Introduction

The use of polymeric materials is increasing in odontology [1,2,3]. Dental composites for dental restoration in odontology consist of a photopolymerizable resin matrix reinforced by mainly mineral particles. This matrix consists of monomers and oligomers with dimethacrylate functions [4, 5], and the most commonly used monomers are Bis-GMA and TEGDMA. The Bis-GMA monomer with aromatic nuclei ensures the rigidity and strength of the composite and presents a low volume contraction during polymerization [6, 7]. This low volume contraction reduces the internal stresses and increases the service life of the restoration [8, 9]. On the other hand, its high viscosity restricts the mobility of the reaction medium, which limits the conversion of methacrylic functions [10, 11]. This results in an alteration of the physical and biological properties of the material obtained [12,13,14]. Therefore, TEGDMA is usually added as a reactive diluent, which improves the conversion [15,16,17,18] and thus the physical and biological properties of the final material, but it also generates a volume contraction in larger material [6, 19, 20].

The photochemical polymerization of methacrylates multifunctional monomers is one of the most effective methods for creating high-density three-dimensional polymer networks [21, 22]. The kinetic behavior of polymerization is a complex process, and a number of characteristic phenomena are observed. These include auto-acceleration and auto-deceleration, incomplete conversion of functional groups, delayed volume shrinkage relative to equilibrium and unequal reactivity of the double bonds of dimethacrylate monomers [23].

One of the main characteristics of the polymerization reactions of multifunctional monomers is the presence of an auto-acceleration phase and an auto-deceleration phase. This result is a dependent on the reaction rate and the conversion with passage through a maximum, of which the position depends on the nature of the monomer and the reaction conditions. This phenomenon is similar to that of autocatalytic reactions, which makes it possible to consider the polymerization of multifunctional monomers as an autocatalytic process (unlike reactions of order n, which predict the maximum reaction rate at the origin of the conversion). Generally, the autocatalytic effect is due to the formation of intermediate species that greatly accelerate the reaction. In the case of the polymerization of multi-vinyl monomers, this effect is due to the decrease in the rate of termination due to the reduced diffusion of the ends of growing reactive chains. Thus, the resulting increase in radical concentration leads to higher curing rates. The autocatalytic model has already been used in the literature to describe the polymerization of multifunctional methacrylic monomers [23,24,25]. The kinetic model of the formulations consists of determining an appropriate kinetic equation for the system analyzed to measure the reaction orders, activation energies, and reaction parameters.

We performed a kinetic study of dental formulation consisting of a mixture of monomers (Bis-GMA/TEGDMA) and a CQ / DAMEMA photoinitiator system, which is the base resin of many dental composites [26]. Previous work was carried out to determine how to control the homogeneity of the materials formed, and we studied the polymerization by subjecting the formulations to several experimental conditions: different concentrations of photoinitiator and diluent, different light intensities, and different temperatures. The influence of various parameters on the evolution of the photopolymerization reaction of Bis-GMA/TEGDMA was studied by photocalorimetry. The results obtained made it possible to determine the optimal polymerization conditions (by weight composition of 75% Bis-GMA/25% TEGDMA and a concentration of the initiator system of 1% by weight (CQ/DMAEMA (1/1))). The optimal final conversion remains below 100% due to the transition to a vitreous state, which causes the blocking of the reactive species and stops the polymerization [27].

Based on the results of previous studies, this article describes kinetic models of the photoinitiated polymerization reaction of this formulation. To this end, we first use a phenomenological model to experimentally determine the various parameters of the kinetic equations, such as the velocity constant k, reaction orders m and n, and activation energy E. Secondly, we will be interested in a mechanistic model that takes into account the diffusional control of the reaction.

Experimental

Materials

The monomers used in this work are commercial monomers of the di-methacrylate type: Bis-GMA (98%) and TEGDMA (95%, containing 200 ppm. of monomethylether hydroquinone). They were used without prior purification. The system used to prime the photopolymerizable formulations was camphorquinone/2-(dimethylamino)ethylmethacrylate (CQ/DMAEMA). The camphorquinone (CQ > 98%) was chosen as photoinitiator, given its wide use in dental formulations [28]. This molecule absorbs light in the visible range, which requires a hydrogen donor molecule to generate radicals by electron and proton transfer from its triplet-excited state. The 2-(dimethylamino)ethylmethacrylate (DMAEMA > 99% from Aldrich) is a co-initiator or photoaccelerator donor of hydrogen. These reagents were purchased from Aldrich (Lyon, France) and were used without further purification.

figure a

Preparation of the photopolymerizable mixture

The photopolymerizable dental formulations were prepared by mixing the monomers (75% Bis-GMA/25%TEGDMA) by weight. The CQ/DMAEMA priming system (1%/1% by weight) was incorporated into the monomer mixtures. The reagents were mixed under magnetic stirring for 30 min at 50 °C in dark conditions. The formulations obtained were kept old and protected from light before their use to avoid any crosslinking reaction.

Analysis technique

The kinetic analysis of the photopolymerization reaction was followed by photocalorimetry (Photo-DSC), which was done using a differential scanning calorimeter (DSC7 Perkin Elmer, Waltham, MA, USA) with an irradiation unit: a dental halogen office lamp emitting quasi-monochromatic radiation at 465 nm (LA500 Blue Light, Philippines Union Commercial Inc., New Manila, Quezon City, Philippines). The heat flow was recorded as a function of time in isothermal mode under nitrogen atmosphere. The optical part of the calorimeter, sample preparation and thermogram processing have been described elsewhere [29]. The calculation of the conversion rate and the reaction rate was carried out according to previously published work [27].

Results and discussion

Photopolymerization of formulations

The photopolymerization of multifunctional monomers (methacrylates) leads to the formation of a strongly cross-linked polymer network. The polymerization reaction of the Bis-GMA/TEGDMA formulation initiated by the CQ/DAMEMA initiation system is schematised in Fig. 1.

Fig. 1
figure 1

Formation of a crosslinked polymer by polymerization of Bis-GMA/TEGDMA formulation

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In general, the process of this radical reaction involves three steps: initiation, propagation and termination.

The first step begins with the breakdown of camphorquinone (CQ) [30, 31]. After absorbing a quantum of light energy, CQ finds itself in an excited state and associates with the amine DMAEMA to form an excimer. The latter eventually decomposes after transfer of hydrogen to α of the amine to form an initiator amine radical and a camphorquinone radical. The amine radical formed, denoted (R•), can form a covalent bond with the double bond of the monomer. The cetyl radical (camphorquinone) does not initiate the polymerization reaction. It generally recombines with another cetyl radical to form a pinacol or returns to its primary form or even forms the secondary alcohol (Fig. 2).

Fig. 2
figure 2

Mechanism of CQ decomposition and formation of the first radical monomer

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The propagation of the polymerization is due to a series of reactions between the radials formed and the vinyl of the starting monomers. Chain growth can lead to bridging and the formation of new polymerization sites (Fig. 3).

Fig. 3
figure 3

Reactions involved in the propagation step

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The termination reaction can be done either by the combination of two macroradical chains to give a new carbon–carbon bond or by disproportionation by the transfer of hydrogen from one macroradical chain to another to give a saturated and an unsaturated macromolecule. The crosslinked material obtained may contain reactive species, which are frozen inside the polymer network [32].

Phenomenological model

The kinetics of the photochemical reaction of multifunctional monomers obeys mainly to two types of kinetics: reaction of order n and autocatalytic reactions. For crosslinks, which follow a reaction of order n, the reaction rate is given by:

$$\frac{{{\text{d}}C}}{{{\text{d}}t}} = k(1 - C)^{n}$$
(1)

where C is the conversion rate, k is the rate constant and n is the order of the reaction.

Autocatalytic reactions are characterized by the expression:

$$\frac{{{\text{d}}C}}{{{\text{d}}t}} = kC^{m} (1 - C)^{n}$$
(2)

where m is also an order of reaction.

For the autocatalytic model, Eq. (2) seems best suited to describe the isothermal crosslinking of multifunctional methacrylic monomers. The shape of the photoinitiated polymerization rate curves is the same as that of the autocatalytic reaction (Fig. 4). Although photoinitiated polymerization is autoaccelerated and not autocatalyzed, this model can be applied to describe these reactions in a purely mathematical manner.

Fig. 4
figure 4

Typical photoinitiated polymerization rate versus time curve at 50 °C. Experimental curve (——) and modeling (♦)

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This work focuses on the Kamal and Sourour approach [33] to take into account the kinetics of cross-linking and access to the energy parameters of Arrhénius.

The heat given off during the cross-linking is proportional to the number of double bonds have reacted. Equation (3) defines the conversion rate.

$$C_{t} = \frac{{\Delta { H}_{t} }}{{\Delta { H}_{{{\text{theorique}}}} }} \times 100$$
(3)

where ΔHt is the enthalpy at time t, and ΔHTheoretical is the theoretical enthalpy that all methacrylate functions should release.

The integral form of relation (3) leads to the expression (4) [33]:

$$C = \frac{{At^{B} }}{{1 + At^{B} }}$$
(4)

where t is the time elapsed since the start of the isothermal reaction, and A and B are coefficients related to the parameters m, n, and k of Eq. (2). Equation (4) can be arranged in an equivalent form:

$$\frac{C}{1 - C} = At^{B}$$
(5)

The derivative of Eq. (4) in relation to time gives:

$$\frac{{{\text{d}}C}}{{{\text{d}}t}} = \frac{{At^{B - 1} }}{{(1 + At^{B} )^{2} }}$$
(6)

Moreover, the combination of Eqs. (2) and (4) makes it possible to obtain at a different form of the reaction rate:

$$\frac{{{\text{d}}C}}{{{\text{d}}t}} = kC^{m} (1 - C)^{n} = k\left[ {\frac{{At^{B} }}{{(1 + At^{B} )}}} \right]^{m} \cdot \left[ {\frac{1}{{1 + At^{B} }}} \right]^{n}$$
(7)

or:

$$\frac{{{\text{d}}C}}{{{\text{d}}t}} = k\frac{{A^{m} t^{Bm} }}{{(1 + At^{B} )^{m + n} }}$$
(8)

Equations (6) and (8) can be directly compared and allow us to deduce the relationships between the parameters (m, n, and k) and the coefficients (A and B).

By analogy, we have m + n = 2 and B-1 = B.m, from which we can deduce:

$$m = 1 - \frac{1}{B}$$
(9)
$$n = 1 + \frac{1}{B}$$
(9')
$$AB = kA^{m}$$
(10)

The photocalorimetric measurements made continuously throughout the reaction make it possible to obtain the heat ΔHt released at time t and to calculate the Ct conversion from Eq. (3). Once the conversion rate is determined as a function of time, we can obtain Eq. (5) in its logarithmic form:

$${\text{ln}}\frac{C}{1 - C} = {\text{ln }}A + B\;{\text{ln }}t$$
(11)

Figure 5 shows a graphical representation of Eq. (11); i.e., the evolution of ln (C/(1-C)) as a function of ln (t) for the photopolymerization of the formulation Bis-GMA/TEGDMA (T = 30 °C, I = 40 mW/cm2, and 1% by mass CQ/DMAEMA (1/1)).

Fig. 5
figure 5

Curve of ln(C/(1 − C)) as a function of ln(t) for the 75% Bis-GMA/25% TEGDMA formulation

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According to the curve in Fig. 5, as long as the reaction system has not reached its gel point, the reaction follows the autocatalytic model perfectly [34]. Analysis of the linear part of the curve allows us to obtain the values of the coefficients m = 0.37 and n = 1.63, which are deduced from the value of the slope, which corresponds to the parameter B. The value of the rate constant k as a function of the conversion before the gel point can then be determined by substituting the calculated m and n values into Eq. (2) or (10).

Some studies have been carried out on the dependence of the orders m and n on these factors [23, 24]. The orders m and n can change considerably with the temperature, surrounding atmosphere, and type of functionality. In this work, we studied the effect of temperature on the m, n, and k values. The photopolymerization kinetics of the 75% Bis-GMA/25% TEGDMA formulation were followed over a temperature range of 20 and to 60 °C (Fig. 6).

Fig. 6
figure 6

Evolution of the conversion as a function of the temperature

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From the conversion curves and the graphical representation of Eq. (11), Table 1 shows the values of the kinetic parameters for the Bis-GMA/TEGDMA system obtained with 1% CQ/DMAEMA by weigh at different reaction temperatures. The values of n and m were found to be constant and equal to 1.6 and 0.4, respectively, regardless of the reaction temperature. Only the rate constant k varies with temperature (Fig. 7).

Table 1 Values of the kinetic reaction parameters
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Fig. 7
figure 7

Evolution of the rate constant k as a function of the reaction temperature

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The application of the Arrhenius law \(k = Ae^{{ - \left( {{\raise0.7ex\hbox{${E_{a} }$} \!\mathord{\left/ {\vphantom {{E_{a} } {RT}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${RT}$}}} \right)}}\) to the values of k makes it possible to determine the activation energy value as 4.04 kJ/mol and the pre-exponential factor A as 0.53 s−1, which are in agreement with literature. In a study of commercial dental composites based on methacrylate, Morancho et al. [25] showed that the activation energy varies between 4 and 10 kJ/mol.

Mechanistic model

In radical photochemistry, a kinetic model was first proposed by Tryson and Shultz [35], and then extensively developed by Decker et al. [36], and Andrzejewska [37]. This model allows us to access the propagation and termination rate constants with simple knowledge of the evolution of the reaction rate over time. This model involves the classical steps of the polymerization reaction.

Initiation:

$$A_{2} \xrightarrow{{hv}}2A^{ \cdot } \quad R_{i} = 2,3\phi \varepsilon I_{0} \left[ {A_{2} } \right]$$
$$A^{ \cdot } + M \to AM^{ \cdot } \quad \left[ {A_{2} } \right] = \left[ {A_{2} } \right]_{0} e^{{ - 2,3\Phi _{0} \varepsilon I_{0} t}}$$

Propagation:

$$\sim ~M^{ \cdot } + M\xrightarrow{{k_{p} }}\sim ~MM^{ \cdot } \quad R_{p} = - \frac{{d[M]}}{{dt}} = k_{p} [M^{ \cdot } ][M]$$

Bimolecular termination:

$$\sim ~M^{ \cdot } + ^\cdot{M}\sim \xrightarrow{{k_{t} }}~\sim MM\sim \quad R_{t} = - \frac{{d[M^{ \cdot }]}}{{dt}} = 2k_{t} [M^{ \cdot }]^{2}$$

The principle of this model is to determine the ratio kp/kt0.5 ratio and the energy of the photopolymerizable formulation by studying the polymerization kinetics under quasi-stationary conditions. The approximation of the quasi-stationary state and the combination of the equations expressing the initiation (Ri), propagation (Rp), and termination (Rt) rates lead to the expression of the following photopolymerization rate:

$$R_{p} = \frac{{k_{p} }}{{\sqrt {k_{t} } }}\sqrt {2,3\varphi \varepsilon I_{0} \left[ {A_{2} } \right]} \left[ M \right]$$
(14)
$$\frac{{k_{p} }}{{\sqrt {k_{t} } }} = \frac{{R_{p} }}{\left[ M \right]}\sqrt {\frac{2}{{2,3\varphi \varepsilon I_{0} \left[ {A_{2} } \right]}}}$$
(15)

where kp and kt are the propagation and termination rate constants, [A2] and [M] are the photoinitiator and monomer concentrations, respectively, ϕ is the initial quantum yield, ε is the molar extinction coefficient of the photoinitiator, and I0 is the intensity of the incident radiation.

In order to determine the evolution of \({{k_{p} } \mathord{\left/ {\vphantom {{k_{p} } {k_{t}^{0,5} }}} \right. \kern-0pt} {k_{t}^{0,5} }}\), the rate of polymerization must first be followed as a function of the time under continuous irradiation. In our case, from the photo-DSC experiments, Rp is defined as follows:

$$R_{p} = \frac{{\left[ M \right]_{0} }}{{\Delta { H}_{{{\text{theor}}}} }}\frac{{{\text{d}}(\Delta { H}_{t} )}}{{{\text{d}}t}}$$

The plot of \(\frac{{R_{p} }}{\left[ M \right]}\sqrt {\frac{2}{{2,3\varphi \varepsilon I_{0} \left[ {A_{2} } \right]}}}\) plot as a function of the conversion makes it possible to determine the evolution of kp/kt0.5 during the reaction. The values of [M] and Rp and the conversion to times are obtained directly from the DSC thermograms. This study examined the 75%Bis-GMA/25%TEGDMA formulation while assuming an initial quantum yield of ϕ = 1.

Figure 8 shows the evolution of kp/kt0.5 as a function of conversion for different reaction temperatures carried out with a radiation intensity of Io = 40 mW/cm2. The evolution of these curves indicates that the photopolymerization of Bis-GMA/TEGDMA can be explained as follows. During the first seconds of the reaction, the viscosity of the medium increases rapidly, the segmental or translational diffusion of reactive species is greatly reduced, and it is increasingly difficult for two radicals to meet and reach the termination stage. Therefore, the rate constant kt is relatively low during this step, which explains the strong increase in the ratio kp/kt0,5.

Fig. 8
figure 8

Variation of kp/kt0.5 as a function of the conversion of the 75% Bis-GMA/25% TEGDMA system

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The termination mechanism is then no longer controlled by segmental diffusion but by reactive diffusion and assumes a propagation mechanism. From the top of the curve, the viscosity continues to increase, and the mobility of the reactive groups is increasingly reduced. This influences the propagation reaction, which then becomes controlled by diffusion. This step is characterized by the rapid decrease in the ratio kp/kt0.5. At the end of the reaction, kp and kt become very weak and the ratio tends towards 0.

The activation energy of the mechanistic model is calculated from the substitution of the rate constants kp and kt by their Arrhenius expressions in the rate Eq. (15). We then obtain the following equation:

$$\begin{aligned}& \ln \left[ { - \frac{{{\text{d}}[M]}}{{{\text{d}}t}}} \right] = \ln \left( {\frac{{A_{p} }}{{\sqrt {A_{t} } }}} \right) + \ln \left( {[M]\sqrt {2,3\Phi \varepsilon I_{0} [A_{2} ]} } \right) - \frac{{E_{R} }}{{{\text{RT}}}} \\ & \quad \text{with} \quad {E}_{R}={E}_{p}-{E}_{t}/2 \end{aligned}$$
(16)

According to Fig. 6, when the temperature increases, there is first a slight increase in the initial polymerization rate \(R_{Pi}\), which is assimilated to the maximum slope of the tangents to the conversion curves at the beginning of the reaction \(R_{P}^{{{\text{max}}}}\).

From Eq. (16), we can plot the evolution of ln (\(R_{P}^{{{\text{max}}}}\)) as a function of 1/T (Fig. 9). The slope of the line gives us a total reaction activation energy of 8.15 kJ/mol. In the case of the phenomenological model, however, the activation energy is 4.04 kJ/mol. This difference can be explained by the fact that the mechanistic model takes into account the reactions that take place during the polymerization up to the maximum rate, while the phenomenological model is based on the reaction process up to the gel point.

Fig. 9
figure 9

Evaluation of reaction rate as a function of 1/T of the formulation 75% Bis-GMA/25% TEGDMA

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Conclusion

The progress of the photoinitiated polymerization reaction of the Bis-GMA/TEGDMA system was followed by photocalorimetry (Photo-DSC). The study of the kinetics at different temperatures made it possible to determine the kinetic parameters of the photochemical reaction using two models. The application of the first autocatalytic model based on the relationship

$$\frac{{{\text{d}}C_{{\text{t}}} }}{{{\text{dt}}}} = kC_{t}^{m} \left( {{1 - }C_{{\text{t}}} } \right)^{n}$$

led to the determination of reaction orders independent of the reaction temperature as 0.4 and 1.6 for m and n, respectively. The temperature dependence was observed in the evolution of the global rate constant k, which followed the Arrhenius law up to 60 °C.

In order to truly take into account the chemical phenomena occurring during the reaction, a second mechanistic model was based on the kinetic equations of the different steps involved in the photopolymerization reaction. This model determined the evolution of the ratio of the propagation rate and termination rate constants during the reaction. The increase in the viscosity of the polymerizing medium had a more marked effect on the termination reactions than on the propagation reactions, which resulted in the passing of kp/kt0.5 by a maximum.