Reducing model complexity by means of the Optimal Scaling: Population Balance Model for latex particles morphology formation

TL;DR

This work addresses the computational challenge of predicting multiphase latex particle morphology by applying Optimal Scaling (OS) to nondimensionalize the Population Balance Model (LPMF PBM). The authors introduce Optimal Scaling with Constraints (OSC) to enforce a regime in which aggregation integrals are negligible, yielding a reduced-complexity, dimensionless PBM; for certain parameter choices, they further derive the r-LPMF PBM that decouples time-dependent rate factors from the distributions via finite-moment ODEs. They provide a rigorous justification, detailed formulas, and numerical evidence showing that OSC/OSC-enabled reduced models achieve orders-of-magnitude faster computations with minimal loss of accuracy in slow-aggregation regimes (characterized by $\pi_0<1$). The practical impact lies in enabling efficient exploration and design of multiphase polymer morphologies, potentially guiding synthesis conditions and accelerating materials discovery. Key contributions include: (i) a constrained OS framework to tailor dimensionless coefficients; (ii) a provably valid reduction by discarding integral terms under the $\lambda_a\to0$ regime; (iii) a closed-form, finite-dimensional moment system (r-LPMF PBM) for certain $b$ values that uncouples dynamics from the full PBM; and (iv) comprehensive numerical demonstrations of accuracy and substantial computational gains.

Abstract

Rational computer-aided design of multiphase polymer materials is vital for rapid progress in many important applications, such as: diagnostic tests, drug delivery, coatings, additives for constructing materials, cosmetics, etc. Several property predictive models, including the prospective Population Balance Model for Latex Particles Morphology Formation (LPMF PBM), have already been developed for such materials. However, they lack computational efficiency, and the accurate prediction of materials' properties still remains a great challenge. To enhance performance of the LPMF PBM, we explore the feasibility of reducing its complexity through disregard of the aggregation terms of the model. The introduced nondimensionalization approach, which we call Optimal Scaling with Constraints, suggests a quantitative criterion for locating regions of slow and fast aggregation and helps to derive a family of dimensionless LPMF PBM of reduced complexity. The mathematical analysis of this new family is also provided. When compared with the original LPMF PBM, the resulting models demonstrate several orders of magnitude better computational efficiency.

Figures (12)

• Figure 1: Reaction mechanisms driving morphology development in a single polymer particle [$\bullet$]: \ref{['fig1a']}polymerization of Monomer 2 [] into Polymer 2 chains [$\,$$\,$], \ref{['fig1b']}nucleation of Polymer 2 agglomerates [$\,$$\,$] into non-equilibrium clusters [$\bullet$], \ref{['fig1c']}growth of equilibrium and non-equilibrium clusters in volumes, \ref{['fig1d']}aggregation of equilibrium and non-equilibrium clusters with sizes $v$ and $u$ into a cluster of a size $v+u$, \ref{['fig1e']}migration of non-equilibrium clusters [$\bullet$] to equilibrium positions [$\bullet$].
• Figure 2: Coefficients $\lambda(\tilde{\theta}^{N_x}_{\mathrm{opt}},\tilde{p})$\ref{['eqn:lambdas_computed_OSC']}, with $\kappa_1,\dots,\kappa_7$ defined in \ref{['tab:PBE_lambdas_def']}, $\tilde{p}=\tilde{p}_{\mathrm{exp}}$ given in \ref{['tab:p_exp_def']}, i.e. $N_{\mathrm{p}}=2.8 \times 10^{17}$, and $q_1 \in \mathbb{R}$. The black vertical dotted lines delimit the ranges of admissibility for solutions corresponding to $(S_1)$ and $(S_2)$ respectively, while the black vertical dashed lines correspond to $\pi_0=1$. The coefficients $\lambda$\ref{['eqn:lambdas_computed_OSC']} achieve the regime \ref{['eqn:lambdas_desired_with_v0ge1']} when they arise from $(S_1)$ and $\pi_0$\ref{['eqn:Pi_a_def']} is $<1$. The values of $k_{\mathrm{a}}$ are shown in $\mathrm{L}^{1/3}$$\mathrm{s}^{-1}$, with $\mathrm{L}$ for Litres and $\mathrm{s}$ for seconds.
• Figure 3: Solutions $m(v,t)$ and $w(v,t)$ of the r-LPMF PBM \ref{['eqn:PBE_latex_approx']}-\ref{['eqn:ODE_M,Wx_PBE_latex_no_IT_x=k(1-b)']} (left column) and the LPMF PBM\ref{['eqn:PBE_latex']}-\ref{['eqn:Sigma_m,w']}, with coefficients $\lambda(\theta,\tilde{p})$ (\ref{['tab:PBE_lambdas_def']}), $\theta=\{\nu_0,t_0,d_0\}$, $\nu_0=1/d_0=1 \, \mathrm{L}$, $t_0=1 \, \mathrm{s}$ and $\tilde{p}=\tilde{p}_{\mathrm{exp}}$ (\ref{['tab:p_exp_def']}). The values of $k_{\mathrm{a}}$ are chosen in such a way that both scenarios, $(i)$$\pi_0$\ref{['eqn:Pi_a_def']}$<1$ ($k_{\mathrm{a}}$ and $\pi_0$ are shown in green) and $(ii)$$\pi_0 \ge 1$ ($k_{\mathrm{a}}$ and $\pi_0$ in red), are present. The tested values of $k_{\mathrm{a}}$ are reported in $\tilde{\mathrm{L}} := \mathrm{L}^{1/3} \, \mathrm{s}^{-1}$. The shown $y=m,w$ are scaled as $y(v,t) \gets [ y(v,t) - \min_{v,t} y(v,t) ] / [ \max_{v,t} y(v,t) - \min_{v,t} y(v,t) ]$, whereas the time $t$ is normalized to $T=10^6$. Computational settings and parameter values are detailed in \ref{['tab:settings_feas_Pia<1']}.
• Figure 4: Dependence of the metric $e_y(t)$\ref{['eqn:metric_comp_rLPMF-LPMF']}, for $y=m$ (left) and $y=w$ (right), on the aggregation rate $k_{\mathrm{a}}$ (\ref{['tab:p_exp_def']}) and, thus, $\pi_0$\ref{['eqn:Pi_a_def']}$\propto k_{\mathrm{a}}$ (see \ref{['tab:PBE_lambdas_def']}). Values of $k_{\mathrm{a}}$ leading to $\pi_0<1$ ensure $e_m(t),e_w(t) \lesssim 10^{-3}$ for all tested times $t$. The plots show $k_{\mathrm{a}}$ in $\mathrm{L}^{1/3}$$\mathrm{s}^{-1}$, while the remaining physical parameters are provided in \ref{['tab:p_exp_def']}. Computational settings and parameter values are reported in \ref{['tab:settings_feas_Pia<1']}.
• Figure 5: Solutions $m(v,t)$ and $w(v,t)$ of the r-LPMF PBM \ref{['eqn:PBE_latex_approx']}-\ref{['eqn:ODE_M,Wx_PBE_latex_no_IT_x=k(1-b)']} and the LPMF PBM \ref{['eqn:PBE_latex']}-\ref{['eqn:Sigma_m,w']}, with coefficients $\lambda(\theta,\tilde{p})$ (\ref{['tab:PBE_lambdas_def']}), $\theta=\{\nu_0,t_0,d_0\}$, $\nu_0=1/d_0=1 \, \mathrm{L}$, $t_0=1 \, \mathrm{s}$ and $\tilde{p}$ given as \ref{['eqn:feas_param_exp1']}. The values of $k_{\mathrm{d}}$ are chosen in such a way that both scenarios, $(i)$$\pi_0$\ref{['eqn:Pi_a_def']}$< 1$ ($k_{\mathrm{d}}$ and $\pi_0$ are shown in green) and $(ii)$$\pi_0 \ge 1$ ($k_{\mathrm{d}}$ and $\pi_0$ in red), are present. The plot provides $k_{\mathrm{d}}$ in $\tilde{\mathrm{L}} := \mathrm{L}^{1/3} \, \mathrm{s}^{-1}$. The shown $y=m,w$ are scaled as $y(v,t) \gets [ y(v,t) - \min_{v,t} y(v,t) ] / [ \max_{v,t} y(v,t) - \min_{v,t} y(v,t) ]$, whereas the time $t$ is normalized to $T=5 \times 10^5$. Computational settings and parameter values are detailed in \ref{['tab:settings_feas_Pia<1']}.

Theorems & Definitions (10)

• Proposition B.1
• proof
• Proposition C.1
• proof
• Proposition C.2
• proof
• Proposition C.3
• proof
• Proposition D.1
• proof