# Modeling the vulcanization reaction of silicone rubber.

INTRODUCTION

Over the past 50 years, silicone rubbers have become increasingly popular. This thermoset elastomer maintains its mechanical and electrical properties over a wide range of temperatures, and is therefore a natural choice for everything from aerospace applications to medical devices [1, 2]. It is used for the production of seals in the automotive and aerospace industry, connectors, and cables for appliances and telecommunications, implants and devices for medical purpose, and packaging and baking pans for the food industry [3]. Silicone rubber is a family of thermoset elastomers that have a backbone of alternating silicone and oxygen atoms and methyl or vinyl side groups. The solidification process of all heat-activated cure thermosets, including silicone rubber, is dominated by an exothermic and irreversible chemical reaction called vulcanization, or cure. The vulcanization process forms a three dimensional network that improves the properties of the final product [4-6]. Two types of reactions can lead to vulcanized silicone rubber: peroxide and platinum catalyzed cross-linking [7, 8].

Silicone rubbers consist mainly of silicone polymers and fillers. They can be classified according to the polymer employed and the vulcanization process as low temperature vulcanizable rubbers (LTV or RTV) and high temperature vulcanizable rubbers (HTV). RTV silicone rubbers come as a soft paste or a viscous liquid. HTV silicone rubbers come in two different physical states: liquid and solid.

Because they exhibit a high viscosity, solid silicone rubbers (commonly referred to as high consistency silicon rubbers) are processed in the same way as normal organic rubbers. Solid silicone rubbers are formed using linear polymers with molecular weights between 400,000 and 600,000 g [mol.sup.-1]. These polymers contain an average of 6000 siloxy units and are water clear newtonian liquids with viscosities between 15,000 and 30,000 Pa s. Solid silicone rubbers are usually vulcanized using a two kinds of peroxide catalyst: aroyl-peroxides and alkyl-peroxides [7, 8]. Aroyl-peroxides are used when the vulcanization process can be accomplished without pressure and they allow high reaction rates. Alkyl-peroxides are only used for vulcanization under pressure, as they do not form carbonic acids as decomposition products [1, 8]. The scheme of the reaction is shown in Fig. 1. The reactive vinyl double group of the polymer (Fig. 1a) reacts with the oxygen free-radical, already produced in the peroxide group (Fig. 1b), during the first stage of the reaction (Fig. 1c). The vulcanization is achieved when free radicals (Fig. 1c) attach themselves to another polymer chain and forms bridges. Peroxide catalysts allow for high reaction rates. Solid silicone rubbers are preprocessed materials that are vulcanized up to a certain degree and then stored in the form of foil-wrapped bars and packed cartons. To process solid rubbers, a sheet is then cut or stamped to the required size and placed in a compression or transfer molder. After vulcanization, the part is removed from the mold, deflashed, and freed of the peroxide decomposition products in a post-curing process.

Liquid silicone rubbers (LSR) have the same structure as solid silicone rubbers. However, the chain length of the polydimethylsiloxane used for LSR is lower by a factor of about 6. Therefore, the viscosity of the polymer is reduced by a factor of about 1000 [1-3, 7]. The vulcanization of liquid silicone rubber is almost exclusively carried out with a platinum-catalyzed hydrosilylation reaction, depicted in Fig. 2, in a reaction that does not generate by-products. Similar to solid silicone rubber, the vinyl double bond on the polymer (Fig. 2a) interacts with the platinum center, which has on free coordination site (Fig. 2b), and activates the double bond. The vinyl group cross-links by transforming the double bond, creating a single bond to a polymer chain; in this case, to a crosslinker molecule containing Si-H groups (Fig. 3c). The catalyst becomes free and is again available for further cross-linking. Liquid silicone rubbers are supplied in barrels or hobbocks. Because of their low viscosity, these rubbers can be pumped through pipelines and tubes to the vulcanization equipment. The two components (A and B component) are pumped to a static mixer by a metering machine were the vulcanization process may start taking place; loss of material in the feed lines is avoided using cold runners [8].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The individual steps of the vulcanization process of silicone rubber can be found in literature [1-4, 7], but none have a kinetic model that describes the cross-linking process. The vulcanization process can be described as the reaction between two chemical groups denoted by A and B which link two segments of a polymer chain. The reaction can be followed by tracking the concentration of unreacted As or Bs, [C.sub.A] or [C.sub.B]. The degree of cure can be defined by [4, 5],

[FIGURE 3 OMITTED]

c = [[C.sub.A,0] - [C.sub.A]]/[C.sub.A,0] (1)

where [C.sub.A,0] is the initial concentration of A. The degree of vulcanization is zero when there has been no reaction and equals 1 when all As have reacted and the reaction is complete. It is difficult to physically monitor reacted and unreacted As and Bs during the reaction without costly spectroscopy analysis. However, the degree of vulcanization can be related to the heat released during the reaction because the vulcanization of elastomers is an exothermic process. The energy released in the exothermal reaction is proportional to the cross-linked bonds formed, as it is assumed that each bond releases the same amount of energy [4-6]. The heat released during vulcanization can be calculated by,

Q = [[integral].sub.0.sup.[tau]] [dot.Q]dt (2)

where Q is the heat released up to time [tau] and [dot.Q] is the instantaneous rate of heat released by the sample. The total heat of reaction, [Q.sub.T], is therefore equal to,

[Q.sub.T] = [[integral].sub.0.sup.[[tau].sub.final]] [dot.Q]dt (3)

where [[tau].sub.final] is the time at which the reaction is complete. The reaction rate, dc/dt, is then calculated as,

[dc/dt] = [dot.Q]/[Q.sub.T]. (4)

The degree of vulcanization is obtained by integration of Eq. 4,

c = Q/[Q.sub.T] (5)

Kinetic models relate the reaction rate to temperature and degree of reaction. Two different approaches can be used to characterize a vulcanization reaction, mechanistic and phenomenological [9]. The first one studies the cross-linking process as a series of individual reactions and models each of them. The phenomenological approach considers the vulcanization as a whole process. This paper uses the phenomenological approach to model the kinetics of vulcanization for different types of silicone rubbers, using Kissinger [10, 11] and Kamal-Sourour models [12, 13].

The challenge of modeling the vulcanization is to take the DSC data and fit models that describe this process. With this information, time-temperature-transformation diagrams [14-16], for each resin, can be obtained and processes that involve vulcanization can be simulated [17-21]. Typically, isothermal DCS scans were used to fit the models that describe the curing reactions [20, 21]. However, several problems arise in the isothermal experiments. First, when performing isothermal scans it is challenging to reach the desired temperature as fast as possible. Second, at realistic processing temperatures, that is, 150[degrees]C for LSR, the reaction rate is too fast to be accurately captured in the measurement. One misses a great part of the reaction by the time the isothermal temperature is reached. Hence, one is left to perform the isothermal tests at temperatures that are significantly lower, that is, 40-60[degrees]C lower, that the processing temperature. This results in models where the process simulation is a gross extrapolation of DSC test results. In addition, all the techniques that use isothermal DSC data solved the fitted constants for each individual temperature, and later used a cumbersome fit to determine the temperature dependence [16, 20, 21].

Hernandez-Ortiz and Osswald [22] developed a technique to use the Kamal-Sourour model to fit dynamic DSC data. The kinetic parameters were expanded with a quadratic dependence on temperature. These fitted parameters do not necessarily have physical meaning; they were simply mathematical expressions that fit the experimental data, which can be used directly in numerical simulations.

In this paper, the Kissinger model is used in conjunction with the Kamal-Sourour in order to give more insight into the reaction and find a physically meaningful value for the activation energy. The use of the combined models results in a more realistic and robust model for the reaction. The Kissinger model is used to determine the activation energy of the silicone rubber from the experimental data. The other parameters of the reaction are determined with the Kamal-Sourour model, using the same methodology proposed by Hernandez-Ortiz and Osswald [22, 23]. The kinetic parameters are used to follow the degree of vulcanization as time and temperature change.

EXPERIMENTAL PROCEDURES

Liquid silicone rubber (LSR) is a two-component system. Component A contains a platinum catalyst and component B contains methylhydrogensiloxane for cross-linking, and an alcohol inhibitor [8, 23]. The A and B components of the liquid silicone rubbers were stored in a two component cartridge. Cartridges were placed at room temperature with room humidity of 35% in a dark cabinet for storage. The two components were mixed in a 1:1 ratio using a static mixer with 13 mixing elements.

Five types of liquid silicone rubber, denoted as LSR-a, LSR-b, LSR-c, LSR-d, and LSR-e, and one type of solid silicone rubber, expressed as HRS, were analyzed via thermal analysis to determine the progression of the vulcanization processes. The specific properties being studied were peak temperature, heat of reaction, and extent of reaction.

Differential scanning calorimeter [24] equipment manufactured by Netzsch (Phox DSC 200 PC) was used to measure the heat of reaction for the samples. Sealed aluminum pans were used to analyze all reactions. The mass of the samples ranged from 10 to 30 mg. A sealed empty pan was used as a reference.

The total heat of reaction was measured by a dynamic scan from 20 to 150[degrees]C using heating rates of 1.0, 2.5, 5.0, and 10.0 K/min. Multiple scanning rates were used to gain insight into the effect of time and temperature on the vulcanization reaction. Repeatability was obtained for each heating rate. All experiments were performed under nitrogen purge.

CURE KINETICS MODEL

Kinetic models relate the reaction rate to temperature and extent or degree of reaction. To study the cross-linking mechanism of elastomers and thermosets resins, two distinct approaches are used: phenomenological and mechanistic [5, 6, 9]. The phenomenological approach studies the reaction from a macroscopic level, considering the reaction as a whole process. The mechanistic approach observes the cross-linking process from a microscopic level and studies the reaction as a series of individual steps.

[FIGURE 4 OMITTED]

Mechanistic models are based on stoichiometric balances of the reactants involved in the elementary reactions, and the mechanics of each reaction. Therefore these models are more difficult to obtain but they can represent the kinetics of vulcanization better than the phenomenological models. Phenomenological models are semiempirical and do not provide a clear description of the vulcanization process and the chemistry behind it. These models are simpler to compute and can be widely used to describe the cross-linking process of different resin systems. They are based on the rate of reaction as a function of the amount of reacted resin and the rate constants, as described by [4-6, 9],

dc/dt = k(T)f(c) (6)

where c is the degree of vulcanization, k is the rate constant as a function of temperature, and f(c) is a function representing the amount of reacted resin. The rate constant is calculated with an Arrhenius equation. Different expressions for f(c) can be found in the literature but the most common are the nth-order reaction and the autocatalytic reaction.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Arrhenius Model

The Arrhenius model is used to describe the kinetics of an activated process. This model assumes that an energy barrier hinders the progress of the reaction. Forward progress of the reaction requires an activation energy to be supplied in order to overcome this barrier. The rate constant can be defined by an Arrhenius model, as follows,

k(T) = a exp (- [E/RT]) (7)

where a is the frequency factor that indicates how many collisions have the correct orientation to lead to products, E is the activation energy, T is the processing temperature in Kelvin, and R is the universal gas constant.

Kissinger Model

The Kissinger model [10, 11] uses an nth-order expression to model the kinetic reaction, defined as

dc/dt = k(1 - c)[.sup.n] (8)

where k is the rate constant defined by the Arrhenius equation, c is the extent of vulcanization, and n is the order of the reaction. The three parameters of the model (a, E, and n) can be fitted to the experimental DSC data.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Heat-activated reactions show a variation of the position of the peak with varying heating rate. If the temperature rises during reaction, the reaction rate, dc/dt, will rise to a maximum value and then return to zero as the reactant is consumed. The temperature at which the reaction rate is the maximum is also the temperature of maximum deflection in differential thermal analysis. Dynamic DSC measurements at different heating rates are used to determine the activation energy of the material, because they show the effects both time and temperature on the reaction [10, 11].

The maximum rate of reaction occurs when the derivative of dc/dt is zero. If the temperature is raised at a constant rate, [dot.T], the maximum rate of reaction can be defined as

[d/dt](dc/dt) = [dc/dt]([[E[dot.T]]/[R[T.sup.2]]] - an(1 - c)[.sup.n] exp(- [E/RT])) (9)

which is obtained by differentiating Eq. 8 and using k from Eq. 7. The maximum rate of reaction occurs at the peak temperature of the heat flow from the thermal analysis, [T.sub.peak]. The peak temperature can be defined by setting Eq. 9 to zero, that is,

[E[dot.T]]/[R[T.sup.2]] = an(1 - c)[.sub.peak.sup.n-1] exp (- [E/[R[T.sub.peak]]]). (10)

The amount of unreacted material (1 - c)[.sub.peak] is not determined directly through DSC analysis. Integration of Eq. 8, assuming a constant heating rate, gives an expression for the extent of reaction as a function of temperature. Approximations of this expression were obtained by Murray and White [25] through integration by parts. The expression for Eq. 8 after using the Murray and White approximations is given as [25],

[1/[n - 1]]([1/[(1 - c).sup.n-1]] - 1) = [[aR[T.sup.2]]/[E[dot.T]]](1 - [2RT/E])exp(- [E/RT]). (11)

The expression for the amount of unreacted material at [T.sub.peak] for reaction orders other than zero or unity, can be simplified to

n(1 - c)[.sub.peak.sup.n-1] = 1 + (n - 1) [[2R[T.sub.peak]]/E]. (12)

An expression for the activation energy can be found by substituting Eq. 12 into Eq. 9 and omitting negligible quantities

-[E/R] = [d(ln[T/[T.sub.peak.sup.2]])]/[d (1/T)]. (13)

The activation energy can be easily calculated by performing DSC tests at different scanning rates regardless of the order of the reaction.

Kamal-Sourour Model or Autocatalytic Model

The Kamal-Sourour reaction model is one of the first models to accurately define autocatalytic cross-linking reactions; it is defined as [12, 13],

dc/dt = ([k.sub.1] + [k.sub.2][c.sup.m]) (1 - c)[.sup.n] (14)

where [k.sub.1] and [k.sub.2] are the rate constants described by the Arrhenius equation, and m and n are the reaction orders.

The six parameters in the model ([a.sub.1], [a.sub.2], [E.sub.1], [E.sub.2], m, and n) can be fit to the experimental DSC data via a least-squares estimation algorithm developed by Marquardt [26, 27]. Karkanas et al. [28] found that the Kamal-Sourour model accurately predicts the degree of cure during early stage of reaction; however, at the later stage, it over-predicts the degree of cure [20, 21]. This deviation is due to the diffusion effects. Thus, it is necessary to modify the Kamal-Sourour model to include the diffusion-limited part of the reaction. However, for elastomeric materials it is not necessary to modify the constants of the reaction due to diffusion, because the materials vulcanize above their glass transition temperatures. For materials that vulcanize before the vitrification point, the model accurately predicts the total process. This condition gives sufficient free volume between the molecules to allow for freedom of movement during the molecular cross-linking process, and therefore diffusion does not play a large role [4, 5].

MODEL DESCRIPTION AND NUMERICAL ALGORITHM

The Kamal-Sourour diffusion model has six parameters that need to be fitted. Let x be the unknown vector parameter defined as,

x = {m n [a.sub.1] [E.sub.1] [a.sub.2] [E.sub.2]}. (15)

The proposed technique uses one or more dynamic DSC scans, avoiding the problems that arise by using isothermal DSC data. The activation energy [E.sub.1] is determined through Kissinger's model, which is then used to mathematically determine the other five parameters for the Kamal-Sourour model. This novel technique finds a physically meaningful value of [E.sub.1]. The activation energy, [E.sub.2], is fitted as a constant. The four remaining parameters (m, n, [a.sub.1], [a.sub.2]) in the cure model are expanded into a power series of the temperature [22], that is,

[FIGURE 9 OMITTED]

[x.sub.i] = [a.sub.i1] + [a.sub.i2]T + [a.sub.i3][T.sup.2] + O([T.sup.3]) (16)

where i = 1,..., 4 and [a.sub.ij] are the new goal of the fitting. In other words, instead of looking for the six parameters using isothermal data, the new technique looks for the 13 components of the A matrix defined in such a way that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

According to Eq. 17, the higher terms in the expansion are neglected; accordingly the fitting must be performed in such a way that the coefficient accompanying the second order term is small. If this condition does not satisfy a specific set of data, the expansion cannot assure that the higher order terms are small; consequently, higher order terms must be included in the expansion (i.e. third, fourth). The parameters in the model are fitted via a least-squares estimation algorithm developed by Marquardt [26, 27] (see also Press et al. [29] or Dennis and Schnabel [30]). Details of the numerical method to find the A matrix can be found in Hernandez-Ortiz and Osswald [22].

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

RESULTS

Repeatability of vulcanization was obtained for each liquid silicone rubber and heating rate, as can be seen in Tables 1-5. The data of heating rate and the temperature at which the maximum rate of reaction occurs, [T.sub.peak], was plotted, and fitted to a linear model. The activation energy, [E.sub.1], of each silicone rubber was calculated with the data from the four dynamic scanning rates tested. The slope of the line corresponds to the negative ratio of the activation energy and the universal gas constant R (8.3145 J g [mol.sup.-1] [K.sup.-1]), as can be seen in Fig. 3 and Table 6. Kelvin temperatures are used to avoid negative temperatures in the fitted model.

The data obtained from the dynamic DSC scans was used to determine the kinetic constants of the Kamal-Sourour model. This process was done by fitting the instantaneous vulcanization rate and the percentage of vulcanization at each specific temperature into a model that describes the reaction. The technique uses one or more dynamic DSC scans to determine a set of kinetic parameters that model the vulcanization process for all the heating rates tested. The experimental data and fitted Kamal-Sourour model for liquid silicone rubber are graphed together in Figs. 4-8. The fitted models are in good agreement with the DSC data. The fitted parameters are listed in Tables 7-11. The orders of the reaction m and n as well as the frequency factors [a.sub.1] and [a.sub.2] are fitted as variables depending on temperature.

[FIGURE 12 OMITTED]

The parameters found with the Kamal-Sourour model are mathematical expressions that fit well the experimental data. However, as can be seen from Tables 7-11, they do not have physical meaning. These parameters are highly temperature dependent. They model the experimental results well when the initial temperature conditions used for modeling are the same as the experimental data. However, when the fitted curves are generated at lower temperatures than the initial temperature condition of the experimental data, the fitted model is shifted from the experimental curves. This effect can be seen in Fig. 9. In addition to the susceptibility to temperature variations, the model is very sensitive to the precision of the parameters. Small changes in some parameters can have large effects on the model. This effect can be seen in Figs. 10 and 11.

Solid silicone rubbers present lower repeatability than liquid silicone rubber, because the material is pre-vulcanized up to a certain degree before storing. During storage, the rubber continues to cross-link at different proportions. The results of solid silicone rubber and one type of liquid silicone rubber were also compared. Various material samples were taken from one HSR batch and tested [23]. Due to the variation in the initial degree of cure between the samples, only one was chosen to fit the model. Figure 12 presents the fitted data and compares it to a LSR for similar applications. The fitted parameters for HSR are taken from a previous work [22] and are summarized in Table 12. It should be noted that for this fitting the Kissinger equation was not used to compute the activation energy. As can be seen, solid silicone rubber vulcanizes at higher temperatures than liquid silicone rubber leading to larger cycle times and energy costs. In addition, the vulcanization rate of liquid silicone rubber is higher than the one of hard silicone rubber. A higher reaction rate will significantly reduce the cycle time.

ACKNOWLEDGMENTS

The authors thank Julian Salguero for assisting in the experimental work. The authors also thank SIMTEC Silicone Parts for providing the material used in this research and ComProTec, Inc. for donating samples of the static mixers and cartridge.

REFERENCES

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L.M. Lopez, A.B. Cosgrove, J.P. Hernandez-Ortiz, T.A. Osswald

Polymer Engineering Center, Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706

Correspondence to: Tim A. Osswald; e-mail: osswald@engr.wise.edu

Contract grant sponsor: Partnership for Innovation (PFI), National Science Foundation.

Over the past 50 years, silicone rubbers have become increasingly popular. This thermoset elastomer maintains its mechanical and electrical properties over a wide range of temperatures, and is therefore a natural choice for everything from aerospace applications to medical devices [1, 2]. It is used for the production of seals in the automotive and aerospace industry, connectors, and cables for appliances and telecommunications, implants and devices for medical purpose, and packaging and baking pans for the food industry [3]. Silicone rubber is a family of thermoset elastomers that have a backbone of alternating silicone and oxygen atoms and methyl or vinyl side groups. The solidification process of all heat-activated cure thermosets, including silicone rubber, is dominated by an exothermic and irreversible chemical reaction called vulcanization, or cure. The vulcanization process forms a three dimensional network that improves the properties of the final product [4-6]. Two types of reactions can lead to vulcanized silicone rubber: peroxide and platinum catalyzed cross-linking [7, 8].

Silicone rubbers consist mainly of silicone polymers and fillers. They can be classified according to the polymer employed and the vulcanization process as low temperature vulcanizable rubbers (LTV or RTV) and high temperature vulcanizable rubbers (HTV). RTV silicone rubbers come as a soft paste or a viscous liquid. HTV silicone rubbers come in two different physical states: liquid and solid.

Because they exhibit a high viscosity, solid silicone rubbers (commonly referred to as high consistency silicon rubbers) are processed in the same way as normal organic rubbers. Solid silicone rubbers are formed using linear polymers with molecular weights between 400,000 and 600,000 g [mol.sup.-1]. These polymers contain an average of 6000 siloxy units and are water clear newtonian liquids with viscosities between 15,000 and 30,000 Pa s. Solid silicone rubbers are usually vulcanized using a two kinds of peroxide catalyst: aroyl-peroxides and alkyl-peroxides [7, 8]. Aroyl-peroxides are used when the vulcanization process can be accomplished without pressure and they allow high reaction rates. Alkyl-peroxides are only used for vulcanization under pressure, as they do not form carbonic acids as decomposition products [1, 8]. The scheme of the reaction is shown in Fig. 1. The reactive vinyl double group of the polymer (Fig. 1a) reacts with the oxygen free-radical, already produced in the peroxide group (Fig. 1b), during the first stage of the reaction (Fig. 1c). The vulcanization is achieved when free radicals (Fig. 1c) attach themselves to another polymer chain and forms bridges. Peroxide catalysts allow for high reaction rates. Solid silicone rubbers are preprocessed materials that are vulcanized up to a certain degree and then stored in the form of foil-wrapped bars and packed cartons. To process solid rubbers, a sheet is then cut or stamped to the required size and placed in a compression or transfer molder. After vulcanization, the part is removed from the mold, deflashed, and freed of the peroxide decomposition products in a post-curing process.

Liquid silicone rubbers (LSR) have the same structure as solid silicone rubbers. However, the chain length of the polydimethylsiloxane used for LSR is lower by a factor of about 6. Therefore, the viscosity of the polymer is reduced by a factor of about 1000 [1-3, 7]. The vulcanization of liquid silicone rubber is almost exclusively carried out with a platinum-catalyzed hydrosilylation reaction, depicted in Fig. 2, in a reaction that does not generate by-products. Similar to solid silicone rubber, the vinyl double bond on the polymer (Fig. 2a) interacts with the platinum center, which has on free coordination site (Fig. 2b), and activates the double bond. The vinyl group cross-links by transforming the double bond, creating a single bond to a polymer chain; in this case, to a crosslinker molecule containing Si-H groups (Fig. 3c). The catalyst becomes free and is again available for further cross-linking. Liquid silicone rubbers are supplied in barrels or hobbocks. Because of their low viscosity, these rubbers can be pumped through pipelines and tubes to the vulcanization equipment. The two components (A and B component) are pumped to a static mixer by a metering machine were the vulcanization process may start taking place; loss of material in the feed lines is avoided using cold runners [8].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The individual steps of the vulcanization process of silicone rubber can be found in literature [1-4, 7], but none have a kinetic model that describes the cross-linking process. The vulcanization process can be described as the reaction between two chemical groups denoted by A and B which link two segments of a polymer chain. The reaction can be followed by tracking the concentration of unreacted As or Bs, [C.sub.A] or [C.sub.B]. The degree of cure can be defined by [4, 5],

[FIGURE 3 OMITTED]

c = [[C.sub.A,0] - [C.sub.A]]/[C.sub.A,0] (1)

where [C.sub.A,0] is the initial concentration of A. The degree of vulcanization is zero when there has been no reaction and equals 1 when all As have reacted and the reaction is complete. It is difficult to physically monitor reacted and unreacted As and Bs during the reaction without costly spectroscopy analysis. However, the degree of vulcanization can be related to the heat released during the reaction because the vulcanization of elastomers is an exothermic process. The energy released in the exothermal reaction is proportional to the cross-linked bonds formed, as it is assumed that each bond releases the same amount of energy [4-6]. The heat released during vulcanization can be calculated by,

Q = [[integral].sub.0.sup.[tau]] [dot.Q]dt (2)

where Q is the heat released up to time [tau] and [dot.Q] is the instantaneous rate of heat released by the sample. The total heat of reaction, [Q.sub.T], is therefore equal to,

[Q.sub.T] = [[integral].sub.0.sup.[[tau].sub.final]] [dot.Q]dt (3)

where [[tau].sub.final] is the time at which the reaction is complete. The reaction rate, dc/dt, is then calculated as,

[dc/dt] = [dot.Q]/[Q.sub.T]. (4)

The degree of vulcanization is obtained by integration of Eq. 4,

c = Q/[Q.sub.T] (5)

Kinetic models relate the reaction rate to temperature and degree of reaction. Two different approaches can be used to characterize a vulcanization reaction, mechanistic and phenomenological [9]. The first one studies the cross-linking process as a series of individual reactions and models each of them. The phenomenological approach considers the vulcanization as a whole process. This paper uses the phenomenological approach to model the kinetics of vulcanization for different types of silicone rubbers, using Kissinger [10, 11] and Kamal-Sourour models [12, 13].

The challenge of modeling the vulcanization is to take the DSC data and fit models that describe this process. With this information, time-temperature-transformation diagrams [14-16], for each resin, can be obtained and processes that involve vulcanization can be simulated [17-21]. Typically, isothermal DCS scans were used to fit the models that describe the curing reactions [20, 21]. However, several problems arise in the isothermal experiments. First, when performing isothermal scans it is challenging to reach the desired temperature as fast as possible. Second, at realistic processing temperatures, that is, 150[degrees]C for LSR, the reaction rate is too fast to be accurately captured in the measurement. One misses a great part of the reaction by the time the isothermal temperature is reached. Hence, one is left to perform the isothermal tests at temperatures that are significantly lower, that is, 40-60[degrees]C lower, that the processing temperature. This results in models where the process simulation is a gross extrapolation of DSC test results. In addition, all the techniques that use isothermal DSC data solved the fitted constants for each individual temperature, and later used a cumbersome fit to determine the temperature dependence [16, 20, 21].

Hernandez-Ortiz and Osswald [22] developed a technique to use the Kamal-Sourour model to fit dynamic DSC data. The kinetic parameters were expanded with a quadratic dependence on temperature. These fitted parameters do not necessarily have physical meaning; they were simply mathematical expressions that fit the experimental data, which can be used directly in numerical simulations.

In this paper, the Kissinger model is used in conjunction with the Kamal-Sourour in order to give more insight into the reaction and find a physically meaningful value for the activation energy. The use of the combined models results in a more realistic and robust model for the reaction. The Kissinger model is used to determine the activation energy of the silicone rubber from the experimental data. The other parameters of the reaction are determined with the Kamal-Sourour model, using the same methodology proposed by Hernandez-Ortiz and Osswald [22, 23]. The kinetic parameters are used to follow the degree of vulcanization as time and temperature change.

EXPERIMENTAL PROCEDURES

Liquid silicone rubber (LSR) is a two-component system. Component A contains a platinum catalyst and component B contains methylhydrogensiloxane for cross-linking, and an alcohol inhibitor [8, 23]. The A and B components of the liquid silicone rubbers were stored in a two component cartridge. Cartridges were placed at room temperature with room humidity of 35% in a dark cabinet for storage. The two components were mixed in a 1:1 ratio using a static mixer with 13 mixing elements.

Five types of liquid silicone rubber, denoted as LSR-a, LSR-b, LSR-c, LSR-d, and LSR-e, and one type of solid silicone rubber, expressed as HRS, were analyzed via thermal analysis to determine the progression of the vulcanization processes. The specific properties being studied were peak temperature, heat of reaction, and extent of reaction.

Differential scanning calorimeter [24] equipment manufactured by Netzsch (Phox DSC 200 PC) was used to measure the heat of reaction for the samples. Sealed aluminum pans were used to analyze all reactions. The mass of the samples ranged from 10 to 30 mg. A sealed empty pan was used as a reference.

The total heat of reaction was measured by a dynamic scan from 20 to 150[degrees]C using heating rates of 1.0, 2.5, 5.0, and 10.0 K/min. Multiple scanning rates were used to gain insight into the effect of time and temperature on the vulcanization reaction. Repeatability was obtained for each heating rate. All experiments were performed under nitrogen purge.

CURE KINETICS MODEL

Kinetic models relate the reaction rate to temperature and extent or degree of reaction. To study the cross-linking mechanism of elastomers and thermosets resins, two distinct approaches are used: phenomenological and mechanistic [5, 6, 9]. The phenomenological approach studies the reaction from a macroscopic level, considering the reaction as a whole process. The mechanistic approach observes the cross-linking process from a microscopic level and studies the reaction as a series of individual steps.

[FIGURE 4 OMITTED]

Mechanistic models are based on stoichiometric balances of the reactants involved in the elementary reactions, and the mechanics of each reaction. Therefore these models are more difficult to obtain but they can represent the kinetics of vulcanization better than the phenomenological models. Phenomenological models are semiempirical and do not provide a clear description of the vulcanization process and the chemistry behind it. These models are simpler to compute and can be widely used to describe the cross-linking process of different resin systems. They are based on the rate of reaction as a function of the amount of reacted resin and the rate constants, as described by [4-6, 9],

dc/dt = k(T)f(c) (6)

where c is the degree of vulcanization, k is the rate constant as a function of temperature, and f(c) is a function representing the amount of reacted resin. The rate constant is calculated with an Arrhenius equation. Different expressions for f(c) can be found in the literature but the most common are the nth-order reaction and the autocatalytic reaction.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

Arrhenius Model

The Arrhenius model is used to describe the kinetics of an activated process. This model assumes that an energy barrier hinders the progress of the reaction. Forward progress of the reaction requires an activation energy to be supplied in order to overcome this barrier. The rate constant can be defined by an Arrhenius model, as follows,

k(T) = a exp (- [E/RT]) (7)

where a is the frequency factor that indicates how many collisions have the correct orientation to lead to products, E is the activation energy, T is the processing temperature in Kelvin, and R is the universal gas constant.

Kissinger Model

The Kissinger model [10, 11] uses an nth-order expression to model the kinetic reaction, defined as

dc/dt = k(1 - c)[.sup.n] (8)

where k is the rate constant defined by the Arrhenius equation, c is the extent of vulcanization, and n is the order of the reaction. The three parameters of the model (a, E, and n) can be fitted to the experimental DSC data.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Heat-activated reactions show a variation of the position of the peak with varying heating rate. If the temperature rises during reaction, the reaction rate, dc/dt, will rise to a maximum value and then return to zero as the reactant is consumed. The temperature at which the reaction rate is the maximum is also the temperature of maximum deflection in differential thermal analysis. Dynamic DSC measurements at different heating rates are used to determine the activation energy of the material, because they show the effects both time and temperature on the reaction [10, 11].

The maximum rate of reaction occurs when the derivative of dc/dt is zero. If the temperature is raised at a constant rate, [dot.T], the maximum rate of reaction can be defined as

[d/dt](dc/dt) = [dc/dt]([[E[dot.T]]/[R[T.sup.2]]] - an(1 - c)[.sup.n] exp(- [E/RT])) (9)

which is obtained by differentiating Eq. 8 and using k from Eq. 7. The maximum rate of reaction occurs at the peak temperature of the heat flow from the thermal analysis, [T.sub.peak]. The peak temperature can be defined by setting Eq. 9 to zero, that is,

[E[dot.T]]/[R[T.sup.2]] = an(1 - c)[.sub.peak.sup.n-1] exp (- [E/[R[T.sub.peak]]]). (10)

The amount of unreacted material (1 - c)[.sub.peak] is not determined directly through DSC analysis. Integration of Eq. 8, assuming a constant heating rate, gives an expression for the extent of reaction as a function of temperature. Approximations of this expression were obtained by Murray and White [25] through integration by parts. The expression for Eq. 8 after using the Murray and White approximations is given as [25],

[1/[n - 1]]([1/[(1 - c).sup.n-1]] - 1) = [[aR[T.sup.2]]/[E[dot.T]]](1 - [2RT/E])exp(- [E/RT]). (11)

The expression for the amount of unreacted material at [T.sub.peak] for reaction orders other than zero or unity, can be simplified to

n(1 - c)[.sub.peak.sup.n-1] = 1 + (n - 1) [[2R[T.sub.peak]]/E]. (12)

An expression for the activation energy can be found by substituting Eq. 12 into Eq. 9 and omitting negligible quantities

-[E/R] = [d(ln[T/[T.sub.peak.sup.2]])]/[d (1/T)]. (13)

The activation energy can be easily calculated by performing DSC tests at different scanning rates regardless of the order of the reaction.

Kamal-Sourour Model or Autocatalytic Model

The Kamal-Sourour reaction model is one of the first models to accurately define autocatalytic cross-linking reactions; it is defined as [12, 13],

dc/dt = ([k.sub.1] + [k.sub.2][c.sup.m]) (1 - c)[.sup.n] (14)

where [k.sub.1] and [k.sub.2] are the rate constants described by the Arrhenius equation, and m and n are the reaction orders.

The six parameters in the model ([a.sub.1], [a.sub.2], [E.sub.1], [E.sub.2], m, and n) can be fit to the experimental DSC data via a least-squares estimation algorithm developed by Marquardt [26, 27]. Karkanas et al. [28] found that the Kamal-Sourour model accurately predicts the degree of cure during early stage of reaction; however, at the later stage, it over-predicts the degree of cure [20, 21]. This deviation is due to the diffusion effects. Thus, it is necessary to modify the Kamal-Sourour model to include the diffusion-limited part of the reaction. However, for elastomeric materials it is not necessary to modify the constants of the reaction due to diffusion, because the materials vulcanize above their glass transition temperatures. For materials that vulcanize before the vitrification point, the model accurately predicts the total process. This condition gives sufficient free volume between the molecules to allow for freedom of movement during the molecular cross-linking process, and therefore diffusion does not play a large role [4, 5].

MODEL DESCRIPTION AND NUMERICAL ALGORITHM

The Kamal-Sourour diffusion model has six parameters that need to be fitted. Let x be the unknown vector parameter defined as,

x = {m n [a.sub.1] [E.sub.1] [a.sub.2] [E.sub.2]}. (15)

The proposed technique uses one or more dynamic DSC scans, avoiding the problems that arise by using isothermal DSC data. The activation energy [E.sub.1] is determined through Kissinger's model, which is then used to mathematically determine the other five parameters for the Kamal-Sourour model. This novel technique finds a physically meaningful value of [E.sub.1]. The activation energy, [E.sub.2], is fitted as a constant. The four remaining parameters (m, n, [a.sub.1], [a.sub.2]) in the cure model are expanded into a power series of the temperature [22], that is,

[FIGURE 9 OMITTED]

[x.sub.i] = [a.sub.i1] + [a.sub.i2]T + [a.sub.i3][T.sup.2] + O([T.sup.3]) (16)

where i = 1,..., 4 and [a.sub.ij] are the new goal of the fitting. In other words, instead of looking for the six parameters using isothermal data, the new technique looks for the 13 components of the A matrix defined in such a way that,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

According to Eq. 17, the higher terms in the expansion are neglected; accordingly the fitting must be performed in such a way that the coefficient accompanying the second order term is small. If this condition does not satisfy a specific set of data, the expansion cannot assure that the higher order terms are small; consequently, higher order terms must be included in the expansion (i.e. third, fourth). The parameters in the model are fitted via a least-squares estimation algorithm developed by Marquardt [26, 27] (see also Press et al. [29] or Dennis and Schnabel [30]). Details of the numerical method to find the A matrix can be found in Hernandez-Ortiz and Osswald [22].

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

RESULTS

Repeatability of vulcanization was obtained for each liquid silicone rubber and heating rate, as can be seen in Tables 1-5. The data of heating rate and the temperature at which the maximum rate of reaction occurs, [T.sub.peak], was plotted, and fitted to a linear model. The activation energy, [E.sub.1], of each silicone rubber was calculated with the data from the four dynamic scanning rates tested. The slope of the line corresponds to the negative ratio of the activation energy and the universal gas constant R (8.3145 J g [mol.sup.-1] [K.sup.-1]), as can be seen in Fig. 3 and Table 6. Kelvin temperatures are used to avoid negative temperatures in the fitted model.

The data obtained from the dynamic DSC scans was used to determine the kinetic constants of the Kamal-Sourour model. This process was done by fitting the instantaneous vulcanization rate and the percentage of vulcanization at each specific temperature into a model that describes the reaction. The technique uses one or more dynamic DSC scans to determine a set of kinetic parameters that model the vulcanization process for all the heating rates tested. The experimental data and fitted Kamal-Sourour model for liquid silicone rubber are graphed together in Figs. 4-8. The fitted models are in good agreement with the DSC data. The fitted parameters are listed in Tables 7-11. The orders of the reaction m and n as well as the frequency factors [a.sub.1] and [a.sub.2] are fitted as variables depending on temperature.

[FIGURE 12 OMITTED]

The parameters found with the Kamal-Sourour model are mathematical expressions that fit well the experimental data. However, as can be seen from Tables 7-11, they do not have physical meaning. These parameters are highly temperature dependent. They model the experimental results well when the initial temperature conditions used for modeling are the same as the experimental data. However, when the fitted curves are generated at lower temperatures than the initial temperature condition of the experimental data, the fitted model is shifted from the experimental curves. This effect can be seen in Fig. 9. In addition to the susceptibility to temperature variations, the model is very sensitive to the precision of the parameters. Small changes in some parameters can have large effects on the model. This effect can be seen in Figs. 10 and 11.

Solid silicone rubbers present lower repeatability than liquid silicone rubber, because the material is pre-vulcanized up to a certain degree before storing. During storage, the rubber continues to cross-link at different proportions. The results of solid silicone rubber and one type of liquid silicone rubber were also compared. Various material samples were taken from one HSR batch and tested [23]. Due to the variation in the initial degree of cure between the samples, only one was chosen to fit the model. Figure 12 presents the fitted data and compares it to a LSR for similar applications. The fitted parameters for HSR are taken from a previous work [22] and are summarized in Table 12. It should be noted that for this fitting the Kissinger equation was not used to compute the activation energy. As can be seen, solid silicone rubber vulcanizes at higher temperatures than liquid silicone rubber leading to larger cycle times and energy costs. In addition, the vulcanization rate of liquid silicone rubber is higher than the one of hard silicone rubber. A higher reaction rate will significantly reduce the cycle time.

ACKNOWLEDGMENTS

The authors thank Julian Salguero for assisting in the experimental work. The authors also thank SIMTEC Silicone Parts for providing the material used in this research and ComProTec, Inc. for donating samples of the static mixers and cartridge.

REFERENCES

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L.M. Lopez, A.B. Cosgrove, J.P. Hernandez-Ortiz, T.A. Osswald

Polymer Engineering Center, Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706

Correspondence to: Tim A. Osswald; e-mail: osswald@engr.wise.edu

Contract grant sponsor: Partnership for Innovation (PFI), National Science Foundation.

TABLE 1. Repeatability of vulcanization of LSR-a. Heating rate Sample [T.sub.peak] Total heat of (K [min.sup.-1]) mass (mg) ([degrees]C) reaction (J [g.sup.-1]) 1.0 19.2 94.9 3.887 1.0 13.3 94.7 3.824 2.5 23.0 102.0 3.856 2.5 18.6 101.3 3.799 5.0 13.9 107.9 3.847 5.0 11.9 108.3 3.826 10.0 15.8 115.8 3.861 10.0 23.4 115.8 3.740 Average 3.830 Standard deviation 0.040 TABLE 2. Repeatability of vulcanization of LSR-b. Heating rate Sample [T.sub.peak] Total heat of (K [min.sup.-1]) mass (mg) ([degrees]C) reaction (J [g.sup.-1]) 1.0 22.6 97.9 4.054 1.0 18.9 97.8 4.034 2.5 18.9 105.8 4.078 2.5 21.6 105.9 4.071 5.0 19.4 112.0 4.044 5.0 16.1 112.2 4.022 10.0 23.1 119.3 4.034 10.0 15.1 119.3 4.043 Average 4.050 Standard deviation 0.020 TABLE 3. Repeatability of vulcanization of LSR-c. Heating rate Sample [T.sub.peak] Total heat of (K [min.sup.-1]) mass (mg) ([degrees]C) reaction (J [g.sup.-1]) 1.0 13.4 101.0 7.344 2.5 19.7 108.7 7.060 2.5 11.9 108.3 7.391 5.0 16.9 115.8 6.780 5.0 14.2 115.8 7.308 10.0 12.4 122.8 7.105 10.0 11.6 122.7 7.121 Average 7.158 Standard deviation 0.211 TABLE 4. Repeatability of vulcanization of LSR-d. Heating rate Sample [T.sub.peak] Total heat of (K [min.sup.-1]) mass (mg) ([degrees]C) reaction (J [g.sup.-1]) 1.0 17.3 91.9 7.038 2.5 8.4 99.9 7.032 2.5 13.3 99.9 7.036 5.0 12.1 105.6 7.086 5.0 13.5 105.6 7.060 10.0 13.9 115.8 7.042 10.0 16.8 116.0 7.042 Average 7.048 Standard deviation 0.019 TABLE 5. Repeatability of vulcanization of LSR-e. Heating rate Sample [T.sub.peak] Total heat of (K [min.sup.-1]) mass (mg) ([degrees]C) reaction (J [g.sup.-1]) 1.0 16.3 98.7 3.834 2.5 21.2 106.2 4.010 2.5 18.5 105.6 3.930 5.0 15.3 110.2 3.706 5.0 18.9 110.3 3.756 10.0 19.9 116.6 3.820 10.0 20.6 116.6 3.723 Average 3.826 Standard deviation 0.112 TABLE 6. Activation energy [E.sub.1] with Kissinger model. Silicone reference [E.sub.1] (J [g.sup.-1] [mol.sup.-1]) LSR-a 124850.53 LSR-b 123886.05 LSR-c 119662.28 LSR-d 110932.06 LSR-e 150134.92 TABLE 7. Fitted parameters for LSR-a. Parameter O(I) O(T) O([T.sup.2]) M 1.5533E + 02 -0.7588E + 00 9.3349E - 04 N 1.7499E + 02 -0.8911E + 00 1.1426E - 03 [a.sub.1] 1.6554E + 15 2.8889E + 12 -1.8655E + 10 [a.sub.2] -5.8582E - 05 2.4736E - 07 -2.3529E - 10 [E.sub.2] -36800.93 0 0 [E.sub.1] 124850.53 0 0 TABLE 8. Fitted parameters for LSR-b. Parameter O(I) O(T) O([T.sup.2]) M -1.8453E + 02 1.0083E + 00 -1.3623E - 03 N -1.2026E + 02 0.6317E + 00 -8.2040E - 04 [a.sub.1] 3.9820E + 14 -1.9886E + 10 -2.4269E + 09 [a.sub.2] -5.4154E + 08 2.8439E + 06 -3.7228E + 04 [E.sub.2] 50873.69 0 0 [E.sub.1] 123886.05 0 0 TABLE 9. Fitted parameters for LSR-c. Parameter O(I) O(T) O([T.sup.2]) M 1.2687E + 00 2.0867E - 02 -5.1606E - 05 N 1.1561E + 01 -5.1847E - 02 6.4931E - 05 [a.sub.1] 8.9623E + 15 -4.7584E + 13 6.3233E + 10 [a.sub.2] -2.8385E + 03 1.2438E + 00 -1.2693E - 03 [E.sub.2] 14019.46 0 0 [E.sub.1] 119662.28 0 0 TABLE 10. Fitted Parameters for LSR-d. Parameter O(I) O(T) O([T.sup.2]) M -2.0297E + 01 1.4441E - 01 -2.2472E - 04 N -8.3737E + 01 4.5245E - 01 -6.0081E - 04 [a.sub.1] 1.5053E + 03 5.8359E + 04 2.1521E + 07 [a.sub.2] -1.2951E + 06 6.8422E + 03 -9.0071E + 00 [E.sub.2] 31023.05 0 0 [E.sub.1] 110932.06 0 0 TABLE 11. Fitted Parameters for LSR-e. Parameter O(I) O(T) O([T.sup.2]) M 1.8749E + 01 5.6195E - 02 -2.6206E - 04 N 8.7104E + 00 1.8726E - 02 -9.6343E - 05 [a.sub.1] -1.9259E + 21 1.0250E + 19 -1.3634E + 16 [a.sub.2] 2.9573E + 03 -1.4983E + 01 1.8986E - 02 [E.sub.2] 6978.32 0 0 [E.sub.1] 150134.92 0 0 TABLE 12. Fitted Parameters for HSR [22]. Parameter O(I) O(T) O([T.sup.2]) M 8.4455E + 01 -5.5926E - 01 9.0122E - 04 N 2.0456E + 01 -1.0317E - 01 1.2983E - 04 [a.sub.1] 1.6682E - 22 -7.1679E - 21 1.8524E - 23 [a.sub.2] -1.5320E + 03 6.6209E + 00 -7.1566E - 03 [E.sub.1] 1.1447E + 05 -6.5515E + 02 2.1787E - 01 [E.sub.2] -2.2411E + 05 -1.4717E + 01 1.0666E + 00

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Author: | Lopez, L.M.; Cosgrove, A.B.; Hernandez-Ortiz, J.P.; Osswald, T.A. |
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Publication: | Polymer Engineering and Science |

Geographic Code: | 1USA |

Date: | May 1, 2007 |

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