# Reactivity ratio models for MMA/ alpha-MS at elevated temperatures.

The MMA/[Alpha]-MS system has long been used as a theoretical study
case due to its unusual behaviour above 60 [degrees] C, influenced
mainly by the low ceiling temperature of [Alpha]-MS. Researchers have
discussed reactivity ratios and copolymer composition data, but recent
industrial interest has prompted a thorough review of the models
proposed to handle the reversible propagation steps.

The work on the Methyl Methacrylate/[Alpha]-Methyl Styrene (MMA/[Alpha]-MS) copolymer system started as a pure experimental investigation, but quickly expanded to include theoretical work, as people became increasingly interested in the properties of the system.

In the first step of our investigation, we collected experimental data for five temperature levels (60, 80, 100, 120, and 140 [degrees] C). Ampoules with 2 mL volumes of monomer and initiator (AIBN or Triginox B) were prepared with feeds varying from 10-90 wt% MMA. In all cases, the conversion was kept below five per cent by using strict time control in an isothermal heating medium. The control on conversion is essential, as the models used to predict copolymer composition are differential in form. The small amount of polymer being produced allows estimation of the derivative properties.

To determine copolymer composition, 1H NMR was used. Good peak resolution was obtained by operating on a Bruker AM-300 MHz machine at room temperature. Deuteroacetone and deuterochloroform were both used as solvents for the copolymer.

Theoretical Development

What makes the MMA/[Alpha]-MS system interesting is the presence of a back-propagation or depropagation step. The details are rooted in simple thermodynamics.

As with every chemical reaction, the propagation mechanism commonly associated with free radical polymerization not only has a forward rate constant ([k.sub.p]), but a reverse rate constant ([k.sub.dp]) as well. The proper way to describe the addition of a monomer unit (M) to a growing radical ([R.sup.*]) is:

[Mathematical Expression Omitted] (1)

In most cases the reverse step (or depropagation) can be ignored. However, the driving force in any reaction is the desire to reduce the Gibbs free energy in a system (G). The change in Gibbs free energy for a polymerization is related to the change in enthalpy ([Delta][H.sub.p]) and the change in entropy ([Delta][S.sub.p]) by the relationship:

[Delta]G = [Delta][H.sub.p] - T[Delta][S.sub.p] (2)

To be favoured, the [Delta]G of a reaction must be negative. For every monomer there exists a temperature, known as the ceiling temperature ([T.sub.c]), for which this is not the case. Once the [T.sub.c] is reached, the propagation step is no longer favoured, and the reverse reaction dominates. For [Alpha]-MS, this temperature is listed as 61 [degrees] C, making homopolymerization impossible. Copolymerization is possible, with a monomer such as MMA, with a [T.sub.c] of about 220 [degrees] C. The reverse reaction also plays a role at temperatures below the [T.sub.c] of a monomer, but the effect is limited.

Historically, the composition of a copolymer has been related to the values of parameters known as reactivity ratios. The Mayo-Lewis equation uses the definition of these ratios to yield a differential form that predicts the ratio of the uptake of one monomer (M1) to another (M2) in the copolymer based on the molar concentrations [M] in the feed.

d[[M.sub.1]]/d[[M.sub.2]] = [[M.sub.1]]([r.sub.1][[M.sub.1]] + [[M.sub.2]])/[[M.sub.2]]([[M.sub.1]] + [r.sub.2][[M.sub.2]]) (3)

The Mayo-Lewis equation works very well for systems that follow irreversible behaviour. For the [Alpha]-MS/MMA system at temperatures above 60 [degrees] C, it is expected that more robust models would be required. The first attempt at taking the reversible nature of a copolymerization system into account was by Lowry [2]). He developed a model that took into account only the depropagation of one monomer in a copolymer system. Further to his work, Wittmer [4] produced a very comprehensive set of equations that took the frilly reversible nature of the MMA/[Alpha]-MS system into account. The general form of his equation is as follows:

[Mathematical Expression Omitted] (4)

with

[r.sub.1] = [k.sub.11]/[k.sub.12]; [r.sub.2] = [k.sub.22]/[k.sub.21] (5)

[Mathematical Expression Omitted] (6)

[r.sub.1] and [r.sub.2] still represent the same ratio of rate constants as in the Mayo-Lewis equation, and have the same physical significance (reactivity ratios). [q.sub.1] and [q.sub.2] are included to account for the reversibility of the cross-propagation reactions.

More recently, Kruger et al. [1]. developed a new set of equations designed to tackle the same problem. The equations use f to represent the molar ratio of the monomers in the copolymer, and [A] and [B] to indicate the concentration of the monomers in solution. They take on the following form:

[Mathematical Expression Omitted] (7)

with

[r.sub.A] = [k.sub.11]/[k.sub.12]; [r.sub.B] = [k.sub.22]/[k.sub.21] (8)

[Mathematical Expression Omitted] (9)

Both sets of equations also have corollary expressions that have not been included here for the sake of brevity. In both cases an iterative approach is required to solve for the copolymer composition given all the reactivity ratios and feed compositions.

Experimental Data

Figure 1 shows copolymer composition data measured at three temperature levels. The abscissa shows the mole fraction of the [Alpha]-MS in the feed, while the ordinate is the molar ratio of [Alpha]MS to MMA in the copolymer. Note that the amount of [Alpha]-MS incorporated into the copolymer drops significantly as the temperature increases.

The first attempt to find reactivity ratio data for the system involved fitting the experimental data to the Mayo-Lewis equation using the RREVM program (Polic et al.,)[3]. While the program gave reasonable values for the reactivity ratios at 60 [degrees] C, as the temperature increased it became obvious that there was a problem. Figure 2 shows the best fit that could be obtained to the 140 [degrees] C data using the Mayo-Lewis equation. Not only is the fit not acceptable, but also it led to negative values for the [Alpha]-MS reactivity ratio. This indicated that the model structure was not satisfactory, and hence the more complete models by Kruger et al. [1] and Wittmer [4] were investigated next.

The Wittmer equations were applied first, and the results were understandably better. The model is essentially a 6-parameter set of equations, meaning it is much more flexible. The equations can be simplified to handle cases where some of the propagation reactions are irreversible. Non-linear least squares (NLLS) techniques were required to come up with reasonable parameter values. Figure 3 shows two examples of how the Wittmer equations are able to describe the behaviour of the copolymer composition data.

The Kruger equations were applied next. While the newer model was developed more from a probabilistic standpoint, the end result appears to be identical with the Wittmer equations. In all cases there is no difference in the predicted copolymer composition given a particular set of reactivity ratio values. However, there is a big advantage to using the Kruger equations. While the results are the same as with the Wittmer model, the Kruger model is much more stable and easier to use over a broad range of feed conditions. The auxiliary equations included in the Wittmer paper are rather ill-conditioned and will not converge at high [Alpha]-MS feed levels. Although there is no difference in the two models, use of the Kruger model is strongly recommended.

Sources of Experimental Error

There are always sources of experimental error when performing any study, but in this particular case the errors proved to be large enough to cause severe problems when it came to finding reliable reactivity ratio estimates.

There were two main sources of error that had to be considered. The first was simple NMR measurement error. Previous experience had led us to believe that 5 per cent is a reasonable magnitude for the error in any given sample measurement. based on this concern, replicate trials were done at every feed level in an attempt to mitigate this problem. In some cases, certain feeds were redone if a problem was suspected. Overall, the effect of NMR error can most likely be absorbed into the non-linear regression error structure.

The second source of error was far more of a problem. A careful analysis of the data showed that the copolymer composition was not only a function of the feed composition, but also of the conversion level obtained, even at levels below 5%. For the 90 mol % [Alpha]-MS feeds, it was possible for the cumulative copolymer composition to change by almost three per cent in the first five per cent of conversion. Such an error creates a false indication of the differential behaviour of the system, and can have a big impact on the parameters predicted by non-linear least squares.

A sensitivity study highlights why this error could not be ignored. Two cases were run at 120 [degrees] C, using the Wittmer equations, with the only difference being the copolymer composition at a single point (out of nine in total). Table 1 shows the results of NLLS when the point being used is from a copolymer sample measured at 1 per cent conversion. Table 2 shows the results when the sample was taken at five per cent conversion. The same feed composition is used in both cases. Not only is there a large difference in the magnitude of the parameter values, but also the set for the one per cent conversion case was much more in line with the predictions at the other temperature levels. based on this finding we were able to obtain a much more accurate set of reactivity ratio values by imposing even stricter controls on the level of conversion that was allowed.

Future Steps

Now that it has been determined that either the Kruger or Wittmer equations give satisfactory results for the [Alpha]-MS/MMA copolymer system, the experimental investigation will be expanded to repeat the experimental data collection for the solution phase (30 wt% toluene). It is expected that this will yield similar parameter values as for the bulk phase investigation.

The final step will be a complete study of the behaviour of this system at high conversion levels at elevated temperatures. It is hoped that the reactivity ratio values obtained at low conversions can be used in a computer simulation to predict full conversion composition results.

Concluding Remarks

An analysis of possible models that could be used to describe the copolymer composition behaviour of the MMM/[Alpha]-MS copolymer system has been conducted. It was found that both the Wittmer and Kruger models are able to predict the behaviour of the system at low conversion levels. The Kruger equations are preferred, as they tend to be more stable and easier to work with.

The main source of error in the experimental data is composition drift with conversion. Unlike in most irreversible systems, the conversion levels must be kept well below 5 per cent (the lower the better), to give an accurate representation of the differential equations.

Table 1. Parameter estimates obtained from sample with 1% conversion.

[q.sub.1] = 0.0000 [q.sub.2] = 13.6266

[r.sub.1] = 0.0093 [r.sub.2] = 0.3402

Table 2. Parameter estimates obtained from sample with 5% conversion.

[q.sub.1] = 2.2749 [q.sub.2] = 3.7000

[r.sub.1] = 0.0010 [r.sub.2] = 0.5790

Acknowledgements

Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada and ICI, worldwide for this research is gratefully acknowledged. Special thanks to Marc Dube, MCIC for his initial screening work in the reactivity ratio parameter estimation.

References

1. Kruger, H., J. Bauer, J. Rubner, 'Ein Modell zur Beschreibung reversibler Copolymerisationen', Makromol. Chem., 188:2163-2175, 1987.

2. Lowry, G.G., 'The Effect of Depropagation on Copolymer Composition. I. General Theory for One Depropagating Monomer', J. Poly. Sci., 42:463-477, 1960.

3. Polic, A.L., T.A. Duever and A. Penlidis, 'Case Studies and Literature Review on the Estimation of Copolymerization Reactivity Ratios', J. Poly. Sci., Poly. Chem., 36:813-822, 1998.

4. Wittmer, P., 'Copolymerization in the Presence of Depolymerization Reactions', Adv. Chem., 99:140-174, 1971.

D.E. Palmer is a graduate student in the department of chemical engineering at the University of Waterloo, Waterloo, ON. N.T. McManus, MCIC and Alexander Penlidis, FCIC are professors of chemical engineering at the University of Waterloo.

This article was selected from the Polymer Science and Engineering Sessions at the London CSChE Conference (1998) as one of the two "best papers given by a graduate student co-author".

The work on the Methyl Methacrylate/[Alpha]-Methyl Styrene (MMA/[Alpha]-MS) copolymer system started as a pure experimental investigation, but quickly expanded to include theoretical work, as people became increasingly interested in the properties of the system.

In the first step of our investigation, we collected experimental data for five temperature levels (60, 80, 100, 120, and 140 [degrees] C). Ampoules with 2 mL volumes of monomer and initiator (AIBN or Triginox B) were prepared with feeds varying from 10-90 wt% MMA. In all cases, the conversion was kept below five per cent by using strict time control in an isothermal heating medium. The control on conversion is essential, as the models used to predict copolymer composition are differential in form. The small amount of polymer being produced allows estimation of the derivative properties.

To determine copolymer composition, 1H NMR was used. Good peak resolution was obtained by operating on a Bruker AM-300 MHz machine at room temperature. Deuteroacetone and deuterochloroform were both used as solvents for the copolymer.

Theoretical Development

What makes the MMA/[Alpha]-MS system interesting is the presence of a back-propagation or depropagation step. The details are rooted in simple thermodynamics.

As with every chemical reaction, the propagation mechanism commonly associated with free radical polymerization not only has a forward rate constant ([k.sub.p]), but a reverse rate constant ([k.sub.dp]) as well. The proper way to describe the addition of a monomer unit (M) to a growing radical ([R.sup.*]) is:

[Mathematical Expression Omitted] (1)

In most cases the reverse step (or depropagation) can be ignored. However, the driving force in any reaction is the desire to reduce the Gibbs free energy in a system (G). The change in Gibbs free energy for a polymerization is related to the change in enthalpy ([Delta][H.sub.p]) and the change in entropy ([Delta][S.sub.p]) by the relationship:

[Delta]G = [Delta][H.sub.p] - T[Delta][S.sub.p] (2)

To be favoured, the [Delta]G of a reaction must be negative. For every monomer there exists a temperature, known as the ceiling temperature ([T.sub.c]), for which this is not the case. Once the [T.sub.c] is reached, the propagation step is no longer favoured, and the reverse reaction dominates. For [Alpha]-MS, this temperature is listed as 61 [degrees] C, making homopolymerization impossible. Copolymerization is possible, with a monomer such as MMA, with a [T.sub.c] of about 220 [degrees] C. The reverse reaction also plays a role at temperatures below the [T.sub.c] of a monomer, but the effect is limited.

Historically, the composition of a copolymer has been related to the values of parameters known as reactivity ratios. The Mayo-Lewis equation uses the definition of these ratios to yield a differential form that predicts the ratio of the uptake of one monomer (M1) to another (M2) in the copolymer based on the molar concentrations [M] in the feed.

d[[M.sub.1]]/d[[M.sub.2]] = [[M.sub.1]]([r.sub.1][[M.sub.1]] + [[M.sub.2]])/[[M.sub.2]]([[M.sub.1]] + [r.sub.2][[M.sub.2]]) (3)

The Mayo-Lewis equation works very well for systems that follow irreversible behaviour. For the [Alpha]-MS/MMA system at temperatures above 60 [degrees] C, it is expected that more robust models would be required. The first attempt at taking the reversible nature of a copolymerization system into account was by Lowry [2]). He developed a model that took into account only the depropagation of one monomer in a copolymer system. Further to his work, Wittmer [4] produced a very comprehensive set of equations that took the frilly reversible nature of the MMA/[Alpha]-MS system into account. The general form of his equation is as follows:

[Mathematical Expression Omitted] (4)

with

[r.sub.1] = [k.sub.11]/[k.sub.12]; [r.sub.2] = [k.sub.22]/[k.sub.21] (5)

[Mathematical Expression Omitted] (6)

[r.sub.1] and [r.sub.2] still represent the same ratio of rate constants as in the Mayo-Lewis equation, and have the same physical significance (reactivity ratios). [q.sub.1] and [q.sub.2] are included to account for the reversibility of the cross-propagation reactions.

More recently, Kruger et al. [1]. developed a new set of equations designed to tackle the same problem. The equations use f to represent the molar ratio of the monomers in the copolymer, and [A] and [B] to indicate the concentration of the monomers in solution. They take on the following form:

[Mathematical Expression Omitted] (7)

with

[r.sub.A] = [k.sub.11]/[k.sub.12]; [r.sub.B] = [k.sub.22]/[k.sub.21] (8)

[Mathematical Expression Omitted] (9)

Both sets of equations also have corollary expressions that have not been included here for the sake of brevity. In both cases an iterative approach is required to solve for the copolymer composition given all the reactivity ratios and feed compositions.

Experimental Data

Figure 1 shows copolymer composition data measured at three temperature levels. The abscissa shows the mole fraction of the [Alpha]-MS in the feed, while the ordinate is the molar ratio of [Alpha]MS to MMA in the copolymer. Note that the amount of [Alpha]-MS incorporated into the copolymer drops significantly as the temperature increases.

The first attempt to find reactivity ratio data for the system involved fitting the experimental data to the Mayo-Lewis equation using the RREVM program (Polic et al.,)[3]. While the program gave reasonable values for the reactivity ratios at 60 [degrees] C, as the temperature increased it became obvious that there was a problem. Figure 2 shows the best fit that could be obtained to the 140 [degrees] C data using the Mayo-Lewis equation. Not only is the fit not acceptable, but also it led to negative values for the [Alpha]-MS reactivity ratio. This indicated that the model structure was not satisfactory, and hence the more complete models by Kruger et al. [1] and Wittmer [4] were investigated next.

The Wittmer equations were applied first, and the results were understandably better. The model is essentially a 6-parameter set of equations, meaning it is much more flexible. The equations can be simplified to handle cases where some of the propagation reactions are irreversible. Non-linear least squares (NLLS) techniques were required to come up with reasonable parameter values. Figure 3 shows two examples of how the Wittmer equations are able to describe the behaviour of the copolymer composition data.

The Kruger equations were applied next. While the newer model was developed more from a probabilistic standpoint, the end result appears to be identical with the Wittmer equations. In all cases there is no difference in the predicted copolymer composition given a particular set of reactivity ratio values. However, there is a big advantage to using the Kruger equations. While the results are the same as with the Wittmer model, the Kruger model is much more stable and easier to use over a broad range of feed conditions. The auxiliary equations included in the Wittmer paper are rather ill-conditioned and will not converge at high [Alpha]-MS feed levels. Although there is no difference in the two models, use of the Kruger model is strongly recommended.

Sources of Experimental Error

There are always sources of experimental error when performing any study, but in this particular case the errors proved to be large enough to cause severe problems when it came to finding reliable reactivity ratio estimates.

There were two main sources of error that had to be considered. The first was simple NMR measurement error. Previous experience had led us to believe that 5 per cent is a reasonable magnitude for the error in any given sample measurement. based on this concern, replicate trials were done at every feed level in an attempt to mitigate this problem. In some cases, certain feeds were redone if a problem was suspected. Overall, the effect of NMR error can most likely be absorbed into the non-linear regression error structure.

The second source of error was far more of a problem. A careful analysis of the data showed that the copolymer composition was not only a function of the feed composition, but also of the conversion level obtained, even at levels below 5%. For the 90 mol % [Alpha]-MS feeds, it was possible for the cumulative copolymer composition to change by almost three per cent in the first five per cent of conversion. Such an error creates a false indication of the differential behaviour of the system, and can have a big impact on the parameters predicted by non-linear least squares.

A sensitivity study highlights why this error could not be ignored. Two cases were run at 120 [degrees] C, using the Wittmer equations, with the only difference being the copolymer composition at a single point (out of nine in total). Table 1 shows the results of NLLS when the point being used is from a copolymer sample measured at 1 per cent conversion. Table 2 shows the results when the sample was taken at five per cent conversion. The same feed composition is used in both cases. Not only is there a large difference in the magnitude of the parameter values, but also the set for the one per cent conversion case was much more in line with the predictions at the other temperature levels. based on this finding we were able to obtain a much more accurate set of reactivity ratio values by imposing even stricter controls on the level of conversion that was allowed.

Future Steps

Now that it has been determined that either the Kruger or Wittmer equations give satisfactory results for the [Alpha]-MS/MMA copolymer system, the experimental investigation will be expanded to repeat the experimental data collection for the solution phase (30 wt% toluene). It is expected that this will yield similar parameter values as for the bulk phase investigation.

The final step will be a complete study of the behaviour of this system at high conversion levels at elevated temperatures. It is hoped that the reactivity ratio values obtained at low conversions can be used in a computer simulation to predict full conversion composition results.

Concluding Remarks

An analysis of possible models that could be used to describe the copolymer composition behaviour of the MMM/[Alpha]-MS copolymer system has been conducted. It was found that both the Wittmer and Kruger models are able to predict the behaviour of the system at low conversion levels. The Kruger equations are preferred, as they tend to be more stable and easier to work with.

The main source of error in the experimental data is composition drift with conversion. Unlike in most irreversible systems, the conversion levels must be kept well below 5 per cent (the lower the better), to give an accurate representation of the differential equations.

Table 1. Parameter estimates obtained from sample with 1% conversion.

[q.sub.1] = 0.0000 [q.sub.2] = 13.6266

[r.sub.1] = 0.0093 [r.sub.2] = 0.3402

Table 2. Parameter estimates obtained from sample with 5% conversion.

[q.sub.1] = 2.2749 [q.sub.2] = 3.7000

[r.sub.1] = 0.0010 [r.sub.2] = 0.5790

Acknowledgements

Financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada and ICI, worldwide for this research is gratefully acknowledged. Special thanks to Marc Dube, MCIC for his initial screening work in the reactivity ratio parameter estimation.

References

1. Kruger, H., J. Bauer, J. Rubner, 'Ein Modell zur Beschreibung reversibler Copolymerisationen', Makromol. Chem., 188:2163-2175, 1987.

2. Lowry, G.G., 'The Effect of Depropagation on Copolymer Composition. I. General Theory for One Depropagating Monomer', J. Poly. Sci., 42:463-477, 1960.

3. Polic, A.L., T.A. Duever and A. Penlidis, 'Case Studies and Literature Review on the Estimation of Copolymerization Reactivity Ratios', J. Poly. Sci., Poly. Chem., 36:813-822, 1998.

4. Wittmer, P., 'Copolymerization in the Presence of Depolymerization Reactions', Adv. Chem., 99:140-174, 1971.

D.E. Palmer is a graduate student in the department of chemical engineering at the University of Waterloo, Waterloo, ON. N.T. McManus, MCIC and Alexander Penlidis, FCIC are professors of chemical engineering at the University of Waterloo.

This article was selected from the Polymer Science and Engineering Sessions at the London CSChE Conference (1998) as one of the two "best papers given by a graduate student co-author".

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Title Annotation: | Methyl Methacrylate/alpha-Methyl Styrene |
---|---|

Comment: | Reactivity ratio models for MMA/ alpha-MS at elevated temperatures.(Methyl Methacrylate/alpha-Methyl Styrene) |

Author: | Palmer, D.E.; McManus, N.T.; Penlidis, A. |

Publication: | Canadian Chemical News |

Geographic Code: | 1CANA |

Date: | May 1, 1999 |

Words: | 2130 |

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