Analytical representation of bimodality in bivariate distribution of chain length and chemical composition of copolymers

Hassanian-Moghaddam, Davood; Moattari, Mahan M; Rezania, Ali; Sharif, Farhad; Ahmadi, Mostafa

doi:10.1016/j.cej.2021.133229

Cited by 4 publications

(8 citation statements)

References 45 publications

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“…At any reaction time and in all simulated experiments, the calculated MWDs coincide with Flory's distribution, whose equation is: [32][33][34][35] MWD s,Flory = s…”

Section: Resultsmentioning

confidence: 83%

“…Besides, as expected, the 1-hexene content is higher for higher initial comonomer concentrations. It has been argued that the composition distribution should follow the simplified Stockmayer's equation, [32][33][34][35][36] obtained integrating over the chain length dimension the bivariate Stockmayer distribution: [34][35][36] w…”

Section: Resultsmentioning

confidence: 99%

“…At any reaction time and in all simulated experiments, the calculated MWDs coincide with Flory's distribution, whose equation is: [ 32‐35 ]

\begin{equation} {\rm{MWD}}_{{\rm{s}},{\rm{Flory}}} = s{\left({\frac{1}{{{{\overline {{x}_{\rm{n}}} }}_{,{\rm{Flory}}}}}} \right)}^2\exp \left( { - s\frac{1}{{{{\overline{{x}_{\rm{n}}}}}_{,{\rm{Flory}}}}}}\right) \end{equation}

…”

Section: Resultsmentioning

confidence: 99%

“…It has been argued that the composition distribution should follow the simplified Stockmayer's equation, [ 32‐36 ] obtained integrating over the chain length dimension the bivariate Stockmayer distribution: [ 34‐36 ]

\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} \displaystyle w\left( {{F}_{{\rm{hex}}}} \right) = \frac{{0.75}}{{\sqrt {2\gamma } {{\left( {1 + \frac{{\left( {{F}_{{\rm{hex}}} - {{\bar{F}}}_{{\rm{hex,Stm}}}} \right)}}{{2\gamma }}} \right)}}^{2.5}}},\ {\rm{where}}\\[12pt] \displaystyle \gamma = \frac{\beta }{{{{\overline {{x}_{\rm{n}}} }}_{{\rm{,Flory}}}}}, \\[12pt] \displaystyle \beta = {{\bar{F}}}_{{\rm{hex,Stm}}}\left({1 - {{\bar{F}}}_{{\rm{hex,Stm}}}}\right)\sqrt{1 - 4{{\bar{F}}}_{{\rm{hex,Stm}}}\left( {1 - {{\bar{F}}}_{{\rm{hex,Stm}}}} \right)\left( {1 - {r}_1{r}_2} \right)} \end{array} \nonumber\\ \end{eqnarray}

where w is the weight–fraction density (equivalent to the CCD in our model), F hex is the molar 1‐hexene composition of a given polymer chain,

{\bar{F}}_{{\rm{hex,Stm}}}

is the instantaneous average molar composition of 1‐hexene, r 1 = k p,AA / k p,AB , and r 2 = k p,BB / k p,BA , and

{\bar{x_{normaln}}}_{}

…”

Section: Resultsmentioning

confidence: 99%

“…The MWD-CCD provides a deeper insight into the molecular structure of the copolymer. In Case 2, the MWD-CCD has the typical shape of the bivariate Stockmayer distribution, [34,36] centered around the instantaneous (and invariant) average copolymer composition of 20% 1-hexene, narrow at high molecular weights and broadening toward lower molecular weights. The MWD-CCD of Case 1 is wider in composition, showing an average of around 12% 1-hexene.…”

Section: 𝛾 = 𝛽mentioning

confidence: 99%

See 4 more Smart Citations

Modeling of the Copolymerization of Propylene with 1‐Hexene to Predict the Copolymer Molecular Weight and Composition

Herrero

Brandolin

Sarmoria

et al. 2023

Macro Reaction Engineering

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A detailed mathematical model of the propylene‐1‐hexene copolymerization based on the two‐dimensional probability generation function technique is developed. It calculates the joint molecular weight‐copolymer composition distribution (MWD‐CCD) of the copolymer, as well as the average copolymer composition distribution, the molecular weight distribution (MWD), the copolymer composition distribution (CCD), average molecular weights and composition, and yield. The parallelized execution of the model code allows for obtaining the different copolymer microstructure distributions efficiently. The model allows for reaching a thorough understanding of the copolymer microstructure under different operating conditions of a semibatch reactor. It also has the potential to become a powerful tool for selecting operating conditions to obtain a material with target molecular properties.

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“…At any reaction time and in all simulated experiments, the calculated MWDs coincide with Flory's distribution, whose equation is: [32][33][34][35] MWD s,Flory = s…”

Section: Resultsmentioning

confidence: 83%

Section: Resultsmentioning

confidence: 99%

“…At any reaction time and in all simulated experiments, the calculated MWDs coincide with Flory's distribution, whose equation is: [ 32‐35 ]

\begin{equation} {\rm{MWD}}_{{\rm{s}},{\rm{Flory}}} = s{\left({\frac{1}{{{{\overline {{x}_{\rm{n}}} }}_{,{\rm{Flory}}}}}} \right)}^2\exp \left( { - s\frac{1}{{{{\overline{{x}_{\rm{n}}}}}_{,{\rm{Flory}}}}}}\right) \end{equation}

…”

Section: Resultsmentioning

confidence: 99%

\begin{eqnarray} \def\eqcellsep{&}\begin{array}{l} \displaystyle w\left( {{F}_{{\rm{hex}}}} \right) = \frac{{0.75}}{{\sqrt {2\gamma } {{\left( {1 + \frac{{\left( {{F}_{{\rm{hex}}} - {{\bar{F}}}_{{\rm{hex,Stm}}}} \right)}}{{2\gamma }}} \right)}}^{2.5}}},\ {\rm{where}}\\[12pt] \displaystyle \gamma = \frac{\beta }{{{{\overline {{x}_{\rm{n}}} }}_{{\rm{,Flory}}}}}, \\[12pt] \displaystyle \beta = {{\bar{F}}}_{{\rm{hex,Stm}}}\left({1 - {{\bar{F}}}_{{\rm{hex,Stm}}}}\right)\sqrt{1 - 4{{\bar{F}}}_{{\rm{hex,Stm}}}\left( {1 - {{\bar{F}}}_{{\rm{hex,Stm}}}} \right)\left( {1 - {r}_1{r}_2} \right)} \end{array} \nonumber\\ \end{eqnarray}

where w is the weight–fraction density (equivalent to the CCD in our model), F hex is the molar 1‐hexene composition of a given polymer chain,

{\bar{F}}_{{\rm{hex,Stm}}}

is the instantaneous average molar composition of 1‐hexene, r 1 = k p,AA / k p,AB , and r 2 = k p,BB / k p,BA , and

{\bar{x_{normaln}}}_{}

…”

Section: Resultsmentioning

confidence: 99%

Section: 𝛾 = 𝛽mentioning

confidence: 99%

See 3 more Smart Citations

Modeling of the Copolymerization of Propylene with 1‐Hexene to Predict the Copolymer Molecular Weight and Composition

Herrero

Brandolin

Sarmoria

et al. 2023

Macro Reaction Engineering

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Circular Polyolefins: Advances toward a Sustainable Future

2023

Self Cite

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Plastics offer several advantages, but their production and disposal processes have severe environmental implications. To overcome these issues, there is a need to switch from the linear to a circular economy by recycling plastic waste and by utilizing renewable resources to create bioplastics. However, this is challenging in the case of nonbiodegradable polyolefins (POs), which form the largest fraction of produced polymers and the least recycled one. Mechanical recycling, chemical recycling, and PO bioplastics are the three pillars of PO circular economy. Although mechanical recycling is an environmentally and economically viable option, it often results in the degradation and downgrading of POs. Nonetheless, innovations in mechanical recycling, such as the use of (nano)fillers or compatibilization with olefin block copolymers, attempt to mitigate these issues. Furthermore, the development of covalent adaptable networks improves the mechanical properties of recycled PO thermoplastics and provides recyclable PO elastomers. If mechanical recycling fails to meet the desired characteristics of the recyclate PO, chemical recycling to other chemicals is a potential alternative. Although retrieving the monomer is ideal for achieving a closed-loop circular economy, traditional approaches for the noncatalytic chemical recycling of POs are energy-intensive and lack specificity. This has been tried to be addressed by advancements in catalytic approaches. Finally, biobased polyolefins, especially those produced through emerging nonbiochemical approaches, offer attractive alternatives that can be integrated into existing petrochemical plants. With this comprehensive perspective on POs circular economy academic and industrial researchers of the field can better contribute to a more sustainable future.

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Beauty of Explicit Dispersity (Đ) Equations in Controlled Polymerizations

Wang,

Zhou,

Luo

et al. 2023

ACS Macro Lett.

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Dispersity (Đ) as a critical parameter indicates the level of uniformity of the polymer molar mass or chain length. In the past several decades, the development of explicit equations for calculating Đ experiences a continual revolution. This viewpoint tracks the historical evolution of the explicit equations from living to reversible-deactivation polymerization systems. Emphasis is laid on displaying the charm of explicit Đ equations in batch reversible-deactivation radical polymerization (RDRP), with highlights of the relevant elegant mathematical manipulations. Some representative emerging applications enabled by the existing explicit equations are shown, involving nitroxide-mediated polymerization (NMP), atom transfer radical polymerization (ATRP), and reversible addition–fragmentation chain transfer (RAFT) polymerization systems. Stemming from the several outlined challenges and outlooks, sustained concerns about the explicit Đ equations are still highly deserved. It is expected that these equations will continue to play an important role not only in traditional polymerization kinetic simulation and design of experiments but also in modern intelligent manufacturing of precision polymers and classroom education.

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Analytical representation of bimodality in bivariate distribution of chain length and chemical composition of copolymers

Cited by 4 publications

References 45 publications

Modeling of the Copolymerization of Propylene with 1‐Hexene to Predict the Copolymer Molecular Weight and Composition

Modeling of the Copolymerization of Propylene with 1‐Hexene to Predict the Copolymer Molecular Weight and Composition

Circular Polyolefins: Advances toward a Sustainable Future

Beauty of Explicit Dispersity (Đ) Equations in Controlled Polymerizations

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