Research Letters in Organic Chemistry

Volume 2008, Article ID 404936, 5 pages

http://dx.doi.org/10.1155/2008/404936

## Dynamical Model to Describe the Interactions between the Chemical Components in Environment of Photopolymerization of MMA by Dye/Amine Systems

Engineering Biomedical Research Group, Universidade Camilo Castelo Branco, UNICASTELO, R. Carolina Fonseca, 08230-030 Itaquera, Brazil

Received 24 June 2008; Accepted 12 August 2008

Academic Editor: Daniel Little

Copyright © 2008 Marcio Magini and Máira R. Rodrigues. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work discusses the model that explains the aspects of photopolymerization of methyl methacrylate initiated by dye/amine systems. This model is based on a simulation that uses differential equations. A similar model following the hypothesis presented here was used with success in a preliminary work, by Magini and Rodrigues (2005), to describe the cationic photopolymerization of THF in the presence of sensitizers/sulfonium salt systems. Using the same structure was possible to generate a straight correlation between experimental and theoretical results for this system, free radically initiated, opening an important theoretical understanding about the photopolymerization systems and their chemical relations during the reaction.

#### 1. Introduction

Polymer science has been the subject of many investigations [1, 2]. This occurs owing to the potential applications of these materials, from basic research to applicable context. The applications commonly appear in industry of biomaterials and manufactured products [3, 4]. On the other hand, basic studies such as polymer formation, theoretical and experimental, or studies around the physical chemistry properties of polymers can be cited [5–16].

Theoretical studies about process of photopolymerization are described by statistical methods [17–19], analysis by interpolation, or similar methods [20, 21]. However, these works do not present evidences of predictability. In statistical mechanics, the results are focused in the dynamical formation of the polymer and simulated by numerical methods like Monte Carlo [18, 22]. The second approach, curves analysis, has a more straight connection with the experimental behavior of the polymerization, but the character of prediction is not considered. Normally, the prediction is a consequence of the model that uses differential equations to determine the behavior of parameters and chemicals involved [23–26].

This work presents a set of mathematical supposition that determines how the different chemicals react in a photopolymerization process in the presence of sensitizers. These hypotheses and models were demonstrated in a preliminary work [27] for a specific problem: the cationic photopolymerization of tetrahydrofuran initiated by systems sensitizer-sulfonium salt. The present work shows that the first model can be applied with the same success for dye/amine systems. Moreover, new experimental results about these systems are showed.

#### 2. Methodology

Measurements
were performed according to previous descriptions [10, 12] and , the rate of
polymerization, was calculated using the standard expression below [10, 12]:
where *h* is the contraction in volume in the
capillary at time *t*(s), relates the volume contraction with
the molar densities of the polymer and the monomer in solution, [M], is the
concentration of monomer, and *f* is
the volume fraction of monomer in the solution.

##### 2.1. Model Development

###### 2.1.1. Radical Chain Polymerization Mechanism

For radical chain polymerization, the initiation mechanism is based on results of photophysical and photochemical behaviors of the initiator. The dye absorbs light in the visible region or UV in the presence of the coinitiator and monomer. In this way, the elemental reactions involved at the initiation process must be evaluated independently.

Many mechanistic studies were reported for radical chain polymerization of MMA by thionine dye () in the presence of an amine A (triethanolamine, TO, or triethylamine, TE) [10–12]. A complete and general mechanism for these systems has been described (Scheme 1) [10–12].

and refer to dyeamine complexes which may be formed in the ground state, or in the excited states as singlet and triplet exciplexes. An alternative pathway including a triplex (intermediate present in this kind of reactions due to the high concentration of amine present) is also introduced in the scheme [12]. In most studies, the radicals originated from amines have been proved to be responsible for the chain initiation in many photoinitiated polymerizations. Also, the interaction between the singlet state of and MMA does not lead to polymerization and the decrease of the polymerization rates, as well as that of the dye semireduced radical yield, observed at high amine concentrations, indicates that excited dye molecules will be increasingly quenched by the amine, precluding their conversion to triplets which originate the active radicals leading to polymerization. Due to multiplicity restrictions, singlet exciplexes are not supposed to lead to separated radicals. The semithionine species () appears in the reaction scheme after both the proton and electron transfer stages which generate the amine radical by deprotonation of the cation radical of the amines, [11, 12].

Similarly to preliminary work [27], the starting point for the
mathematical approach is the variation of amine radical concentration (
in time, which interacts with the monomer (MMA) and initiates a sequence of
events leading to polymerization. Consequently, the values of chemical
variables related with this process time dependent are as follows: amine and
monomer concentrations, the conversion of monomer to polymer. From this point,
it was supposed that the reduction of rate of the monomer concentration is
related with the initial value of conversion, of the amine concentration, and
of the monomer in accordance with where is an nondimensional
constant, resulted from the solution of the differential equation, *fc*, and are conversion
function amine concentration and monomer concentration,
respectively. The conversion function can be interpreted as
the behavior of the polymerization or the proportionality of conversion that
determines how the
monomer is increased
or decreased. Solving this equation leads to (5). This supposition will be explained in
the next section.

#### 3. The Modeling

The chemical assumptions of the model presented here follow the same mathematical structure of the dynamical model proposed by Magini and Rodrigues [27] and use the same equations for radical chain photopolymerization systems. The intention of this study is to achieve a global approach for photopolymerization systems and understand how the interaction between the chemical variables is.

To explain the radical chain photopolymerization, here reported, a set of equations that describes the conversion function, through variation of time for two different systems of dye-amine in diverse concentrations, was used.

*Assumption 1. *The polymerization
reaction is initiated from interaction between
species and the monomer MMA (M), which causes a decreasing of amine radical
concentration. This species
is produced by the interaction between light, and amine.
Independently of amine (A) and thionine () initial
concentrations, in the final portion of process and immediately before the
interaction of with monomer (Scheme 1), the
concentrations of and are
numerically equal.
The process of polymerization is connected with [ and *fc*. Equation (3) comes from the
information above:

It has been assumed that species concentration has an exponential variation (directly dependent of the
initial value of the conversion function, represented here by ) with a
scale parameter ,
which determines how fast is the variation of in time. Notice that this variable is consumed
when time () is increasing. The term provides a smooth decreasing of . This variable is analogous to the introduced in the work of Magini and
Rodrigues [27].

*Assumption 2. *Amine is the chemical
variable that determines the efficiency of the photopolymerization processes.
By analyzing the value of and its influence on
the polymerization induced by light, it can be concluded that when assumes null value for
, the photopolymerization processes do not occur, mathematically represented in (4).
This hypothesis results in the final form of the conversion function given by
(6), where the term multiplies all terms. If
the value of is null for , (4)
assumes the value zero and (5) assumes a constant value:

*Assumption 3. *The variation of monomer
concentration has exponential dependency with the values of ,
the variable that limits this reaction, and .
This equation is consequence of the solution of (1) for a range of time values
: where is a parameter of , and is the value of monomer concentration when the process of polymerization is
started. This supposition is correlated with the complexity of the mechanism of
photopolymerization, Scheme 1, and with the work proposed by Magini and
Rodrigues [27] that follows the same complexity; when the function *fc* grows,
the monomer and the amine are consumed; when the values of *fc* and are maximum, the value achieves its minimum
value.

The values of and parameters are resulted from solution of each differential ordinary equation that models the variation of species and [A]. The result from (1) is related with the initial values of amine concentration and . The differential equation below is the final form proposed to determine the conversion function, in time, give in :

The constant has dimension of mol-second per Einstein and can be interpreted as the number of mol that absorbs radiation in time. The term (Einstein per liter-second), called quantum yield of polymerization, is defined as the number of monomer units polymerized per absorbed photon and incorporates the effects of the light intensity in (6). This equation is the simplest form assumed to explain the conversion function. The direct sum of the terms can be considered the most basic construction for (6) and the multiplication by the term reveals that has to be null when amine concentration is zero. The amine concentration is the precursor of the entire process, and consequently its value is directly connected with the initial value of . For , , and as a result the value of is null.

The results obtained here allow generalizing the model for two distinct systems: cationic and free radical. This is a new way to see these systems, and the comments about this correlation will be done in the discussions. Values of [M], [A], and were computed simultaneously with the result given by the solution of the differential equation (6).

#### 4. Results and Discussions

Table 1 lists the polymerization rate , the polymerization quantum yield (), and the monomer conversions for the photopolymerization of MMA obtained at different TO concentrations. Values were obtained as previously described [12].

It can be seen that the polymerization rate increases and reaches a maximum value when TO concentrations are and , decreasing afterward. This decrease is ascribed mainly to the quenching of the singlet excited state, which decreases the amount of triplets produced by intersystem crossing and, therefore, the yield of initiating radicals. The same behavior has been described for various monomer/initiator/amine systems [10–12, 28, 29] and it suggests that the polymerization is originated from the interaction of the amine with the triplet excited state of the dye. The amine radical , resulted from the chain initiation, comes form bleaching of the dye during the photoinitiated polymerization and it originates from reactions of the semithionine species () after both the proton and electron transfer stages.

The simulations of the polymerization rate for TO using the experimental values for concentration and polymerization quantum yield, Table 1, are showen in Figure 1, where reproduced by the approach follows an approximated pattern when compared with the experimental results. Similarly, in Figure 1, it is possible to see an optimum value of the concentration of TO and quantum yield to achieve higher values for these measures.

In Figure 1, the values were calibrated to accept the numerical discrepancy between the values of , conversion, and time. The is resulted from the rate between the conversion function and the monomer concentration. The results show the behavior through the time variation. The results presented in Figure 1 confirm the coherence of this approach to classify and determine the general behavior of chemical variables, which are directly influenced by the photopolymerization processes.

Numerical tests were realized for TO with different values of polymerization quantum yield. In Figure 2, the theoretical behaviors of the conversion function for different values of quantum yields and time are showed. The starting values of monomer concentration for all curves were the same, (experimental value). This initial value was fixed to study the influence of monomer variation under of the quantum yield. It is possible to see the relation between high values of directly related with the high values of quantum yield, which is experimentally expected.

Figure 3 describes the conversion function versus the variation of monomer concentration for different values of [TO], which is fixed for each curve. The results showed are in a straight accordance with the experimental data, Table 1. The maximum value of the conversion function is obtained when the . For values bigger than , the value of decay. In Table 1, this fact is verified, for , the conversion is 0.043 (maximum), for , , and , the values of conversion are 0.037, 0.025, and 0.016, respectively. The curve in Figure 3 follows the same pattern.

The previous experimental results obtained to polymerization of MMA
photoinitiated by ThH^{+}/TE [12] are here comparing with theoretical behavior.
The initial study was completed generating a set of theoretical tests that show
the relation between values of amine concentration, polymerization quantum
yield and monomer starting values. Figure 4 shows that theoretical and
experimental values for of TE are well correlated.

#### 5. Conclusions

Both experimental and simulated data were compared, and the correlation between their values was verified (Figures 1, 2, and 4). The was theoretically calculated, and it value is in accordance with the experimental results proving the capacity of adaptation of this simulation for these systems (free radical). The same occurs for the conversion function at different values of [TO], where the theoretical fit reproduces the behavior of the experimental data, as shown in Table 1. , and are adjust parameters, related the initial concentration of chemicals. The chemical parameters were completely described in this dynamical approach, as the results show. The assumptions show clearly how the concentrations vary through the time.

As in the previous work [27], an only function describes the system, and in a deeper view was proved that it is possible to establish a theoretical adjust with two systems of polymerization: cationic and free radical. The mechanism logic that induces the polymerization process follows similar patterns for these two systems. As a consequence, the model constructed for cationic systems can be adjusted for free radical systems.

#### Acknowledgment

The authors would like to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) for financial support

#### References

- D. R. Randall,
*Radiation Curing of Polymers II*, Royal Society of Chemistry, Cambridge, UK, 1991. - D.C. Sherrington and P. Hodge,
*Synthesis and Separations Using Functional Polymers*, John Wiley & Sons, New York, NY, USA, 1988. - J. P. Fouassier,
*Photoinitiation, Photopolymerization and Photocuring: Fundamentals and Applications*, Hanser, Munich, Germany, 1995. - P. Pappas,
*UV Curing Science and Technology*, Technology Marketing, New York, NY, USA, 1992. - F. Higashi and S. Koyanagi, “The effect of compositions and combinations of the reactants upon the copolycondensation of isophthalic acid/terephthalic acid with a combination of hydroquinones and bisphenols by tosyl chloride/dimethylformamide/pyridine,”
*Journal of Polymer Science Part A*, vol. 42, no. 5, pp. 1100–1106, 2004. View at Publisher · View at Google Scholar - X. Allonas, J. Lalevée, and J.-P. Fouassier, “Investigation of cleavage processes in photoinitiators: from experiments to molecular modeling,”
*Journal of Photochemistry and Photobiology A*, vol. 159, no. 2, pp. 127–133, 2003. View at Publisher · View at Google Scholar - M. G. Neumann and M. R. Rodrigues, “A study of the elemental reactions involved in the initiation of the polymerization of tetrahydrofuran inducted by the photosensitization of a triphenylsulphonium salt by perylene,”
*Journal of the Brazilian Chemical Society*, vol. 14, no. 1, pp. 76–82, 2003. View at Publisher · View at Google Scholar - M. R. Rodrigues and M. G. Neumann, “Cationic photopolymerization of tetrahydrofuran: a mechanistic study on the use of a sulfonium salt-phenothiazine initiation system,”
*Journal of Polymer Science Part A*, vol. 39, no. 1, pp. 46–55, 2001. View at Publisher · View at Google Scholar - M. R. Rodrigues and M. G. Neumann, “Mechanistic study of tetrahydrofuran polymerization photoinitiated by a sulfonium salt/thioxanthone system,”
*Macromolecular Chemistry and Physics*, vol. 202, no. 13, pp. 2776–2782, 2001. View at Publisher · View at Google Scholar - M. G. Neumann and M. R. Rodrigues, “The mechanism of the photoinitiation of the polymerization of MMA by the thionine-triethanolamine system,”
*Polymer*, vol. 39, no. 8-9, pp. 1657–1661, 1998. View at Publisher · View at Google Scholar - M. R. Rodrigues, F. Catalina, and M. G. Neumann, “Photoinitiation of MMA polymerization by thionine/amine systems in the presence of SDS micelles,”
*Journal of Photochemistry and Photobiology A*, vol. 124, no. 1-2, pp. 29–34, 1999. View at Publisher · View at Google Scholar - M. R. Rodrigues, F. Catalina, and M. G. Neumann, “MMA photopolymerization initiated by thionine/triethylamine,”
*Journal of Photochemistry and Photobiology A*, vol. 127, no. 1–3, pp. 147–152, 1999. View at Publisher · View at Google Scholar - M. H. Gehlen, S. F. Buchviser, M. R. Rodrigues, and M. G. Neumann, “Photoinitiation of vinyl polymerization,”
*Química Nova*, vol. 21, no. 6, pp. 794–801, 1998. View at Google Scholar - M. R. Rodrigues and M. G. Neumann, “Fotopolimerização: princípios e métodos,”
*Polímeros: Ciência e Tecnologia*, vol. 13, no. 4, pp. 277–287, 2003. View at Google Scholar - P. Bosch, A. Fernández, E. F. Salvador, T. Corrales, F. Catalina, and C. Peinado, “Polyurethane-acrylate based films as humidity sensors,”
*Polymer*, vol. 46, no. 26, pp. 12200–12209, 2005. View at Publisher · View at Google Scholar - G. Odian,
*Principles of Polymerization*, John Wiley & Sons, Hoboken, NY, USA, 2004. - R. Dickman,
*Nonequilibrium Statiscal Mechanics in One Dimension*, Cambridge University Press, Cambridge, UK, 1997. - J. A. Corzo E, N. Davidenko, and R. Alvarez, “Photoinitiated bulk polymerization of furfuryl methacrylate. Stochastic simulation results using the new model CORUB,”
*Computational and Theoretical Polymer Science*, vol. 11, no. 2, pp. 89–94, 2001. View at Publisher · View at Google Scholar - J. Lange, J. Rieumont, N. Davidenko, and R. Sastre, “Kinetics modelling of the crosslinking in the photopolymerization of furfuryl methacrylate in bulk,”
*Computational and Theoretical Polymer Science*, vol. 9, no. 1, pp. 63–72, 1999. View at Publisher · View at Google Scholar - O. L. Bottecchia, “A model of the growth of copper selenide thin films controlled by diffusion and chemical reaction,”
*Journal of the Brazilian Chemical Society*, vol. 9, no. 6, pp. 515–520, 1998. View at Publisher · View at Google Scholar - J. R. Pliego Jr., A. F. de C. Alcântara, D. P. Veloso, and W. B. de Almeida, “Theoretical and experimental investigation of the formation of $E$- and $Z$-aldimines from the reaction of methylamine with acetaldehyde,”
*Journal of the Brazilian Chemical Society*, vol. 10, no. 5, pp. 381–388, 1999. View at Publisher · View at Google Scholar - L. Krentsel, F. Chaubet, A. Rebrov et al., “Anticoagulant activity of functionalized dextrans. Structure analyses of carboxymethylated dextran and first Monte Carlo simulations,”
*Carbohydrate Polymers*, vol. 33, no. 1, pp. 63–71, 1997. View at Publisher · View at Google Scholar - L. M. Racz, L. Li, and B. Abedian, “Cure kinetics of light-activated polymers,”
*Journal of Polymer Science Part B*, vol. 36, no. 16, pp. 2887–2894, 1998. View at Publisher · View at Google Scholar - G. A. Miller, L. Gou, V. Narayanan, and A. B. Scranton, “Modeling of photobleaching for the photoinitiation of thick polymerization systems,”
*Journal of Polymer Science Part A*, vol. 40, no. 6, pp. 793–808, 2002. View at Publisher · View at Google Scholar - L. Lecamp, B. Youssef, C. Bunel, and P. Lebaudy, “Photoinitiated polymerization of a dimethacrylate oligomer: 2. Kinetic studies,”
*Polymer*, vol. 40, no. 6, pp. 1403–1409, 1999. View at Publisher · View at Google Scholar - T. M. Lovestead, J. A. Burdick, K. S. Anseth, and C. N. Bowman, “Understanding multivinyl monomer photopolymerization kinetics through modeling and GPC investigation of degradable networks,”
*Polymer*, vol. 46, no. 16, pp. 6226–6234, 2005. View at Publisher · View at Google Scholar - M. Magini and M. R. Rodrigues, “A dynamical system to describe the cationic photopolymerization of tetrahydrofuran initiated by systems sensitizer-sulfonium salt,”
*Polymer*, vol. 46, no. 10, pp. 3489–3495, 2005. View at Publisher · View at Google Scholar - J. Lalevée, X. Allonas, and J.-P. Fouassier, “N-H and $\alpha $(C-H) bond dissociation enthalpies of aliphatic amines,”
*Journal of the American Chemical Society*, vol. 124, no. 32, pp. 9613–9621, 2002. View at Publisher · View at Google Scholar - T. Corrales, F. Catalina, C. Peinado, and N. S. Allen, “Free radical macrophotoinitiators: an overview on recent advances,”
*Journal of Photochemistry and Photobiology A*, vol. 159, no. 2, pp. 103–114, 2003. View at Publisher · View at Google Scholar