# Dynamical Model to Describe the Interactions between  the Chemical

Authors: Marcio Magini; Máira R. Rodrigues
Journal: Research Letters in Organic Chemistry
 (2008)

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## 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.


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## Body

## 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 Rp, the rate of
polymerization, was calculated using the standard expression below [10, 12]:
(1)Rp=hF⋅t⋅f[M](mol⋅L−1⋅s−1), where h is the contraction in volume in the
capillary at time t(s), F=(100_(dp−dm)/dp) 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 (ThH+) 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].Scheme 1[ThH+⋅A] and [Th+⋯A] 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
[ThH⋯(A)2] (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 ThH+← 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 (ThH•) appears in the reaction scheme after both
the proton and electron transfer stages which generate the amine radical A•(–H+) by deprotonation of the cation radical of the
amines, A•+ [11, 12].Similarly to preliminary work [27], the starting point for the
mathematical approach is the variation of amine radical concentration (A•(–H+)
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 (2)−d[M]dt=δfc[A][M], where δ   is an nondimensional
constant, resulted from the solution of the differential equation, fc, [A] and [M] 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.


## 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 (ThH+) 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].Scheme 1[ThH+⋅A] and [Th+⋯A] 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
[ThH⋯(A)2] (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 ThH+← 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 (ThH•) appears in the reaction scheme after both
the proton and electron transfer stages which generate the amine radical A•(–H+) by deprotonation of the cation radical of the
amines, A•+ [11, 12].Similarly to preliminary work [27], the starting point for the
mathematical approach is the variation of amine radical concentration (A•(–H+)
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 (2)−d[M]dt=δfc[A][M], where δ   is an nondimensional
constant, resulted from the solution of the differential equation, fc, [A] and [M] 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.


## 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 (ThH+) 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].Scheme 1[ThH+⋅A] and [Th+⋯A] 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
[ThH⋯(A)2] (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 ThH+← 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 (ThH•) appears in the reaction scheme after both
the proton and electron transfer stages which generate the amine radical A•(–H+) by deprotonation of the cation radical of the
amines, A•+ [11, 12].Similarly to preliminary work [27], the starting point for the
mathematical approach is the variation of amine radical concentration (A•(–H+)
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 (2)−d[M]dt=δfc[A][M], where δ   is an nondimensional
constant, resulted from the solution of the differential equation, fc, [A] and [M] 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 betweenA•(–H+)
species and the monomer MMA (M), which causes a decreasing of amine radical
concentration. This species
is produced by the interaction between ThH+, light, and amine.
Independently of amine (A) and thionine (ThH+) initial
concentrations, in the final portion of process and immediately before the
interaction of A•(–H+) with monomer (Scheme 1), the
concentrations of ThH• and A•(–H+) are
numerically equal.
The process of polymerization is connected with [A•(–H+)] and fc. Equation (3) comes from the
information above: (3)[ThH•]t+1=[ThH•]tfc×exp(−ρt).
It has been assumed thatThH• species concentration has an exponential variation (directly dependent of the
initial value of the conversion function, represented here by fc) with a
scale parameter ρ,
which determines how fast is the variation of [ThH•] in time. Notice that this variable is consumed
when time (t) is increasing. The term fc provides a smooth decreasing of [ThH•]. This variable is analogous to the [Sens+•] 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[A] and its influence on
the polymerization induced by light, it can be concluded that when [A] assumes null value for
t=0, 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 ([A]t)1/2 multiplies all terms. If
the value of [A] is null for t=0, (4)
assumes the value zero and (5) assumes a constant value: (4)[A]t+1=[A]t×exp(−βt).Assumption 3.
The variation of monomer
concentration has exponential dependency with the values of[A],
the variable that limits this reaction, and fc.
This equation is consequence of the solution of (1) for a range of time values
t<t′<t+1: (5)[M]t+1=[M]t×exp(−fc[A]tδ), where δ  is a parameter of adjustii, and [M]t 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 [A]t are maximum, the [M]t value achieves its minimum
value.The values ofρand βparameters are resulted from solution of each
differential ordinary equation that models the variation of species [ThH•]
and [A]. The result
from (1) is related with the initial values of amine concentration and fc. The differential equation below is the final form proposed to determine the
conversion function, in time, give in mol⋅L−1: (6)dfcdt=([ThH•]t+αΦm+[M]t+fc)1/2([A]t)1/2.The constantαhas dimension of mol-second per Einstein and
can be interpreted as the number of mol that absorbs radiation in time. The
term Φm (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 [A]t reveals that fc 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 fc. For t=0, [A]t=0, and as a result the value
of fc 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[ThH•] were computed
simultaneously with the result given by the solution of the differential equation
(6).


## 4. Results and Discussions

Table1 lists
the polymerization rate (Rp), the polymerization
quantum yield (Φm), and the monomer conversions for the photopolymerization of
MMA obtained at different TO concentrations. Values were obtained as previously
described [12].Table 1
Experimental data for polymerization
of MMA(4.8mol⋅L−1) in methanol (1:1) at 30C°
photoinitiated by ThH+(0.8×10−5mol⋅L−1)/TO.
TO(mol⋅L−1)Rp(mol⋅L−1⋅s−1)×105ΦmConversion0.0502.14660.0130.103.021400.0390.203.821460.0430.253.281260.0370.302.33900.0250.351.63700.016It can be seen that the polymerization rate increases and reaches a
maximum value when TO concentrations are0.10mol⋅L−1 and 0.20mol⋅L−1,
decreasing afterward. This decrease is ascribed mainly to the quenching of the
ThH+ 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 A•(–H+), resulted from the chain
initiation, comes form bleaching of
the dye during the photoinitiated polymerization and it originates from
reactions of the semithionine species (ThH•) after both the proton and electron
transfer stages.The simulations of the polymerization rate(Rp) for TO using
the experimental values for concentration and polymerization quantum yield,
Table 1, are showen in Figure 1, where Rp 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.Figure 1
Simulated values for
velocity of polymerization of MMA,Rp, using data of Table 1 for
different concentrations of TO.In Figure1, the Rp values were calibrated to accept the
numerical discrepancy between the values of Rp, conversion, and
time. The Rp 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 Figure2, 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, 4.8 mol⋅L−1 (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 fc directly
related with the high values of quantum yield, which is experimentally
expected.Figure 2
Theoretical results for thefc of MMA
at different concentrations of TO and quantum yields Φm where Φ1<Φ2<⋯<Φ6.Figure3 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 [TO]=0.20mol⋅L−1. For values bigger than 0.20mol⋅L−1, the value of fc decay. In Table 1, this fact is
verified, for [TO]=0.20mol⋅L−1,
the conversion is 0.043 (maximum), for [TO]=0.25mol⋅L−1, [TO]=0.30mol⋅L−1, and [TO]=0.35mol⋅L−1, the values of conversion are 0.037, 0.025, and
0.016, respectively. The curve in Figure 3 follows the same pattern.Figure 3
Values offc through the decreasing of monomer
concentration. Different values of [TO] are plotted.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 0.1mol⋅L−1 of TE are well correlated.Figure 4
fc of MMA
photopolymerization in the presence of ThH+(0.8×10−5mol⋅L−1) and TE (0.1mol⋅L−1) systems. Experimental data
is represented by the symbol (•) and the numerical by the continuous line.


## 5. Conclusions

Both
experimental and simulated data were compared, and the correlation between
their values was verified (Figures1, 2, and 4). The Rp 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. α,ρ,δ, and βare 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.



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*Source: 404936-2009-01-25.xml*