CALCULATION OF TIMES AND TEMPERATURES FOR PRESS VULCANIZATION OF THICK RUBBER PADS
ABSTRACT
Rubber manufacturers need simple rules to enable them to find the correct cure time and temperature for thick articles. To this end finite element analysis, controlled by Python scripting, has been used to calculate press times that meet specifications for adequate cure for uniaxial heat flow through molded rubber pads over a range of thicknesses. Three press temperatures, the effect of first preheating the rubber, and the rate of cooling were investigated. A low press temperature, preheating of the rubber, and fast cooling were required to achieve a reasonably even state of cure through thick moldings. The heat of vulcanization was included in some of the models. Its effect was small. Unified plots of the simulation results for five commercial rubber compounds are presented, and simple equations are provided to enable the cure time at any cure temperature to be estimated. The applicability of the results to fast cures of thin sections is discussed.
INTRODUCTION
Rubber components are often manufactured by compression molding in a press. Owing to the low thermal diffusivity of rubber, it can take several hours for the center of a large rubber component to reach the press temperature. Such items are usually cured at low temperatures for a long time so that the ratio of thermal diffusion time to cure time is sufficiently low to avoid excessive overcure of the outer parts of the component while ensuring that the center is sufficiently cured. A helpful approach for determining whether a particular cure is satisfactory is to use numerical techniques to solve the differential equations for heat flow and combine these with calculations of non-isothermal cure. Examples include Claxton and Liska,1 Gehman,2 Prentice and Williams,3 Toth et al.,4 Steen et al.,5 Gregory et al.,6 Han et al.,7 Warley,8 Vergnaud and Rosca,9 Ghoreishy,10 El Labban et al.,11 and Ghoreishy et al.12 Recently, the widespread availability of commercial finite element (FE) codes has made such calculations reasonably straightforward, although the accuracy of the analyses is still limited by the quality of the data available for the thermal properties of the rubber; their dependence on temperature, pressure, and state of cure; and the accuracy of the boundary conditions used to model factors such as the heat transfer from the press and convection losses to the surroundings.
The differential equation governing uniaxial heat flow through the rubber is
where θ is the temperature, t is time, x is position, K is the thermal conductivity, c is the specific heat capacity, ρ is the density, and Q is the volumetric power generated by the material such as that due to a heat of vulcanization. In the case that K, ρ, and c are constant and there is no heat of vulcanization, Eq. 1 may be simplified to
where κ = K/(ρc) is the thermal diffusivity.
The simplest method used to calculate the non-isothermal state of cure2,3,6,13,14 assumes that, over the temperature range in which the rubber cures, the cure rate increases with increasing cure temperature according to the relationship
where t1 and t2 are a characteristic cure time at temperatures θ1 and θ2, respectively. C is termed the temperature coefficient of vulcanization. Equation 3 is consistent with the Arrhenius law provided that the temperature range is small compared with the absolute temperature.2 For a non-isothermal cure, the equivalent isothermal cure time at the press temperature, θp, may be derived from Eq. 3 as
The integration is normally carried out numerically. To obtain the final equivalent cure time, the integral is carried out over the time that the component is curing in the press and while it is cooling when it will continue to cure. In practice the integral is truncated when all of the rubber has cooled to near ambient temperature and the cure is deemed to have finished. Since the rubber is only curing very slowly at this stage, the exact cut-off point has little effect on the final value for teq.
The isothermal cure times can be determined experimentally using a moving die rheometer or rubber curemeter. These machines apply small rotational oscillations to a shallow conical sample of initially uncured rubber and measure the increase in torque with time as it cures at a constant temperature. An example of the behavior of a sulfur-cured natural rubber is shown in Figure 1. Parameters such as tn, the time to n% of maximum torque, are calculated. C is then determined from such measurements made at different temperatures over the range of interest. Because n is a percentage of maximum torque, this method normalizes the results to accommodate differences in the final modulus at different cure temperatures. The underlying assumption in Eq. 3 is equivalent to the normalized rheometer torque curves remaining the same shape but becoming compressed in time with increasing temperature. It is not necessary to make any further assumption about the cure kinetics or the shape of the cure plots. The equivalent cure time, teq, calculated at any time or location may be used to estimate a “degree of cure” value from the rheometer plot at the same temperature, expressed, for example, as a percentage of maximum rheometer torque.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
The properties of the vulcanizate depend to some extent on the temperature at which they were cured. This was addressed by Gregory et al.,6 who used Eq. 3 to define and calculate a representative cure temperature from the non-isothermal cure temperature history and found good correlation between the properties of non-isothermally cured samples and those cured at the corresponding representative temperature. Their work provides experimental validation of Eq. 4.
An alternative approach, adopted by many workers (for example, Vergnaud and Rosca,9 Ghoreishy,10 and Isayev and Deng15), is to determine a specific model for the cure kinetics of the rubber. They acknowledge that the cure chemistry involves a series of chemical reactions, but usually simplifying assumptions are made and a small number of parameters are determined for different stages of the cure process, such as induction, cure, and postcure. In this case specific assumptions about the shape of the cure plot are made.
Manufactures need simple reliable rules to enable them to determine the optimum cure procedure for a variety of shapes and sizes of moldings. Various nomograms, such as that of Conant et al.,14 were published from which “incubation times” may be read off for different sizes, shapes, and thermal diffusivities. From the solution of Eq. 1 or from Einstein's diffusion equation,16
where
> is the root mean square distance traveled, it has been noted that the heating time of a body is proportional to the square of the distance from its center to the surface.6,13,14,17 Gregory et al.6 therefore proposed that te, the extra cure time beyond that required to cure the rubber isothermally at the press temperature, is proportional to the Einstein diffusion time and thus proposed the rule:
where l is the half thickness of the rubber pad, tp is the press time, and teqt is the required equivalent cure time at the press temperature. Equation 6 applies to cases in which the cure temperature is sufficiently low and/or the pads sufficiently thin that the ratio of press time to diffusion time is less than unity. The constant of proportionality, k, was approximately unity for curing at 100 °C above ambient and decreased slightly with decreasing temperature jumps.
Some workers have combined numerical simulation with an optimization algorithm to find a cure time or temperature that meets certain criteria. Steen et al.5 proposed a method to find a vulcanization time such that the equivalent cure at all points within a component was between pre-defined limits, based on the physical properties of isothermally cured testpieces. Han et al.7 optimized the bladder temperature history for a tire cure. El Labban et al.11 attempted to find the mold temperature that achieved an optimized compromise between uniformity of cure throughout the part and productivity. These studies were limited to a single component size. Warley8 found the preheat temperature that minimized cure variation throughout sheets of various thicknesses while meeting a target value for state of cure after cooling.
This work extends these ideas by using a commercial FE code in conjunction with Python scripting to find the minimum cure time for rubber pads of thicknesses varying between 12.7 and 152.4 mm under several different heating and cooling procedures. Only uniaxial heat flow, through the thickness of the pads, was modeled, and thus the results are applicable to pads in which the plan dimensions are at least an order of magnitude larger than their thickness. For pads with smaller plan dimensions, lateral heat flow through the sides of the mold is likely to be significant, but these results would serve as a conservative (upper bound) estimate of the required press time. Five rubber compounds were simulated, and the resulting data were normalized in order to provide a simple estimate of the required press time for any similar compound or thickness.
DETERMINATION OF THERMAL PROPERTIES
Five commercial sulfur-cured filled natural rubber compounds were provided by Mason Industries Inc. To avoid deterioration during shipment, curatives were added at TARRC. The compounds are denoted A to E in this work.
The cure characteristics of the compounds were determined using a Monsanto moving die rheometer at 100, 120, 130, and 140 °C.18 These temperatures were chosen to provide data over the full range of temperatures at which a significant rate of cure occurs under the non-isothermal conditions expected within a thick pad of rubber during a press cure. The rheometer curves for Compound B are shown in Figure 1; the other compounds showed similar behavior. The coefficients of vulcanization were normally calculated from the gradient of plots of the logarithm of t95 against temperature, where t95 is the time to 95% of maximum rheometer torque. For Compound C the cure rate became very slow as it approached t95, so C was obtained from the t90 values instead. The plots are given in Figure 2; in all cases, good straight lines were observed, confirming the validity of Eq. 3. Values for t50 and t95 at 104, 121, and 138 °C were calculated from the rheometry data at the nearest rheometer temperature using Eq. 3 for use in the modeling work. The values obtained are given in Table I.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
Test samples for thermal conductivity and density measurements were molded at 130 °C to maximum rheometer torque. The thermal conductivity was measured using a LaserComp Fox 50 heat flow apparatus over the temperature range of 80 to 150 °C. The thermal conductivities varied linearly with temperature, and values calculated from the best fit straight lines were used in the modeling work. The densities of the cured samples of the compounds were measured at 23 °C by the method of hydrostatic weighings.19
The specific heat capacity was calculated from the specific heats of the compound ingredients:2
where w1, w2, w3, and so on are the weight fractions of the ingredients, and c1, c2, c3, and so on are their specific heat capacities. Values for the specific heat capacities of the ingredients were taken from Gehman.2 For some of the minor ingredients no values were available so the assumption was made that their weighted average specific heat capacity was the same as that of the known ingredients.
The thermal properties of the rubber are reported in Table II. The values used for the steel mold are also given.
NUMERICAL MODEL
The commercial code MSC.Marc was used for the FE analysis. A model was created representing a 25.4 mm thick steel mold plate in perfect thermal contact with a rubber pad. Since only uniaxial heat flow was modeled, a single row of elements through the thickness was sufficient. It was assumed that the top and bottom halves of the mold were identical, so only half the thickness of the pad was modeled, and the behavior of the full pad was obtained by symmetry. The number of elements through the thickness of the steel and rubber remained constant for all thicknesses of rubber pad. Eight-node quadratic planar heat transfer elements were used. The model is shown in Figure 3.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
The rubber and steel mold were modeled with the thermal properties given in Table II. The thermal conductivity was assumed to vary linearly with temperature. The values used for the specific heat capacity and density are appropriate for room temperature and ambient pressure, and their variation with temperature and pressure was neglected in the analyses. Since the heat transfer across the mold is very rapid, the simulation is not very sensitive to the values used for the steel.
The initial temperature of the rubber and mold was set at either 25 °C or at 80 °C. The latter temperature was used to model the case when the sheet rubber was preheated before being loaded into the (preheated) mold. The platen temperature was modeled with a film boundary condition on the surface of the mold with a film coefficient of 10 000 W m−2 K−1, which represents very good thermal contact between the press platens and the mold. The FE mesh was continuous at the interface between the rubber and the steel, and hence perfect thermal contact was modeled here.
Because the cure reactions are exothermic, some heat is generated by the compound during curing. A large range of values for heat of vulcanization is reported in the literature for various rubber compounds, varying from about 1 to over 20 Jg−1 (refs 2, 4, 7, 15, 21). It was not measured for this work, and for most of the analyses it was taken as zero, but, in order to assess its effect, a few analyses were run in which a heat of vulcanization of total magnitude 10 Jg−1 was introduced. This value would give an adiabatic temperature rise of about 5 °C. It was modeled as a volumetric flux, defined as
where ΔH(teq) is the heat of vulcanization and the meanings of the other symbols are defined for Eq. 1. Although no detailed analysis of the cure chemistry was undertaken it is reasonable to suppose that the rate of emission of the heat of vulcanization, Q, is approximately proportional to the rate of increase of rheometer torque, as both are due to the formation of chemical crosslinks. Two shapes of the cure exotherm were modeled:
- 1.
Step function
- 2.
Triangular function
where a and b are constants, such that ΔHtotal = 10 Jg−1. These functions are illustrated in Figure 4. The curves of Figure 4b, particularly for the triangular function, are reminiscent of a plot of rheometer torque versus time as required. A user subroutine was written to calculate and apply the heat of vulcanization during the simulation.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
The analyses were run for platen temperatures of 104 and 121 °C. For Compound B the analyses were also run at 138 °C. Two conditions were specified to ensure that the cure was adequate:
- 1.
If the press is opened before the rubber is sufficiently cured, gases dissolved in the rubber at molding pressures are liable to come out of solution and blow holes in the uncured rubber leading to porosity. Therefore the first requirement for a satisfactory cure is that a reasonable level of cure has been achieved at the center of the rubber pad before the press is opened. Preliminary work suggested that a cure to t30 is likely to be sufficient to avoid porosity; for the current investigation a more conservative requirement of t50 was used.
- 2.
When the pad has cooled a cure level of at least t95 (or t90 for Compound C) was reached throughout. Since the pad continues to cure after it is removed from the press, the final state of cure of the pad is dependent on the rate of cooling. The rate of heat loss to the surroundings is difficult to estimate. Two extremes were considered:
- i.
Slow-cooled: the pad cooled in the mold in air by natural convection. An empirical formula for heat convection from an upward facing horizontal surface in still air is22
where R is the film coefficient in W m−2 K−1 and Δθ is the temperature difference between the mold surface and the surroundings in degrees Celsius. This condition was implemented in the FE model as a film boundary condition at the top and bottom mold surfaces. - ii.
Fast-cooled: the pad was cooled quickly in the mold by conduction through an efficient heat sink, for example, by placing it on a cold metallic table and placing a cold thick metal plate on top. This was implemented in the FE model with a film boundary condition, with a high value of 10 000 W m−2 K−1 for the film coefficient.
The ambient temperature for cooling was set at 25 °C, and the analysis ended when the temperature throughout the pad fell below 35 °C. The calculation of teq is weakly dependent on when the analysis ends because cure is assumed to continue even at low temperatures, but its rate is so slow compared with the rate at curing temperatures that its contribution to teq is insignificant.
A Python script was written to modify and submit the models, to iterate to the minimum press time required to meet these two conditions, and to process the results. Various criteria were used to define the incremental time steps during different stages of the analysis to provide an efficient but accurate solution. Except for a short period at the start of the heating and cooling stages, when there were very high temperature gradients on the surface of the mold, the maximum permitted temperature change was 2 °C. Small time increments were used at the end of the heating stage to ensure that the press time was accurate to within 5 s. The model was run for 12.7 mm increments of rubber thickness from 12.7 mm to 152.4 mm. The variable teq was evaluated numerically from Eq. 4 from:
where C, the coefficient of vulcanization, is given in Table I.
RESULTS AND DISCUSSION
The results are first presented for a single compound to illustrate the main phenomena and probe the effect of various factors such as preheating, cooling rate, and the heat of vulcanization. A subset of the analyses for all five compounds is then presented in a unified way that leads to a new method for calculating a satisfactory cure time and temperature.
For comparison of results at different press temperatures, it was helpful to define a dimensionless equivalent cure time, Teq, as
This parameter is a measure of the state of cure; a value of one indicates a cured pad, less than one an undercured pad, and significantly more than one an overcured pad.
results for compound b
The press times required to meet the two criteria for adequate cure defined in the “Numerical Model” section are shown in Figure 5. The press time increases substantially as the pad thickness increases. It is only affected by the rate of cooling for thin pads or low cure temperatures; otherwise it is determined by the need to reach a sufficient level of cure when the press is opened and so cannot be reduced even though the pad continues to cure after it is removed from the press. Preheating the rubber enables a substantial reduction in press time for thick pads.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
In most cases the lowest state of cure (after cooling) occurred at the center of the pad. However, when the rubber was preheated and the press temperature was low or the pads were thick, and in a few cases where a non-zero heat of vulcanization was modeled, the lowest state of cure occurred at a location between the center and the surface. This is illustrated in Figure 6 for the fast-cooled rubber cured at 104 °C with a preheat to 80 °C and arises from the contribution to cure during the cooling phase. A similar observation was reported by Warley.8 The effect is greatest for the thicker pads, where the cure during cooling is most significant, but in these cases the press time is determined by the need to achieve a sufficient level of cure before the rubber is removed from the press, at which time the state of cure is always lowest at the center of the pad. For the slow-cooled pads the additional cure, which occurred during cooling, was always sufficient to ensure that the total state of cure throughout was sufficient. Therefore, a calculation of the press time such that the pad meets the conditions for adequate cure only at its center would be unlikely to give a final cure anywhere that is significantly below that required.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
Although all the cures shown in Figure 5 meet the two criteria for adequate cure, in many cases parts of the pad are very overcured. This is shown in Figure 7. The most overcured pads reached a cure of five times t95 at their center and 45 times t95 at the surface. Such cures are unlikely to be satisfactory in practice. Lower cure temperatures and preheating should be used to reduce the ratio between the diffusion time and cure time and hence the degree of overcure.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
The temperature at the center of the pad only reached the press temperature for the thinnest pads as shown in Figure 8. The difference between the press temperature and cure temperature at the center was greater for high press temperatures but did not depend significantly on whether or not the rubber was preheated. The effect of a preheat is shown in Figure 9; it shortens the heating time and thus enables a shorter press time to be used. However, because the peak temperature and rate of change of temperature are similar with and without a preheat, the final extent of cure is not much affected. The effect can be visualized as a horizontal translation of the curve in Figure 9. Ideally the rubber should be preheated in the form of thin sheets or extrudate before it is placed in the mold so that heat transfer is fast. Alternatively, the filled mold could be preheated by placing it in the press at a lower temperature than the cure temperature to allow the rubber at the center to heat up without significant cure occurring at the surface, but this process makes heavy demands on press time.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
The effect of the heat of vulcanization is shown in Figures 10 and 11 for a press temperature of 121 °C. Inclusion of the heat of vulcanization gives only a very small reduction in press time; for thick pads this is because the press time is controlled by the need to achieve sufficient cure at the center when the press is opened, which is largely determined by the thermal diffusion time. The evolved heat does, however, cause an increase in the final state of cure at the center of the pad, so it gives a more uniform, though higher, final state of cure. Since the effect is quite small and not very sensitive to the shape of the exotherm, the use of more sophisticated kinetic models to capture it would not be worthwhile.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
unified results for five compounds
The extra press time, te, defined as tp – t95(θp), is plotted against l2/κ in Figure 12 for five rubber compounds under four cure regimes: press temperatures of 104 and 121 °C, with and without preheating to 80 °C, all with fast cooling and zero heat of vulcanization. No results were plotted for a cure temperature of 138 °C, and only thinner sheets were included for a cure temperature of 121 °C without preheating in order to eliminate the most unrealistic cure cycles. Also shown is Eq. 6 with k = 1. Equation 6 works well in its region of applicability: l2/κ < t95, and for k = 1, a large temperature jump,6 but substantially overestimates the cure time required for thicker pads. The results in Figure 12 fall onto two curves: one for the preheated pads and one for the pads that were not preheated.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
In order to unify the results an approach similar to that of Gregory et al.6 was adopted. The dimensionless half thickness, L, and dimensionless extra press time, Tpe, were defined as
where teqt was taken as t95 at the press temperature. In the simplest case of a constant thermal diffusivity and no heat of vulcanization, the governing differential equation (Eq. 2) then becomes
where the dimensionless time, T, and position, X, are defined as
Noting that the data of Figure 12 suggest that the cure time initially increases with the square of the thickness but then increases less rapidly, plots of Tpe against L2 (for small L) and L (for large L) are presented in Figures 13 and 14, respectively.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
The results fall approximately onto four lines, according to the value of the temperature jump, but there is also some dependence on compound. From the form of Eq. 16 and the definition of L (Eq. 14), we might suppose that the plots in Figures 13 and 14 are only weakly dependent on values used for κ and teqt, and thus it is more likely that the compound dependence arises from the different values of the coefficient of vulcanization for the different rubbers. The dependence on coefficient of vulcanization is likely to be most significant for thick sheets because a larger proportion of the cure takes place at temperatures well below the press temperature and, thus, as seen from Eq. 3, the effect of the value of coefficient of vulcanization on the contributions to teq is larger.
Figures 13 and 14 indicate that, for thin sheets,
where k1 is a constant that depends on the temperature jump. For thicker pads, when the center temperature does not reach the press temperature, the quadratic dependence on L breaks down and a linear relationship is more appropriate:
Best fit values for k1 and k2 were obtained for each of the temperature jumps, and reasonable empirical temperature dependences of k1 and k2 were found to be
where Δθ is the temperature jump between the initial rubber temperature and the press temperature in °C. Equations 20 and 21 are expected to be valid in the range 20 °C < Δθ < 110 °C. Equation 20 is not plausible for larger temperature jumps, since its quadratic form means that k1 passes through a maximum. Equations 18 and 19 with values of k1 and k2 obtained from Eqs. 20 and 21 are plotted on Figures 12, 13, and 14. The percentage difference between the value of Tpe calculated from the equations and that obtained from the simulation is shown in Figure 15. In most cases the values are within 20%.
Citation: Rubber Chemistry and Technology 90, 1; 10.5254/rct.16.83774
Equations 18 and 19 may be used directly to estimate the extra press time required for pads of different thickness. Also, they may be useful for estimating the required press time for a pad from a known cure time of another pad of a different thickness or compound when rearranged to give
where the subscripts A and B refer to the two pads.
application to other cure procedures and compounds
Figures 13 and 14 can be used to estimate press times for any combination of values for l, κ, and teqt for which Eq. 16, with the boundary conditions used in the analyses, is applicable. Temperature dependent thermal properties, or a non-zero heat of vulcanization, change the form of the governing differential equation such that it cannot be presented in the simple dimensionless form of Eq. 16. However, since these effects are normally small, the plots are still expected to be satisfactory in most cases.
For example, Figures 13 and 14 could be used to estimate press times for thinner, preheated sheets at higher press temperatures, such as might be of interest for optimizing an injection molding process. However, there would be a few anomalies in such an extrapolation owing to the conditions used in the analyses:
- 1.
In transforming to the new scenario, the mold thickness would scale with L in the same way as the rubber thickness and hence would be modeled as thinner at the higher press temperature. Also, these analyses assume that the mold is initially at the same temperature as the rubber, whereas in injection molding the mold would usually be preheated to the nominal cure temperature. These effects would result in errors that would tend to cancel. Furthermore, these errors would not be large, since heat transfer through the mold is rapid and thus has little effect on the calculated press time.
- 2.
The analyses were terminated when the temperature fell to 10 °C above ambient. If analyses without a preheat were used for a scenario with the same temperature jump where the rubber was preheated, this translates to 10 °C above the preheat temperature. However, the consequent underestimation of the final state of cure would be minimal because the strong dependence of cure rate of temperature means that the contribution to cure at temperatures well below the maximum is very small.
- 3.
The boundary condition governing the heat transfer at the surface during heating and cooling is, in dimensionless form,6
where θ0 is the external temperature. Thus, the film coefficient R scales inversely with l (for a constant thermal conductivity K). This means that higher film coefficients are assumed in the new scenario. Injection molded parts are usually demolded quickly but then cooled slowly in air. For both these reasons the contribution to cure during cooling would be underestimated in the new scenario.
Modifications could be made to the simulations to correct for these issues, in particular it would be sensible to omit the mold from the model and to use a lower film coefficient to model convection cooling in air. Nevertheless, the results presented here should provide a reasonable estimate of the press time for injection molded parts. The reasoning above demonstrates the applicability of the data to scenarios very different from those that formed the original motivation for the work.
All of the rubbers investigated in this study have a coefficient of vulcanization, C, close to 2, which is typical at least of sulfur-cured vulcanizates. The normalization used to generate Figures 13 and 14 should correct for variations in cure time (at a given temperature) and thermal diffusivity but not for significant variation in C. Rubbers with values of C significantly greater than 2 are likely to require longer cures than would be estimated from Figures 13 and 14, particularly if they are thick, because much of the cure occurs at temperatures below the press temperatures where the cure rate would be correspondingly slower. It should be possible to modify the normalization procedure to include a suitable dependence on C, but in the absence of simulation data for such rubbers this has not yet been attempted.
SUMMARY AND CONCLUSIONS
Python scripting has been used with MSC.Marc FE software to calculate press times meeting criteria for adequate cure for a wide range of thicknesses of rubber pads. In this work the criteria chosen were (i) sufficient cure when the press was opened to avoid porosity and (ii) a cure of at least t95 after the pad had cooled. The expected quadratic relationship between extra press time and thickness was confirmed for pads for which the ratio of half thickness to thermal diffusivity times cure time was less than one. The extra cure times of thicker pads show a linear dependence with thickness; in these cases the center of the pad does not reach the press temperature. In both regimes the constants of proportionality depend strongly on the temperature jump between the initial rubber temperature and the press temperature, and empirical relationships for these were calculated. These equations can be used to make estimates of the required press time for pads of any thickness for any temperature jump for rubbers for which the coefficient of vulcanization is around 2. The results are also approximately applicable to, for example, fast, high-temperature cures of preheated sheets up to moderate thickness.
A heat of vulcanization can be included in the model through a user subroutine. Its effect is to reduce slightly the required press time for thick pads.
Sensible approaches to achieving a satisfactory cure for thick pads include
- 1.
Moderate preheating of the rubber, ideally of thinner sheets, but otherwise by an initial procure at a lower temperature to the cure temperature. Excessive preheating is likely to result in poor quality moldings due to premature cure of the rubber and may result in the center of the molding becoming excessively overcured.
- 2.
A low cure temperature.
- 3.
Fast cooling to avoid excessive overcure occurring while the rubber cools.
- 4.
Cooling the pad while maintaining pressure. This would enable the heating times for the thick pads to be reduced below those reported here, and reduce the final level of overcure, since it would avoid the need to meet a minimum level of cure at the center before cooling.
So far only uniaxial heat flow has been modeled, but the methodology developed is very versatile and can easily be extended to more complex shapes, including moldings with inserts. It has been implemented in Abaqus as well as MSC.Marc. Plans are underway to extend the work to simple two- and three-dimensional shapes, such as cylinders and spheres, and to consider compounds with very different cure characteristics. The criteria for adequate cure could also be modified to include additional requirements, such as a maximum permissible state of cure. It should be possible to optimize the preheat and press temperatures in addition to the press time.
Further work is needed to provide reliable values for the thermal properties of rubber, taking into account their dependence on temperature and pressure and to establish appropriate rates of surface cooling under various practical scenarios. This would be a useful prerequisite to an experimental study of the practical utility of the cure regimes developed in this work.
Rheometer plots for Compound B.
Plot of logarithm of time to n% of maximum rheometer torque, where n is 95% for Compounds A, B, D, and E, and 90% for Compound C, as a function of temperature. The slope of the plot is −log C/10, where C is the coefficient of vulcanization.
Finite element analysis (FEA) model for 50.8 mm thick rubber pad (white elements) in a 25.4 mm thick mold (gray elements). There is a symmetry plane at the right-hand edge of the model. The arrow represents the boundary with the press platens (during heating) or atmosphere/cooling medium (during cooling). Heat flow occurs only in the direction of (or opposite to) the arrow.
Plots of exotherms used to model the heat of vulcanization. (a) Differential form, (b) integration of (a).
Press times required to reach a cure of t50 when the pad is removed from the press and t95 after it has cooled (ΔH = 0).
Distribution of the final state of cure through the thickness of pads preheated to 80 °C, cured at 104 °C and cooled quickly (ΔH = 0). The legend gives the pad thickness in millimeters.
Final state of cure at (a) the center and (b) the surface of rubber pads (ΔH = 0). The legend gives the press temperature and cooling and preheat conditions (no preheat unless stated).
Pad temperature at the center when removed from the press, relative to the press temperature for fast-cooled pads (ΔH = 0).
Temperature history at the center of a fast-cooled pad cured at 138 °C with and without a preheat to 80 °C (ΔH = 0).
Effect of a heat of vulcanization of 10 Jg−1 on the required press time of fast-cooled pads at a press temperature of 121 °C with and without preheating to 80 °C.
Effect of a heat of vulcanization of 10 Jg−1 on the final amount of cure at the center of fast-cooled pads cured at a press temperature of 121 °C with and without preheating to 80 °C.
Relationship between extra press time, te , and thermal diffusion time, l2/κ. Fast-cooled pads, ΔH = 0. Points: FEA results. Legend indicates compound and temperature jumps (in °C) of 104 − 80 = 24, 121 − 80 = 41, 104 − 25 = 79, and 121 − 25 = 96. Lines: fits to Eqs. 18 and 19, with the constants given by Eqs. 20 and 21, and Eq. 6.6
Relationship between dimensionless extra press time, Tpe, and dimensionless half thickness squared, L2, for thin pads. Points, FEA results; lines, fits to Eqs. 18 and 19 with the constants given by Eqs. 20 and 21. Legend indicates compound and temperature jumps (in °C) of 104 − 80 = 24, 121 − 80 = 41, 104 − 25 = 79, and 121 − 25 = 96.
Relationship between dimensionless extra press time, Tpe, and dimensionless half thickness, L. Points, FEA results; lines, fits to Eqs. 18 and 19 with the constants given by Eqs. 20 and 21. Legend indicates compound and temperature jumps (in °C) of 104 − 80 = 24, 121 − 80 = 41, 104 − 25 = 79, and 121 − 25 = 96.
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