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Pulverization of rubber under high compression and shear
Powder Technology 102 Ž1999. 207–214
Pulverization of rubber under high compression and shear D. Schocke, H. Arastoopour), B. Bernstein Department of Chemical and EnÕironmental Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA Received 1 May 1998; accepted 28 September 1998
Abstract Non-cryogenic pulverization of rubber material was obtained under high compression and shear using a modified Bridgman Anvil apparatus. The effects of operating variables, such as temperature, normal and shear forces, shear rate, and residence time were examined, and optimum conditions for obtaining desired particle size distribution with minimum agglomeration were identified. Based on our strain energy storage theory, a criterion of pulverization was obtained and computational analysis of deformation of rubber using a Mooney-type equation for stored energy was performed. The numerical values for strain energy distribution in a rubber disk, which indicate potential pulverization under different compression and shear forces, were obtained using the ANSYS computer program. This information was used as a guide to obtain optimum operating conditions and design parameters for optimum design of the solid state shear extrusion ŽSSSE. pulverization process wH. Arastoopour, Single Screw Extruder for Solid State Shear Extrusion Pulverization, U.S. Patent No. 8,101,468 Ž1998.x. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Pulverization; Rubber; Extruder; Shear deformation; Agglomeration
1. Introduction The combined effects of hydrostatic pressure and shearing forces on metals were first studied by Bridgman w3x. He created an apparatus with two fixed pistons with a movable anvil in between that could be used to deform a thin disk of metals or other materials. At that time, he discovered that the combined effects of hydrostatic pressure and shearing stress caused some compounds that were stable under normal conditions to become unstable and violently explode or to react with other substances to which they were usually inert. The high normal loading and shear stress were also found to cause irreversible chemical changes and polymorphic transitions in several elements. Bridgman also suggested that disintegration could be accomplished under high shear and normal stresses and was successful in pulverizing metals and ceramics. Studies of high pressure and shear deformation were then extended by Enikolopian w4x to include the pulverization of polymeric materials. It was determined that pulverization was the result of the dissipation of the elastic energy stored in the material under shear deformation caused by the formation of new surfaces. Shearing and
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hydrostatic pressure were also found to promote mixing and diffusion and therefore reaction in the solid state. Through this research, the solid state shear extrusion ŽSSSE. process, which uses an extruder to impart shearing and normal forces in order to pulverize polymeric materials, was developed. Although SSSE has been successfully used to pulverize low density polyethylene, polyurethane foam, and some forms of rubber at IIT’s Center of Excellence in Polymer Science and Engineering ŽCEPSE., the process is still not well understood. Most of the research performed with the process has focused on the optimization of the pulverization process using a twin screw extruder w8x and a single screw extruder w1x. The primary goal of this research is to establish insight into the effects of compression and shear on the pulverization of natural rubber. In order to gain a better understanding of the pulverization process, a Bridgman Anvil was used to apply various deformations to samples of natural rubber. The rubber powder produced in the Bridgman Anvil was then analyzed to determine the effects of normal forces and shearing on pulverization. Thermal analysis and particle size analysis of the pulverized rubber were performed to investigate possible chemical and physical changes. A computational analysis of the deformation of a rubber disk under conditions similar to those used in the
0032-5910r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 0 3 2 - 5 9 1 0 Ž 9 8 . 0 0 2 1 3 - 7
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Bridgman Anvil was also performed to compare experimental observations with theory.
2. Equipment The Bridgman Anvil consists of two hardened steel surfaces: one of the surfaces is held stationary while the other is rotated to achieve the desired shearing effect. A press is used to squeeze the two surfaces together in order to exert pressure on a sample placed between them. A schematic diagram of the major components of the apparatus is given in Fig. 1. The Bridgman Anvil designed and constructed at the CEPSE consists of two hardened steel cylinders, 1.9 cm in diameter, with accurately ground plane ends. These upper and lower anvils are mounted in steel surrounds designed to add the support needed to keep the hardened steel from yielding under high stress conditions, and are removable so that pins of different diameters or surface roughness can be installed. A steel collar is used to confine the anvils and to prevent loss of sample. An electric heater band was placed around the collar and the upper anvil so that the temperature of the apparatus could be controlled. The lower anvil has no temperature control. A 1 hp variable speed geared DC electric motor manufactured by General Electric is used to rotate the lower anvil. The maximum torque output of the motor is 116 Nm, and the rate of rotation may be varied from 0 to 600 rpm using a speed controller manufactured by Glas Col. The two anvils and motor assembly are mounted in a Fred Carver, four posted hydraulic laboratory press with a capacity of 11.8 tons.
3. Experimental procedure and measurements Since the lower anvil has no temperature control, the two pins were first brought in contact with one another and the whole apparatus was allowed to preheat for 1 h to ensure that a steady control temperature was reached. A disk of vulcanized natural rubber with a diameter 0.8 cm and a thickness of 0.16 cm was then weighed and the mass recorded. The heated apparatus was then opened and the rubber sample placed on the lower anvil and centered. The lower pin and sample were then raised to come in contact with the upper anvil and the rubber was allowed to heat for 5 min. After heating the sample, the press was used to apply the desired normal force. A pressure gauge on the hydraulic ram of the press indicated the amount of force applied to the sample. The motor was then turned on to provide the prescribed rotation rate. Once the sample had been subjected to the desired deformation, the normal load was released and the rotation was stopped. The apparatus was then opened so that the deformed sample could be collected from the anvils. In order to systematically study the effects of high normal forces on the pulverization of natural rubber, thin disks of the material were subjected to one of five normal loads: 3700, 11,120; 22,240; 44,480; or 66,720 Newton Žas measured on a 0.88 cm diameter ram.. Rotation rates of 2, 4, 8, 16, or 32 rpm were also prescribed to each sample in order to study the effect of different shearing deformations. Additionally, the deformation of each rubber sample was carried for a specific residence time in the Bridgman Anvil of 5, 10, 20, or 40 s. While the vast majority of runs were conducted at a constant temperature of 808C, several runs at temperatures of 408C, 608C, 1008C, 1208C, and 1408C were also conducted. The collection of the powder produced in the Bridgman Anvil was difficult due to the presence of strong electrostatic attraction between the particles and the anvil. Therefore, the collected samples may not be an exact representation of the produced particles. Although the results provided in the following section are usually given in a quantitative fashion, the conclusion was drawn only from the qualitative trends of the effects of normal and shearing forces on the pulverization of natural rubber.
4. Results and discussion 4.1. General obserÕations
Fig. 1. Schematic diagram of the Bridgman Anvil.
In the runs where no powder was produced, the rubber disk was observed to have tiny fractures along the radial edge of the top plane Ži.e., the surface in contact with the fixed upper anvil.. The rest of the sample remained intact except for a single fracture along the entire circumference of the central plane. When very little powder was produced in the Bridgman Anvil, pulverization occurred at the outer
D. Schocke et al.r Powder Technology 102 (1999) 207–214
Fig. 2. Minimum load required for pulverization of rubber in the Bridgman Anvil as a function of residence time.
radial edge of the top plane in runs with short residence time. As the residence time was extended beyond 10 s, some powder was also formed at the outer radial edge of the bottom plane. When the normal loading and rotational rate were increased and larger quantities of powder were produced, pulverization still occurred at the outer radial edge of the sample, with the center remaining intact. As residence times were further increased, it was observed that the fractures along the outer edge spread toward the center of the disk. Agglomeration of the particles was not apparent until the top and bottom planes of the sample were highly fractured and a sizable quantity of powder had been produced. The sizes of the particles produced were independent of residence time: both large and very fine particles were produced in all runs. In runs where very little powder was produced, the particles were typically extremely fine and difficult to collect. It was therefore concluded that the smaller particles are not produced progressively from large particles, but instead are produced simultaneously with the larger particles. 4.2. Effects of normal loading and shearing on pulÕerization As mentioned previously, most of the experimental runs performed with the Bridgman Anvil were conducted at a constant temperature of 808C. This was done so that the effects of normal loading, shearing, and residence time could be studied independent of temperature effects. At 808C, all combinations of the five normal loads, five rotation rates, and four residence times were investigated. The residence time required for pulverization at various conditions was evaluated. Our experimental data showed that the residence time needed in the Bridgman Anvil to produce powder is significantly reduced by increasing the normal loading on the rubber sample. This can be explained by the increase in stored elastic energy caused by an increase in hydrostatic pressure.
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Our experimental data also showed that the minimum residence time in the anvil required for pulverization was reduced as the rate of rotation was increased. This again can be explained in terms of the time required to store energy in the sample. If it is assumed that the amount of energy stored in the sample due to shearing is related to the angular displacement of the lower anvil, it would be expected that strain energy would be stored faster in the rubber sample at higher rotation rates. If energy is stored in the sample more quickly at higher rotation rates, then the dissipation of energy through creation of new surface Ži.e., pulverization. must also happen faster. Fig. 2 shows the minimum normal load required for pulverization of rubber vs. rate of rotation at different residence times. The normal load required to pulverize the rubber sample decreased as the rate of rotation increased at a constant residence time. This was expected and can be explained in terms of energy being stored in the material faster at a higher rotational rate. At a given residence time, more energy must be supplied through compression if the rate at which shear is introduced is low. Fig. 2 also shows that the minimum normal force needed to produce powder in the Bridgman Anvil is reduced as residence time is increased. It is worth mentioning that, irrespective of the magnitude of shear force, a minimum normal force is required to create the necessary friction for storing strain energy sufficient for initiation of the pulverization process. Fig. 3 shows a summary of all of the data obtained at a rotational rate of 2 rpm. The weight percent of particles with diameters less than 800 mm produced in the Bridgman Anvil is plotted against the residence time at different normal loads. In addition to particles less than 800 mm, larger agglomerates Že.g., particles larger than 800 mm., also were obtained. Fig. 3 also shows that, in general, as the residence time is increased, the weight percent of powder that is produced is also increased at a given normal load. This is expected from the standpoint of more energy being stored in the sample as residence time is increased. Fig. 3 also shows that the amount of pulverized material increases with an increase in normal load, with the excep-
Fig. 3. Weight percent of powder Žparticles smaller than 800 mm. at different compression and rpm of 2.
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tion of a load of 66,720 N at a residence time of 40 s. As already stated, a higher normal force imparts greater strain on the sample and therefore increases the amount of stored energy in the sample. However, higher normal forces also provide more favorable conditions for particle agglomeration immediately after pulverization. Thus, the reduction of weight percent of powder produced at a residence time of 40 s may be attributed to the agglomeration of fine particles and a thin sintered film which was formed on the surface of the lower anvil. Indeed, in some cases, an increase in normal force in the Bridgman Anvil caused the contact forces between particles to exceed the shearing forces, and the particles to agglomerate. The same behavior shown in Fig. 3 was also observed at a rotational rate of 4 rpm. When the rate of rotation was increased to 8 rpm ŽFig. 4., the decrease in powder production due to agglomeration was no longer observed. This can be attributed to significant strain energy storage and in turn more pulverization at higher shear rate. With normal loads of 44,480 and 66,720 N, the amount of powder produced increased sharply after a residence time of 10 s, showing that residence time can also greatly affect energy storage and pulverization. When the rate of rotation was increased to 32 rpm, both the trends of increasing powder production with increasing normal force and with increasing residence time seem to disappear except at the lowest two normal loads of 3700 and 11,120 N. At a normal load of 22,240 N, no appreciable difference in powder production was observed as residence time was increased. Normal loads of 44,480 and 66,720 N showed distinct maximums in the amount of powder produced vs. residence time. Agglomeration into thin films and clumps Žparticles or agglomerates larger than 800 mm. was observed at all residence times at normal loads of 44,480 and 66,720 N at 32 rpm. In summary, we may conclude that there exists an optimum, normal force, rate of rotation and residence time in which pulverization is maximized and agglomeration is minimized. It is important to note that the lack of precision in setting the parameters of the apparatus and the difficulty in
Fig. 4. Weight percent of powder as a function of residence time at different compression and rpm of 8.
collecting the powder produced allow only qualitative interpretation of the above results. Great care was taken, however, in collecting the entire deformed sample so that experimental observations could be made. When the experiments were repeated, nearly all of the data were found to be reproducible within a precision of "2%. Runs that were determined to be non-reproducible with a precision of "2% were repeated about ten times so that an average value of weight percent of powder produced could be obtained. 4.3. Effect of temperature A series of experiments were conducted at temperatures ranging from 408C, the lowest temperature at which the apparatus could be effectively controlled, to 1408C, the temperature at which rubber degradation begins to become a problem. Our experimental data showed that the weight percent of powder produced was practically identical Žwithin experimental error. at temperatures ranging from 408C to 1408C. This does not mean the temperature has no effect on pulverization. In fact, it was observed that the powders comprised different portions of fine and larger particles. This indicates probable agglomeration at higher temperatures. It was also observed that a comparatively smaller particle size was achieved at temperatures about 1008C. This can most likely be explained by the weakening of the crosslinking disulfide bonds at higher temperatures, allowing for an easier pulverization into smaller particles. There was also a discoloration of the rubber at temperatures above 1008C Že.g., 1208C–1508C. accompanied by the smell of burning rubber, indicating that degradation of the rubber had occurred.
5. Analysis of the rubber powder The results of the thermogravimetric analysis ŽTGA. on the original natural rubber and the powder obtained from the Bridgman Anvil showed no change in composition or degradation of the natural rubber at high shear and normal forces. Differential scanning calorimetry ŽDSC. analysis was conducted using both the original rubber sample and the powder produced using the Bridgman Anvil. The results were not identical for both samples. Although two peaks were obtained for both the original and pulverized rubber, the first peak, which appears around 1058C, is noticeably shorter for the powder sample. Since the disulfide crosslinking bonds in vulcanized rubber are known to weaken above 1008C, we believe that the decrease in peak height can be attributed to the breakage of vulcanization bonds during the pulverization process. The size and shape of particles produced by pulverization under high shear and normal forces play an important
D. Schocke et al.r Powder Technology 102 (1999) 207–214
role in the reprocessing or recycling of the material. If the size of the particles is not comparable to those produced by other methods, the product may not find use as a filler in high grade rubber goods. The shape of the particles might also affect the recycling capabilities of the process, which is an important factor in determining the tendency for agglomeration. A base set of operating conditions Ž22,240 N normal loading, 8 rpm rotation rate, 20 s residence time, 808C. was selected. Based upon this, samples of powder were produced with: Ž1. the base operating conditions, Ž2. twice the normal loading, Ž3. twice the rotation rate, Ž4. twice the residence time, and Ž5. at a temperature of 1008C. In an attempt to minimize the inaccuracy caused by difficulties in sample collection, the experiment for each set of operating conditions was repeated ten times and the powder produced was mixed together for the size and shape analysis. An optical microscope was used to view the particles suspended in methanol under magnification. The pulverized rubber had an irregular shape and the majority of the particles could not be described with a single dimension. Fig. 5 shows a representative photograph of the pulverized rubber obtained from the Bridgman Anvil operated with a normal loading of 22,240 N, a rotation rate of 8 rpm, a residence time of 20 s, and a temperature of 808C. The particles are shown at a magnification that is approximately 160 = their normal size. As can be seen from the photographs, there is a wide distribution of particle sizes in the rubber powder. The rubber particles produced at a temperature of 1008C were observed to be slightly smaller than those formed at 808C. The overall particle size generally increased at higher normal loads, and the large particle seems to be a cluster of smaller particles. The formation of clusters of particles
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Table 1 Summary of the particle sizes produced with the Bridgman Anvil Normal load ŽN.
rpm
Residence time Žs.
Temperature Ž8C.
Minimum size Žmm.
Maximum size Žmm.
22,240 22,240 44,480 22,240 22,240
8 8 8 8 16
20 20 20 40 20
80 100 80 80 80
5 5 15 5 5
290 245 615 290 525
is evidence of the agglomeration that takes place under higher normal loads. The powder produced at a residence time of 40 s was not significantly different from that produced at a residence time of 20 s at the normal load of 22,240 N. The powder produced at a rotational rate of 16 rpm contained several large particles in the form of agglomerates. This shows that the increase in rotational rate for rotational rate of less than 16 rpm also contributes to particle agglomeration, but not as significantly as an increase in normal force. Table 1 provides a summary of the results obtained in the study of particle size. For example, a typical volume average size of produced powders for the runs at 16 rpm was about 350 mm, with more than 50% of particles less than 250 mm.
6. Theoretical and computational analysis 6.1. PulÕerization hypothesis Although the theory for pulverization under high shear and normal forces is not yet well developed, the studies of Enikolopian w4x have provided a basic understanding of the phenomenon. The formation of new surfaces Ži.e., pulverization. is thought to be the means by which elastic energy stored in the polymer sample during deformation is dissipated. The process is described as having a branched chain behavior in which the energy released in the creation of each new fracture is in turn spent on the formation of the next fracture. The propagation of fractures in the polymer sample has therefore become the focus of these theoretical studies. In order to understand how the fractures form and spread through the polymer sample, it is helpful to look at the growth of a single crack. In order to determine whether a crack within a bounded system Žthe rubber sample. will increase in size, a simple energy balance can be performed: Energy Transferred into the Systemy Energy Dissipated s Stored Energyq Kinetic Energy
Fig. 5. Photograph of the rubber powder produced at 808C, 8 rpm, 20 s residence time and a normal load of 22,240 N.
Ž 1.
The kinetic energy term can be dropped from the equation because our sample is not in motion. It was assumed that no energy is transferred into the system. If
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6.2. Numerical calculation of stored energy
Fig. 6. Configuration of finite element grids used in the model.
the only way for energy to be dissipated is through the formation of cracks and new surface area, the criterion for fracture propagation in a sample of elastic material with fixed boundaries can be represented as follows: dW y dc
Gg
dA dc
Ž 2.
Eq. Ž2. is known as the Griffith Fracture Criterion in which W is the total strain energy stored in the sample, c is the crack length, A is the fracture area, and g is a constant that represents the energy associated with free surface area for a linear elastic material w5x. If the elastic behavior of the material is nonlinear Žas it is with rubber., g is no longer constant and cannot be easily associated with free surface energy. We also believe that pulverization under high shear and normal forces occurs as a result of excessive shear stresses in the sample that initiate fracture. The compressive forces are thought to hold the fractured material together via friction w2x. This, in turn, allows the transfer of shearing forces from one section of the material to the next without relative movement between the surfaces along the fractures. Holding the sample together with high frictional forces allows the shear force increases in the sample and creates a vast network of microcracks and fractures. The energy stored in the sample during the deformation then reaches a threshold at which the material can no longer be held together by friction. It is at this threshold that the sample pulverizes, forming new free surfaces through the dissipation of energy. According to this theory, the criterion for pulverization under high shear and normal forces can be expressed as: Es y Ef G 0
Ž 3.
where Es is the energy stored in the material and Ef is the energy dissipated by the frictional forces. The important factors needed in order to gain an understanding of the pulverization phenomenon are therefore, the amount of energy stored in the deformed rubber sample, and the magnitude of the stress resulting from compression and shear deformation w6x. Therefore, numerical simulations of a rubber disk under compression and shear deformations were conducted using ANSYS, a finite element code.
ANSYS, a finite element analysis software package, was used to calculate the stress and strain energy associated with various deformations of thin cylindrical disks of rubber material according to finite elastic strain theory. The size of the rubber disk was similar to the samples used for experiments in the Bridgman Anvil, having a diameter of 1.0 cm and a thickness of 0.21 cm. The top of the cylinder Ž z s 0.21 cm. was held fixed in all three directions Ž r, u , and z .. This is an approximation of what is experienced by the face of the rubber sample that is in contact with the upper anvil, as movement in the r and u directions is restricted by high friction and movement in the z direction is prevented by the hardened steel surface. The bottom of the rubber cylinder is subjected to one of three types of deformation: angular displacement, compression, and a combination of angular displacement and compression. Similar to our Bridgman experiment, movement in the radial direction was not allowed on the bottom face. There was a total of 924 eight-node brick elements defined in the model, arranged in the configuration given in Fig. 6. In order to calculate the stress and strain energy in the disk during deformation, the material was given a Mooney type equation for stored energy w7x. This equation, which is used to describe the behavioral characteristics of incompressible nonlinear elastic material like natural rubber, has the following form: `
ws
Ý
i
Ci j Ž I1 y 3 . Ž I2 y 3 .
j
Ž 4.
is0, js0
where w is the strain energy per unit volume if the material is considered to be isotropic and incompressible, where I1 and I2 are the first and second principle invariants of the Cauchy strain tensor. The constants Cij must be determined experimentally for a given material. In order to determine the approximate constants for the natural rubber material used in the Bridgman Anvil experiments, a simple extension test was performed using a Rheometrics ŽRSA II. Solids Analyzer to obtain stress–strain data. These data were then used to calculate the constants needed for the Mooney equation using the least squares regression routine available in ANSYS. The results of this calculation are given in Table 2. The root mean square error and linear correlation coefficient values for the regression were 0.25 Table 2 The Mooney constants for natural rubber Constant
ŽNrm2 .
C10 C01 C20 C11 C02
1.78=10 5 7.36=10 4 1.29=10 5 y3.26=10 4 y1.16=10 5
D. Schocke et al.r Powder Technology 102 (1999) 207–214
and 0.9999, respectively. Because the constants in Table 2 are all roughly the same order of magnitude, more parameters are probably needed for an accurate description of the material. As with all nonlinear elastic materials, when shear was applied to the rubber sample, a force was exerted by the material normal to the direction of the shear force. This phenomenon, which is known as the Poynting effect, is realized in the Bridgman Anvil as a force in the direction opposite to the compressive force. The displacement of the lower face of the sample will therefore decrease as shear is applied. The compression in the computational analysis must therefore be applied with a surface force acting on the bottom of the sample, allowing the displacement to vary according to relative amounts of shear and compressive force. 6.3. Energy stored during deformation Fig. 7 is a plot of the strain energy per unit volume vs. radial position at the central plane of the rubber disk when only an angular displacement of 28 is applied to the bottom face. Because of the cylindrical shape, there is no real displacement at the center of the disk when the bottom face is rotated. The strain energy profile increases radially in a generally parabolic shape by approximately 1400 Jrm3 as one goes from the center to the edge of the disk. This is not surprising because the relative displacement of the bottom face at the outer radial edge of the disk is much larger than that observed near the center of the disk. Fig. 7 also shows the strain energy profile for a rubber disk subjected to a compressive force of 350,000 Nrm2 and no angular displacement of the bottom plane. It can be seen from Fig. 7 that compression of the rubber disk results in the storage of a large amount of energy at the center of the disk Žjust above 19,000 Jrm3 .. The amount of energy stored in the sample decreases radially to its minimum at the edge of the disk where the strain energy drops to a value less than 14,000 Jrm3. This is most likely the result of the boundary conditions applied to the top and bottom faces, where no radial movement was allowed.
Fig. 7. Strain energy profiles for a rubber disk subjected to a rotation of 28, a compression force of 350,000 Pa, and a combination of 28 rotation and 350,000 Pa compression.
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Fig. 8. Strain energy profiles for a rubber disk subjected to a compressive force of 350,000 Pa, and rotations of 38, 48, 58, and 78.
Because the sample in the numerical simulation was allowed to expand freely in the radial direction at the central plane of the disk, the rubber bulged out, causing less energy to be stored near the outer radial edge. Now that the individual effects of compression and shear on the storage of energy have been established, the combined effects of the two deformations can be studied. Fig. 7 also shows a plot of strain energy density vs. radial position for a rubber disk that is subjected to both a compression force of 350,000 Nrm2 and an angular displacement of 28. Because of the relatively large magnitude of the compressive force, the profile follows the general trend established for the case where there is only compression with the maximum stored energy at the center of the disk and the minimum at the outer radial edge. The curve is different from the pure compression case, however, as the stored energy remains nearly constant until the radius is approximately 0.3 cm, where it begins to decrease sharply. This change in the strain energy profile is a reflection of how shear affects the sample. Fig. 8 shows the strain energy profiles for a rubber disk subject to the same 350,000 Nrm2 compressive force and angular displacements of 38, 48, 58, and 78 so that the development of shearing effects can be seen. This graph shows that an angular displacement of 38 no longer has the maximum strain energy at the center of the disk, but instead at a radius around 0.3 cm. The minimum amount of strain energy was still observed to be at the outer radial edge, however. When the angular displacement was increased to 48, the maximum was again moved outward Žnear r s 0.35 cm., and the minimum occurred at the center of the disk. This general trend continues with angular displacements of 58 and 78 where the maximums were observed to be at r s 0.4 cm and r s 0.5 cm, respectively, and the minimums were at r s 0. The strain energy density profile for the case in which the angular displacement is 78 was nearly identical to that of only shear Žno compression., except for the fact that there was energy stored at the center of the disk as well, and the magnitude of strain energy stored in the entire sample was greatly increased.
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So, when the rubber sample was subjected to both compression and shear, the compressive force allowed the energy to be stored everywhere in the sample, and the shear force caused more energy to be stored. From these strain energy curves it can be concluded that, in the Bridgman Anvil, the rubber disk originally had most of its stored energy concentrated at its center when the initial compression was applied. When the rotation of the lower anvil was started, however, this maximum was gradually shifted outward as the angular displacement of the bottom surface was increased. This finding is in line with our experimental observation that the powder is initially formed at the outer edge of the disk. The fact that an increase in angular displacement causes an increase in stored energy at the edge can also help to explain our experimental observation that increasing the rate of rotation resulted in a decrease in the required normal load for pulverization. Although the numerical analysis was conducted at lower compression and deformation, the qualitative agreements validate our pulverization hypothesis and the potential for our theory and numerical code to be used as a design and scale-up tool.
7. Conclusion Non-cryogenic pulverization of rubber can be achieved at high shear and normal forces. An optimum operation condition Žshear and normal forces, temperature and residence time. in which to control reagglomeration of produced particles is needed to achieve desired particle size distribution. The strain energy distribution prior to pulverization was calculated using a Mooney-type equation for stored energy. The qualitative agreement between our numerical values and Bridgman Anvil experimental observations, such as pulverization occurrence at the outer edge of the disk, validates the potential of our theory in providing
guidance to obtain optimum pulverization operating conditions, apparatus selection, and design parameters. For example, based on our Bridgman Anvil data, it seems that the single screw extruder is an effective apparatus for pulverization of rubber with continuous pulverization expected to occur at the outer edge of the screw between the screw and the barrel; and in addition, the extruder should be designed such that it provides the needed shear and compression.
Acknowledgements The authors would like to thank the NSF Fluid, Particulate and Hydraulic Systems Division for financial support of this work under NSF Grant No. CTS-9629650.
References w1x H. Arastoopour, Single Screw Extruder for Solid State Shear Extrusion Pulverization, U.S. Patent No. 8,101,468 Ž1998.. w2x S.H. Benabdallah, Shear strength resulting from static friction of some thermoplastics, Journal of Materials Science 28 Ž1993. 3149– 3154. w3x P.W. Bridgman, Effects of high shearing stress combined with high hydrostatic pressure, Physical Review 48 Ž1935. 825–847. w4x N.S. Enikolopian, Some aspects of chemistry and physics of plastic flow, Pure and Applied Chemistry 57 Ž11. Ž1985. 1707–1711. w5x A.A. Griffith, The phenomena of rupture and flow on solids, Philosophical Transaction of the Royal Society of London 221 Ž1921. 163–197. w6x A. Riahi, Study of Pulverization of Low Density Polyethylene Using Solid State Shear Extrusion, PhD Thesis, Illinois Institute of Technology, July 1996. w7x R.S. Rivlin, D.W. Saunders, Large elastic deformations of isotropic materials: VII. Experiments of the deformation of rubber, Philosophical Transaction of the Royal Society of London 243 Ž1951. 251–288. w8x F. Shutov, G. Ivanov, H. Arastoopour, U.S. Patent No. 5,415,354 Ž1995..
Pulverization of rubber under high compression and shear D. Schocke, H. Arastoopour), B. Bernstein Department of Chemical and EnÕironmental Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA Received 1 May 1998; accepted 28 September 1998
Abstract Non-cryogenic pulverization of rubber material was obtained under high compression and shear using a modified Bridgman Anvil apparatus. The effects of operating variables, such as temperature, normal and shear forces, shear rate, and residence time were examined, and optimum conditions for obtaining desired particle size distribution with minimum agglomeration were identified. Based on our strain energy storage theory, a criterion of pulverization was obtained and computational analysis of deformation of rubber using a Mooney-type equation for stored energy was performed. The numerical values for strain energy distribution in a rubber disk, which indicate potential pulverization under different compression and shear forces, were obtained using the ANSYS computer program. This information was used as a guide to obtain optimum operating conditions and design parameters for optimum design of the solid state shear extrusion ŽSSSE. pulverization process wH. Arastoopour, Single Screw Extruder for Solid State Shear Extrusion Pulverization, U.S. Patent No. 8,101,468 Ž1998.x. q 1999 Elsevier Science S.A. All rights reserved. Keywords: Pulverization; Rubber; Extruder; Shear deformation; Agglomeration
1. Introduction The combined effects of hydrostatic pressure and shearing forces on metals were first studied by Bridgman w3x. He created an apparatus with two fixed pistons with a movable anvil in between that could be used to deform a thin disk of metals or other materials. At that time, he discovered that the combined effects of hydrostatic pressure and shearing stress caused some compounds that were stable under normal conditions to become unstable and violently explode or to react with other substances to which they were usually inert. The high normal loading and shear stress were also found to cause irreversible chemical changes and polymorphic transitions in several elements. Bridgman also suggested that disintegration could be accomplished under high shear and normal stresses and was successful in pulverizing metals and ceramics. Studies of high pressure and shear deformation were then extended by Enikolopian w4x to include the pulverization of polymeric materials. It was determined that pulverization was the result of the dissipation of the elastic energy stored in the material under shear deformation caused by the formation of new surfaces. Shearing and
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hydrostatic pressure were also found to promote mixing and diffusion and therefore reaction in the solid state. Through this research, the solid state shear extrusion ŽSSSE. process, which uses an extruder to impart shearing and normal forces in order to pulverize polymeric materials, was developed. Although SSSE has been successfully used to pulverize low density polyethylene, polyurethane foam, and some forms of rubber at IIT’s Center of Excellence in Polymer Science and Engineering ŽCEPSE., the process is still not well understood. Most of the research performed with the process has focused on the optimization of the pulverization process using a twin screw extruder w8x and a single screw extruder w1x. The primary goal of this research is to establish insight into the effects of compression and shear on the pulverization of natural rubber. In order to gain a better understanding of the pulverization process, a Bridgman Anvil was used to apply various deformations to samples of natural rubber. The rubber powder produced in the Bridgman Anvil was then analyzed to determine the effects of normal forces and shearing on pulverization. Thermal analysis and particle size analysis of the pulverized rubber were performed to investigate possible chemical and physical changes. A computational analysis of the deformation of a rubber disk under conditions similar to those used in the
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Bridgman Anvil was also performed to compare experimental observations with theory.
2. Equipment The Bridgman Anvil consists of two hardened steel surfaces: one of the surfaces is held stationary while the other is rotated to achieve the desired shearing effect. A press is used to squeeze the two surfaces together in order to exert pressure on a sample placed between them. A schematic diagram of the major components of the apparatus is given in Fig. 1. The Bridgman Anvil designed and constructed at the CEPSE consists of two hardened steel cylinders, 1.9 cm in diameter, with accurately ground plane ends. These upper and lower anvils are mounted in steel surrounds designed to add the support needed to keep the hardened steel from yielding under high stress conditions, and are removable so that pins of different diameters or surface roughness can be installed. A steel collar is used to confine the anvils and to prevent loss of sample. An electric heater band was placed around the collar and the upper anvil so that the temperature of the apparatus could be controlled. The lower anvil has no temperature control. A 1 hp variable speed geared DC electric motor manufactured by General Electric is used to rotate the lower anvil. The maximum torque output of the motor is 116 Nm, and the rate of rotation may be varied from 0 to 600 rpm using a speed controller manufactured by Glas Col. The two anvils and motor assembly are mounted in a Fred Carver, four posted hydraulic laboratory press with a capacity of 11.8 tons.
3. Experimental procedure and measurements Since the lower anvil has no temperature control, the two pins were first brought in contact with one another and the whole apparatus was allowed to preheat for 1 h to ensure that a steady control temperature was reached. A disk of vulcanized natural rubber with a diameter 0.8 cm and a thickness of 0.16 cm was then weighed and the mass recorded. The heated apparatus was then opened and the rubber sample placed on the lower anvil and centered. The lower pin and sample were then raised to come in contact with the upper anvil and the rubber was allowed to heat for 5 min. After heating the sample, the press was used to apply the desired normal force. A pressure gauge on the hydraulic ram of the press indicated the amount of force applied to the sample. The motor was then turned on to provide the prescribed rotation rate. Once the sample had been subjected to the desired deformation, the normal load was released and the rotation was stopped. The apparatus was then opened so that the deformed sample could be collected from the anvils. In order to systematically study the effects of high normal forces on the pulverization of natural rubber, thin disks of the material were subjected to one of five normal loads: 3700, 11,120; 22,240; 44,480; or 66,720 Newton Žas measured on a 0.88 cm diameter ram.. Rotation rates of 2, 4, 8, 16, or 32 rpm were also prescribed to each sample in order to study the effect of different shearing deformations. Additionally, the deformation of each rubber sample was carried for a specific residence time in the Bridgman Anvil of 5, 10, 20, or 40 s. While the vast majority of runs were conducted at a constant temperature of 808C, several runs at temperatures of 408C, 608C, 1008C, 1208C, and 1408C were also conducted. The collection of the powder produced in the Bridgman Anvil was difficult due to the presence of strong electrostatic attraction between the particles and the anvil. Therefore, the collected samples may not be an exact representation of the produced particles. Although the results provided in the following section are usually given in a quantitative fashion, the conclusion was drawn only from the qualitative trends of the effects of normal and shearing forces on the pulverization of natural rubber.
4. Results and discussion 4.1. General obserÕations
Fig. 1. Schematic diagram of the Bridgman Anvil.
In the runs where no powder was produced, the rubber disk was observed to have tiny fractures along the radial edge of the top plane Ži.e., the surface in contact with the fixed upper anvil.. The rest of the sample remained intact except for a single fracture along the entire circumference of the central plane. When very little powder was produced in the Bridgman Anvil, pulverization occurred at the outer
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Fig. 2. Minimum load required for pulverization of rubber in the Bridgman Anvil as a function of residence time.
radial edge of the top plane in runs with short residence time. As the residence time was extended beyond 10 s, some powder was also formed at the outer radial edge of the bottom plane. When the normal loading and rotational rate were increased and larger quantities of powder were produced, pulverization still occurred at the outer radial edge of the sample, with the center remaining intact. As residence times were further increased, it was observed that the fractures along the outer edge spread toward the center of the disk. Agglomeration of the particles was not apparent until the top and bottom planes of the sample were highly fractured and a sizable quantity of powder had been produced. The sizes of the particles produced were independent of residence time: both large and very fine particles were produced in all runs. In runs where very little powder was produced, the particles were typically extremely fine and difficult to collect. It was therefore concluded that the smaller particles are not produced progressively from large particles, but instead are produced simultaneously with the larger particles. 4.2. Effects of normal loading and shearing on pulÕerization As mentioned previously, most of the experimental runs performed with the Bridgman Anvil were conducted at a constant temperature of 808C. This was done so that the effects of normal loading, shearing, and residence time could be studied independent of temperature effects. At 808C, all combinations of the five normal loads, five rotation rates, and four residence times were investigated. The residence time required for pulverization at various conditions was evaluated. Our experimental data showed that the residence time needed in the Bridgman Anvil to produce powder is significantly reduced by increasing the normal loading on the rubber sample. This can be explained by the increase in stored elastic energy caused by an increase in hydrostatic pressure.
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Our experimental data also showed that the minimum residence time in the anvil required for pulverization was reduced as the rate of rotation was increased. This again can be explained in terms of the time required to store energy in the sample. If it is assumed that the amount of energy stored in the sample due to shearing is related to the angular displacement of the lower anvil, it would be expected that strain energy would be stored faster in the rubber sample at higher rotation rates. If energy is stored in the sample more quickly at higher rotation rates, then the dissipation of energy through creation of new surface Ži.e., pulverization. must also happen faster. Fig. 2 shows the minimum normal load required for pulverization of rubber vs. rate of rotation at different residence times. The normal load required to pulverize the rubber sample decreased as the rate of rotation increased at a constant residence time. This was expected and can be explained in terms of energy being stored in the material faster at a higher rotational rate. At a given residence time, more energy must be supplied through compression if the rate at which shear is introduced is low. Fig. 2 also shows that the minimum normal force needed to produce powder in the Bridgman Anvil is reduced as residence time is increased. It is worth mentioning that, irrespective of the magnitude of shear force, a minimum normal force is required to create the necessary friction for storing strain energy sufficient for initiation of the pulverization process. Fig. 3 shows a summary of all of the data obtained at a rotational rate of 2 rpm. The weight percent of particles with diameters less than 800 mm produced in the Bridgman Anvil is plotted against the residence time at different normal loads. In addition to particles less than 800 mm, larger agglomerates Že.g., particles larger than 800 mm., also were obtained. Fig. 3 also shows that, in general, as the residence time is increased, the weight percent of powder that is produced is also increased at a given normal load. This is expected from the standpoint of more energy being stored in the sample as residence time is increased. Fig. 3 also shows that the amount of pulverized material increases with an increase in normal load, with the excep-
Fig. 3. Weight percent of powder Žparticles smaller than 800 mm. at different compression and rpm of 2.
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tion of a load of 66,720 N at a residence time of 40 s. As already stated, a higher normal force imparts greater strain on the sample and therefore increases the amount of stored energy in the sample. However, higher normal forces also provide more favorable conditions for particle agglomeration immediately after pulverization. Thus, the reduction of weight percent of powder produced at a residence time of 40 s may be attributed to the agglomeration of fine particles and a thin sintered film which was formed on the surface of the lower anvil. Indeed, in some cases, an increase in normal force in the Bridgman Anvil caused the contact forces between particles to exceed the shearing forces, and the particles to agglomerate. The same behavior shown in Fig. 3 was also observed at a rotational rate of 4 rpm. When the rate of rotation was increased to 8 rpm ŽFig. 4., the decrease in powder production due to agglomeration was no longer observed. This can be attributed to significant strain energy storage and in turn more pulverization at higher shear rate. With normal loads of 44,480 and 66,720 N, the amount of powder produced increased sharply after a residence time of 10 s, showing that residence time can also greatly affect energy storage and pulverization. When the rate of rotation was increased to 32 rpm, both the trends of increasing powder production with increasing normal force and with increasing residence time seem to disappear except at the lowest two normal loads of 3700 and 11,120 N. At a normal load of 22,240 N, no appreciable difference in powder production was observed as residence time was increased. Normal loads of 44,480 and 66,720 N showed distinct maximums in the amount of powder produced vs. residence time. Agglomeration into thin films and clumps Žparticles or agglomerates larger than 800 mm. was observed at all residence times at normal loads of 44,480 and 66,720 N at 32 rpm. In summary, we may conclude that there exists an optimum, normal force, rate of rotation and residence time in which pulverization is maximized and agglomeration is minimized. It is important to note that the lack of precision in setting the parameters of the apparatus and the difficulty in
Fig. 4. Weight percent of powder as a function of residence time at different compression and rpm of 8.
collecting the powder produced allow only qualitative interpretation of the above results. Great care was taken, however, in collecting the entire deformed sample so that experimental observations could be made. When the experiments were repeated, nearly all of the data were found to be reproducible within a precision of "2%. Runs that were determined to be non-reproducible with a precision of "2% were repeated about ten times so that an average value of weight percent of powder produced could be obtained. 4.3. Effect of temperature A series of experiments were conducted at temperatures ranging from 408C, the lowest temperature at which the apparatus could be effectively controlled, to 1408C, the temperature at which rubber degradation begins to become a problem. Our experimental data showed that the weight percent of powder produced was practically identical Žwithin experimental error. at temperatures ranging from 408C to 1408C. This does not mean the temperature has no effect on pulverization. In fact, it was observed that the powders comprised different portions of fine and larger particles. This indicates probable agglomeration at higher temperatures. It was also observed that a comparatively smaller particle size was achieved at temperatures about 1008C. This can most likely be explained by the weakening of the crosslinking disulfide bonds at higher temperatures, allowing for an easier pulverization into smaller particles. There was also a discoloration of the rubber at temperatures above 1008C Že.g., 1208C–1508C. accompanied by the smell of burning rubber, indicating that degradation of the rubber had occurred.
5. Analysis of the rubber powder The results of the thermogravimetric analysis ŽTGA. on the original natural rubber and the powder obtained from the Bridgman Anvil showed no change in composition or degradation of the natural rubber at high shear and normal forces. Differential scanning calorimetry ŽDSC. analysis was conducted using both the original rubber sample and the powder produced using the Bridgman Anvil. The results were not identical for both samples. Although two peaks were obtained for both the original and pulverized rubber, the first peak, which appears around 1058C, is noticeably shorter for the powder sample. Since the disulfide crosslinking bonds in vulcanized rubber are known to weaken above 1008C, we believe that the decrease in peak height can be attributed to the breakage of vulcanization bonds during the pulverization process. The size and shape of particles produced by pulverization under high shear and normal forces play an important
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role in the reprocessing or recycling of the material. If the size of the particles is not comparable to those produced by other methods, the product may not find use as a filler in high grade rubber goods. The shape of the particles might also affect the recycling capabilities of the process, which is an important factor in determining the tendency for agglomeration. A base set of operating conditions Ž22,240 N normal loading, 8 rpm rotation rate, 20 s residence time, 808C. was selected. Based upon this, samples of powder were produced with: Ž1. the base operating conditions, Ž2. twice the normal loading, Ž3. twice the rotation rate, Ž4. twice the residence time, and Ž5. at a temperature of 1008C. In an attempt to minimize the inaccuracy caused by difficulties in sample collection, the experiment for each set of operating conditions was repeated ten times and the powder produced was mixed together for the size and shape analysis. An optical microscope was used to view the particles suspended in methanol under magnification. The pulverized rubber had an irregular shape and the majority of the particles could not be described with a single dimension. Fig. 5 shows a representative photograph of the pulverized rubber obtained from the Bridgman Anvil operated with a normal loading of 22,240 N, a rotation rate of 8 rpm, a residence time of 20 s, and a temperature of 808C. The particles are shown at a magnification that is approximately 160 = their normal size. As can be seen from the photographs, there is a wide distribution of particle sizes in the rubber powder. The rubber particles produced at a temperature of 1008C were observed to be slightly smaller than those formed at 808C. The overall particle size generally increased at higher normal loads, and the large particle seems to be a cluster of smaller particles. The formation of clusters of particles
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Table 1 Summary of the particle sizes produced with the Bridgman Anvil Normal load ŽN.
rpm
Residence time Žs.
Temperature Ž8C.
Minimum size Žmm.
Maximum size Žmm.
22,240 22,240 44,480 22,240 22,240
8 8 8 8 16
20 20 20 40 20
80 100 80 80 80
5 5 15 5 5
290 245 615 290 525
is evidence of the agglomeration that takes place under higher normal loads. The powder produced at a residence time of 40 s was not significantly different from that produced at a residence time of 20 s at the normal load of 22,240 N. The powder produced at a rotational rate of 16 rpm contained several large particles in the form of agglomerates. This shows that the increase in rotational rate for rotational rate of less than 16 rpm also contributes to particle agglomeration, but not as significantly as an increase in normal force. Table 1 provides a summary of the results obtained in the study of particle size. For example, a typical volume average size of produced powders for the runs at 16 rpm was about 350 mm, with more than 50% of particles less than 250 mm.
6. Theoretical and computational analysis 6.1. PulÕerization hypothesis Although the theory for pulverization under high shear and normal forces is not yet well developed, the studies of Enikolopian w4x have provided a basic understanding of the phenomenon. The formation of new surfaces Ži.e., pulverization. is thought to be the means by which elastic energy stored in the polymer sample during deformation is dissipated. The process is described as having a branched chain behavior in which the energy released in the creation of each new fracture is in turn spent on the formation of the next fracture. The propagation of fractures in the polymer sample has therefore become the focus of these theoretical studies. In order to understand how the fractures form and spread through the polymer sample, it is helpful to look at the growth of a single crack. In order to determine whether a crack within a bounded system Žthe rubber sample. will increase in size, a simple energy balance can be performed: Energy Transferred into the Systemy Energy Dissipated s Stored Energyq Kinetic Energy
Fig. 5. Photograph of the rubber powder produced at 808C, 8 rpm, 20 s residence time and a normal load of 22,240 N.
Ž 1.
The kinetic energy term can be dropped from the equation because our sample is not in motion. It was assumed that no energy is transferred into the system. If
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6.2. Numerical calculation of stored energy
Fig. 6. Configuration of finite element grids used in the model.
the only way for energy to be dissipated is through the formation of cracks and new surface area, the criterion for fracture propagation in a sample of elastic material with fixed boundaries can be represented as follows: dW y dc
Gg
dA dc
Ž 2.
Eq. Ž2. is known as the Griffith Fracture Criterion in which W is the total strain energy stored in the sample, c is the crack length, A is the fracture area, and g is a constant that represents the energy associated with free surface area for a linear elastic material w5x. If the elastic behavior of the material is nonlinear Žas it is with rubber., g is no longer constant and cannot be easily associated with free surface energy. We also believe that pulverization under high shear and normal forces occurs as a result of excessive shear stresses in the sample that initiate fracture. The compressive forces are thought to hold the fractured material together via friction w2x. This, in turn, allows the transfer of shearing forces from one section of the material to the next without relative movement between the surfaces along the fractures. Holding the sample together with high frictional forces allows the shear force increases in the sample and creates a vast network of microcracks and fractures. The energy stored in the sample during the deformation then reaches a threshold at which the material can no longer be held together by friction. It is at this threshold that the sample pulverizes, forming new free surfaces through the dissipation of energy. According to this theory, the criterion for pulverization under high shear and normal forces can be expressed as: Es y Ef G 0
Ž 3.
where Es is the energy stored in the material and Ef is the energy dissipated by the frictional forces. The important factors needed in order to gain an understanding of the pulverization phenomenon are therefore, the amount of energy stored in the deformed rubber sample, and the magnitude of the stress resulting from compression and shear deformation w6x. Therefore, numerical simulations of a rubber disk under compression and shear deformations were conducted using ANSYS, a finite element code.
ANSYS, a finite element analysis software package, was used to calculate the stress and strain energy associated with various deformations of thin cylindrical disks of rubber material according to finite elastic strain theory. The size of the rubber disk was similar to the samples used for experiments in the Bridgman Anvil, having a diameter of 1.0 cm and a thickness of 0.21 cm. The top of the cylinder Ž z s 0.21 cm. was held fixed in all three directions Ž r, u , and z .. This is an approximation of what is experienced by the face of the rubber sample that is in contact with the upper anvil, as movement in the r and u directions is restricted by high friction and movement in the z direction is prevented by the hardened steel surface. The bottom of the rubber cylinder is subjected to one of three types of deformation: angular displacement, compression, and a combination of angular displacement and compression. Similar to our Bridgman experiment, movement in the radial direction was not allowed on the bottom face. There was a total of 924 eight-node brick elements defined in the model, arranged in the configuration given in Fig. 6. In order to calculate the stress and strain energy in the disk during deformation, the material was given a Mooney type equation for stored energy w7x. This equation, which is used to describe the behavioral characteristics of incompressible nonlinear elastic material like natural rubber, has the following form: `
ws
Ý
i
Ci j Ž I1 y 3 . Ž I2 y 3 .
j
Ž 4.
is0, js0
where w is the strain energy per unit volume if the material is considered to be isotropic and incompressible, where I1 and I2 are the first and second principle invariants of the Cauchy strain tensor. The constants Cij must be determined experimentally for a given material. In order to determine the approximate constants for the natural rubber material used in the Bridgman Anvil experiments, a simple extension test was performed using a Rheometrics ŽRSA II. Solids Analyzer to obtain stress–strain data. These data were then used to calculate the constants needed for the Mooney equation using the least squares regression routine available in ANSYS. The results of this calculation are given in Table 2. The root mean square error and linear correlation coefficient values for the regression were 0.25 Table 2 The Mooney constants for natural rubber Constant
ŽNrm2 .
C10 C01 C20 C11 C02
1.78=10 5 7.36=10 4 1.29=10 5 y3.26=10 4 y1.16=10 5
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and 0.9999, respectively. Because the constants in Table 2 are all roughly the same order of magnitude, more parameters are probably needed for an accurate description of the material. As with all nonlinear elastic materials, when shear was applied to the rubber sample, a force was exerted by the material normal to the direction of the shear force. This phenomenon, which is known as the Poynting effect, is realized in the Bridgman Anvil as a force in the direction opposite to the compressive force. The displacement of the lower face of the sample will therefore decrease as shear is applied. The compression in the computational analysis must therefore be applied with a surface force acting on the bottom of the sample, allowing the displacement to vary according to relative amounts of shear and compressive force. 6.3. Energy stored during deformation Fig. 7 is a plot of the strain energy per unit volume vs. radial position at the central plane of the rubber disk when only an angular displacement of 28 is applied to the bottom face. Because of the cylindrical shape, there is no real displacement at the center of the disk when the bottom face is rotated. The strain energy profile increases radially in a generally parabolic shape by approximately 1400 Jrm3 as one goes from the center to the edge of the disk. This is not surprising because the relative displacement of the bottom face at the outer radial edge of the disk is much larger than that observed near the center of the disk. Fig. 7 also shows the strain energy profile for a rubber disk subjected to a compressive force of 350,000 Nrm2 and no angular displacement of the bottom plane. It can be seen from Fig. 7 that compression of the rubber disk results in the storage of a large amount of energy at the center of the disk Žjust above 19,000 Jrm3 .. The amount of energy stored in the sample decreases radially to its minimum at the edge of the disk where the strain energy drops to a value less than 14,000 Jrm3. This is most likely the result of the boundary conditions applied to the top and bottom faces, where no radial movement was allowed.
Fig. 7. Strain energy profiles for a rubber disk subjected to a rotation of 28, a compression force of 350,000 Pa, and a combination of 28 rotation and 350,000 Pa compression.
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Fig. 8. Strain energy profiles for a rubber disk subjected to a compressive force of 350,000 Pa, and rotations of 38, 48, 58, and 78.
Because the sample in the numerical simulation was allowed to expand freely in the radial direction at the central plane of the disk, the rubber bulged out, causing less energy to be stored near the outer radial edge. Now that the individual effects of compression and shear on the storage of energy have been established, the combined effects of the two deformations can be studied. Fig. 7 also shows a plot of strain energy density vs. radial position for a rubber disk that is subjected to both a compression force of 350,000 Nrm2 and an angular displacement of 28. Because of the relatively large magnitude of the compressive force, the profile follows the general trend established for the case where there is only compression with the maximum stored energy at the center of the disk and the minimum at the outer radial edge. The curve is different from the pure compression case, however, as the stored energy remains nearly constant until the radius is approximately 0.3 cm, where it begins to decrease sharply. This change in the strain energy profile is a reflection of how shear affects the sample. Fig. 8 shows the strain energy profiles for a rubber disk subject to the same 350,000 Nrm2 compressive force and angular displacements of 38, 48, 58, and 78 so that the development of shearing effects can be seen. This graph shows that an angular displacement of 38 no longer has the maximum strain energy at the center of the disk, but instead at a radius around 0.3 cm. The minimum amount of strain energy was still observed to be at the outer radial edge, however. When the angular displacement was increased to 48, the maximum was again moved outward Žnear r s 0.35 cm., and the minimum occurred at the center of the disk. This general trend continues with angular displacements of 58 and 78 where the maximums were observed to be at r s 0.4 cm and r s 0.5 cm, respectively, and the minimums were at r s 0. The strain energy density profile for the case in which the angular displacement is 78 was nearly identical to that of only shear Žno compression., except for the fact that there was energy stored at the center of the disk as well, and the magnitude of strain energy stored in the entire sample was greatly increased.
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So, when the rubber sample was subjected to both compression and shear, the compressive force allowed the energy to be stored everywhere in the sample, and the shear force caused more energy to be stored. From these strain energy curves it can be concluded that, in the Bridgman Anvil, the rubber disk originally had most of its stored energy concentrated at its center when the initial compression was applied. When the rotation of the lower anvil was started, however, this maximum was gradually shifted outward as the angular displacement of the bottom surface was increased. This finding is in line with our experimental observation that the powder is initially formed at the outer edge of the disk. The fact that an increase in angular displacement causes an increase in stored energy at the edge can also help to explain our experimental observation that increasing the rate of rotation resulted in a decrease in the required normal load for pulverization. Although the numerical analysis was conducted at lower compression and deformation, the qualitative agreements validate our pulverization hypothesis and the potential for our theory and numerical code to be used as a design and scale-up tool.
7. Conclusion Non-cryogenic pulverization of rubber can be achieved at high shear and normal forces. An optimum operation condition Žshear and normal forces, temperature and residence time. in which to control reagglomeration of produced particles is needed to achieve desired particle size distribution. The strain energy distribution prior to pulverization was calculated using a Mooney-type equation for stored energy. The qualitative agreement between our numerical values and Bridgman Anvil experimental observations, such as pulverization occurrence at the outer edge of the disk, validates the potential of our theory in providing
guidance to obtain optimum pulverization operating conditions, apparatus selection, and design parameters. For example, based on our Bridgman Anvil data, it seems that the single screw extruder is an effective apparatus for pulverization of rubber with continuous pulverization expected to occur at the outer edge of the screw between the screw and the barrel; and in addition, the extruder should be designed such that it provides the needed shear and compression.
Acknowledgements The authors would like to thank the NSF Fluid, Particulate and Hydraulic Systems Division for financial support of this work under NSF Grant No. CTS-9629650.
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