Mathematical modelling of polymerization kinetics

Analysis of the mass balance equations that describe a reaction system may be useful to provide information about its dynamics, such as the restricted set of compositions that can be achieved from a given set of initial compositions and the effect of feeding reactants to the reaction environment along the reaction course. Since these results may be important for the formulation of reaction policies, this work presents the properties of a matrix polymerization model previously developed and extended to describe transient conditions. This model is based on the definitions of two matrices: the consumption matrix (A À Kt), which contains information about chemical transformations among the many active polymer species in the system, and the propagation matrix Kp, which contains information about chain growth. It is shown that the set of mass balance equations that describes the dynamics of active chemical species in polymerization reactions has a stable and unique solution, which is bounded if feed rates are also bounded. It is also shown that the set of compositions that may be reached through manipulation of the feed rates is restricted and may not include all possible chemical compositions. Finally, it is shown that the obtained molecular weight distributions are special multiple time convolutions of the initiation rates.

Section: Introductionmentioning

confidence: 99%

Matrix Representation of Polymer Chain Size Distributions, 2

Sanches

Melo

Pinto

2007

“…However, the PBM procedure also suffers from many deficiencies and limitations compared to popular probabilistic methods such as Monte Carlo (MC) and Markov chain. [11,12] Mathematical treatment often becomes tedious with the task of endless solving sets of integro-differential equations pre-requisite in determining the fragment evolution from the various moment average data. This may force many approximation steps, lead to assume a form of distribution truncated by the method of moments or numerical solution.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modelling and Computer Simulation of Linear Polymer Degradation: Simple Scissions

Bose

Git

2004

Summary: Degradation of a polymer in a reactor by the degrading agent(s) follows a distinct pattern, primarily influenced by structural integrity and reactor environment. This distinct pattern is recorded in the changes in the evolved molecular weight distribution (MWD) or polymer chain length distribution (PCLD) curve characteristics from the initial intact state. Modern size exclusion chromatography (SEC) is the best laboratory‐based method that can clearly provide these plots in the form of chromatogram; however, detailed molecular information is not available. The nature of molecular destruction can be well‐characterised if the distinct MWD shift patterns can be simulated to fingerprint the different chain scission dynamics. This is investigated by our current research using the power of computer simulation techniques to gain insight into the polymer ageing processes. One such technique for studying simple decay processes is presented here, and the results are compared with experimental findings. The concept of a binary tree scission model is introduced to show chain rupture as a sequence of probabilistic events and as a non‐linear function of time. Two new mathematical algorithms, an iterative Monte Carlo structured probability scheme and a semi‐iterative algebraic exact statistical formulation method, are investigated to implement this model and simulate the evolution of resultant temporal MW distribution. The latter, an innovative approach to mathematical modelling, has the potential to generate a statistically perfect instant MWD decay curve. A statistical comparison of the product yield is presented from the data obtained using a wide variety of simulated scission regimes to determine the sources of variability.Simulated MWD lateral shift for percent cut scission model showing deviation from the initial MWD (red) over degradation time zones Tj(0 ≥ j ≤ 9) with bimodal and curve broadening effect due to accumulation of varied percent cut range 5–30%.imageSimulated MWD lateral shift for percent cut scission model showing deviation from the initial MWD (red) over degradation time zones Tj(0 ≥ j ≤ 9) with bimodal and curve broadening effect due to accumulation of varied percent cut range 5–30%.

“…This is a dynamic feature of copolymerization mechanisms that has not been found previously. If c 1 < 0, p > 0 for every f. Then, it is necessary to verify the existence of complex eigenvalues with the use of Condition II of Equation (18). Substituting the expressions of p and q in Equation (15)- (16) (18), one can rewrite the latter as:…”

Section: Dynamic Behavior Of a Copolymerization Problemmentioning

confidence: 99%

“…Therefore, the sign of p Ã can be used to check condition I of Equation (18). The coefficients c 0 , c 1 and c 2 of the polynomial in Equation (19) are:…”

Section: Dynamic Behavior Of a Copolymerization Problemmentioning

confidence: 99%

“…This is related to the common ''eigenzeit'' transformation M dt ¼ du, which has been used to facilitate the analytical solution of the system of mass balance equations, by transforming second-order reaction systems into pseudofirst-order reaction systems. [1,18] Then, after introduction of the ''eigenzeit'' or characteristic time u t; t ð Þ ¼ R t t M ds, Equation (46) becomes:…”

Section: Analysis Of Relative Frequencies In a Homopolymerization Promentioning

confidence: 99%

See 1 more Smart Citation

Matrix Representation of Polymer Chain Size Distributions, 3 – Case Studies for Linear Polymerization Mechanisms at Transient Conditions

Sanches

Melo

Pinto

2008

The mass balance equations used to describe different polymerization systems are represented and analyzed with the help of a general matrix framework previously developed. Some of its properties were presented in previous works and are illustrated here in three examples. The first example regards the dynamics of a copolymerization reactor. It is shown that terminal and penultimate models may predict qualitatively different responses in batch reactions. The second example also regards the dynamics of copolymerization reactions. It is shown how manipulation of initial conditions and monomer feed rates can be used to keep copolymer composition constant throughout the batch. The third example concerns the existence of oscillating chain size distributions when there are multiple monomer insertion steps.magnified image