Nonisothermal Degradation of Zetaplus Impression Material: Kinetic

Jul 13, 2009 - It was found that the first degradation stage of Zetaplus impression material can be described by the three-dimensional diffusion (D3) ...
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Ind. Eng. Chem. Res. 2009, 48, 7044–7053

Nonisothermal Degradation of Zetaplus Impression Material: Kinetic Aspects Bojan Jankovic´* Faculty of Physical Chemistry, UniVersity of Belgrade, Studentski trg 12-16, P.O. Box 137, 11001 Belgrade, Serbia

Kinetic analysis of the nonisothermal degradation process of Zetaplus impression material has been performed. Kinetic data for the investigated degradation process were obtained from the thermogravimetric (TG) and differential thermal analysis (DTA) measurements conducted at the three different heating rates (β ) 10, 20, and 25 °C min-1) in an air atmosphere. For determination of the kinetic model function (f(R) or g(R)), the isoconversional and Master-plot methods were used. It was found that the first degradation stage of Zetaplus impression material can be described by the three-dimensional diffusion (D3) and with the following values of kinetic parameters: Ea ) 99.5 kJ mol-1 and A ) 1.05 × 108 min-1. Kinetic analysis shows that the first degradation stage corresponds to the bond cleavages and polydimethylsiloxane thermal volatilization, which is contolled by the rate of diffusion and evaporation of oligomers produced on decomposition. The observed second degradation step is more complex; it includes two parallel independent reactions with different values of the apparent activation energy. By applying the nonlinear regression (NLR) method, it was found that the second degradation stage can be described by two independent parallel first-order reactions (nth-order reaction model). 1. Introduction The construction of the usual oral appliances (such as dentures, crowns, inlays, bridges, etc.) requires the preparation of a model of the patient’s oral tissues. A great number of impression materials have therefore been developed including hydrocolloids and elastomeric or nonelastomeric materials, and many authors have studied their application and properties.1,2 The general requirement for an impression material concerns its working characteristics and compatibility with oral tissues. On the other hand, the dimensional stability (short and longterm) of the impression is of primary importance for the acceptance of a material.3 Major advances have occurred in the past decade in the area of elastic impression materials, probably the most important being the development of the addition and condensation silicone systems.4-8 The conventional silicone impression materials are also known as condensation reaction silicones.9 The condensation silicones have more shrinkage on setting than other rubber impression materials. Their dimensional stability is less than that of polysulfide although greater than that of reversible hydrocolloid. Condensation silicone and polysulfide have a dimensional instability that is due to their mode of polymerization.10,11 Silicone impression materials are supplied in viscosities labeled as light-, regular-, and heavy-bodied and silicone putty.12 Silicones are macromolecular substances with chains composed of SidO groups and alcil radicals (methyl, vinyl, halogen, or hydrogen groups).13,14 In the medical and pharmaceutical industries, the term “silicone” typically encompasses materials based on the dimethylsiloxane structure. The term “silicones” was first coined due to the structural resemblance between R2SiO and “ketones” (R2CO). However, the SidO bond is very unstable, unlike the CdO bond in ketones. The term “polysiloxane” was derived from the structure of the repeating unit in the polymer backbone (SisOsSi). These polymers can be considered as intermediate chemical structures between organic and inorganic compounds, specifically between silicates and * To whom correspondence should be addressed. Phone/Fax: ++381-11-2187-133. E-mail: [email protected].

organic polymers. Two types of silicones are recognized, depending on the chemical reaction during polymerization, addition, and condensation types. Condensation-type silicones are polydimethylsiloxanes with OH groups in terminal positions. When mixed with a catalyst (tetrafunctional ethylsilicate), they react with terminal OH groups of the silicone chain, with the separation of an alcohol (evaporates) and other chains entering into chemical reaction until a three-dimensional net is finally formed. On the other hand, the cross-linking of the addition type silicones is based on their unsaturated (vinyl) side groups. The setting of these materials includes an addition reaction of the double bond (promoted by several special catalysts), and no byproducts are produced.15 Thermal analysis methods are often used for determination of the degradation processes that take place in the progressive heating of a polymer or a polymeric material as well as for evaluation of the kinetic parameters of each degradation step. The kinetic parameters can be correlated with the thermal or thermo-oxidative stability of the investigated material and/or can be used for prediction of the degradation behavior of the material in various conditions (i.e., other heating rates than those used for the evaluation of kinetic parameters; isothermal conditions). In general, the degradation mechanisms experienced by polymers are free-radical processes initiated by bond dissociation at the pyrolysis temperature. The specific pathway followed by a particular polymer is related to the relative strength of the polymer bonds and the structure of the polymer chain. These mechanisms are generally grouped into three categories: random scission, unzipping, and side-group elimination. The aim of this study was to observe the degradation kinetics under nonisothermal conditions of high viscosity condensationtype silicone material (Zetapluss“putty” silicone), in terms of kinetic parameters (apparent activation energy (Ea), preexponential factor (A)) and reaction mechanism function (f(R) or g(R)). For estimation of kinetic parameters and mechanism function of the investigated process, the isoconversional (“modelfree”) method16 and the Master-plot method17 were used. To check the established kinetic results of nonisothermal degrada-

10.1021/ie900104b CCC: $40.75  2009 American Chemical Society Published on Web 07/13/2009

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Figure 1. TG curves (mass (%) versus T (°C)) of the nonisothermal degradation of Zetaplus impression material recorded in air (solid line) and in nitrogen (N2) (dotted line) at a heating rate of 20 °C min-1.

tion of the Zetaplus impression material, the nonlinear leastsquares regression method18 was used. 2. Experimental Section 2.1. Materials and Methods. The silicone impression material (ZetaplussZhermack SpA, 100 45021 Badia Polesine (RO), Italy), the condensation-type putty body silicone, was studied. According to ISO-4823,19 the investigated material belongs to the type Ishigh viscosity impression silicone materialsscategory. The green color Zetaplus packed impression sample was obtained for the thermal measurements. The experimental samples for thermogravimetry (TG) and differential thermal analysis (DTA) measurements were cut into small pieces of an approximately cubic shape. Simultaneous differential thermal analysis (DTA) and thermogravimetric analysis (TGA) were performed on a STA-1000 simultaneous thermal analyzer, Stanton Redcroft, UK. The Zetaplus samples with approximately m ≈ 13.23 ( 0.05 mg were heated at heating rates of β ) 10, 20, and 25 °C min-1 in an air atmosphere (air flow φ ) 20 mL min-1) in the temperature range of 20-1000 °C. The heating rate is an important factor in thermal analysis, because it influences the shape of the thermoanalytical curves. Low heating rates (5-10 °C min-1) favor the separation of different steps in the TG curves and thus improve the accuracy of the quantitative determination. High heating rates (20-25 °C min-1), on the other hand, improve the definition of the peaks in the DTA curves. This fact has led us to prefer the second alternative because it has the best chance for the characterizations. An added bonus of the high heating rate is that almost twice as many analyses may be carried out at the same time, a decisive factor when the number of samples is very large. Typical mass loss (TG) and derivative of mass loss (DTG) curves of Zetaplus impression material at a heating rate of 20 °C min-1 under air and N2 atmospheres are shown in Figures 1 and 2. In the DTG curves, the temperature of maximum rate of mass loss or the peak temperature (Tp) is designated. It can be seen from Figure 1 that the TG curve of the Zetaplus degradation process in N2 follows the same trend as the TG curve in air up to approximatelly 352 °C. After this temperature, the slope of TG curve in N2 is changed in comparison with slope of the TG curve in an air atmosphere. Also, the total mass loss is a little higher for the TG curve in N2 than the total mass loss for the TG curve in air (Figure 1). The TG curves of the Zetaplus degradation process for both, air and nitrogen atmospheres, indicate two reaction stages (Figure 1), which are reflected as double peaks in the DTG curves (Figure 2). The

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Figure 2. DTG curves of the nonisothermal degradation of Zetaplus impression material recorded in air (solid line) and in nitrogen (N2) (dotted line) at a heating rate of β ) 20 °C min-1. The corresponding temperature at the maximum rates of mass loss Tp1 and Tp2 for the individual degradation stages were determined from the peaks of the appropriate differential thermogravimetric (DTG) curves.

difference in peak temperature (Tp) of the DTG curves between the two atmospheres is 0 °C for the first and 47.81 °C for the second degradation stage, respectively (Figure 2). Figure 2 shows that the DTG peaks for the first degradation stage in both considered atmospheres have similar wides and shapes. On the other hand, the DTG peak for the second degradation stage in the nitrogen atmosphere is much wider than the corresponding DTG peak in the air atmosphere, while the Tp2 value is shifted to the higher temperature range (511.31 °C). This behavior showed that the reaction atmosphere had an important effect on the mechanism of thermal and oxidative degradation of Zetaplus impression material. In the present investigation, the nonisothermal degradation kinetics of Zetaplus impression material was studied by TG and DTA in an air atmosphere. 3. Kinetic Analysis Though in many cases the thermal and thermo-oxidative degradation of a polymer or a polymeric material is a complex process involving successive and/or parallel steps, the kinetic analysis of nonisothermal data is generally performed by using a single step kinetic equation:20

( )

Ea dR dR ≡β ) A exp f(R) dt dT RT

(1)

where R is the degree of conversion, t is the time, T is the temperature, β is the constant heating rate, A is the preexponential factor, Ea is the apparent activation energy, R is the gas constant, and f(R) is the differential conversion function (kinetic model). The degree of conversion (R) may be defined as the ratio of the actual mass loss to the total mass loss corresponding to a given step of the degradation process (R ) (m0 - mT)/(m0 - mf) where m0, mT, and mf are the initial, actual, and final masses of the sample, respectively). The use of eq 1 supposes that a kinetic triplet (A, Ea, and f(R)) describes the time evolution of a physical or a chemical change. Starting from eq 1, various methods of kinetic triplet evaluation were developed, and some recent papers21-24 contain critical analyses of these methods. Especially, such analyses showed the importance of the isoconversional (“model-free”) methods, which require R-T curves to be recorded at several heating rates. These methods allow the dependence of the apparent activation energy on the conversion to be obtained. If Ea does not depend on R, the

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investigated process is simple and can be described by a unique kinetic triplet. If Ea changes with R, the process is complex (multistep reaction), and the shape of the Ea ) Ea(R) curve offers indications concerning the reaction mechanism. In some recent papers,25-27 the dependencies of the apparent activation energy on the degree of conversion are reported for the degradation of some polymeric materials. As was mentioned above, the dependence of Ea on R shows that the investigated process is a complex one (which may include the consecutive, parallel, and reversible reactions). Consequently, the rate equation is more complicated than eq 1. 3.1. Isoconversional (Model-Free) Approach. Rearranging eq 1 and integrating both sides of the equation leads to the following expression: A g(R) ) β

( )

Ea A exp dT ≈ T0 RT β

∫

T

∫

T

0

( ) ( )

Ea exp dT ) RT AEa p(x) βR

(2)

where p(x) ) -∫∞x (exp(-x)/x2) dx and x ) Ea/RT. The degradation reactions are very slow at the subambient temperatures, so that the lower limit of the integral on the right-hand side of eq 2 (T0), can be approximated to zero. p(x) also called the “temperature integral” cannot be analytically integrated. In this study, to solve this problem, an approximate formula reported by Wanjun et al.28 is introduced into eq 2, and taking the logarithms of both sides, eq 3 is obtained in the following form:

(

ln

β T

1.894 661

) [ ]

AEa + 3.635 041 - 1.894 661 ln Ea ) ln Rg(R) Ea (3) 1.001 450 RT

where g(R) represents the integral conversion function. Equation 3 is the basis of the isoconversional (model-free) method for the nonisothermal data.28 Assuming that the kinetic model, g(R), must be invariant for all considered runs, analysis of TA measurements related to a given degree of conversion (R) at the different heating rates allows us to evaluate the apparent activation energy without knowledge of the true g(R) function. If this basic assumption is not fulfilled, an apparent Ea value would be calculated, which differs from the actual value. Analysis of the invariance of the apparent activation energy value will provide important clues about the reaction mechanism.29 Normally, a constant Ea value is assumed in the case of a single step reaction. 3.2. Determination of Kinetic Model (f(r)) and Preexponential Factor (A). The objective of the kinetic modeling lies in the determination of an f(R) (or g(R)) function, which must fulfill the mathematical requirement of eq 1. The simple kinetic models are mostly derived on the basis of a formal physical description of a single reaction mechanism (physicalgeometrical assumptions of regularly shaped bodies). However, it is rational to assume that these numerical models may accommodate the complex solid-state reactions with multiple steps using an appropriate deconvolution method where each isolated region is controlled by its own rate-limiting reaction. An appropriate deconvolution of the experimental DTG curves allows the range for R to be restricted within regions corresponding to the kinetics for each individual stage and therefore the complexity of the process to be readily discovered. Hence, within these limits, the derivatives for the individual stage can be distinguished from each other, and the overall kinetic curves

coincide with one of the corresponding partial kinetic curves. Each mechanism may be therefore individually treated by the specific reaction models. The f(R) function can be extracted from both the knowledge of Ea and the recorded TG data, using the method introduced by Ma´lek et al.30 The authors described a reference equation based on the rate of process, dR/dt, which is derived in the form of a special factor y(R) proportional to the f(R) function by simply plotting the term dR/dt exp(x) (where x ) Ea/RT) as a function of R: y(R) )

( dRdt )exp(x) ) Af(R)

(4)

According to the shape of the as-plotted curves, the f(R) function for the related mechanism may be deduced. However, it is important to mention here that the kinetic parameters can deviate from the true values as a simple mathematical consequence of eq 1 due to the complexity of the process. Therefore, TG curves cannot be only described by the f(R) function.31 According to the expected complexity of the investigated degradation process, we can considered the Master-plot method as an alternative route to eq 4. The Master-plot method consists of the introduction of an empirical function z(R) defined by eq 5 containing the smallest possible number of constancy leading to some flexibility sufficient to describe the real process as closely as possible, and the following equation is valid: z(R) )

T dR π(x) ) f(R)g(R) β dt

( )

(5)

where π(x) is an approximation of the temperature integral. The y(R) and z(R) functions display each maximum Ry and R∞z values for corresponding kinetic models.30 Because the y(R) and z(R) functions should be invariable with respect to the temperature or heating rate, the combination of the curve shape with the values of Ry and R∞z allows the most appropriate kinetic model to be determined for each individual degradation stage. It should be mentioned that under dynamic conditions, the shapes of y(R) and z(R) strongly depend on Ea. As a consequence, such functions should be normalized within the (0,1) conversion range. To provide a complete kinetic view, the reliability of ln A was also checked using an alternative method based on the second derivative of R (second derivative method,30 eq 6). The pre-exponential factor A (eq 6) can be obtained for every mechanism, from the derivative of f(R) function, f′(R), assuming the validity of f(R):30 A)-

βxp exp(xp) f'(Rp)Tp

(6)

where Tp represents the peak temperature on the DTG or DTA curve and xp ) Ea/RTp. In addition, the pre-exponential factor can be calculated from eq 2, if the integral kinetic model is properly selected (g(R)). Taking into account the true expression for g(R), experimental data, and the average apparent activation energy value which can be introduced into eq 2, the following equation was obtained: ln

[ ]

βR - ln[p(x)] ) ln A - ln[g(R)] Ea

(7)

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Figure 3. TG curves (mass (%) versus T (°C)) for the nonisothermal degradation process of Zetaplus impression material in an air atmosphere at three different heating rates: (a) 10, (b) 20, and (c) 25 °C min-1.

A group of lines can be obtained by plotting ln[βR/Ea] - ln[p(x)] against ln[g(R)]. The lines corresponding to various heating rates should be superposed on each other, and from these lines, we can obtain the logarithmic values of the pre-exponential factor (ln A). 4. Results and Discussion 4.1. Experimental Results. The experimental thermogravimetric (TG) curves at the different heating rates (β ) 10, 20, and 25 °C min-1) for the degradation process of Zetaplus impression material in an air atmosphere are given in Figure 3. It can be observed from Figure 3 that the nonisothermal degradation of Zetaplus impression material represents the complex process, which consists of, at least, two degradation stages. The TG curves also show that the small mass loss exists in the temperature range of 100-150 °C. This can be attributed to the desorption of small amounts of water and the low temperature decomposition process attributable to the presence of residual polymerization catalyst. These processes do not represent the main degradation stages of the investigated polymer material, and they do not affect the further kinetic analysis. A much clearer picture can be obtained from the corresponding DTA curves of the investigated process (Figure 4). The inspection of DTA curves (Figure 4) for Zetaplus samples shows one endothermic peak in the temperature range of approximately T ) 175-375 °C, at all considered heating rates. The observed endothermic peak can probably be associated with the bond cleavages and recombination effects in the polymer network.32 In addition, the second, exothermic peak can be observed in the temperature range of approximately T ) 390-530 °C, at all heating rates (Figure 4). This degradation stage can be directly connected with thermo-oxidative stability of the investigated Zetaplus sample. Andrianov has stated that many of the properties in siloxane elastomers are the result of molecular structure.33 Namely, the polydimethylsiloxane molecules are in a spiral form with six to eight links in the spiral. At high temperatures, warping of

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sections of chain links has resulted in closure of the chains into rings, where the chain links in the spiral are close together. Baney34 indicated in a review that the high temperature degradation consists of siloxane rearrangements and ring-chain equilibrium. The silicon-carbon bond can withstand 600 °C, while 200-350 °C is the temperature limit within which silicon-oxygen bond rearrangements occur. The equilibrium between cyclic and polymer structure is shifted in favor of cyclic components at elevated temperatures. Because of the above-mentioned facts, we can attributed the exothermic peak on DTA curves (Figure 2) to the formation of cyclosiloxanes probably via depolymerization of the polysiloxane backbone.35-39 Once a C-H bond is broken and a backbone radical is formed, scission of C-C links in the backbone adjacent to the radical site becomes much more favorable because of the 63-170 kJ mol-1 drop in C-C bond energy in the radical, as compared with the saturated backbone. A radical site on the chain end of an addition polymer may lead to a progressive depolymerization.40 Table 1 shows the effect of heating rates on the characteristic temperatures and total mass losses in the corresponding TG and DTA curves of the investigated Zetaplus degradation process. It is evident from Table 1 that the values of characteristic degradation temperatures for both stages increase with an increase in the heating rate (β). On the other hand, the corresponding values of total mass loss for both stages show the decreasing behavior with an increase in heating rate. From the average value of total mass loss (〈∆mt〉), it can be observed that the value of total mass loss is a little higher for the second degradation stage, with respect to the first degradation stage. In addition, the intensity of the DTA peaks for both degradation stages increases with an increase in the heating rate (Figure 4). As it is seen in Figure 4 and Table 1, when increasing the heating rate, the peaks on DTA curves systematically shift toward higher temperatures. This behavior is typical of the thermally activated phenomena. 4.2. Isoconversional Kinetic Analysis. The isoconversional approach can be used to evaluate both simple and complex chemical reactions. For the evaluation of data with this method, no kinetic rate expression is assumed a priori. The isoconversional methods are advocated for their ability to provide estimates of the apparent activation energy independent of reaction mechanisms.41 If the isoconversional analysis results in a constant apparent activation energy, as a function of the degree of conversion, the reaction rate is presumably limited by a single rate-determining step and it should be easy to discover the pre-exponential factor and the reaction mechanism. On the other hand, if the resulting apparent activation energy varies with the conversion, then most simple mechanistic explanations are ruled out and it is also useless to carry out model fitting with a single step unidirectional reaction rate model. Evidently then, isoconversional methods do not contribute to an understanding of the underlying chemical phenomena. On the basis of the above facts, we will first establish the complexity of the investigated process, i.e. dependence of Ea on R. In order to evaluate the dependence of Ea on R for the nonisothermal degradation of Zetaplus impression material, Wanjun’s isoconversional method was used. The dependencies of Ea on R evaluated by means of Wanjun’s isoconversional method, for the first and second degradation stages, are shown in Figures 5 and 6, respectively. For the first degradation stage (Figure 5), in the conversion range 0.10 e R e 0.90, the Ea values obtained are practically constant, and therefore, the investigated degradation stage can

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Figure 4. Nonisothermal differential thermal analysis (DTA) curves for the degradation process of Zetaplus impression material in an air atmosphere at three different heating rates: (a) 10, (b) 20, and (c) 25 °C min-1. The double exothermic peak at all heating rates was designated by an arrow symbol. Table 1. Values of Characteristic Temperatures and Mass Losses Coresponding to Two Steps of the Nonisothermal Degradation Process of Zetaplus Impression Material degradation stage

β (°C min-1)

I

10 20 25 10 20 25

Ti (°C)a

Tf (°C)a

Tp (°C)b

∆mt (%)c

〈∆mt〉 (%)d

8.24 7.96 7.94 10.41 8.86 8.29

8.05

Data from TG-DTA Curves

II

154.78 157.81 162.68 351.58 375.57 386.34

351.58 375.57 386.34 698.70 699.23 699.98

290.21 300.74 308.05 453.99 477.27 484.39

9.19

a Ti and Tfstemperatures at the beginning and at the end of each decomposition step, according to TG or DTA curves, respectively. b Tpspeak temperature corresponding to the “maximum” in the DTA curve. c ∆mtsTotal mass loss corresponding to each step. d Average value of ∆mt.

Figure 5. Dependence of Ea on R evaluated by means of Wanjun’s isoconversional method for the first degradation stage of Zetaplus impression material. The full square symbols (9) designate the apparent activation energy values, while the empty square symbols (0) designate the apparent isoconversional intercept values.

be characterized as simple (one step process) and can be described by a unique kinetic triplet, i.e. Ea, A, and f(R). The average value of Ea in the conversion range 0.10 e R e 0.90 (Figure 5), obtained by the integral isoconversional (model-

Figure 6. Dependence of Ea on R evaluated by means of Wanjun’s isoconversional method for the second degradation stage of Zetaplus impression material. The full circle symbols (•) designate the apparent activation energy values, while the empty circle symbols (O) designate the apparent isoconversional intercept values.

free) method, amounts to 〈Ea〉 ) 99.5 kJ mol-1. The corresponding average value of the isoconversional intercept calculated from Figure 5, in the same conversion range, was found to be 〈ln[AEa/Rg(R)]〉 ) 30.17 (A in min-1). From the shape of dependence of Ea on R for the second degradation stage of Zetaplus impression material (Figure 6), we can conclude that the considered degradation step is a complex, which probably includes two parallel independent reactions with different values of the apparent activation energy.42,43 Vyazovkin et al.42,43 showed that the shape of such dependence is determined by the ratio of the rates of individual reactions and their partial contribution to the gross transformation degree. It can be pointed out that the relative error of calculated apparent activation energies for the first and second degradation stage does not exceed 5 kJ mol-1 (which represents the satisfactory error value for Ea), because the temperature reproducibility in all performed thermoanalytical measurements is Rep ) (0.5 °C. The mentioned value of Rep represents a good reproducibility of the results, which unambiguously causes the small error in Ea.

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Figure 7. Experimental y(R) function vs R for the first degradation stage of Zetaplus impression material at different heating rates (10, 20, and 25 °C min-1).

A variation in the pre-exponential factors with activation energies in the higher temperature zone (for the second degradation stage (Figure 6)) indicates the existence of a compensation effect for two parallel reactions. The apparent compensation effect44 was found for the second degradation stage and can be expressed through the following relationship:

[ ]

ln

AEa ) 10.988 59 + 0.157 01Ea a b Rg(R)

(8)

where a and b represents the compensation parameters (or coefficients). From the slope of the linear relationship, b ) 1/RTiso, the isokinetic temperature (Tiso) was calculated and amounts to Tiso ) 492.9 °C. This value of temperature lying in the temperature range which corresponds to the second degradation stage of the investigated material (Figure 4). The observed apparent compensation effect can be assigned to the dependence of the kinetic parameters on the degree of conversion. Also, it was found that the variation of the pre-exponential factor and activation energy values (the right-hand side of eq 8) in the higher temperature zone are consistent and show a trend toward a nonuniform activation energy value for all the heating rates. This could be due to the significance effect of temperature on the degradation rate in the higher temperature zone (second degradation stage, Figure 6). 4.3. Evaluation of Kinetic Model and the Pre-exponential Factor of the Degradation Process. Once the activation energy has been determined, it is possible to find the kinetic model which best describes the measured set of thermoanalytical data. It can be shown that for this purpose it is useful to define two special functions y(R) and z(R), which can easily be obtained by the simple transformation of experimental data.30 The y(R) function is proportional to the f(R) function. Thus, by plotting the y(R) dependence, normalized within (0,1) interval, the shape of the function f(R) is obtained. The f(R) function is therefore characteristic for a given kinetic model.45 Figure 7 presents the y(R) kinetic plots of (dR/dt) exp(x) versus R obtained for the first degradation stage of Zetaplus impression material at different heating rates (10, 20, and 25 °C min-1). For construction of the y(R) kinetic plots at all heating rates for the first degradation stage, the apparent activation energy (Ea) calculated from the isoconversional method was used.

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Figure 8. Experimental z(R) function vs R for the first degradation stage of Zetaplus impression material at different heating rates (10, 20, and 25 °C min-1).

Figure 7 shows that the representative curves of y(R) display identical concave profiles with a maximum value Ry ) 0 for all considered heating rates, being invariant with respect to heating rate. The concave profile of the y(R) functions obtained from the experimental data correspond to the diffusion-type of kinetic models.45 As a consequence, the first degradation stage of the investigated material is clearly governed by a diffusioncontrolled thermal process in which every individual transport mechanism is of the deceleration (Dn) type. Namely, D1, D2, D3, and D4 type mechanisms are equally possible only by considering the y(R) function (from eq 4). The nature of the diffusion-contolled transport mechanisms may be clarified using eq 5 based on the definition of the z(R) function. The z(R) method consists of the plot of T/β(dR/dt)π(x) as a function of R in a normalized R interval as represented in Figure 8 for all considered heating rates. For corresponding z(R) plots shown in Figure 8, the following maximum values R∞z were calculated: 0.715 (10 °C min-1), 0.705 (20 °C min-1) and 0.704 (25 °C min-1). These values are in agreement with position of Rz∞ maximum for the D3 type mechanism (the three-dimensional diffusion (Jander equation)).30 The curve profile even remained almost independent of the heating rate. The above results suggest that the first nonisothermal degradation stage of Zetaplus impression material can be modeled by a Dn type equation, where the three-dimensional diffusion-type mechanism is the rate-limiting step. These results allow us to determine the pre-exponential factor (A) associated with the D3 reaction model. By assuming the three-dimensional diffusion mechanism, the corresponding experimental data, the expression of the D3 model (g(R) ) [1 - (1 - R)1/3]2), and the average apparent activation energy, the group of lines can be obtained by applying the eq 7. As shown in Figure 9, the lines corresponding to various heating rates superpose each other nearly completely. From these lines, the logarithmic values of the pre-exponential factor for the actual values of the heating rates may be easily calculated. The values of the pre-exponential factor (A) obtained from eq 7 at different heating rates, for the first degradation stage of the investigated impression material are presented in Table 2. The pre-exponential factor is calculated using the isoconversional method from the y-intercept of the isoconversional lines,

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Figure 9. Plotting ln[βR/Ea] - ln[p(x)] against ln{[1 - (1 - R)1/3]2} (D3 model) for the first degradation stage of Zetaplus impression material at the different heating rates and their linear-fit drawing (solid lines). Table 2. Determination of the Pre-exponential Factor for the First Degradation Stage of Zetaplus Impression Material Using the Isoconversional, Second-Derivative, and Linear Methods β (°C min-1) 10 20 25 average

isoconversional method A × 108 (min-1)

second-derivative method (eq 6) A × 108 (min-1)

linear method (eq 7) A × 108 (min-1)

1.05

1.49 2.41 2.56 2.15

5.77 5.01 7.11 5.96

taking into account both the integral form of model function (g(R)) and calculated apparent activation energies (Figure 5). The average value of A estimated from the isoconversional analysis is presented in Table 2. It can be pointed out that the ln A values are derived from the y-intercept term of the isoconversional plot with a logarithm form, which may produce a large variation in its measurement from a small deviation of values for the y-intercept. Therefore, the second-derivative method estimated from eq 6 was alternatively used to check the reliability of the isoconversional results for a given reaction mechanism (D3). Obviously, if results in the pre-exponential values are corroborating using both equations, then the physical nature of the above-postulated reaction mechanism can be validated. The corresponding results are reported in Table 2. Table 2 shows that the isoconversional and second-derivative methods are in very good agreement with respect to the order of A (× 108). Also, both results are in good agreement with the average value of A calculated using eq 7 (for the same order of A (× 108)). This points to the fact that the diffusion-type model (the three-dimensional diffusion (D3)) and calculated value of Ea are validated for the first nonisothermal degradation stage of Zetaplus impression material. Accordingly, for the first degradation stage of Zetaplus impression material, we can expected the “diffusion-limited kinetics” of the polydimethylsiloxane thermal degradation, which can be ultimately described by means of evaporation rates of the degradation products. 4.4. Proposed Stage I: Reaction Mechanism. Three reaction mechanisms are proposed for the thermal degradation of polydimethylsiloxane:46 (a) the unzipping mechanism of the silanol terminated polymer, (b) the random chain scission

Figure 10. Distribution function (f(Ea)) of the apparent activation energy values for the second degradation stage of Zetaplus impression material and corresponding approximate distributions.

mechanism of the trimethylsilyl end-terminated polymer, and (c) the externally catalyzed mechanisms for the polymer contained ionic impurities. All these mechanisms involve siloxane rearrangement reactions, but they differ fundamentally in the location of site in the polymer chain at which the degradation process begins. The unzipping mechanism is based on the reactions of the polymer end-group as the reactive group responsible for the start of the degradation process. Polysiloxanes which are terminated by silanol end-groups (as the Zetaplus impression material) can “backbite” to undergo a siloxane exchange and rearrangement reaction, which leads to the formation of low molecular cyclic compounds (haxamethylcyclotrisiloxane). It can be pointed out that the six- and eight-membered cyclic compounds would be expected to be the most prominent components because of their higher rate of formation.46 Among the cyclic products of polydimethylsiloxane degradation, the most abundant is haxamethylcyclotrisiloxane, with an enthalpy of evaporation of 55.2 kJ mol-1.47 This can therefore be considered to be the lowest limit of the apparent activation energy of diffusion-evaporation of the polydimethylsiloxane degradation products (Figure 5 for the first degradation stage), which represents the basic molecule in the composition of Zetaplus impression material. As the polymer is heated, its molecular weight first sharply increases with an apparent activation energy of 30-35 kJ mol-1, which is typical for the silanol condensation reactions and hence indicative of an intermolecular reaction between the polymer chain ends. This reaction reaches its maximum intensity at about 250-260 °C, above which the depolymerization takes over, as manifested by the subsequent decrease in the polymer molecular weight and formation of the volatile cyclics.32 In the case of the nonisothermal degradation process of Zetaplus impression material, this reaction can be detected on DTA curves (Figure 4) as the double exothermic peak at all heating rates (designated by an arrow in the narrow temperature range 215 e T e 265 °C). Also, the same reaction can be observed in Ea-R dependence for the first degradation stage (Figure 5), which is manifested in the increase of Ea value for ∆Ea ) 27.6 kJ mol-1, in the narrow conversion range (0.05 e R e 0.10). On the basis of the above kinetic treatment, results show that the polydimethylsiloxane thermal volatilization becomes domi-

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Table 3. Determination of the Ea and ln A Values for the Two Degradation Steps of Zetaplus Impression Material Using a Modified Marquardt Algorithm from Matlab Software kinetic parameters

d

degradation steps

E (kJ mol )

〈Ea〉 (kJ mol-1)

ln Ac

〈ln A〉d

first step second step

99.265 < EaI < 100.988 132.921 < EaII,1 < 133.811 172.554 < EaII,2 < 173.271

〈EaI〉 ) 100.126 〈EaII,1〉 ) 133.366 〈EaII,2〉 ) 172.913

18.598 < ln AI < 18.910 23.523 < ln AII,1 < 23.754 25.712 < ln AII,2 < 25.892

〈ln AI〉 ) 18.754 〈lnAII,1〉 ) 23.640 〈ln AII,2〉 ) 25.805

a a

-1

b

a Range of calculating Ea values for 95% confidence interval. b Average value of Ea. c Range of calculating ln A values for 95% confidence interval. Average value of ln A.

nated by the rate of three-dimensional diffusion (model D3) and evaporation of oligomers produced on the decomposition. 4.5. Proposed Stage II: Reaction Mechanism. In an air atmosphere, the typical TG-DTA experiments of the Zetaplus degradation process are manifested as a two-step mechanism process, in which the second step starts at about 350-390 °C (Table 1 and Figures 2 and 4), which is characterized by an apparent activation energy of about 125.5 kJ mol-1.32 This value of Ea (125.5 kJ mol-1) is in good agreement with estimated results presented in Figure 6, for the initial part of the second degradation stage of the Zetaplus impression material (for conversion value R ) 0.15, the value of Ea ) 121.9 kJ mol-1 was obtained (Figure 6)). We can assume, that in the early stages, this degradation occurs through the formation of side-chain peroxides, followed by the splitting of the polymer side groups.48 It is well-known that during thermo-oxidative degradation, the polymer becomes insoluble and the structure of the residue changes over the course of the reaction, apparently because of the cross-linking through formation of the Si-O-Si bridges. The volatile products of the thermo-oxidative degradation of polydimethylsiloxane consist mainly of carbon monoxide (CO) and water, but they also include smaller amounts of carbon dioxide (CO2), formaldehyde, methanol, and traces of formic acid.49,50 It can be pointed out that methanol and formic acid volatilized during degradation are a source of fuel in a combustion process. On the basis of these results, we can assume that the free radical mechanisms can be proposed for the thermo-oxidative degradation process in which the initial step consists of the reaction of oxygen with pendant organic groups to form a polymeric hydroperoxide. In addition, the hydroperoxide decomposes to form the corresponding hydroxyl and silyl radicals and formaldehyde. Lately, the two different radicals then can recombine to form the silanol group, which, in turn, can undergo a silanol condensation reaction to form the cross-links proposed and to yield water as a byproduct. At the higher temperatures, thermal decomposition of formaldehyde yields carbon monoxide and hydrogen. These mechanisms of Zetaplus thermo oxidative degradation may be predominant at relatively lower temperatures (370-400 °C), because above these temperatures siloxy bonds can rupture and the degradation process further can become much more complicated (Figure 6). Because of the cross-linking reactions, which occur during thermo-oxidative degradation, the thermal stabilization of the investigated polymer exists, which can be manifested in the additionally increasing apparent activation energy values in the second part of the Ea-R curve in Figure 6. As the degradation progresses, the structure and composition of the residue change, resulting in pure SiO2 above about 600 °C (Figure 3). It can be pointed out that the reduction in cyclic silicone evolution with increasing temperature occurs because of in situ oxidation, as evidenced by the white powder remaining at the end of the experiment. The above facts can be accepted as the prediction assumptions based on the shape of the Ea-R dependence in Figure 6 and

established values of the apparent activation energy (Ea). In particular, this warning applies to all processes exhibiting a complex behavior. Therefore, it seems that meaningful conclusions concerning the real mechanism of the process should be based on the other types of complementary evidence, including microscopic observations, gas chromatography, or Fourier transform infrared (FTIR) spectroscopy. It can be pointed out that the conversion-dependent apparent activation energy obtained by the isoconversional method results from the distribution of the apparent activation energy,51,52 and the kinetics of the second degradation stage cannot be described by a single reaction but by a number of parallel, irreversible, first-order reactions with different activation energies occuring simultaneously. In this paper, Miura’s method51 was used to estimate the distribution of the apparent activation energies (f(Ea)) for the second degradation stage of Zetaplus impression material. Figure 10 was obtained by differentiating the R versus Ea relationship by Ea (the second degradation stage) and also differentiating the curve obtained by a Gaussian and log-normal fits as separate fit functions, which were obtained from f(Ea). Figure 10 shows the distribution of the apparent activation energies (f(Ea)) for the second degradation stage, with two clearly separated maxima, where each of them can be attributed to one parallel irreversible reaction with a characteristic value of the apparent activation energy (Ea,p). Distribution f(Ea) has peaks at 0.0078 mol kJ-1 (reaction I) and 0.0089 mol kJ-1 (reaction II) and has corresponding apparent activation energies at 132.7 kJ mol-1 (Ea,pI) and 172.2 kJ mol-1 (Ea,pII), respectively. The f(Ea) distribution for the second degradation stage can be approximated by the mixture of normal and log-normal distribution functions. Due to the complexity of the second degradation stage of Zetaplus impression material, the use of a single reaction could not efficiently describe the degradation curve, and thus, multiple parallel or even competative reactions have to be employed. From the above presented results, the model of two independent parallel reactions53 could be utilized to efficiently describe the TG and DTG curves corresponding to the second degradation stage of Zetaplus impression material. This model implies that each component reacts independently and the thermal behavior of the sample could be simulated by the sum of its individual components. 4.6. Model Simulation. Here, we are interested in determining if both our kinetic parameters and our kinetic models for the degradation process of Zetaplus impression material may predict the mass loss data with respect to temperature. The aim in this subsection is to obtain a representation as simple as possible to describe the degradation process producing accurate results. Bearing in mind the above established results, the overall mass loss can be expressed through eq 9:

( dR dT )

) overall

dR +( ) ( dR dT ) dT I

II

(9)

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It can be observed that the first degradation step is characterized by the low kinetic parameters. This stage also presents the lowest contribution to the overall degradation kinetics. It can be pointed out that the steps with low apparent activation energies are followed by proportionally low pre-exponential factors.54 Furthermore, the comparison between the calculated kinetic parameters and the experimental DTG curves leads to the conclusion that the reaction with low apparent activation energy and the pre-exponential factor is represented by the lower and wider DTG curve (Figure 2). 5. Concluding Remarks Figure 11. Simulated (full and dash lines) and experimental (empty square, circle, diamond, and triangle symbols) curves of the degradation of Zetaplus impression material as a function of the temperature during TG experiments at β ) 10 and 25 °C min-1.

where (dR/dT)I and (dR/dT)II are the reaction rates for the first and the second degradation stage of the investigated impression material. To calculate the kinetics parameters, an adjustment to the experimental data by nonlinear least-squares regression was carried out. A modified Marquardt algorithm from Matlab software was used. The fitting was done on the experiments at 10 and 25 °C min-1. The simulated curves were calculated by introducing the three-dimensional diffusion mechanism (D3 model) for the firstand nth-order reaction mechanism for the second degradation stage. Two independent, first-order, parallel reactions were applied for the second degradation stage. The two processes can be presented by the following kinetic laws:

( )

AI EaI 3(1 - RI)2/3 dR exp ) dT I β RT 2[1 - (1 - R )1/3] I

( ) ( dR dT )

) II

(

)

AII,i EaII,i exp (1 - RII,i), i ) 1, 2 β RT

(10)

(11)

where the subscript i represent the ordinal number of the firstorder reaction. The corresponding results of kinetic parameters are reported in Table 3. It should be mentioned that the values were averaged for simulation. The kinetic results show that the average value of Ea (Table 3) for the first step is close to the average value of Ea estimated by the isoconversional analysis (99.5 kJ mol-1). In addition, for the second step, the results of Ea are closer to the values of Ea established by Miura’s method assuming a parallel first-order reaction model (132.7 and 172.2 kJ mol-1). In this paper, the formal resolution of eqs 10 and 11 with Matlab software was used. The ordinary differential equation system was solved with the average values reported in Table 3. The resulting degree of conversion was simulated on the whole temperature domain. Simulation was reported for heating rates at β ) 10 and 25 °C min-1, but it should be underlined that simulations were tested and validated for all experimental heating rates. Figure 11 shows the comparison between the total simulated conversions and the experimental curves at 10 and 25 °C min-1. It can be seen that the simulations matched all experimental data at the considered heating rates. From the obtained results, we can see that the proposed models predict very well the experimental degradation data of the investigated Zetaplus impression material.

The kinetics of the nonisothermal degradation of Zetaplus impression material was accurately determined from a series of thermoanalytical (TG-DTA) experiments at different constant heating rates. For calculation of the kinetic parameters and mechanism function of the investigated process, the isoconversional (model-free) method and the Master-plot method were used. It was found that the investigated degradation process is a complex one, which consists the two degradation stages in different temperature regions. The obtained results from kinetic analysis suggested that the first degradation stage can be described by the three-dimensional diffusion (D3) (Jander equation), f(R) ) (3/2)(1 - R)2/3[1 - (1 - R)1/3]-1. For the first degradation stage of Zetaplus impression material, the following values of kinetic parameters were obtained: Ea ) 99.5 kJ mol-1 and A ) 1.05 × 108 min-1. The first degradation stage of Zetaplus impression material can be attached by the evaporation of the degradation products from the polydimethylsiloxane, which usually have the same chemical composition as the original polymer. From the shape of dependence of Ea on R for the second degradation stage of Zetaplus impression material, it was concluded that the considered degradation step includes the two parallel independent reactions with different values of the apparent activation energy. This degradation stage probably occurs through the cleavage of organo-groups bound to the silicon atoms resulting in the formation of gaseous products, which is accompanied by the cross-linking reactions. Finally, a kinetic law expressions for the both degradation stages has been derived by a numerical modeling that appropriately predicted the experimental data at all investigated heating rates. Acknowledgment This study was partially supported by the Ministry of Science and Development of Serbia, under the following Project 142025. The author wishes to thank MSc Dragana Aran{elovic´ for performed thermal measurements. Literature Cited (1) Brown, D. An Update on Elastomeric Impression Materials. Brit. Dent. J. 1981, 150, 35–40. (2) Craig, R. G. Review of Dental Impression Materials. AdV. Dent. Res. 1988, 2, 51–64. (3) Bell, J. W.; Davies, E. H.; Von Fraunhofer, J. A. The Dimensional Changes of Elastomeric Impression Materials under Various Conditions of Humidity. J. Dent. 1976, 4, 73–82. (4) Craig, R. G. A Review of Properties of Rubber Impression Materials. Mich. Dent. Assoc. J. 1977, 59, 254–261. (5) Craig, R. G. Evaluation of An Automatic Mixing System for An Addition Silicone Impression Material. J. Am. Dent. Assoc. 1985, 110, 213– 215. (6) Harcourt, J. K. A Review of Modern Impression Materials. Aust. Dent. J. 1978, 23, 178–186.

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ReceiVed for reView January 20, 2009 ReVised manuscript receiVed June 20, 2009 Accepted June 25, 2009 IE900104B