Simultaneous long-chain branching and random scission: I. Monte Carlo simulation

Abstract

In free‐radical olefin polymerizations, the polymer transfer reactions could lead to chain scission as well as forming long‐chain branches. For the random scission of branched polymers, it is virtually impossible to apply usual differential population balance equations because the number of possible scission points is dependent on the complex molecular architecture. On the other hand, the present problem can be solved on the basis of the probability theory by considering the history of each primary polymer molecule in a straightforward manner. The random sampling technique is used to solve this problem and a Monte Carlo simulation method is proposed. In this simulation method, one can observe the structure of each polymer molecule formed in this complex reaction system, and virtually any structural information can be obtained. In the illustrative calculations, the full molecular weight distribution development, the gel point determination, and examples of two‐ and three‐dimensional polymer structure are shown. © 2001 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 39: 391–403, 2001

Simultaneous long-chain branching and random scission. II. Analytic expression for the weight-average molecular weights

“…The estimated values at N → ∞ are shown in Figure 11, and these values are interpolated by a quadratic regression curve. The gel point x c = 0.468 is obtained, which agrees reasonably well with x c = 0.47 that was obtained from the MC simulation 1…”

“…The gel point is simply described as a point at which the largest eigenvalue of the matrix, M , which defines chain connection statistics, reaches unity. The illustrative calculations showed that the calculated results agree satisfactorily with those from the Monte Carlo simulation method developed in the first part of this series 1. The present approach provides a newer insight into the complex phenomena occurring during branched polymer formation.…”

“…The star symbol shows the P̄ w value at N = 150, and good agreement with the extrapolated curve shows that the present estimation method works well. Note that under condition BS2, gelation occurs around x = 0.47, as shown in the first part of this series 1. It is shown here that the extrapolation method can be applied satisfactorily even when the weight‐average DP becomes fairly large.…”

Simultaneous long-chain branching and random scission. II. Analytic expression for the weight-average molecular weights

“…The estimated values at N → ∞ are shown in Figure 11, and these values are interpolated by a quadratic regression curve. The gel point x c = 0.468 is obtained, which agrees reasonably well with x c = 0.47 that was obtained from the MC simulation 1…”

“…The gel point is simply described as a point at which the largest eigenvalue of the matrix, M , which defines chain connection statistics, reaches unity. The illustrative calculations showed that the calculated results agree satisfactorily with those from the Monte Carlo simulation method developed in the first part of this series 1. The present approach provides a newer insight into the complex phenomena occurring during branched polymer formation.…”

“…The star symbol shows the P̄ w value at N = 150, and good agreement with the extrapolated curve shows that the present estimation method works well. Note that under condition BS2, gelation occurs around x = 0.47, as shown in the first part of this series 1. It is shown here that the extrapolation method can be applied satisfactorily even when the weight‐average DP becomes fairly large.…”

Simulation of size exclusion chromatography for branched polymers formed by simultaneous long-chain branching and random scission

“…However, we assumed constant values for simplicity. The parameters for BS1 are set to be close to the production of low-density polyethylene [5]. B1 is the case without random scission (C s = 0), and the value of C s is made larger in BSh1.…”

Mathematical Modeling of the Molecular Weight Distribution in Low Density Polyethylene. I. Steady‐State Operation of Multizone Autoclave Reactors

“…Mathematical modeling has been an important tool for the clarification and understanding of certain physicochemical phenomena involved in the polymerization processes in general. This tool has been in particular applied to the LDPE production process for almost 50 years now, although the models developed throughout these years have been steadily increasing in detail and complexity …”

Simultaneous long-chain branching and random scission. II. Analytic expression for the weight-average molecular weights

“…The estimated values at N → ∞ are shown in Figure 11, and these values are interpolated by a quadratic regression curve. The gel point x c = 0.468 is obtained, which agrees reasonably well with x c = 0.47 that was obtained from the MC simulation 1…”

“…The gel point is simply described as a point at which the largest eigenvalue of the matrix, M , which defines chain connection statistics, reaches unity. The illustrative calculations showed that the calculated results agree satisfactorily with those from the Monte Carlo simulation method developed in the first part of this series 1. The present approach provides a newer insight into the complex phenomena occurring during branched polymer formation.…”

“…The star symbol shows the P̄ w value at N = 150, and good agreement with the extrapolated curve shows that the present estimation method works well. Note that under condition BS2, gelation occurs around x = 0.47, as shown in the first part of this series 1. It is shown here that the extrapolation method can be applied satisfactorily even when the weight‐average DP becomes fairly large.…”

Simulation of size exclusion chromatography for branched polymers formed by simultaneous long-chain branching and random scission

“…However, we assumed constant values for simplicity. The parameters for BS1 are set to be close to the production of low-density polyethylene [5]. B1 is the case without random scission (C s = 0), and the value of C s is made larger in BSh1.…”

Mathematical Modeling of the Molecular Weight Distribution in Low Density Polyethylene. I. Steady‐State Operation of Multizone Autoclave Reactors

“…Mathematical modeling has been an important tool for the clarification and understanding of certain physicochemical phenomena involved in the polymerization processes in general. This tool has been in particular applied to the LDPE production process for almost 50 years now, although the models developed throughout these years have been steadily increasing in detail and complexity …”

Simultaneous long-chain branching and random scission. II. Analytic expression for the weight-average molecular weights

“…The estimated values at N → ∞ are shown in Figure 11, and these values are interpolated by a quadratic regression curve. The gel point x c = 0.468 is obtained, which agrees reasonably well with x c = 0.47 that was obtained from the MC simulation 1…”

“…The gel point is simply described as a point at which the largest eigenvalue of the matrix, M , which defines chain connection statistics, reaches unity. The illustrative calculations showed that the calculated results agree satisfactorily with those from the Monte Carlo simulation method developed in the first part of this series 1. The present approach provides a newer insight into the complex phenomena occurring during branched polymer formation.…”

“…The star symbol shows the P̄ w value at N = 150, and good agreement with the extrapolated curve shows that the present estimation method works well. Note that under condition BS2, gelation occurs around x = 0.47, as shown in the first part of this series 1. It is shown here that the extrapolation method can be applied satisfactorily even when the weight‐average DP becomes fairly large.…”

Simulation of size exclusion chromatography for branched polymers formed by simultaneous long-chain branching and random scission

“…However, we assumed constant values for simplicity. The parameters for BS1 are set to be close to the production of low-density polyethylene [5]. B1 is the case without random scission (C s = 0), and the value of C s is made larger in BSh1.…”

Mathematical Modeling of the Molecular Weight Distribution in Low Density Polyethylene. I. Steady‐State Operation of Multizone Autoclave Reactors

“…Mathematical modeling has been an important tool for the clarification and understanding of certain physicochemical phenomena involved in the polymerization processes in general. This tool has been in particular applied to the LDPE production process for almost 50 years now, although the models developed throughout these years have been steadily increasing in detail and complexity …”