Bubble Removal in Centrifugal Casting: Combined Effects of Buoyancy and Diffusion Aaron D. Mazzeo, Michelle E. Lustrino, David E. Hardt Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 This article models bubble removal in centrifugal casting of the thermosetting silicone polydimethylsiloxane (PDMS). Given spin speeds, mold geometry, and resin properties for centrifugal casting, it is possible to predict the amount of spin time required to produce bubble-free parts. For the particular conditions and materials considered in this work, required spin times range from seconds to minutes. The model itself balances two complementary physical behaviors: bubble dissolution and buoyancy. Diffusion dominates the removal of small bubbles below a critical size, while buoyancy dominates the removal of bubbles above a critical size. To simulate the removal of bubbles from a spinning solution, two first-order nonlinear differential equations are solved numerically. To verify simulated results, designed sets of experiments related the number of remaining bubbles in cured parts to spin times and speeds. Dow Corning Sylgard 184, Sylgard 186, and 3-6121 were the silicones tested in this work. POLYM. ENG. SCI., 52:80–90, 2012. ª 2011 Society of Plastics Engineers INTRODUCTION This work is motivated by the desire for rapid production of bubble-free silicone components with micro/nano features (e.g. microfluidic devices) [1, 2]. This is achieved by the use of a centrifugal casting technique. Centrifugal or spin casting techniques have molded a variety of materials including metals, ceramics, thermoplastics, and thermosets. With these materials, centrifugal casting has produced steel tubes, optical telecommunication fibers, polyester and polyvinyl pipes, functional gradient metalceramic materials, porous ceramic supports for membrane applications, gears, rubber tires, and hollow rubber dolls [3–6]. Centrifugal casting has also formed objects from rubbery polyurethanes [7, 8] and can even mold metals with a rubber mold [9]. Correspondence to: D.E. Hardt; e-mail: hardt@mit.edu Contract grant sponsor: Singapore-MIT Alliance. DOI 10.1002/pen.22049 Published online in Wiley Online Library (wileyonlinelibrary.com). C 2011 Society of Plastics Engineers V Centrifugal casting is a recognized method for eliminating bubbles in polymer solutions. In some cases, buoyant transport of the bubbles toward the center axis of rotation is the physical explanation for removal [4, 7, 8]. At high enough spin speeds with a polymer with low enough viscosity, buoyant forces aid in removing bubbles. In other cases, diffusion is the physical explanation for bubble removal [8, 10–14]. These are cases in which rotational speeds are not high enough to generate gravitational/buoyant forces sufficient to overcome the high viscosity of a polymer and significantly move bubbles toward the center axis of rotation. In diffusion-based bubble removal, there is a gradient of dissolved gas between the bubble wall and the polymer surrounding the bubble. If the concentration of gas dissolved within the bubble wall is greater than the concentration of gas in the surrounding solution, the bubble will lose gas. The aspects of diffusion described in this work are similar to those present in devolatilization of polymers. Nevertheless, there are at least two key differences between devolatilization and centrifugal casting as presented in this work. In devolatilization, the volatiles— gaseous or low-molecular weight undesirables—are removed from the processed polymer. In centrifugal casting, the gases from the bubbles are either removed through buoyancy or dissolved into the polymer. Another difference between these two processes involves the relevant sets of boundary conditions. Machines for devolatilization are designed to bring a portion of the processed polymer in contact with atmospheric pressure or vacuum through an outlet. In centrifugal casting as described in this article, there is only one surface exposed to atmospheric pressure, which is the liquid-air interface near the center of the spinning mold cavity. In this work, the centrifugal accelerations are on the order of 1000g’s (10,000 m/s2) with spinning speeds of thousands of revolutions per minute and resulting fluid pressures on the order of 700 kPa (100 psi). At these pressures, air bubbles in PDMS can readily dissolve into solution, however, the low dynamic viscosity of the polymer (1 or 10 Pa s) combined with high centrifugal accelerations also leads to buoyancy-driven transport of bubbles out of the fluid. Given the manufacturing POLYMER ENGINEERING AND SCIENCE—-2012 c ¼ kD p; FIG. 1. Mold cavity for centrifugal casting being spun about an axis of rotation. The parameters gr and gb represent centrifugal and gravitational accelerations, respectively. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] objectives of quality (bubble-free) and production rate (time), the key issue is the length of time required to remove all bubbles by both mechanisms. Centrifugal Casting and Bubble Dissolution In centrifugal casting, a reservoir receives a quantity of PDMS, which is then spun to fill the mold cavity. The specific geometry used in this work is shown in Fig. 1. During spin-up of the mold, the PDMS solution traps some air from the mold cavity, which causes the PDMS to be slightly more saturated with air relative to its initial state. Two things then occur during spinning: (1) the PDMS attempts to return to a normal saturated state via diffusion at all liquid–air interfaces and (2) bubbles move toward the center of the centrifuge. The movement of bubbles toward the center (larger ones faster than smaller ones) combined with the dissolution of some bubbles leaves a distribution of smaller and smaller bubbles in the PDMS as time progresses. To achieve a bubble-free part, all bubbles, regardless of size, must be removed either by buoyancy-driven transport or by dissolution. As will be discussed below, there is a critical bubble diameter that is defined by the longest time for removal either by buoyancy-driven transport (above the critical diameter) or by dissolution (below the critical diameter). Bubbles shrink or grow depending on the difference between the air concentration inside the bubble wall and the air concentration in the PDMS solution surrounding the bubble. If the concentration inside the bubble wall is less than the concentration in the surrounding solution, the bubble will grow as air diffuses into the bubble through its wall. If the concentration of air in the wall of the bubble is greater than the concentration of air in the solution surrounding the bubble, the bubble will tend to shrink. The concentration of air in the bulk of the PDMS solution is initially slightly supersaturated from the centrifuge’s startup, and during spinning, the air concentration remains nearly constant. However, the air concentration c in the polymer of the bubble’s wall varies linearly with internal pressure p of the bubble according to Henry’s Law: DOI 10.1002/pen (1) where kD is Henry’s coefficient. Higher internal pressures within a bubble will correspond to higher concentrations of air in the wall of a bubble and greater diffusion of the gas into the polymer. This pressure will increase not only as the pressure of surrounding polymer increases, but also as the diameter of the bubble decreases because of the effects of surface tension. Thus, the air concentration in the bubble wall will be larger for smaller bubbles, and if the air concentration in the bubble wall is larger than the air concentration of the surrounding PDMS fluid, the bubble will shrink and dissolve completely given sufficient time. The goal here is to determine the minimum time required to remove all bubbles from a mold cavity whether by diffusion of the air within the bubbles into the polymer or by buoyancy-driven transport of the bubbles to the free surface at the center of the centrifuge. For the geometries considered here, it will be shown that centrifugal casting will remove a large bubble, (1 mm in diameter) from a solution spinning at thousands of revolutions per minute by buoyant forces before the bubble has sufficient time to dissolve completely, whereas smaller bubbles, (10 lm in diameter) will dissolve into solution before buoyant forces drive them to the liquid–air interface. The basic geometry for analyzing the bubble dynamics is shown in Fig. 2. For description of the parameters, please see Nomenclature. Bubble Diffusion In 1950, Epstein and Plesset [15] published a seminal work describing diffusional growth or dissolution of an air bubble in water. Using their results and including both concentration gradients and surface tension effects, the following relationship for the rate of change in bubble radius can be derived [1]: ( ) dR kðcs1 ðcf 1Þ 2kRD sÞ 1 1 ¼ þ ; (2) r1 þ 2t=ð3RÞ dt R ðpktÞ1=2 FIG. 2. Gas bubble in a mold cavity of fluid being spun in a centrifuge. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] POLYMER ENGINEERING AND SCIENCE—-2012 81 where: s ¼ 2Mr/(BT). For descriptions of the parameters, please see Nomenclature. To simulate the full dynamics of a bubble, it is also necessary to track the change in mass, and this is given by a modified form of Eq. 2:
dm 2 ¼ 4pR kkD pfluid ðcf dt 1Þ 2s R ( ) 1 1 þ : R ðpktÞ1=2 (2a) In Eq. 2, if cf  1, the bubble will always shrink regardless of its radius. However, if cf [ 1 (supersaturation of the fluid with dissolved air), the following condition must hold for the bubble to shrink: R< 2s pfluid ðcf 1Þ : (3) fuge. The buoyant force in the radial direction (driven by the high value of gr) is FBr ¼ ðrf ra ÞVr gr ; (4) where rf is the density of the fluid, rf is the density of the air within the bubble, and Vr is the volume of the bubble. The drag force can be modeled according to the Hadamard-Rybczynski relationship [14, 15, 19]: FDr ¼ 3pmf dur 2mf þ 3mb ; 3mf þ 3mb (5) where lb is the dynamic viscosity of the air within the bubble and lf is the viscosity of the fluid. For air (lb 1025 Pa s) and for PDMS Sylgard 184 (lf  1 Pa s), the drag force acting on an air bubble in PDMS reduces to the following: Depending on the diameter of the bubble, pressure, and concentration of air dissolved in solution, calculated times for dissolution of a bubble in stagnant PDMS may require only a fraction of a second or may take longer than an hour [1]. Regardless of these times, it is assumed here that the bubble velocity will not alter the rate of gas diffusion, and therefore the quiescent fluid condition of the Epstein-Plesset model. For the small bubbles encountered in this system, this is a reasonable assumption. By balancing the buoyant and drag forces on a bubble and assuming negligible inertial contributions, the following relationship holds for the velocity of a bubble vr at a given location r being spun at a speed x: Viscous Effects on Dissolution Simulations for Bubble Transport and Diffusion Kloek et al. [16], Zana and Leal [17], Kim et al. [18], and others have derived relationships for growth of bubbles by diffusion or dissolution including the viscosity of the fluid. Kontopoulou and Vlachopoulos [12] applied models of diffusive dissolution of bubbles with viscosity to rotational molding. They modeled bubble dissolution in polyethylene using viscosity values of 500 Pa s, 5000 Pa s, and 10,000 Pa s (all of which are at least a factor of 5 greater than the viscosity of PDMS used in this study) and determined that for these values ‘‘the effect of viscosity is marginal and is more prominent at longer times, when bubble diameter becomes small.’’ Gogos [14] also addressed dissolution of bubbles in rotational molding, cited the work of Kontopoulou and Vlachopoulos [12], and ignored the effects of viscosity in his calculations. In both works by Kontopoulou and Gogos, the authors assumed the bubble was stationary. They also modeled the case of a single bubble in an infinite medium. Given that the effect of the material’s viscosity has only ‘‘marginal effects’’ on bubble growth/dissolution, the modeling in this work ignores the effects of viscosity on bubble dissolution. The removal of bubbles from the polymer involves both the change in radius and mass of the air bubble described by Eq. 2a and the transport by buoyancy described by Eq. 7. Since both sets of equations are nonlinear, they are best solved numerically, and the following describes a simple Euler-integration scheme used to obtain simulated results. This approach also allows for modeling a nonconstant centrifuge speed owing to finite start-up times. Initially, the bubble diameter is determined by using the ideal gas law for the calculated mass of air within a bubble. The mass of bubble at time interval i þ 1 is found by integrating dm dt from Eq. 2a: Buoyancy Effects Unless or until the bubble dissolves into the polymer, it is subjected to buoyant and drag forces in the centri82 POLYMER ENGINEERING AND SCIENCE—-2012 FDr  2pmf dur : ur  (6) ðra rf Þ 2 2 ro d : 12mf miþ1 ¼ mi þ dm j Dt; dt i (7) (8) Once the diameter is known, the radial position of the bubble at time interval i þ 1 is given by riþ1 ¼ ri þ ui Dt; (9) where vi is the radial velocity given by Eq. 7 at time interval i. The numerical solution of the combined model for a single bubble of specified initial radius and radial position has the following steps: DOI 10.1002/pen Initialization The initial position, radial distance, and all fluid properties in Eqs. 2 and 7 are specified, and the initial concentration of air within the liquid is determined from ci ¼ cf init kD pinit ; (10) where cf_init is the initial amount of fractional supersaturation (i.e. cf_init ¼ 1.05 would indicate a supersaturation of 5%) caused by pouring, mixing, or the sloshing of fluid at the start of spinning. The parameter pinit is the initial pressure (atmospheric for this work). Additional parameters including chamber pressure, temperature, maximum spin speed, slew rate (centrifuge’s angular acceleration), and time step for the numerical integration (Dt) are specified. It is important to note that the simulation is for a single bubble, and as will be shown, the initial bubble diameter do with the mold cavity at rest is a key initial condition. First Iteration Number of Moles of Gas. Given the initial value do of the bubble, the initial number of moles n within the bubble is determined with the following expressions: Pb ¼ 4s þ Pcham ; d0 (11) pPb d03 ; 6BT (12) n¼ where B is the Universal Gas Constant (287 J/(kg K)). Fluid Pressure. Using the current spin speed (can include a startup or ramp profile as well as the steadystate speed), the pressure of the liquid Pfluid at the bubble’s location is determined from:  1 Pfluid  Pcham þ rf o2 r 2 2  rf2 : (13) Bubble Pressure. The internal pressure Pb of the air within the bubble is found from:  4pPb Pb ¼ 2s 3nBT 1=3 þPfluid : (14) Bubble Diameter. Using Pb and the ideal gas law, a new diameter d for the bubble is determined. Repetitive Iterations Bubble Speed. Given the diameter, the speed vi of the bubble at the current location and time can be found from DOI 10.1002/pen Eq. 7, and using the speed, the new location ri found by Eq. 9. þ 1 is Nondimensional Degree of Saturation cf. Using the fluid pressure and the Henry’s coefficient, the value for cf is given by: cf ¼ ci : kD pfluid (15) Mass of the Bubble. From the current number of moles and molecular mass of the gas, the current mass of the bubble m is determined. Equation 8 is then used in conjunction with Eq. 2a to update the mass for time interval i þ 1. Using the new mass of air within the bubble and the molecular weight of the air (28.97 g/mol), the new number of moles within the bubble is determined. Update the Internal Bubble Pressure. Using the updated number of moles of gas within the bubble and the solution to Eq. 14, along with the fluid pressure at the current bubble location, the updated internal bubble pressure can be computed. From this resulting internal bubble pressure, a new diameter of the bubble is found using the ideal gas law. The algorithm cycles through the above sequence of repetitive iterations until the bubble diameter goes to zero or the bubble position reaches the desired end point. Simulation Results Using the above algorithm, it is possible to simulate bubble travel in a viscous polymer given a set of centrifugal casting parameters. The following three figures show a variety of outputs for simulations using the values in Table 1. The termination point of a line signifies that the bubble has either reached the desired end position (the liquid–air interface in this case) or dissolved into solution (d  0). The selected parameter values are representative of the designed and tested system. Figure 3 shows the value of cf plotted against time for different spin speed profiles and initial bubble diameters. The initial value of cf starts at 1.05 and then decreases because of the acceleration of the centrifuge. As the centrifuge accelerates at a rate of 300 rpm/s, the fluid pressure builds up causing a decrease in cf as described by Eq. 15. The centrifuge then reaches steady-state, and the bubble continues moving toward the liquid–air interface near the center of the centrifuge. As the bubble moves closer to the center of the centrifuge, the fluid pressure surrounding the bubble decreases as described by Eq. 13. This decrease in fluid pressure leads to an increase in the cf parameter, and if the bubble survives without dissolving and arrives at the liquid–air interface, the cf parameter returns to 1.05. POLYMER ENGINEERING AND SCIENCE—-2012 83 TABLE 1. Parameter values used in simulating combined bubble speed and diffusion behavior. Parameter Value Description Bubble_start_pos Bubble_end_pos rf_start T r l qf Pcham Spin_speed Slew_rate Offset j kD M dt cf_init 0.0615 m 0.0343 m 0.0343 m 298 K 0.02 N/m 3.9 Pa s 1030 kg/m3 101.3 kPa 1000–8000 rpm 300 rpm/s 100 rpm 1.47 3 1029 m2/s 1.927 3 1026 kg(gas)/m3(polymer)/Pa 0.02897 kg/mol 0.01 s 1.05 Bubble start distance from center axis of rotation Bubble ending distance from center axis of rotation Liquid–air interface distance from center axis of rotation Temperature during spinning Liquid–gas surface tension Liquid viscosity Liquid density Chamber pressure Set spin speed Spin speed acceleration (slew rate) Spin speed at start Diffusion coefficient Henry’s coefficient Molecular weight of gas Time step Initial supersaturation fraction Figure 4 shows the bubble diameter as a function of bubble position. The bubble starts at the right of each graph (61.5 mm from the axis of rotation) and moves toward the center/left of the graph (34.3 mm from the axis of rotation). Both plots in Fig. 4 show bubbles with do ¼ 100 lm dissolving into solution within 5 mm of the bubble’s starting location. Assuming the original solution is slightly supersaturated, it is also possible for a bubble to shrink by diffusion and then reach a region where the change in the pressure of the fluid causes the gradient in the concentration of gas between the bubble wall and the PDMS surrounding the bubble to flip. This change in the direction of the gradient results in diffusional growth of the bubble as the bubble gets closer to the center of the centrifuge. In general, diffusional shrinking of bubbles competes against the natural growth of bubbles as they move toward the axis of rotation. Without diffusion, the volume of the bubbles would increase as they approach the axis of rotation because of lower pressures near the axis of rotation. Figure 5 shows trajectories of bubbles with different diameters. The bubbles start at the upper horizontal line and progress toward the lower line. The bubbles either dissolve into solution or have buoyancy carry them to the lower horizontal line. Both graphs shown in Fig. 5 have an initial bubble diameter (do ¼ 132.1 lm for a spin speed of 1000 rpm and do ¼ 166.2 lm for a spin speed of 8000 rpm) that results in a maximum amount of time required for a given spin profile. These values of do resulting in maximized times for bubble removal were obtained using an iterative optimization routine. Figure 6 shows the maximum time values for these optimizations plotted against spin speed. near the point of maximum removal time (peak of the curve in Fig. 7 a). For example, near do ¼ 132.1 lm, a change of 1 3 1029 lm corresponds to a difference in time of 151 s for bubble removal. It is possible to avoid spurious bubble removal times with unrealistic sensitivity in the numerical simulation by instead considering the Sensitivity to Initial Diameter FIG. 3. The cf parameter plotted against time for a variety of initial bubble diameters and set spin speeds. The bubble starts 61.5 mm away from the center of rotation, and the liquid–air interface is 34.3 mm from the center of rotation. (a) 1000 rpm, (b) 8000 rpm. The numerical simulation is very sensitive to changes or perturbations to initial diameter do when do is chosen 84 POLYMER ENGINEERING AND SCIENCE—-2012 DOI 10.1002/pen moval time as shown in Fig. 7. That said, the maximum time for bubble removal is associated with a specific initial diameter, which is dependent on the profile of the spinning centrifuge and the bubble’s starting location. To the left of the peak in bubble removal time, diffusion dominates bubble removal whereas to the right of the peak, buoyancy and bubbles reach the liquid–air interface before dissolving. To calculate removal times associated with specified percentages of trapped bubbles, the simulation considers the travel of bubbles with initial diameters from 0 lm to 1 mm in increments of 25 nm. Figure 8 shows the maximums for the amount of time required for bubble removal over the specified domain of do. These values are depicted by the jagged curves just inside the curve showing the maximum exit times for optimized values of do. The lower three curves in Fig. 8 indicate the amount of spin time required to remove approximately 90% (line width FIG. 4. The bubble diameter plotted against position for a variety of initial bubble diameters and set spin speeds. The bubble starts 61.5 mm away from the center of rotation, and the liquid–air interface is 34.3 mm from the center of rotation. (a) 1000 rpm, (b) 8000 rpm. probability of removing certain fractions of bubbles initially trapped within the solution. Assuming a uniform probability distribution of values for do (equal probability of any particular bubble diameter occurring) between 0 lm and 1 mm, domains of values for do can be associated with a probability of occurrence. For example, if a specified domain has a span of 100lm, there is a 10% chance of a bubble existing within the specified domain over the total domain between 0 lm and 1 mm. Accordingly, if it is desirable to remove 99% of all the bubbles between 0 lm and 1 mm, a domain length of 10 lm is permissible. Given a domain length, it is possible to estimate the time required to remove the corresponding fraction of bubbles from solution. Figure 7b and c show the required amounts of time to remove 90% of bubbles with diameters from 0 lm to 1 mm. The cutoff line with a length of 100 lm in Fig. 7b suggests spinning at 1000 rpm for 65 s to remove 90% of the bubbles. A similar cutoff line in Fig. 7c suggests spinning at 8000 rpm for 13 s to remove 90% of the bubbles starting at the outer edge of the mold. Using the maximum times for bubble removal associated with the optimized value of do alone will likely overpredict the amount of time necessary to remove bubbles, especially for simulations with sharp peaks in bubble reDOI 10.1002/pen FIG. 5. The bubble position plotted against time for a variety of initial bubble diameters and two set spin speeds. The bubble starts 61.5 mm away from the center of rotation, and the liquid–air interface is 34.3 mm from the center of rotation. (a) 1000 rpm, (b) 8000 rpm. [Color figure can be viewed in the online issue, which is available at wileyonline library.com.] POLYMER ENGINEERING AND SCIENCE—-2012 85 less than 20 s for bubbles that start at 4 cm from the center axis of rotation. Because the differences in maximum times for bubble removal are minimal at low spin speeds FIG. 6. Top graph shows initial bubble diameter values (do) requiring the longest time to exit the solution of Sylgard 184. Bottom graph shows the optimized times for the bubble to exit. For the simulated data shown, the bubble started 61.5 mm from the center. of 100 lm), 95% (line width of 50 lm), and 99% (line width of 10 lm) of the bubbles that started at the outer edge of the mold. Effect of Starting Position of the Bubble Up to this point, all the simulations shown for Sylgard 184 have started with the bubbles sitting at the outer edge of the mold. In reality, the bubbles requiring the most time for removal do not necessarily start at the outer edge of the mold. In a case study of different geometries [1], it is shown that the slowest bubbles for removal may start in the middle of the part. Using the same parameters as the above simulations, Fig. 9 shows the simulated times required for removal plotted against multiple spin speeds and starting locations. The times for bubble removal are the simulated amount of time required for a bubble at a specified starting location with a specified initial diameter to either dissolve into solution or reach the desired location (edge of the mold or the liquid-air interface). The peaks (simulated maximum times) shown in Fig. 9 are dependent on the discretization of the initial diameters. To estimate actual removal times for bubbles and eliminate the unrealistic sensitivity to the discretization of the initial diameters of the bubbles, specified line widths representing a percentage of the possible diameters between 0 and 1 mm were utilized. As shown in Fig. 10, the maximum time to remove 99% of the bubbles for a set spin speed of 1000 rpm actually occurs with a bubble starting approximately 4.5 cm from the center axis of rotation. However, at higher spin speeds, the maximum time occurs with a bubble starting at the outer edge of the mold. For example, with a set spin speed of 5000 rpm, the bubble removal time is  20 s for bubbles that start at the outer edge but 86 POLYMER ENGINEERING AND SCIENCE—-2012 FIG. 7. (a) Bubble removal time plotted against initial bubble diameters near do ¼ 132.1 lm for a spin speed of 1000 rpm with Sylgard 184. (b) Calculated time to exit for a spin speed of 1000 rpm with do varied in steps of 25 nm. (c) Calculated time to exit for a spin speed of 8000 rpm with do varied in steps of 25 nm. In all these cases, the bubbles started near 61.5 mm from the central axis of rotation. DOI 10.1002/pen FIG. 8. Simulated bubble removal times based on optimization of do and different fractions of removed bubbles. The bubbles started at the outer edge of the molding region (61.5 mm from the axis of rotation). or at multiple steps after filling goes beyond the scope of this work. The agreement between the experimental results and the simulations reported suggests that the assumptions made about the initial distribution of relevant bubbles with diameters less than 1 mm are reasonable. Images of the centrifugally cast parts were acquired using a Canon CanoScan LiDE 600F scanner at a resolution of 1200 dpi. Figure 11a shows an example image of parts scanned without any bubbles in the regions of interest (outlined rectangular regions of images). Figure 11b shows an example image of parts scanned with a number of bubbles left in them. An algorithm (described in Ref. 1) was used to process the images and count the number of bubbles in each part. Figure 12 presents the average number of bubbles found in four parts at each combination of spinning parameters. The numbers next to the asterisk marks are the average number of bubbles meas- (1000 rpm) for bubbles starting in the middle of the part versus the outer edge of the part, it is simpler to use the outer edge as the starting position associated with the maximum bubble removal time. Experimental Validation To verify the predictive capabilities of the described numerical simulation, a series of experiments were performed varying spin time and spin speed. The spin speeds and spin times were chosen to produce approximately half of the resulting parts with bubbles and half without bubbles. The spin times were 0.5, 1, 2, 3, and 5 min, and the spin speeds were 1000, 1600, 2500, and 4000 min. With five discrete spin times and four discrete spin speeds, there were 20 possible combinations. The spin time refers to the total amount of time the centrifuge’s motor was energized and does not include the time for the centrifuge to decelerate and stop. The centrifuge accelerated to the desired spin speed with a slew rate of 300 rpm/s. The rest of the relevant geometric and fluid properties are included in Table 1. Two parts were produced per trial and each set of parameters was run twice. With 20 possible combinations and each combination run twice, there were 40 trials. Since each trial yielded two parts, there were a total of 80 parts produced. The number of bubbles in each part was counted, and the number of bubbles observed for a given combination of spin time and spin speed was calculated by averaging the numbers of bubbles in four parts. Counted numbers of bubbles in the cured parts were compared with the simulated results for percentages of bubbles removed. While a comparison between experimental results and simulations strictly using either numbers of bubbles or percentages of removed bubbles would be more desirable, the careful characterization of the distributions of bubbles in parts immediately after filling the mold DOI 10.1002/pen FIG. 9. Simulated times for bubbles to be removed from PDMS Sylgard 184 plotted against a domain of initial bubble diameters. The lengths indicated at the various peaks indicate the bubble starting position. (a) 1000 rpm, (b) 4000 rpm. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] POLYMER ENGINEERING AND SCIENCE—-2012 87 FIG. 10. Plot of simulated times for 99% removal of bubbles up to 1 mm in diameter from PDMS Sylgard 184 versus bubble starting positions and spin speeds. The dots represent simulated results. ured in four samples. Figure 12 also shows simulated times from Fig. 8 for bubble removal with bubbles starting at the outer edge of the mold. When Fig. 12 is replotted on log-log coordinates as shown in Fig. 13, the simulated curves become straight lines over much of the region of experimental interest. The line with pentagrams (time required to remove 99% of the bubbles) roughly separates the parts with more than one bubble from those that averaged less than one bubble. This dividing line shows agreement between the experimental data and the numerical simulations. Effect of Material Property Changes Finally, the simulations and modeling described in this work also apply to other materials besides Sylgard 184. Figures 14 and 15 show simulated calculations and experimental results for centrifugally casting Dow Corning 36121 and Dow Corning Sylgard 186. The only parameter changed in the simulations for these different materials was the viscosity. Dow Corning 3-6121 has a specified dynamic viscosity of 25 Pa s, the Sylgard 186 has a dynamic viscosity of 65 Pa s, and the Sylgard 184 has a specified dynamic viscosity of 3.9 Pa s. The line with diamonds (simulated time required to remove 99% of the bubbles) in Fig. 14 for Dow Corning 3-6121 roughly separates the parts with more than one bubble from those averaging less than one bubble just as shown in Figs. 12 and 13 for Sylgard 184. Figure 15a with Sylgard 186 shows data exhibiting the same trends as those shown for Sylgard 184 and 3-6121. However, the level of agreement between the simulations for removing 99% of the bubbles and the data for spin speeds less than 2500 rpm is not as pronounced (the simulated line does not sharply delineate the parts with bubbles from those without bubbles). This disagreement 88 POLYMER ENGINEERING AND SCIENCE—-2012 FIG. 11. (a) Scanned image of parts produced in a trial (1000 rpm and 5 min) with no counted bubbles in the regions of interest. (b) Scanned image of parts produced in a trial (1000 rpm and 0.5 min) with 144 bubbles counted within the region of interest in the part on the left and 21 bubbles counted within the region of interest in the part on the right. FIG. 12. Experimental data plotted against simulated predictions for Sylgard 184 with a viscosity of 3.9 Pa s. DOI 10.1002/pen FIG. 13. Log-log plot showing experimental data and predictions for Sylgard 184 with a viscosity of 3.9 Pa s. may partially result from the simulations in Fig. 15a only starting with the bubble at the outer edge of the mold, 61.5 mm from the axis of rotation. For simulations with the bubbles starting at the outer edge of the mold, the required spin time to remove 99% of the trapped bubbles with a set spin speed of 1000 rpm is 1090 s. Additional simulations run with a set spin speed of 1000 rpm and bubbles starting 49.5 mm from the axis of rotation showed a required spin time of 1190 s to remove 99% of the trapped bubbles. Thus, a bubble starting 45 mm away from the center axis of rotation with a set spin speed of 1000 rpm will take longer to exit than a bubble starting at the outer edge. More likely, the disagreement between the simulated results and the experimental data in this case results from unmodeled interactions between bubbles, fluid, and surfaces of the mold, unmodeled effects of vis- FIG. 15. (a) Plot showing experimental data and predictions for Sylgard 186 with a viscosity of 65 Pa s. The numbers represent the average number of bubbles counted in four parts, and the predictions assume bubbles start 61.5 mm from the axis of rotation. (b) Plot of simulated times for 99% removal of bubbles up to 1 mm in diameter from Sylgard 186 versus bubble starting positions and spin speeds. The dots represent simulated results. cosity on bubble dissolution, or variations in the viscosity or Henry’s coefficient. CONCLUSIONS FIG. 14. Plot showing experimental data and predictions for Dow Corning 3-6121 with a viscosity of 25 Pa s. The numbers represent the average number of bubbles counted in four parts. The lines demarcate the simulated results to remove different percentages of bubbles from the regions of interest. DOI 10.1002/pen The model presented in this work captures the combined effects of buoyancy, drag, and diffusion on bubble removal in centrifugal casting. The simulated results predict the time required for bubbles to either dissolve into solution or leave the region of interest with a given spin profile and starting location. For a given set of conditions for centrifugal casting, there is a critical initial size for bubbles. This critical initial size, or diameter, separates the bubbles removed predominantly by diffusion from those primarily removed by buoyancy. The required times for bubble removal are estimated using a numerical POLYMER ENGINEERING AND SCIENCE—-2012 89 solution of a pair of first-order nonlinear differential equations for rate of bubble mass and position change with time. Experiments on silicones with three different viscosities verified the theoretical basis for the model and numerical simulations described in this work. q! cS! cf ACKNOWLEDGMENTS The authors thank the Singapore-MIT Alliance for funding this work. MIT students Eehern Wong, A.J. Schrauth, Matthew Dirckx, Melinda Hale, Hayden Taylor, David Lee Hennan, Vikas Srivastava, and J.P. Urbanski also contributed to the development of this work. In addition, Dr. Brian Anthony and Dr. James Bales were very helpful in getting high-speed video images, which aided in further understanding the physics behind bubble removal and centrifugal casting. NOMENCLATURE The following parameters are pertinent to the geometry in Fig. 2: x gr vr FDr FBr d Pb rf r Pcham T Angular velocity of the centrifuge Radial acceleration owing to angular velocity ¼ rx2 Radial velocity of the bubble Viscous drag on the bubble in the radial direction Buoyant force on the bubble in the radial direction Bubble diameter Bubble pressure Radial distance from the free surface to the axis of rotation Radial distance of the bubble from the axis of rotation Ambient pressure in the centrifuge Polymer temperature (assumed to be uniform and constant). For Eq. 2, we have the following parameters: M B j Molecular weight of the gas (28.97 kg/1000 mol) Universal gas constant (8.314 J/(mol K)) Coefficient of diffusion 90 POLYMER ENGINEERING AND SCIENCE—-2012 ci Density of the gas at the pressure and temperature of the surrounding fluid kDpfluid (the concentration of dissolved gas in the wall of a bubble if the radius of the bubble were infinite) ci cS1 (the normalized saturation of the solution surrounding the bubble) Initial concentration of gas dissolved in the solution REFERENCES 1. A.D. Mazzeo, Centrifugal Casting and Fast Curing of Polydimethylsiloxane (PDMS) for the Manufacture of Micro and Nano Featured Components, PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA (2009). 2. J.C. McDonald and G.M. Whitesides, Acc. Chem. Res., 35, 491 (2002). 3. P.M. Biesheuvel, A. Nijmeijer, and H. Verweij, AIChE. J., 44, 1914 (1998). 4. C.D. Spencer, Soc. Plast. Eng. J., 18, 774 (1962). 5. W.M. 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