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 phase-separated polymers ◮ Morphology : pattern of phase-separated domains. It defines the material’s performance ◮ Practical Interest : multiphase polymers provide performance advantages over particles with uniform composition ◮ Applications : synthetic rubber, latex, cosmetics, drug delivery Examples of particle morphologies: the white and black areas indicate phase-separated domains

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

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 [ • ]

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):  ∂ t m ( v, t ) = − ∂ v ( g ( v, t ) m ( v, t )) + n ( v, t ) − µ ( v, t ) m ( v, t )    � �� � � �� � � �� �    Transport - Growth Source - Nucleation Dissipation - Migration  � v  + 1   a ( v − u, u, t ) m ( v − u, t ) m ( u, t ) du    2  0   � �� �    Integral Term - Aggregation  � ∞ ∀ v, t ∈ R + , − m ( v, t ) a ( v, u, t ) m ( u, t ) du ,   0   � �� �    Integral Term - Aggregation         ∀ v ∈ R + , ∀ t ∈ R + m ( v, 0) = ω 0 ( v ) ≥ 0 , g (0 , t ) m (0 , t ) = 0 ,     � �� � � �� �  Initial Distribution Boundary Condition D. Ramkrishna, Population Balances: Theory and Applications to Particulate Systems in Engineering , Academic Press, 2000

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 ≈ 10 0 , instead of physical quantities, e.g. v ≈ 10 − 17 L, m ≈ 10 32 L − 1 ⇒ guarantee minimal rounding errors ◮ Question : how to define unitless and computationally tractable models?

Nondimensionalization Procedure ◮ General Problem Formulation : scale to unitless and computationally tractable variables the equation f ( x ; p ) = 0 with � f : R N x × (0 , ∞ ) N p → R , expressed in physical units (p.u.) � x ≡ { x 1 , .., x N x } ∈ R N x : independent and unknown variables in p.u. � p ∈ (0 , ∞ ) N p : parameters with experimental values in p.u. ◮ Nondimensionalization Procedure : 1. Define factors θ ≡ { θ 1 , .., θ N x } ∈ (0 , ∞ ) N x with dimensions as x 2. Change of variables x → ˜ x ≡ { ˜ x 1 = x 1 /θ 1 , .., ˜ x N x = x N x /θ N x } 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 ) ≡ { λ 1 ( θ ; p ) , .., λ N d ( θ ; p ) } ∈ (0 , ∞ ) N d ◮ Remark : constants θ estimate characteristic properties of the system ◮ Question : how to select θ ? How to ensure magnitudes ≈ 10 0 ?

State-of-the-art for Selection of Constants θ ◮ Objective : find θ such that unitless variables assume magnitude ≈ 10 0 ◮ State-of-the-art : impose as many λ as possible equal to 1 (Holmes, 2009) ◮ If N d ≤ N x , solve N d equations with N x unknowns, ∃ θ ∈ (0 , ∞ ) N x s.t. λ i ( θ ; p ) = 1 , ∀ i = 1 , .., N d ◮ If N d > N x , it is possible to ensure at most N x coefficients λ equal to 1, while there is no control on the remaining N d − N x > 0 coefficients ◮ Question : what can be a rational choice of θ for an arbitrary equation?

Optimal Scaling: A Rational Choice of θ ◮ Optimal Scaling Factors θ opt ∈ (0 , ∞ ) N x : find θ = θ opt such that all coefficients λ have magnitude ≈ 10 0 , θ opt ≡ argmin C ( θ ) , θ ∈ (0 , ∞ ) Nx N d � � � 2 log 10 ( λ i ( θ ; p )) − log 10 (10 0 ) C ( θ ) ≡ i =1 ◮ C ( θ ) measures the distance of the magnitudes of λ from 10 0 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

Optimal Scaling: Analytical Solution for θ opt ◮ Motivation : nondimensionalization is independent from resolution methods and it should not imply a significant computational effort ◮ Thus, we derive an analytical solution for argument θ opt of minimum: 1. The Buckingham Π -theorem ensures ∀ i = 1 , .., N d and ∀ j = 1 , .., N x α i λ i ( θ ) = κ i θ α i α i Nx 1 .. θ 1 N x , κ i > 0 , j ∈ R 2. Imposing ∂ θ j C ( θ ) = 0 , ∀ j = 1 , .., N x , one obtains a symmetric linear − α i κ j ≡ � N d j system , with ˆ i =1 κ and j = 1 , .., N x , i � N d � N d � � � � α i 1 α i α i N x α i ρ 1 + .. + ρ N x = log 10 (ˆ κ j ) , j j i =1 i =1 whose solution provides θ opt = { 10 ρ j } N x j =1

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 ≈ 10 32 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¨ odinger Equation for an Hydrogen Atom in a Magnetic Field 2. PBE model for Multiphase Particles Morphology Formation

Schr¨ odinger Equation ◮ Consider an hydrogen electron, with wavefunction ψ ( � r, t ) , under a constant magnetic field � B = B� n � r, t ) | 2 d� ◮ Schr¨ odinger Equation in physical units for ψ ( � r, t ) , with | ψ ( � r = 1 : r, t ) = − � 2 ∇ ψ ) + e 2 B 2 e 2 2 µ ∇ 2 ψ − i � eB 8 µ [ r 2 − ( � r × � n ) 2 ] ψ − i � ∂ t ψ ( � 2 µ ( � n · � r · � 4 πǫr ψ ◮ Nondimesionalization : � ρ ≡ � r/α 0 , τ ≡ t/β 0 and φ ≡ ψ/γ 0 � N x = 3 scaling factors θ = { α 0 , β 0 , γ 0 } ∈ (0 , ∞ ) N x � N d = 5 dimensionless coefficients λ ( θ ) = { λ 1 ( θ ) , .., λ 5 ( θ ) } ∈ (0 , ∞ ) N d , λ 3 ( θ ) = e 2 B 2 α 2 e 2 β 0 λ 1 ( θ ) = � β 0 , λ 2 ( θ ) = e B β 0 0 β 0 , λ 5 ( θ ) = α 3 0 γ 2 , λ 4 ( θ ) = 0 µ α 2 2 µ 8 µ � 4 πǫ � α 0 0 ◮ Remark : the effect of B on the unitless model is quantified by λ 2 , 3 ( θ ) ⇒ it may be distorted by the choice of θ

Schr¨ odinger Equation: Atomic Units vs. Optimal Scaling ◮ Performed Experiment : consider a magnetic field � B = B� n and compare λ 2 , 3 ( θ ) for θ = θ atom (atomic units) and θ = θ opt (optimal scaling) ◮ Atomic Units : setting λ 1 , 4 , 5 = 1 leads to often used atomic units and factors θ atom = { α 0 , β 0 , γ 0 } , with α 0 ≈ the radius of an hydrogen atom 16 2 10 10 15 10 θ atom 14 1 10 10 13 10 θ opt 12 10 0 10 11 10 10 10 −1 10 −8 10 −2 −10 10 10 −3 −12 10 10 λ ( θ atom ) −5 −14 10 10 −7 −16 λ ( θ opt ) 10 10 −18 −9 10 10 α * β * γ * λ 1 λ 2 λ 3 λ 4 λ 5 0 0 0 ◮ The effect of B can be better appreciated when θ = θ opt rather than θ = θ atom , since λ 2 , 3 ( θ opt ) ≈ λ 1 , 4 , 5 ( θ opt ) , while λ 2 , 3 ( θ atom ) ≪ λ 1 , 4 , 5 ( θ atom ) ◮ Factors θ opt allow rough estimations of system properties, such as hydrogen radius Notation x ∗ is defined as x ∗ ≡ x/c , for c = 1 X and x = α 0 , β 0 , γ 0 expressed in X