Modeling and Experimental Evaluation of Single Particle Growth in Syndiotactic Polymerization of Styrene

Abstract

A comprehensive mathematical model and experimental study of single particle growth for styrene polymerization over a silica-supported metallocene catalyst were investigated. The model was developed based on the modification of the well-known multigrain model (MGM) by introducing mesoparticle scale limitations. Thereafter, the model was employed to predict the effects of bulk phase temperature and catalyst properties (initial catalyst active site concentration and initial catalyst particle size) on the polymerization rate, degree of polymerization (DP), and the polydispersity index (PDI) of syndiotactic polystyrene (SPS). The simulation results showed a significant radial distribution of styrene concentration across polymer particle growth at different polymerization conditions. It was found that increasing the initial catalyst concentration and bulk phase temperature resulted in polymerization rate enhancement. In context, the polymerization rate decreased as the initial catalyst particle size increased from 20 to 50 μm. The results revealed that a uniform increase in DP of the polymer was obtained by increasing the initial catalyst concentration and the reaction temperature, while resulting in a decrease of the PDI value. Meanwhile, the DP and PDI values varied inversely under the influence of initial catalyst particle size within a period of time similar to the one needed in the catalyst decay. The simulated results in the study agree well with experimental data of SPS.

Appendix

The changes in the shells volume, (∆V _i ) and the location of the grid points (R _i ) with time are given in this section. As shown in Fig. 2. The hypothetical shell can be defined as (R _h, _i−1 ≤ r ≤ R _h,i ) such that the entire polymer produced (since time t = 0) by the catalyst particles of radius (R _c) are accommodated in it. In the interval (t to t + ∆t), the volume of microparticles (V _μ,i) in the first shell are given by:

$$ \frac{{dV_{{{{\upmu}},i}} }}{dt} = \frac{{k_{\text{p}} C^{ *} M_{{{{\upmu}},i}} \left( {\frac{{4{{\uppi}}}}{3}R_{\text{c}}^{3} } \right)\left( {\text{MW}} \right)}}{{{{\uprho}}_{\text{p}} }};\quad i = 1,2, \ldots ,N $$

(1.1)

With V _μ,i (t = 0) being the initial volume of microparticle in the first shell.

$$ V_{{{{\upmu}},i}} \left( {t = 0} \right) = \frac{{4{{\uppi}}}}{3}R_{\text{c}}^{3} $$

(1.2)

The total volume of polymer (V _i ), the volume of mesoparticle (V _s,i ) produced at ith shell are given by:

$$ \frac{{dV_{i} }}{dt} = \frac{{k_{\text{p}} C^{ *} M_{{{\text{s}},i}} \left( {N_{i} \frac{{4{{\uppi}}}}{3}R_{\text{c}}^{3} } \right)\left( {\text{MW}} \right)}}{{{{\uprho}}_{\text{p}} }}\quad i = 2,3, \ldots ,N $$

(1.3)

$$ \frac{{dV_{{{\text{s}},i}} }}{dt} = \frac{{k_{\text{p}} C^{ *} M_{{{\text{s}},i}} \left( {\frac{{4{{\uppi}}}}{3}R_{\text{c}}^{3} } \right)\left( {\text{MW}} \right)}}{{{{\uprho}}_{\text{p}} }} ;\quad i = 2,3, \ldots ,N $$

(1.4)

With V _i (t = 0) and V _s,i (t = 0) being the initial total volume and volume of every polymer mesoparticle of ith volume, respectively.

$$ V_{i} \left( {t = 0} \right) = \frac{{N_{i} \left( {\frac{4\uppi }{3}R_{\text{c}}^{3} } \right)}}{(1 - \upvarepsilon )};\quad i = 2,3, \ldots ,N $$

(1.5)

$$ V_{{{\text{s}},i}} \left( {t = 0} \right) = \frac{{4{{\uppi}}}}{3}R_{\text{c}}^{3} $$

(1.6)

We can now define the hypothetical shells at any time by:

$$ R_{{{\text{h}},1}} = \left( {\frac{3}{{4{{\uppi}}}} \sum \limits_{j = 1}^{i} V_{{{{\upmu}},i}} } \right)^{1/3} $$

(1.7)

$$ R_{{{\text{h}},i}} = \left( {\frac{3}{{4{{\uppi}}}} \sum \limits_{j = 1}^{i} V_{j} } \right)^{1/3} ;\quad i = 2,3, \ldots ,N $$

(1.8)

where $R_{{{\text{h}},{\text{o}}}} = 0$

The radius of meso and microparticle at ith shell being:

$$ R_{{{\text{s}},i}} = \left( {\frac{3}{{4{{\uppi}}}}V_{{{\text{s}},i}} } \right)^{1/3} $$

(1.9)

$$ {\text{R}}_{{{{\upmu}},{\text{i}}}} = \left( {\frac{3}{{4{{\uppi}}}}{\text{V}}_{{{{\upmu}},{\text{i}}}} } \right)^{1/3} $$

(1.10)

The catalyst particles are assumed to be placed at the mid points of each hypothetical shell. Thus:

$$ R_{1,i} = R_{{{\text{h}}.i - 1}} + \left( \frac{1}{2} \right)\left( {R_{{{\text{h}},i}} - R_{{{\text{h}},i - 1}} } \right);\quad i = 2,3 \ldots ,N $$

(1.11)

Then the computational grid points are related to (R _1,i ) by:

$$ R_{1} = 0 $$

(1.12)

$$ R_{2} = R_{\text{c}} $$

(1.13)

$$ R_{i + 1} = R_{1,i} + R_{{{\text{s}},i }} ;\quad i = 2,3, \ldots ,N $$

(1.14)

$$ R_{N + 2} = R_{{{\text{h}},N}} $$

(1.15)

The values of (∆r _i ) to be used in the equation of Table 2 are given by:

$$ \Updelta r_{i} = R_{i + 1} - R_{i} ;\quad i = 1,2, \ldots ,N + 1 $$

(1.16)