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A Computationally Feasible Model for Multiphase Particles Morphology Formation Simone Rusconi∗
∗BCAM - Basque Center for Applied Mathematics, Bilbao, Spain
A Computationally Feasible Model for Multiphase Particles Morphology - - PowerPoint PPT Presentation
A Computationally Feasible Model for Multiphase Particles Morphology - - PowerPoint PPT Presentation
A Computationally Feasible Model for Multiphase Particles Morphology Formation Simone Rusconi BCAM - Basque Center for Applied Mathematics, Bilbao, Spain December 7, 2018 Multiphase Particles Morphology Multiphase Particles : comprise
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State-of-the-art & Our Objective
◮ Current Status:
synthesis of multiphase particles is time and resources consuming it largely relies on heuristic knowledge no general methodology for prediction of morphology formation
◮ Objective: to develop a computationally feasible modelling approach
for prediction of multiphase particles morphology formation
SLIDE 4
Multiphase 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 [•]
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PBE model for Clusters Size Distribution
◮ Morphology Formation is described through time t evolution of the size v
distribution m(v, t) of clusters belonging to a given phase
◮ Distribution m(v, t) satisfies the Population Balance Equation (PBE):
∂tm(v, t) = − ∂v(g(v, t) m(v, t))
- Transport - Growth
+ n(v, t)
Source - Nucleation
− µ(v, t) m(v, t)
- Dissipation - Migration
+ 1 2 v a(v − u, u, t) m(v − u, t) m(u, t) du
- Integral Term - Aggregation
− m(v, t) ∞ a(v, u, t) m(u, t) du
- Integral Term - Aggregation
, ∀v, t ∈ R+, m(v, 0) = ω0(v) ≥ 0
- Initial Distribution
, ∀v ∈ R+, g(0, t) m(0, t) = 0
- Boundary Condition
, ∀t ∈ R+
- D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in
Engineering, Academic Press, 2000
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Nondimensionalization of PBE in Physical Units
◮ Obtained Outcome: well defined PBE model for morphology formation
expressed in physical units, e.g. v in Litres [L], t in seconds [s] and m in L−1
◮ Next Step: nondimensionalization, i.e. define the unitless PBE model ◮ Reasons for Nondimensionalization:
estimate characteristic properties of the system specify dimensionless coefficients
⇒ determine the behaviour of a class of dimensional models
estimate relative magnitude of various terms
⇒ simplify (eventually) the problem
manipulate numerically quantities with magnitude ≈ 100, instead of physical quantities, e.g. v ≈ 10−17 L, m ≈ 1032 L−1
⇒ guarantee minimal rounding errors
◮ Question: how to define unitless and computationally tractable models?
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Nondimensionalization Procedure
◮ General Problem Formulation: scale to unitless and computationally
tractable variables the equation f(x; p) = 0 with f : RNx × (0, ∞)Np → R, expressed in physical units (p.u.) x ≡ {x1, .., xNx} ∈ RNx: independent and unknown variables in p.u. p ∈ (0, ∞)Np: parameters with experimental values in p.u.
◮ Nondimensionalization Procedure:
- 1. Define factors θ ≡ {θ1, .., θNx} ∈ (0, ∞)Nx with dimensions as x
- 2. Change of variables x → ˜
x ≡ {˜ x1 = x1/θ1, .., ˜ xNx = xNx/θNx}
- 3. Plug the change of variables x → ˜
x in f(x; p) = 0
- 4. Rewrite f(x; p) = 0 in unitless form ˜
f(˜ x; λ) = 0 with dimensionless coefficients λ(θ; p) ≡ {λ1(θ; p), .., λNd(θ; p)} ∈ (0, ∞)Nd
◮ Remark: constants θ estimate characteristic properties of the system ◮ Question: how to select θ? How to ensure magnitudes ≈ 100?
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State-of-the-art for Selection of Constants θ
◮ Objective: find θ such that unitless variables assume magnitude ≈ 100 ◮ State-of-the-art: impose as many λ as possible equal to 1 (Holmes, 2009) ◮ If Nd ≤ Nx, solve Nd equations with Nx unknowns,
∃θ ∈ (0, ∞)Nx s.t. λi(θ; p) = 1, ∀i = 1, .., Nd
◮ If Nd > Nx, it is possible to ensure at most Nx coefficients λ equal to 1,
while there is no control on the remaining Nd − Nx > 0 coefficients
◮ Question: what can be a rational choice of θ for an arbitrary equation?
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Optimal Scaling: A Rational Choice of θ
◮ Optimal Scaling Factors θopt ∈ (0, ∞)Nx: find θ = θopt such that
all coefficients λ have magnitude ≈ 100, θopt ≡ argmin
θ∈(0,∞)Nx
C(θ), C(θ) ≡
Nd
- i=1
- log10(λi(θ; p)) − log10(100)
2
◮ C(θ) measures the distance of the magnitudes of λ from 100
- S. Rusconi, D. Dutykh, A. Zarnescu, D. Sokolovski, E. Akhmatskaya, An optimal
scaling to computationally tractable dimensionless models: Study of latex particles morphology formation, in preparation, 2018
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Optimal Scaling: Analytical Solution for θopt
◮ Motivation: nondimensionalization is independent from resolution
methods and it should not imply a significant computational effort
◮ Thus, we derive an analytical solution for argument θopt of minimum:
- 1. The Buckingham Π-theorem ensures ∀i = 1, .., Nd and ∀j = 1, .., Nx
λi(θ) = κi θαi
1
1 .. θ αi
Nx
Nx ,
κi > 0, αi
j ∈ R
- 2. Imposing ∂θjC(θ) = 0, ∀j = 1, .., Nx, one obtains a symmetric linear
system, with ˆ κj ≡ Nd
i=1 κ −αi
j
i
and j = 1, .., Nx, Nd
- i=1
αi
1αi j
- ρ1 + .. +
Nd
- i=1
αi
Nxαi j
- ρNx = log10(ˆ
κj), whose solution provides θopt = {10ρj}Nx
j=1
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A Computationally Feasible PBE Model?
◮ Obtained Outcomes:
PBE model in physical units for Morphology Formation, with infeasible magnitudes, i.e. v ≈ 10−17 L and m ≈ 1032 L−1 Optimal Scaling (OS) for rational definition of unitless variables
◮ Questions:
Does OS provide a unitless PBE model with feasible magnitudes? Performance of Optimal Scaling?
◮ Cases of Study:
- 1. Schr¨
- dinger Equation for an Hydrogen Atom in a Magnetic Field
- 2. PBE model for Multiphase Particles Morphology Formation
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Schr¨
- dinger Equation
◮ Consider an hydrogen electron, with wavefunction ψ(
r, t), under a constant magnetic field B = B n
◮ Schr¨
- dinger Equation in physical units for ψ(
r, t), with
- |ψ(
r, t)|2 d r = 1:
i∂tψ( r, t) = − 2 2µ∇2ψ − ieB 2µ ( n · r × ∇ψ) + e2B2 8µ [r2 − ( r · n)2]ψ − e2 4πǫr ψ
◮ Nondimesionalization:
ρ ≡ r/α0, τ ≡ t/β0 and φ ≡ ψ/γ0 Nx = 3 scaling factors θ = {α0, β0, γ0} ∈ (0, ∞)Nx Nd = 5 dimensionless coefficients λ(θ) = {λ1(θ), .., λ5(θ)} ∈ (0, ∞)Nd
λ1(θ) = β0 µ α2 , λ2(θ) = e B β0 2 µ , λ3(θ) = e2 B2 α2
0 β0
8 µ , λ4(θ) = e2 β0 4πǫ α0 , λ5(θ) = α3
0 γ2
◮ Remark: the effect of B on the unitless model is quantified by λ2,3(θ)
⇒ it may be distorted by the choice of θ
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Schr¨
- dinger Equation: Atomic Units vs. Optimal Scaling
◮ Performed Experiment: consider a magnetic field
B = B n and compare λ2,3(θ) for θ = θatom (atomic units) and θ = θopt (optimal scaling)
◮ Atomic Units: setting λ1,4,5 = 1 leads to often used atomic units and factors
θatom = {α0, β0, γ0}, with α0 ≈ the radius of an hydrogen atom
10
10
10
11
10
12
10
13
10
14
10
15
10
16
10
−18
10
−16
10
−14
10
−12
10
−10
10
−8
α* β* γ* θatom θopt 10
−2
10
−1
10 10
1
10
2
10
−9
10
−7
10
−5
10
−3
λ1 λ2 λ3 λ4 λ5 λ(θatom) λ(θopt)
◮ The effect of B can be better appreciated when θ = θopt rather than θ = θatom,
since λ2,3(θopt) ≈ λ1,4,5(θopt), while λ2,3(θatom) ≪ λ1,4,5(θatom)
◮ Factors θopt allow rough estimations of system properties, such as hydrogen radius Notation x∗ is defined as x∗ ≡ x/c, for c = 1 X and x = α0, β0, γ0 expressed in X
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PBE model for Morphology Formation
◮ PBE model in physical units for Morphology Formation, with infeasible
magnitudes, i.e. v ≈ 10−17 L and m ≈ 1032 L−1
◮ Nondimesionalization:
Nx = 8 scaling factors θ = {θ1, .., θ8} ∈ (0, ∞)Nx Nd = 19 unitless coefficients λ(θ) = {λ1(θ), .., λ19(θ)} ∈ (0, ∞)Nd
◮ Obj. 1: verify if Optimal Scaling (OS) provides feasible magnitudes ◮ Obj. 2: investigate how the choice of θ affects the quality of PBE solution ◮ Performed Experiment: given the same computational effort, integrate the
PBE model for θ = θopt (OS) and θ = θtest
◮ Factors θtest are selected by imposing as many λ as possible equal to 1
- S. Rusconi, D. Dutykh, A. Zarnescu, D. Sokolovski, E. Akhmatskaya, An optimal
scaling to computationally tractable dimensionless models: Study of latex particles morphology formation, in preparation, 2018
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PBE model: Performance of Optimal Scaling (OS)
Optimal Scaling: solution m(v, t) and error εm(t) for PBE with θ = θopt
2 4 6 8 0.5 1 1.5 2 2.5 x 10
−3
v m(v,t)
t=0.00 t=0.02 t=0.04 t=0.05 t=0.07 t=0.09 t=0.11 t=0.13 t=0.15 t=0.16 t=0.18 t=0.20 0.05 0.1 0.15 0.2 10
−12
10
−9
10
−6
10
−3
10
εm(t) t
Traditional Scaling: solution m(v, t) and error εm(t) for PBE with θ = θtest
50 100 150 200 −8 −6 −4 −2 2 x 10
−4
m(v,t)
2000 4000 6000 8000 10000
v
t=0.00 t=0.03 t=0.06 t=0.09 t=0.12 t=0.15 t=0.18 t=0.21 t=0.24 t=0.27 t=0.30 t=0.33 0.05 0.1 0.15 0.2 0.25 0.3 10
−12
10
−9
10
−6
10
−3
10
εm(t) t
◮ OS with θ = θopt provides tractable magnitudes, i.e. v ≈ 101 and m ≈ 10−3 ◮ The integration performed with θopt ensures errors εm(t) up to three orders
- f magnitude smaller than θtest
◮ The simulation performed with θtest is affected by non-physical oscillations
to negative values, whereas no such oscillations are observed for θopt
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Conclusions & Discussion
◮ We have presented:
- 1. a PBE model for Multiphase Particles Morphology Formation
- 2. Optimal Scaling (OS) for rational definition of unitless variables