# (722g) Kinetic Modeling of Atom Transfer Radical Polymerization: Linking Control to Reaction Rates

- Conference: AIChE Annual Meeting
- Year: 2008
- Proceeding: 2008 AIChE Annual Meeting
- Group: Materials Engineering and Sciences Division
- Session:
- Time:
Thursday, November 20, 2008 - 6:00pm-8:30pm

**1. Introduction**

Atom transfer radical polymerization (ATRP) is a recently developed and industrially attractive controlled radical polymerization technique (Matyjaszewski and Xia, 2001). It results in well-defined macromolecules, i.e. macromolecules with a controlled molecular mass, a controlled chain architecture and chain composition, a low polydispersity index and possible end group functionality in their structure. The latter allows further chemical modification of the polymer, such as the synthesis of block copolymers.

Figure 1 presents the principle of ATRP. Macroradicals (*R*) can be deactivated by a transition metal complex (*M _{t}^{n+1}L_{y}X_{2}*, deactivator) into a dormant form (

*RX*) by transfer of the end group functionality (

*X*) of the polymer from the deactivator to the macroradical. The end group functionality

*X*of the polymer is generally a halogen atom. The transfer of

*X*between the deactivator and the macroradical results in the formation of a transition metal complex (

*M*,

_{t}^{n}L_{y}X*activator) with a lower oxidation number for the transition metal. Reaction of*

_{}*RX*and

*M*regenerates the active form

_{t}^{n}L_{y}X*R*and allows an increase of the polymer chain length by propagation of

*R*with monomer (

*M*). Termination of radicals results in the formation of dead polymer molecules with loss of the end group functionality

*X*. However, for high deactivation reaction rates a low radical concentration compared to the concentration of the dormant species, and hence a low termination reaction rate, can be obtained. Therefore ATRP allows to synthesize polymers with a limited loss of end group functionality

*X*during the polymerization.

Figure 1: Principle of ATRP; k_{a}, k_{da}, k_{p} and k_{t} = rate coefficient of activation, deactivation, propagation and termination (m^{3}mol^{-1}s^{-1}); M_{t} = transition metal; n(+1)= oxidation number; L = Ligand; y = number; X = end group functionality of the polymer (halogen atom)

**2. Methodology**

In this contribution a methodology is presented for the kinetic modeling of ATRP that allows to simulate the monomer conversion and important structural characteristics of the polymer as a function of polymerization time and process conditions.

A first important structural characteristic of a polymer is its molecular mass distribution (MMD). The MMD of a polymer determines important physical properties of the polymer, such as the polymer strength and the polymer plasticity. The developed methodology allows to simulate the full MMD of the dormant polymer molecules and averages of the MMD of the polymer (i.e. dead and dormant polymer molecules) as a function of polymerization time and process conditions.

Secondly, for controlled radical polymerizations the living character of the polymer (i.e. the number of polymer molecules with end group functionality *X*) is important. The fraction of the polymer molecules having end group functionality *X,* *f _{p}*

*,*is defined as and calculated from:

_{}

In equation (1) [*RX _{tot}*] and [

*P*]

_{tot}*are the total concentration of respectively the dormant polymer molecules and the dead polymer molecules.*

In the kinetic model considered in this study, the most important reaction steps to describe ATRP (i.e. propagation, termination, activation, deactivation and chain transfer to monomer) are included. For all considered reaction steps diffusional limitations are systematically accounted for by introducing an apparent rate coefficient for each reaction step, consisting of a chemical and a diffusional contribution (de Kock et al., 2001). The required diffusion coefficients for the latter contribution are calculated using the free volume theory (Vrentas and Vrentas, 1998).

To solve the rate equations, a methodology based on the method of moments (Achilias, 1988) and on the quasi-steady state approximation for intermediate reactive species (Monteiro et al, 2001; Russell, 2002) has been developed. An extension of the method of moments is used allowing a better description of the contribution of the polymer molecules with a high molecular mass to the MMD of the polymer (i.e. the right side of the MMD of the polymer). Using this approach, moment dependent average apparent rate coefficients for all reaction steps involving macromolecules are introduced. For example the zeroth order average apparent rate coefficient for termination by recombination (<k_{tc},0>) is defined by:

_{}

In equation (2) *k _{tc}^{ij}*

*is the apparent rate coefficient for termination by recombination between a macroradical with chain length*

*i*and a macroradical with a chain length

*j*and [

*R*] the concentration of the macroradicals with chain length

_{i}*i*. The [

*R*] required for the calculation of the average apparent rate coefficients of the reaction steps can be obtained by application of the quasi-steady state approximation (QSSA) to all the macroradicals (i.e. by solving a set of algebraic equations).

_{i}However, the calculation of [*R _{i}*] requires also the knowledge of the concentration of the dormant polymer molecules with chain length

*i*

*,*[

*R*]. The latter cannot be obtained by application of the QSSA to the dormant polymer molecules, as they are not intermediate reactive species. Instead, [

_{i}X*R*] is calculated by directly solving the corresponding rate equations. Hence, both [

_{i}X*R*] and [

_{i}*R*] are obtained by solving a set of differential algebraic equations.

_{i}X**3. Results**

The developed methodology is applied to the batch isothermal ATRP of methylmethacrylate (MMA). The extent to which control of the ATRP is obtained is analyzed in terms of the rates of the occurring reaction steps. The ATRP of MMA was studied within a representative range of process conditions (polymerization temperature (*T _{pol}*) range: 313-363 K; volume fraction of solvent: 0-0.7; initial

*M*/

*R*molar ratio: 50-200).

_{0}XFigure 2 (a) and Figure 2 (b) present respectively the first three averages of the MMD of the polymer (i.e. *M _{n}^{pol}*,

*M*and

_{m}^{pol}*M*) and the

_{z}^{pol}*f*of the polymer as a function of monomer conversion (

_{p}*x*) and this for the following reaction conditions: a

_{m}*T*of 363 K, a volume fraction of solvent of 0.5, an initial

_{pol}*M*/

*R*/

_{0}X*M*/

_{t}^{n}X*L*molar ratio equal to 100/1/1/1 and no deactivator present at the beginning of the polymerization.

Figure 2: (a) First three averages of the MMD of the polymer (i.e. *M _{n}^{pol}*,

*M*

_{m}^{pol}and*M*;

_{z}^{pol}*full lines)*

*and ideal average number molecular mass (*

*M*; dashed line)

_{n}^{id}*as a function of monomer conversion (*

*x*) (b) Fraction of polymer molecules with end group functionality X (

_{m}*f*;

_{p}*equation (1)) as a function of monomer conversion (*

*x*);

_{m}It can be seen from Figure 2 (a) that all averages of the MMD of the polymer have a similar profile: a decrease at low *x _{m}*, followed by a linear increase at high

*x*. The linear increase of the first average of the MMD of the polymer (

_{m}*M*) at high

_{n}^{pol}*x*approaches the ideal number average molecular mass of the polymer (

_{m}*M*) as a function of

_{n}^{id}*x*(equation (3)).

_{m} _{}

In equation (3) *MM*_{A} (*A=M, R _{0}X*) is the molar mass of the reaction component

*A*and

*PD*the desired degree of polymerization at final monomer conversion.

^{id}From Figure 2(b) it follows that the *f _{p}*

_{}of the polymer (equation (1)) decreases after an initial increase as a function of

*x*. The

_{m}*f*profile is strongly related to the termination reaction rate, as every termination reaction results in loss of the end group functionality

_{p}*X*of the polymer. Figure 3 presents the termination reaction rate (

*R*) as a function of

_{term}*x*. It can be seen from this figure that

_{m}*R*is significant at low

_{term}*x*and then decreases rapidly as a function of

_{m}*x*. However, at high

_{m}*x*

_{m}_{}

*R*is still high enough to result in further loss of end group functionality of the polymer.

_{term}Figure 3: Termination reaction rate (*R _{term}*) as a function of monomer conversion (

*x*)

_{m}**Acknowledgements**

The Institute for the Promotion of Innovation through Science and Technology in

Flanders (IWT Vlaanderen) and the Belgian Government (IAP/IUAP/PAI P6/27: ''Functional Supramolecular Systems'') are acknowledged for financial support.

**References**

Achilias D., Kiparissides C. (1988) Modeling of diffusion-controlled free-radical polymerization reactions. *J. Appl. Polym. Sci*., 35: 1303

de Kock J.B.L., van Herk A.M., German A.L. (2001). Bimolecular free-radical termination at low conversion. *J. Macromol. Sci.-Pol. R*., C41(3):199

Matyjaszewski K. and Xia J.H. (2001). Atom Transfer Radical Polymerization. *Chem. Rev.*, 101: 2921

Monteiro M.J., Subramaniam N., Taylor J.R., Pham B.T.T., Tonge M.P. and Gilbert R.G. (2001). Retardative chain transfer in free radical polymerizations of vinyl *neo*-decanoate in low molecular weight polyisoprene and toluene*.** Polymer*, 42:2403

Russell G.T. (2002) The kinetics of free radical polymerizations: Fundamental aspects*. Aust. J. Chem.*, 55: 399

Vrentas J.S. and Vrentas C.M. (1998). Predictive methods for self-diffusion and mutual diffusion coefficients in polymer-solvent.* Eur. Polym. J.*, 34 (5/6):797