Nonlinear analysis, polymerization

Shult and Volpert performed the linear stability analysis for the same model and confirmed this result (48), Spade and Volpert studied linear stability for nonadiabatic systems (49), Gross and Volpert performed a nonlinear stability for the one-dimensional case (50), Commissiong et al. extended the nonlinear analysis to two dimensions (this volume). In the former analysis, they confirmed that, unlike in SHS (57), uniform pulsations are difficult to observe in frontal polymerization. In fact, no such one-dimensional pulsating modes have been observed. 

A free boundary model is ui d to describe frontal polymerization. Weakly nonlinear analysis is applied to investigate pulsating instabilities in two dimensions, llie analysis produces a pair of Landau equations, which describe the evolution of the linearly unstable modes. Onset and stability of spinning and standing modes is described. 

Off-line analysis, controller design, and optimization are now performed in the area of dynamics. The largest dynamic simulation has been about 100,000 differential algebraic equations (DAEs) for analysis of control systems. Simulations formulated with process models having over 10,000 DAEs are considered frequently. Also, detailed training simulators have models with over 10,000 DAEs. On-line model predictive control (MPC) and nonlinear MPC using first-principle models are seeing a number of industrial applications, particularly in polymeric reactions and processes. At this point, systems with over 100 DAEs have been implemented for on-line dynamic optimization and control. 

We first discuss the materials research which includes monomer synthesis, growth of monomer crystalline structures and polymerization in the solid state, yielding the requisite polymer structures. Next, the nonlinear optical experimental research is discussed which includes a novel experimental technique to measure x (w). Linear and nonlinear optical data obtained for the polydiacetylene films is subsequently presented. Detailed theoretical analysis relating the data to x (< >) and subsequently to its molecular basis will be discussed in a later publication. 

Flory (1941a,b, 1953) and Stockmayer (1943, 1944) laid out the basic relations for establishing the evolution of structure with conversion in nonlinear polymerizations. Their analysis is based on the following assumptions defining an ideal network ... 

In this chapter we deal primarily with experimental results that have been reported dealing with parametric interactions in nonlinear poled polymers, i.e. mostly on second harmonic generation in phase-matched configurations. Because the theoretical analysis associated with these processes has been known for some time and has been independently reviewed many times, we will only briefly overview these basics. Also, the polymeric materials developed for similar applications are reviewed in another section of this book and we refer the reader to that for details. 

In the first place, the averaged model equations are highly nonlinear and require sophisticated numerical analysis for solution. For example, the attempt to obtain numerical solutions for motions of polymeric liquids, based upon simple continuum, constitutive equations, is still not entirely successful after more than 10 years of intensive effort by a number of research groups worldwide [27]. It is possible, and one may certainly hope, that model equations derived from a sound description of the underlying microscale physics will behave better mathematically and be easier to solve, but one should not underestimate the difficulty of obtaining numerical solutions in the absence of a clear qualitative understanding of the behavior of the materials. 

Batch Polymerization. Batch polymerization with this mechanism was first treated by Flory (19) using a statistical development. The same results were obtained by Biesenberger (8) using a kinetic analysis with an analytical solution. This was also one of the cases treated by Kilkson (35) using Z-transforms. In the simple cases, his result reduces to the Flory, or random, MWD with the dispersion index of 2. In more complex cases, he solves directly for the moments of the distribution. The Z-transform is probably the most powerful tool for solving condensation MWD problems the convolution theorem allows the nonlinear product terms in the kinetic equation to be handled conveniently. 

The MC simulation method is particularly suitable for investigating emulsion polymerization that involves various simultaneous kinetic events with a very small locus of polymerization. The MC simulation method will become a standard mathematical tool for the analysis of complex reaction kinetics, both for linear and nonlinear emulsion (co)polymerization. 

D. CoMissiONG, L. Gross, and V. Volpert, Bifurcation analysis of polymerization fronts, in Nonlinear Dynamics in Polymeric Systems, J. Pojman and Q. Tran-Cong-Miyata, eds., no. 869 in ACS Symposium Series, American Chemical Society, Washington, DC, 2003, pp. 147-159. 

L. K. Gross and V. A. Volpert, Weakly nonlinear stability analysis of fronted polymerization, Smdies in Applied Mathematics, 110 (2003), pp. 351-375. 

When a polymer is subject to an intense sinusoidal electric field such as that due to an intense laser pulse, Fourier analysis of the polarization response can be shown to contain not only terms in the original frequency co, but also terms in 2(0 and 3nonlinear response depends on the square of the intensity of the incident beam for 2co, and the third power for 3 . For the second-order effects, the system must have some asymmetry, as discussed previously. For poling, this means both high voltage and a chemical organization that will retain the resulting polarization for extended periods of time. Polymeric systems investigated have been of three basic types ... 

Gross, L.K. and Volpert, VA. (2003) Weakly nonlinear stability analysis of frontal polymerization. Stud. Appl. Math., 110, 351-376. 

Sulfobetaine cydopolymers. The first sulfobetaine cyclopolymer series OGQ) consists of the copolymers of 6 with 8 (Scheme 6) (42,43). Compositional analysis of this series of cyclopolymers is listed in Table 3. Reactivity ratios, determined via nonlinear least squares analysis " of chemical compositions determined from gated-decoupled C NMR, indicate random incorporation of the comonomers (ri=1.14 r2=0.97). The five-membered ring structure common to polymerized diallyl ammonium salts was retained in the sulfobetaine mer unit. Weight average molecular weights for this series range from 3.04 x 10" to 6.03 x 10 g mol. Second virial coefficients decrease from 8.78 x 10"" to 2.50 x lO"" ml mol g as the amount of 8 incorporated into the copolymer increases. 

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