial bubble radius was chosen 1 [micro]m (10), (20), (21). Initial radius 1 [micro]m could also lead to a more agreeable results compared with other initial radiuses. The first term in continuity equation (R R and 3/2 [R.sup.2]) can be neglected because the Reynolds number in polymer flow is very low (4), (10). Combining the equations and using Henry's law, the following equation can be derived (4).
[d/dt]([[rho].sub.g][R.sup.3]) = [[6[R.sup.4]D[[rho].sub.1.sup.2][([c.sub.[infinity]] - [c.sub.w]).sup.2]]/[[R.sup.3][[rho].sub.g] - [R.sub.0.sup.3][[rho].sub.g0]]], (5)
[[rho].sub.g] = [M[P.sub.g]/[bar.R]T], (6)
where [P.sub.g] = [k.sub.h][c.sub.w].
[d/dt]([[rho].sub.g][R.sup.3]) = [d/dt]([[[R.sup.3]M[k.sub.h][c.sub.w]]/[bar.R]T]). (7)
To simplify the computations, it is assumed that parameter k does not change significantly by temperature.
[d/dt]([[rho].sub.g][R.sup.3]) = [1/T][d/dt]([[[R.sup.3]M[k.sub.h][c.sub.w]]/[bar.R]]) - [1/[T.sup.2]]([[[R.sup.3]M[k.sub.h][c.sub.w]]/[bar.R]])[dT/dt]. (8)
Using dimensionless terms
[xi]* = [[[R.sup.3][c.sub.w]]/[[R.sub.0.sup.3][c.sub.[infinity]]]] and R* = [R/[R.sub.0]].
Equation 8 changes to
[d/dt]([[rho].sub.g][R.sup.3]) = [1/T][d/dt]([xi]* [[[R.sub.0.sup.3]M[k.sub.h][c.sub.[infinity]]]/[bar.R]]) - [1/[T.sup.2]]([xi]* [[[R.sub.0.sup.3]M[k.sub.h][c.sub.[infinity]]]/[bar.R]])[dT/dt]. (9)
Combining 5, 6, and 9 and performing proper algebraic operations the following equation is obtained:
[d[xi]*/dt*] = [[6[([R*.sup.3] - [xi]*).sup.2][D(T)/[D.sub.0]]/[R*.sup.2]([xi]* [[T.sub.0]/T] - 1)]] [T/[T.sub.0]] + [1/T][xi]* [dT/dt*] (10)
where t* is the dimensionless time and is equal to [t/[theta]] where [theta] (reference time) is as follows:
[theta] = [([[M[k.sub.h][R.sub.0]]/[[bar.R][[rho].sub.l]T]]).sup.2][1/[D.sub.0]]. (11)
For the case of isothermal condition ([dT/dt] = 0). Eq. 10 will be reduced to the one given by Ramesh et al. (10), (21). The continuity equation can be reduced to the following equation after proper algebraic operations (10):
[dR*/dt*] = R* (1 - [1/n])[([P.sub.0][R.sub.e][[xi]*/[R.sup.*2]] - [P.sub.[infinity]]R* [R.sub.e] - [[R.sub.e]/[W.sub.e]]).sup.[1/n]] (12)
[P.sub.0] = [[[k.sub.h][c.sub.[infinity]][[theta].sup.2]]/[[[rho].sub.l][R.sub.0.sup.2]]], (13a)
[R.sub.e] = [[R.sub.0.sup.2]/[P[[theta].sup.[2-n]]]], (13b)
P = [[4k[(2[square root of 3]).sup.[n-1]]]/[n[[rho].sub.l]]], (13c)
[W.sub.e] = [[[R.sub.0.sup.3][[rho].sub.l]]/[2[sigma][[theta].sup.2]]], (13d)
[P.sub.[infinity]] = [[[P.sub.a][[theta].sup.2]]/[[R.sub.0.sup.2][[rho].sub.l]]]. (13e)
In the earlier relations, parameters including diffusivity, surface tension, density, Henry's law constant, and viscosity are functions of temperature. Temperature profile in cavity in injection molding process can be estimated by the following equation (22). However, for simplicity, average temperature of the cavity is considered in this study:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
[a.sup.2] = [k/[rho][C.sub.p]].
If only first term (n = 1) is considered then
[T.sub.avg] = [T.sub.w] + [[8([T.sub.m] - [T.sub.w])]/[[pi].sup.2]]exp ([[-[[pi].sup.2][a.sup.2]]/[h.sup.2]]t). (15)
Other physical properties are temperature dependent, and thus vary via cooling the melt in the cavity. The gas diffusivity into polymers increases with temperature and can be expressed as (19) follows:
D = [D.sub.0] exp([-[E.sub.d]/[bar.R]T]). (16)
Surface tension of the melt can also be estimated by the following expression (23):
[sigma] = [[sigma].sub.0][(1 - [T/[T.sub.c]]).sup.[11/9]]. (17)
Density of polymers is temperature dependent and can be expressed by the following relations (24):
[rho](T) = [rho](298) [[[(1.42[T.sub.g] + 44.7)]/[(1.57[T.sub.g] + 0.3(T - [T.sub.g])]]]T > [T.sub.g], (18a)
[rho](T) = [rho](298) [[[(1.42[T.sub.g] + 44.7)]/[(1.42[T.sub.g] + 0.15T)]]] T [less than or equal to] [T.sub.g]. (18b)
Note that as the holding pressures 10-18 MPa is low, so the influence of pressure on density variation is neglected in these equations. Another important property that is required to complete the numerical simulation is Henry's law constant. This parameter shows strong dependency on the temperature and can be expressed by the following expression (19):
[k.sub.h](T) = [k.sub.h] (298) exp[[[DELTA][H.sub.s]/[bar.R]([1/298] - [1/T])] (19)
where [k.sub.h](298) represents Henry's law constant at T = 298 K, and [DELTA][H.sub.s] and [R.sub.g] denotes the heat of solution (J/mol) and gas constant, respectively.
The rheological properties of PS-[CO.sub.2] system is strongly affected by the dissolution of [CO.sub.2] gas into the PS melt. This issue has been widely studied by the researchers (25-27). Noticeably, it was shown that depending on the experimental conditions, the viscosity of polystyrene can be lowered as much as 80% by [CO.sub.2] which acts as a plasticizer (25). To obtain power law parameters (K and n) in the experiment condition, data given in research work of Royer et al. (25) were used. Using the following shift factors, the [eta] - [gamma] diagram can be obtained in different gas concentration, pressure, and temperature:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20b)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20c)
where [c.sub.1] and [c.sub.2] are WLF constants and can be determined experimentally for most common polymer systems (25), (28). Consistency index is temperature dependent and can be estimated by the following equation (29);
K = [K.sub.0] exp (- [alpha](T - [T.sub.0])). (21)
RESULTS AND DISCUSSION
Figure 5 shows the images of filling sequences in the cavity. As the gas charged molten polymer enters the mold cavity, melt pressure drops below the solubility pressure, and thus myriad of bubbles nucleate. However, if the cavity is completely filled by the melt solution, via a full shot, the instant increase in pressure causes the majority of the nucleated bubbles to collapse instantly (Fig. 5d). It then takes some time to observe bubble growth in the molten polymer to form a foamed sample as shown in Fig. 5c-g.
[FIGURE 5 OMITTED]
To study the growth behavior of bubbles in more details, a target cell was chosen in the melt as shown in Fig. 5e. To observe the effect of the pressure on the bubble growth and collapse, different holding pressure was exerted to the gas-charged molten polymer after any given specific (delay) time. Longer lime promotes further cooling of the melt, which in turn causes an increase in the viscosity. When the pressure was applied, the bubble collapse was observed.
Figure 5h-k show the collapse behavior of bubbles for a sample with 25 sec (initial) growing lime under 15 MPa holding pressure. Then, when removing the pressure, the bubble growth was initiated by gas diffusion back into the collapsed bubbles, and thus the second growing stage was observed. Bubbles show smaller radius size compared with those of the first growing stage accounted for the increase in the melt viscosity. The bubble sizes were measured during these three stages. Figure 6 shows bubble radius versus time for the above specimen.
[FIGURE 6 OMITTED]
To fulfill theoretical statement of bubble dynamics, it was necessary to obtain pressure history in the mold cavity. Thus, a pressure measurement transducer, Kistler model 5015, was implemented to obtain pressure-time diagram. Figure 7 shows the schematic view of the prepared set-up. During the experiments, because of too long period of process and so freezing the skin layer of the injected part, no pressure was transferable to the pressure sensor tip after 15 sec. This is an inherent limitation of the transducer to measure the pressure at a longer time. Therefore, only the first few seconds of the pressure measurement was reliable. To estimate the pressure history near the target cell, Moldflow software was used. Figure 8 shows the pressure-time prediction at vicinity of the targeted growing cell for the condition where the holding pressure was 15 MPa and for 25 sec (corresponding to the bubble growth stage).
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
Using MATLAB software, Eqs. 10 and 12 were numerically solved by forth order Runge-Kutta method. Figures 9-11 show the results of the model prediction and their comparison with those of experimental observation. Figure 9 shows the growth stage and collapse behavior of the bubbles under 10 MPa of holding pressure. As the figure reveals, the holding pressure could not completely collapse target cell after 10, 15, and 25 seconds of growing stage. Furthermore, regrowth stage is retarded by removing the holding pressure. Regrowth stage is retarded in the longer growth time in the samples. A holding pressure of 15 MPa was high enough to completely collapse target cell after 10 and 15 sec of the growth stage (Fig. 10), consistent with the prediction of the model that the target cell radius is decreased below visualization limit (~400 [micro]m). As also is shown in Fig. 10c, at this holding pressure, the target cell does not fully collapse after 25 sec of growth stage, justified by a high melt viscosity due to further cooling, and thus opposing the cell growth. Finally, for specimens shown in Fig. 11, with exerting holding pressure of 18 MPa, all target cells were collapsed after 10, 15, and 25 sec of growth stage. It should be pointed out that according to the results obtained from the model, at shorter growth time, R-t diagram shows higher slope at the beginning of the collapse stage. Also, it indicates that a higher holding pressure presents a steeper slope of R-t diagram at collapse stage. Similarly, the slope of the R-t diagram in regrowth stage is higher in a shorter time of growth stage.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
[FIGURE 11 OMITTED]
As seen in Figs. 9-11, there is some difference between power law model predictions and the experimental data obtained from the visualization method. The largest and the smallest differences correspond to the initial growth stage and the regrowth stage, respectively. The error mean value at the initial growth stage is 48%, at the collapse stage is 17%, and at the regrowth stage is ~15%. Although inaccuracy of the data especially of pressure history and rheological properties could be the main causes of the differences, it also may indicate the limitation of the model.
Experimental and theoretical studies were carried out on dynamics of bubble growth and collapse in foam injection molding process for a PS-[CO.sub.2] system. The behavior of bubble growth and collapse under different holding pressure and different growth time intervals was recorded using a high-speed recording system. A model was proposed to predict the behavior of bubble dynamics in a nonisothermal condition. The results show that at a shorter delay time and a higher holding pressure, the cell collapse is more evident. The model prediction shows a fair agreement with that of visual data especially for the first growth and collapse stages. The model shows discrepancy in the second growth stage, although minor.
NOMENCLATURE [a.sub.c] concentration shift factor [a.sub.p] pressure shift factor [a.sub.T] temperature shift factor C mass fraction of [CO.sub.2] gas [C.sub.0] initial mass fraction of [CO.sub.2] gas [C.sub.w] mass fraction of [CO.sub.2] gas at cell wall [C.sub.[infinity]] mass fraction of [CO.sub.2] gas far from cell [C.sub.1], [C.sub.2] WLF constants [C.sub.p] heat specific (J/kg K) D diffusion coefficient ([m.sup.2]/s) [D.sub.0] diffusion coefficient at reference temperature (298 K) ([m.sup.2]/s) [E.sub.d] activation energy for diffusion (kcal/mol) h thickness (m) k thermal conductivity (W/m K) [k.sub.h] Henry's law constant [[cm.sup.3] (STP)/g atm] K power law constant, (Pa [s.sup.n]) [K.sub.0] value of K at [T.sub.0] (Pa [s.sup.n]) M molecular weight of [CO.sub.2] gas (g/mol) n power law index [P.sub.g] pressure inside the bubble gas (Pa) [P.sub.a] pressure at the bubble wall (Pa) R bubble radius at each moment ([mu]m) [R.sub.0] initial bubble radius ([mu]m) [R.bar] dimensionless radius R gas constant (J/K mol) r radial coordinate ([mu]m) T temperature (K) [T.sub.0] reference temperature (K) t time (s) t dimensionless time [T.sub.g] glass transition temperature (K) [T.sub.m] melt temperature (K) [T.sub.w] cavity wall temperature (K) [T.sub.c] critical temperature of the gas (K) [rho]l density of polymeric melt (kg/[m.sup.3]) [rho]g density of gas (kg/[m.sup.3]) [rho]*go initial gas density (kg/[m.sup.3]) [xi] dimensionless variable [theta] reference time (s) [sigma] surface tension (N/m) [[sigma].sub.0] surface tension at reference temperature (N/m) [DELTA][H.sub.s] heat of solution (J/mol) [eta] viscosity (Pa s) [alpha] empirical parameter in Eq. 21
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Mehdi Mahmoodi, Amir Hossein Behravesh, Seyyed Abdol Mohammad Rezavand, Mohammad Golzar
Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran
Correspondence to: Amir Hossein Behravesh; e-mail: firstname.lastname@example.org
Published online in Wiley InterScience (www.interscience.wiley.com).
[C] 2009 Society of Plastics Engineers
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|Author:||Mahmoodi, Mehdi; Behravesh, Amir Hossein; Rezavand, Seyyed Abdol Mohammad; Golzar, Mohammad|
|Publication:||Polymer Engineering and Science|
|Date:||Mar 1, 2010|
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