Population Balance Approach for PredictingPolymer Particles Morphology

S. Rusconi1, D. Dutykh2, A. Zarnescu1,3,4, D. Sokolovski3,5 and E. Akhmatskaya1,3

1BCAM – Basque Center for Applied Mathematics, 48009 Bilbao, Spain2Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France

3IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain4“Simion Stoilow” Institute of the Romanian Academy, 010702 Bucharest, Romania

5Departamento de Química-Física, Universidad del País Vasco, UPV/EHU, Leioa, Spain

BCAM – Bilbao – Basque Country

Basque Coast Pintxos

Multi-phase Particles Morphology (MPM): Motivation & Our Objective

Multi-phase Particles: comprise phase-separated polymers

Morphology: pattern of phase-separated domains. It defines the material’s performance

Practical Interest: multi-phase polymers provide performance advantages over particles with uniform composition

Applications: synthetic rubber, latex, cosmetics, drug delivery

State-of-the-art: synthesis of multi-phase particles is time and resource consuming. Nowadays, it largely relies on heuristic knowledge

Our Objective: to develop a computationally efficient modelling approach for prediction of multi-phase particles morphology formation

Examples of particle morphologies: the white and black areas indicate phase-separated domains

Multi-phase Particles Morphology Formation

Morphology Formation dynamics of phase-separated polymers clusters≡

Reaction Mechanisms driving polymers clusters within a single particle [●]:

(a) Polymerization: conversion of monomers [▪] into polymers chains [▪▪▪]

(b) Nucleation: polymers chains [▪▪▪] agglomerate into clusters [•]

(c) Growth: clusters [•] increase their volume

(d) Aggregation: clusters, with sizes v and u, merge into a newly formed cluster with size v+u

(e) Migration: transition of clusters from a phase [•] to another phase [•]

Idea: morphology formation can be described through the time t evolution ofthe size v distribution, m(v,t), of clusters belonging to a given phase

Population Balance Equation (PBE) for Multi-phase Particles Morphology (MPM)

Polymers clusters in MPM development are subjected to:Nucleation, Growth, Aggregation and Migration

The distribution m(v,t) of clusterssize v satisfies the PBE system:

S. Rusconi, Probabilistic modelling of classical and quantum systems, Ph.D. thesis, UPV/EHU - University of the Basque Country, 2018

The rate functions g(v,t), n(v,t), μ(v,t), a(v,u,t) and the initial condition ω0(v)must be defined to encode the relevant kinetic and thermodynamic effects

PBE: Goals & Challenges

Challenges

Critical Nucleation Size

Transport Term

Integral Terms

Experimental ParametersComputationally IntractableOrders of Magnitude

n(v,t) proportional to δ(v-v0), withv0>0 and δ(x) the Dirac delta

Steep Moving Fronts

Non-Local TermsUnbounded Support of m(v,t)

GoalsTo provide an accurate and efficient solution of PBE model

for prediction of Multi-phase Particles Morphology formation

Highly aggregating processes may lead to:(a) numerical inaccuracies(b) domain errors

Infinitely steep source term

Optimal Scaling: Computationally Tractable Variables

S. Rusconi, D. Dutykh, A. Zarnescu, D. Sokolovski, E. Akhmatskaya, An optimal scaling to computationally tractable dimensionless models: Study of latex particles morphology formation, Computer Physics Communications 247: 106944, 2020

Numerical Study of Latex Particles Morphology

Data provided by POLYMAT led by Prof. Asua

Original Modelv ≈ 10-17 L, t ≈ 103 s, m ≈ 1032 L-1 Range of Orders of Magnitude ≈ 1049

Dimensionless ModelRange of Orders of Magnitude max λi / min λi ≈ 104

Idea: to avoid computationally intractable orders of magnitude throughthe novel optimal non-dimensionalization procedure

PBE integration method: Laplace Transform Technique (LTT)

PBE integration method: Generalised Method Of Characteristics (GMOC)

Novel Implementation of Known Method Of Characteristics (MOC) S. Rusconi et al., Comput. Phys. Commun., 2020

GMOC: Drawbacks

Numerical Oscillations due toMoving Fronts

Drawbacks

Targeted Accuracy: max ε ≈ 10-1

CPU time GMOC: 7.5×103 sec

Approximation of δ(v-v0) leads to h«σ0«v0

Non-Trivial Choice of Curves v=φk(t):We use φk(t)=kh, h>0, since Beneficial for A±

Inefficient Treatment ofSmall Nucleation Size v0 and

Large Volume Domains

Model I

Novel PBE integration method: Laplace Induced Splitting Method (LISM)

Conceptually New Methodology for PBE Systems S. Rusconi, Ph.D. thesis, UPV/EHU, 2018

Drawback: LISM relies on Availability ofAnalytical Solutions for any ω0(v)

LISM: Accurate

LISM does not Suffer from Oscillations as GMOC

LISM: Model I

Targeted Accuracy: max ε ≈ 10-1

CPU time GMOC: 7.5×103 secCPU time LISM: 103 sec

GMOC: Model I

LISM: Efficient

LISM is faster than GMOC by up to 102 times

Targeted Accuracy for Model I: max ε ≈ 10-1

Targeted Accuracy for Models II-III: max ε ≈ 10-2

Benefits LISM efficiently Deals with Small Nucleation Sizesand Large Volume Domains

Divide et Impera Splitting of PBE into Simpler Sub-Problems should

Support Complex Physical Rate Functions

Nucleation Size v0 = 2.7×10-2

Volume Domain = [0,103]Targeted Accuracy: max ε ≈ 10-2

Estimated CPU time GMOC » 106 secCPU time LISM: 1.3×103 sec

Model I

LISM: Accurate & Efficient

Conclusions & Possible Future Developments

Our Objective: to develop a computationally efficient modelling approach for prediction of multi-phase particles morphology formation

1. PBE Model captures the time evolution of the size distribution of polymers clusters composing the morphology of interest

2. Optimal Scaling: rational definition of dimensionless PBE model, allowing for parameters with experimental values

3. LISM: potentially promising methodology for accurate and efficient solution of dimensionless PBE model

Possible Future Developments:

1. Extension of LISM Applicability: address PBE models with physically sound rate functions

2. Tuning of LISM: choice of appropriate numerical schemes for time splitting and/or non-solvable terms

3. Comparison with State-of-the-art Solvers for PBE: Pivot Technique (Kumar and Ramkrishna, 1996, 1997), Monte Carlo methods (Meimaroglou et al., 2006) and Finite Elements methods (Mahoney and Ramkrishna, 2002)

4. Comparison with Experimental Data