Estimating Reactivity Ratios From Triad Fraction Data

Abstract

Reactivity ratio estimation is a non‐linear estimation problem. Typically, reactivity ratios are estimated using the instantaneous copolymer composition equation, otherwise known as the Mayo‐Lewis model, based on low conversion (<5%) copolymer composition data. However, there are other instantaneous models, which can be used to estimate reactivity ratios, such as the instantaneous triad fraction equations. The aim of this paper is to determine the potential improvement in reactivity ratio estimates when triad fraction data is used in place of and in combination with copolymer composition data. The interest in using triad fraction data in parameter estimation, stems from the fact that there are a greater number of responses measured (six triad fractions) compared to composition leading to data with theoretically more information content. In principle this should lead to reactivity ratio estimates having less uncertainty. In this study, the parameter estimates are obtained by employing the error in variables model (EVM), assuming a multiplicative error structure. Several case studies involving published literature data for different copolymer systems are presented. As the case studies demonstrate in general more precise estimates can be obtained from triad fraction data. Combining the triad fraction with composition data leads to little additional improvement. However, discrepancies arise between reactivity ratios estimated from composition data compared with those obtained from triad fraction data depending upon the copolymer system. Those copolymer systems exhibiting more heterogeneity due to phase separation during polymerization may be showing more discrepancy.

Reactivity ratios and copolymer properties of 2‐(diisopropylamino)ethyl methacrylate with methyl methacrylate and styrene

“…Because of their importance for understanding the kinetic and mechanistic aspects of copolymerization as well as the polymerization process and polymer product optimization, reactivity ratios are a constant focus of research. Recently, significant progress in determining the reactivity ratios for binary and ternary systems has been made by Penlidis and co‐workers . They have applied an advantageous error‐in‐variables model framework that enables direct and more reliable parameter estimation.…”

Nitroxide Mediated Polymerization of 1‐(4‐vinylbenzyl)‐3‐butylimidazolium Ionic Liquid Containing Homopolymers and Methyl Methacrylate Copolymers

“…The solving of both the FR and KT methods, along with the linearization plots, can be found in the supporting information (Figure S10 to S15). The Mayo‐Lewis equation is characteristically non‐linear, thus, linearization methods such as FT and FR can be argued as inaccurate . Nonlinear least squares (NILS) methods and more advanced error‐in‐variables (EVM) methods have been developed and shown to be more appropriate for use with the nonlinear Mayo‐Lewis model.…”

“…The Mayo-Lewis equation is characteristically non-linear, thus, linearization methods such as FT and FR can be argued as inaccurate. [94][95][96] Nonlinear least squares (NILS) methods and more advanced error-in-variables (EVM) methods have been developed and shown to be more appropriate for use with the nonlinear Mayo-Lewis model. The EVM method was implemented to determine a more accurate estimation of the reactivity ratios using the values solved by the linearization KT method as first estimates and is displayed in Figure 5.…”

Reactivity Ratios Estimation of the Free‐Radical Polymerization of Itaconic Acid and N‐Vinyl‐2‐Pyrrolidone by the Error‐in‐Variables Methodology

“…The error function used in the procedure is given in Equation (19) and its vector of partial derivatives, B i , has two forms depending on the error structure used, Equation (20) for additive error and Equation (21) for multiplicative error. Since multiplicative error considers the natural logarithm of 1 F to be the EVM variable, the derivative of g must be taken with respect to the ln( ) 1 F , an expansion of the derivative is thus necessary…”

“…This work seeks to lessen the lack of availability of these more statistically sound methods so that others may use and further contribute to their development in a free environment, in particular of EVM. Besides the reports of the groups who worked on the early development of EVM, [ 18–21 ] other groups worked on their own implementations of EVM. [ 22–24 ] None of these codes is easily accessible.…”