Selection of the optimum conditions for the vulcanization of rubber products, particularly of bulky ones, often is quite difficult. Especially in cases of high demand, it is unavoidable to destruct expensive prototypes in order to check important properties of the rubber at various locations throughout the product. Finite element calculations can be used to predict the flow of heat during the vulcanization process. By converting the total heat input at a selected location during a certain time into a so-called 'effective vulcanization time' at a fixed reference temperature, it becomes possible to predict the properties of the rubber at that location. An example of such a procedure is given. However, to secure optimum conditions in this way would require repetitive (expensive) calculations. In this article a method is developed in which the vulcanization conditions are derived from the required properties of the rubber compound. The idea behind it is that optimal vulcanization is reached when at each location within the rubber product the effective vulcanization time at a selected reference temperature lies within predefined limits. The choice of the limits can be derived from the dependence of each of the properties chosen on the vulcanization time at the reference temperature. INTRODUCTION To convert a quantity of unvulcanized rubber into an elastic product, one needs to shape it and to vulcanize it. Vulcanization is accomplished by heating the shaped rubber mass. The heat required usually is transferred by the metal of the mold in which the product is shaped. It is often not easy to estimate beforehand how much heat is needed to be sure that the rubber will be vulcanized properly. Of course, there are methods to estimate how long the mass should be heated given the temperature of the mold. However, this becomes rather difficult for bulky products, especially when such products have complex shapes. In such cases it is common practice to produce prototypes that are then destroyed to allow the measurement of a number of physical-mechanical properties at critical locations. Because bulky products are expensive, this trial and error method can be both costly and time consuming. This article describes possibilities of calculating beforehand the optimum vulcanization procedure by using finite element calculations. This approach is not completely new (1-11). The difference from previous studies is that we have tried to calculate not only the heat flow in the rubber as a function of time, but also some physical-mechanical properties at selected locations within the product. The first problem was to select a finite element method to calculate the heat flow, taking into account the heat produced by the vulcanization reaction as well as the temperature dependence of the specific mass, the specific heat, and the heat conductivity of the rubber. The second problem was to find a method to calculate some selected physical-mechanical properties at predefined locations within the product from the amount of heat transferred to these locations at a certain moment. The third problem was to combine both methods in such a way that optimization becomes possible. This article describes the results of the first two phases of this study. From the third phase, only the approach we have in mind will be indicated. HEAT FLOW DURING VULCANIZATION Calculation of the heat flow from the metal of the mold into the rubber by means of analytical mathematical methods is only possible for products with simple geometries. For products with complex shapes and with, for instance, a number of metal inserts, a finite element calculation is the obvious choice. Such a calculation can also take into account the dependence of the properties of the rubber compound on temperature and the generation of heat from the exothermic vulcanization reaction. The idea behind finite element calculations is to split up a cross section of the product into a number of small elements and to calculate the transfer of heat in a given period from the wall of the mold to the first element, from the first to the second, and so on. There are a number of computer programs for finite element calculations available. We used a program called DIANA, developed by the TNO-Institute for Building Materials and Structures. This program contains a module to describe the hardening of large concrete elements (12-15). The module takes into account the hydration heat released during the hardening process. From a mathematical point of view, the contribution of the heat developed by the vulcanization reaction can be described in almost the same way as that of the heat from the hydration reaction. This program could therefore be used for our purpose after a few small modifications. For the description of the heat flow during the vulcanization the following equation has been used: |Mathematical Expression Omitted~ where |rho~ is the specific mass of the rubber compound used; |lambda~ is the heat conductivity of the compound; |c.sub.p~ is the specific heat of the compound; T is temperature; t is time; and q is the heat produced per unit of time by the vulcanization reaction. This equation expresses that the change in temperature equals the sum of the heat coming from the mold and the heat produced by the exothermic vulcanization reaction itself. To solve this equation, the material parameters |lambda~, |rho~, and |c.sub.p~, the heat production q, and the so-called boundary conditions must be known. Material Parameters We determined the temperature dependence of the specific mass, the heat conductivity, and the specific heat of the rubber compound to be used for the manufacturing of model products (19). For the determination of the specific mass the method of subsequent weighing in air and in a liquid has been used. The liquid in this case was a silicone oil, which could be heated to 125|degrees~C. The heat conductivity was measured by means of the Guarded Plate Method and the Hot Wire Method. The specific heat was determined by differential scanning calorimetry. Each of these properties appeared to show an almost linear dependence on the temperature (16-19): |rho~ = 1133 - 0.54T |kg |center dot~ |m.sup.-3~ |lambda~ = 0.216 + 0.00049T |W |center dot~ |m.sup.-1~ |cente dot~ ||degrees~C.sup.-1~~ |c.sub.p~ = 1635.5 + 3.082T |J |center dot~ |kg.sup.-1~ |center dot~ ||degrees~C.sup.-1~~ Heat Production Next we need to know the contribution to the vulcanization process of the heat developed by the vulcanization reaction itself. The heat production was determined by means of differential scanning calorimetry. To calculate from a DSC-curve how much heat at a given time was generated, the following procedure was followed. The degree of vulcanization (DV) is defined as: |Mathematical Expression Omitted~ where |r.sub.i~ is DV at time |t.sub.i~; Q(|t.sub.i~) is heat produced up to |Mathematical Expression Omitted~ is total heat produced at complete vulcanization. An example of a DSC-curve is shown in Fig. 1. At each time |t.sub.i~ (or temperature |T.sub.i~), |r.sub.i~ can be calculated by dividing the hatched area by the total area above the base line. The heat produced at time |t.sub.i~, |q.sub.i~ depends on the temperature |T.sub.i~ and on |r.sub.i~, i.e., |Mathematical Expression Omitted~ The exotherm shown by this line is caused by the vulcanization. As an example it is demonstrated how the degree of vulcanization is obtained from the curve at 145|degrees~C, viz. by dividing the hatched area by the total area above the base line. We assumed (14) that it is permissible to rewrite this equation as |q.sub.i~ = f(|r.sub.i~) |cente dot~ a(|T.sub.i~, |T.sub.ref~) where f(|r.sub.i~) is a function describing the heat production at a given temperature |T.sub.ref~; and a(|T.sub.i~, |T.sub.ref~) is a shift factor taking into account the acceleration or deceleration of the vulcanization reaction. This shift factor is given by the Arrhenius equation: E/R(|T.sup.-1.sub.ref~ - |T.sup.-1.sub.i~) where E is the activation energy of the vulcanization reaction; and R is Boltzmann's gas constant. The function f(|r.sub.i~) at a reference temperature of 150|degrees~C as derived from a DSC-curve for the compound used for the experimental part of our study is shown in Fig. 2. The function f(|r.sub.i~) can be fitted to an empirical relation; see Ref. 14. The heat generated at any moment and at any temperature during the vulcanization process can thus be calculated from a heat production curve like the one in Fig. 2 in combination with the Arrhenius equation given above. For the activation energy E we used for this particular rubber compound a value of 102 kJ/mol, derived from rheometer curves measured at various temperatures. The same value for E could be obtained from DSC-curves. However, with the equipment we used, the rheometer curves were more accurate. The activation energy has been calculated from the slope of a plot of ln(|t.sub.90~ - |t.sub.2~) as a function of the reciprocal of the absolute temperature. |t.sub.90~ and |t.sub.2~ are the times at which the torque in the rheometer experiment reaches 90% and 2%, respectively, of its maximum value. Boundary Conditions Boundary conditions in fact describe the situation of the 'outside world' during the calculations. In this case we need to know the temperature of the inner walls of the mold as a function of time. Because it is not known beforehand how much time is needed to attain a temperature of, say, 149|degrees~C, the choice has been made to measure the boundary conditions in situ. The process was initiated with the rubber, the mold, and the press all at 22|degrees~C; the mold was filled with the rubber compound and placed in the press. Then the whole was heated to 149|degrees~C, and cooled again to 22|degrees~C. The temperature profile thus measured was used for the calculations Verification The calculations of the heat profiles were carried out with the DIANA finite element program mentioned before. To demonstrate the accuracy of the numerical calculations a number of model products were prepared. The following compound was used.: SBR 1500 100 parts Zinc oxide 5 Stearic acid 2 N-Isopropyl-N|prime~-phenyl-p-phenylenediamine 2.5 Poly-2,2,4-trimethyl -1,2-dihydroquinoline 1.5 Carbon black N375 37.5 High aromatic oil 5 N-Cyclohexyl-2-be nzothiazyl sulfenamide 1.6 N, N|prime~-Diphenylguanidine 0.4 Sulfur 2 At several locations thermocouples were embedded in the unvulcanized rubber mass. In this way a comparison could be made between the calculated temperature profiles and the measured ones. The locations of three of these thermocouples in a cylindrical product with a height of 120 mm and a diameter of 200 mm are shown in Fig. 5. The temperature profiles measured by these thermocouples were compared with profiles calculated in Figs. 6 through 8. The dashed lines are the measured (filled circles) ones, the solid the calculated ones. The agreement between the calculated and the measured profiles during the heating is rather good. Differences to a maximum of 8|degrees~C can be seen in the center of the cylinder (at C). These differences are probably due to small inaccuracies in the material properties and in the calculated heat production. The extra temperature rise due to the pressure exerted on the rubber mass was neglected. The contribution of this will be approximately 2 to 3|degrees~C. PROPERTIES OF THE VULCANIZED RUBBER To estimate some selected physical-mechanical properties at various locations within the model product, so-called 'effective vulcanization times' at these locations were calculated. The effective vulcanization time is defined as the equivalent time needed to vulcanize the rubber at a constant reference temperature. During each time step |Delta~t, the heat transferred to a certain location contributes to the vulcanization process. By converting the time step |Delta~t at the actual temperature T to an equivalent time step |Delta~|tau~ at a reference temperature |T.sub.ref~, the contributions of each step of the iteration process can be summed to the total effective vulcanization time, using the following equation for |Delta~|tau~ |Mathematical Expression Omitted~ For the activation energy E the same value was used as previously mentioned for the calculation of the heat production (102 kJ/mol). In this way, for each location the effective vulcanization time could be calculated. Then, the compression set after 24 h at 100|degees~C (ISO 815) and the tear strength (ISO 816, Delft-method) were determined on test pieces vulcanized at a reference temperature of 150|degrees~C for various times. The results are shown in Figs. 9 and 10. The compression set and the tear strength were chosen because these properties are very sensitive to the degree of vulcanization. From these Figures the values of the compression set and the tear strength corresponding with the effective vulcanization times calculated could be determined. The same properties were measured on test pieces taken at different locations from the test product. The locations where these test pieces were taken are shown in Fig. 11. The calculated values are compared with the measured ones. The results are presented in Tables 1 and 2. For locations near the edges of the product the calculated values are in good agreement with the measured ones. More to the center the calculated temperatures are lower than the measured ones. Because of this the calculated effective vulcanization times will be too short. Especially for short vulcanization times, compression set and tear strength are very time dependent, so larger discrepancies can be expected. Table 1. Compression Set. Effective Compression Set Location Vulcanization Calculat ed Measured time (s) (%) (%) 1 6060 30 30 2 970 65 55 3 590 73 67 4 3530 45 43 Table 2. T ear Strength. Effective Tear Strength Location Vulcanization Calculat ed Measured (s) (N) (N) 1 6060 35 37 2 970 38 38 3 590 49 33 4 3530 35 34 OPTIMIZATION From the previous sections it is evident that the required physical-mechanical properties of the rubber at selected locations within a product after a given vulcanization time can be calculated fairly accurately. Specifications usually require that a number of physical-mechanical properties of a rubber product meet at least minimum values. Because of the fact that the properties are strongly position dependent, especially in voluminous products, it is often difficult to be sure that the requirements are fulfilled at all locations. Theoretically, it is possible to check this by means of the procedure described above by repeated calculations until the required minimum levels of the chosen properties have been reached. However, this is a (computer) time consuming and therefore costly procedure. So we looked for another approach. The question is: Can the optimal procedure to vulcanize a (bulky) rubber product be calculated, taking into account that at each location minimum values for selected physical-mechanical properties have to be attained while maximum values should not be exceeded? We have developed a procedure that makes this possible, at least in principle. First, the limits between which each of the properties in question should fall are selected and transferred into upper and lower limits of the effective vulcanization time. So for each property a time interval has to be defined. The relative importance of the various properties then is taken into account by introducing 'weight factors.' Optimization can now be attained by ensuring that the effective vulcanization times throughout the procedure lie in the interval |Mathematical Expression Omitted~ where |P.sub.i~ is the weight factor for property i; |L.sub.i~ is the lower limit for the effective vulcanization time for property i; |U.sub.i~ is the upper limit for the effective vulcanization time for property i; and C is the width of the interval. The definition of an optimal vulcanization procedure then is: A vulcanization procedure ensuring that the difference in effective vulcanization time between two randomly chosen locations in the product after vulcanization is smaller than or equal to a constant C. Ideally, optimal vulcanization then will be achieved when the differences in effective vulcanization times between two randomly chosen locations in the product for a timestep |Delta~t are equal to 0. Obviously, such conditions can never be met during vulcanization. However, one can require that the largest difference in effective vulcanization time between two randomly chosen locations after n time steps in the calculations does not exceed C, C being the sum of n small differences c (C = nc). That it is possible to calculate in this way the optimal vulcanization conditions will be demonstrated by means of a simple example: A sheet large compared to the thickness is heated. We assume that there is no reaction heat and that |rho~, |lambda~, and |c.sub.p~ are independent of the temperature. We require that for all locations in the sheet the effective vulcanization times remain within the interval |L, L + nc~. The largest difference in temperature will occur between the edge, |T.sub.edge~, and the center, |T.sub.center~. During the first (arbitrarily chosen) time step |Delta~t, the difference in effective vulcanization time between the center and the edge then is also known. |T.sub.center~ equals the ambient temperature, so we can calculate |T.sub.edge~ from: |Mathematical Expression Omitted~ In the next time steps |T.sub.center~ will be the temperatures achieved by heating. This can be repeated so that after n time steps a difference in effective vulcanization time of nc has been reached. This is an implicit method. Then, the interval is | A, A + nc~. However, we need an effective vulcanization time between L and L + nc. Shifting the interval |A, A + nc~ to |L, L + nc~ now can be achieved by repeating the calculations with a different value of the timestep |Delta~t. It is easy to see that |Delta~t should be reduced when A is larger than L, and increased when A is smaller than L. The iteration is continued until the difference between A and L is sufficiently small, i.e. /A - L/ |is less than or equal to~ |epsilon~. This procedure can be formalized in a binary search program. In principle, the same procedure can be followed when the contribution of the reaction heat is taken into account. One should consider then that the location of the maximum temperature is not necessarily on the edges of the product but could be somewhere else in the product. However, we still require that in the timestep |Delta~t the difference in effective vulcanization time between two random locations equals c. If this is not possible, for instance when the contribution of the reaction heat is too large, a new |Delta~t must be chosen so that this difference again equals c. In other words, |Delta~t then becomes variable. We have applied this principle to the simple example of the large sheet mentioned above (no reaction heat). The thickness was set to 20 mm, the temperature of the sheet at the start of the calculations to 25|degrees~C. The reference temperature chosen is 150|degrees~C. We require that the effective vulcanization times for each location in this sheet lie within the intervals |200, 1000~, |500, 1000~, and |800, 1000~, respectively. The results of the calculations are shown in the Figs. 12 through 14. In these Figures the curve with the circles represents the temperature in the center of the sheet and the curve with the triangles that at the edge. The numbers at the end of the curves are the effective vulcanization times calculated. The curves with triangles in fact indicate the temperature of the mold. It is clear that narrow limits are possible by starting with low mold temperatures, of course at the cost of large increases in real vulcanization times. Furthermore, the press used must be able to change the temperature continuously: Ideally the temperature of the press should follow the profile as closely as possible. In practice one may expect that adjusting the temperatures in discrete steps will not influence the final result too much. Application of this procedure will contribute to the development of better methods for the vulcanization of especially bulky rubber materials. CONCLUSIONS It appears that it is possible to calculate the heat flow during vulcanization of a product and from this the properties of the rubber at any location in a product. The agreement between the calculated and the measured levels of these properties is rather good, although the deviations become somewhat large going from the edge of the product to the inside. Optimization is possible by the trial and error method, requiring repeated calculations. The principle of a different procedure is given. 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