(217ap) Nonlinear Dynamics and Hopf Bifurcation in Controlled/"Living" Radical Polymerization of Styrene

Abstract

Approximately 50% of all synthetic polymers are currently made using radical polymerization (RP) processes.¹  However, the major drawback of the conventional radical polymerization is poor control of living chains, which leads to a broad molecular weight distribution.  In recent years, a new “living” free-radical technique has witnessed remarkable progress due to its potential to obtain highly controlled micro-structure, molecular weight, and polydispersity, and superior mechanical properties.

Several publications focus on the “living” nitroxide-mediated radical polymerization (NMRP) of styrene, based upon the experimental work of Georges and co-workers.² Greszta and Matyjaszewski proposed a kinetic scheme based on experimental data of bulk styrene polymerization.³  Matyjaszewski and co-workers sought to increase the rate of nitroxyl-mediated styrene polymerization and discussed the effect of different initiators,^4-6 whereas Georges and co-workers focused on the theoretical aspects of the process in the absence of an additional initiator.^{7, 8}  Cunningham and co-workers preserved the living character of nitroxide-mediated polymerizations through the addition of ascorbic acid or a free radical initiator.^{9, 10}

The aforementioned research improves the understanding of NMRP, however, most of the work focuses on finding new agents to convert a conventional free-radical polymerization system into a living one, or accelerating the polymerization in an acidic environment. 

Compared with previous studies, this work aims at investigating the nonlinear behavior arising due to the special “living” features in the nitroxide-mediated radical polymerization.  The Hopf bifurcation points and limit cycles, directly related to oscillatory behavior in this highly exothermic NMRP system, are identified using bifurcation analysis.  Furthermore, an approach to introducing a desired Hopf bifurcation point or modifying the frequency of the periodic behavior is discussed.

NMRP Kinetics and Multiplicity

The simulation implemented herein is based on the living-free-radical mechanism and mathematical model reported by Bonilla, et al.¹¹  The addition of a nitroxide “controller” molecule results in the reversible reaction of dormant-living exchange (for both monomeric and polymeric alkoxyamine).  However, it should be noticed that Bonilla’s isothermal model neglected energy balances involving heat removal.  Herein, ordinary differential equations for reactor and cooling-jacket temperatures have been included in the continuous stirred-tank reactor (CSTR) model of bulk polymerization using Arrhenius-like kinetics. 

Stationary bifurcation diagrams are obtained utilizing the continuation package, Cl_MATCONT, developed by Dhooge et al.¹²  The cooling water temperature, reactor feed temperature, cooling water flow rate, and reactor feed rate were chosen as bifurcation parameters due to the ease of manipulation.  Numerical strategies to maintain precision and consistency between MATLAB and MATCONT were necessary due to the high degree of nonlinearity, and concentrations ranging from trace over many orders of magnitude; i.e., 10^-12 – 10⁰ mol/L.

Hopf  Bifurcation and Periodic Solution in NMRP

The stationary bifurcation identified the Hopf bifurcation points, and Floquet multiplies to determine the stability of the periodic solutions.  When the cooling water flow rate or cooling water temperature is chosen as the bifurcation parameter, the sign symbol of the first Lyapunov coefficients at the Hopf bifurcation point varies with the initial concentration of  Nitroxyl-ether.  This suggests that the accompanying limit cycles, could be supercritical or orbitally unstable subcritical ones, which are even more dangerous.

Of special interest, is that the Hopf bifurcation points and corresponding limit cycles are at high conversions, while the NMRP process typically leads to slow polymerizations and relatively low monomer conversions despite its positive features.

Artificial Bifurcation Placement in NMRP

So far, the results from bifurcation control theory have been employed to stabilize polymerization reactors¹⁴ – as the presence of Hopf bifurcation points is a source of potential operability problems.  In fact, currently it is not well understood whether the orbitally-stable oscillatory operation of living-free-radical polymerization are advantageous over conventional steady-state operation,¹³ especially when conversions are high.  Ray and co-workers were among the first to discuss the merits of oscillatory behavior in polymerization,¹⁵ and recently Marquardt and co-workers demonstrated an approach to artificially introduce or modify Hopf bifurcation points in simple polymerization processes – to  obtain deliberate stability exchanges.

Herein, the impact of oscillatory operation on living-free-radical polymerization is investigated.  In the meantime, bifurcation placement is performed to consider the possibility of deliberately introducing a desired bifurcation point or modifying the frequency of periodic solution in the NMRP system.

Key words:  nitroxide-mediated radical polymerization; stability; Hopf bifurcation

Literature cited

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