Assumptions: two incompressible and immiscible fluids. Due to small Reynolds numbers, inertial effects are neglected. • Single fluid: Poiseuille flow

• Two-Fluids: Washburn Where is the meniscus position is pressure at the nodes and ; is the effective length and is the effective radii. The average viscosity, average density and capillary pressure are defined, respectively, as is the interfacial tension and is the contact angle

GOALS AND DELIVERABLES

• Consolidation and compaction of powders (tableting) • Understand and model powder processing properties interrelations that

control disintegration behavior upon solvent penetration. • Performance of pharmaceutical solid products (e.g., swelling, disintegration) • Develop a multi-physics model, based on coupled particles mechanics and

pore scale approaches • Particle mechanics strategies – Nonlocal contact formulation for confined

granular systems will provide the pore-network. (models individually describe all particles in the powder bed)

• Flow in porous media – Dynamic pore-network model for two-phase flow. • Swelling and disintegration - Fick’s law for porous particles and Case-II

diffusion

• Predictive constitutive models of inter-particle interactions for a variety of physical mechanisms + Predictability at high levels of confinement remains an open problem

• Concurrent and efficient multi-scale strategies which are fully-descriptive at the granular scale + Based on a particle mechanics description

+ Pore-network models

TESSELLATION OF POLYDISPERSE PARTICLES

Modeling and Simulation of Transport Phenomena in Confined Granular Systems

DISPLACEMENT RULES: HYSTERESIS & TRAPPING MECHANISMS

DISPLACEMENT RULES: DYNAMICS IN A SINGLE THROAT

MODELING APPROACH

PARTICLE MECHANICS STRATEGIES

DYNAMIC PORE-NETWORK MODEL FOR TWO-PHASE FLOW

Dominant mechanisms: - Elastic deformations - Plastic deformations - Bonding - Strain-rate mechanisms - Friction and fracture - Water intake and swelling

National Science Foundation www.nsf.gov

Center for Particulate Products and Processes engineering.purdue.edu/CP3

Pedro Cidreiro, Payam Poorsolhjouy , Prof. Marcial Gonzalez www.marcialgonzalez.net

SWELLING AND DISINTEGRATION Multi-physics model, based on coupled particle mechanics and pore-scale approaches. The release performance depends upon the accomplishment of: • Development of capillarity rise model in confined powders under

increasing loads • Multi-physics swelling, liquid uptake and disintegration model of

compacted powders • A coupled particle mechanics and pore-scale approach to swelling and

disintegration of compacted powders informed by ad-hoc experiments

Pore-network models that consider the morphology of the pore structure within the grain packing and uses Voronoi tessellations to decompose the pore space into throats which are connected to each other at pores. The complex throat's geometry is idealized into straight channels and an effective radii, area and length are assigned to each throat. The pores are considered dimensionless and are filled in instantaneously.

(a) BCC particles arrangement; (b) BCC particles and real throat boundaries; (c) real throat network; (d) BCC particles and idealized throat boundaries; (e) idealized throat network; (f) idealized network filled with water, the inlet is located at the bottom and the outlet at the top of the network; (g)2D view of the tessellation and throat scheme.

(g)

An open source code Voro++ [1] was used to perform the Radical Vonoi Tessellation (RVT) also referred to as the Laguerre tessellation. RVT allows the construction of a throat network system from polydisperse particles, in which the cell boundaries are weighted according to the relative radii of the particles. This distinction ensures that the throats will always be outside the particles themselves.

Radial Voronoi Tessellation of a poly disperse particle system under different compaction states

( )2

ij j irq P P glsin

8 l= − − +ρ φ

µ

( )2

ij j i c

dx (t) rq P P (t) lg sin Pdt 8 (t) lξ= = − − + ρ φ

µ

P i j effleffr

wnw nw w nw c

eff

2 cos(t) x (l x ), (t) x (l x ), Prξ ξ ξ ξ

σ θµ = µ +µ − ρ = ρ +ρ − =

xξ

θwnσ

(a)

Networks used for validation of the code: (a) straight vertical tube; (b) Y tube; (c) 5 nodes frame; (d) 8 nodes frame; (e) 13 nodes frame; (f) cube with central node; (g) BCC; (h) 2D Network.

SWELLING: DIFFUSION INTO POROUS PARTICLES

REFERENCES

;

0;

;

a ac c

a rc c

r rc c

P P P P

p P P P

P P P P

∆ − ∆ <∆ = ≤ ∆ <∆ − ∆ >

( ) sin∆ = − + ρ φi j t lgP P P

Hysteresis: Two different contact angles for the advancing and receding contacts, and , respectively, resulting in two different capillary pressures and .

( )ij ij ij cq g P g PP= ∆ = ∆ −

aθ rθ

acP r

cP

t+∆ t+∆

Trapping Mechanism: Only a single interface is allowed to move in a throat at any time. Therefore, if the throat has more than one interface, these interfaces must be trapped.

(a) (b) (c) (d) (e) (f) (g) (h)

Conservation of mass requires that the sum of fluxes incoming to all pores is zero where is the number of throats that are connected at the joint jth

jcN

iji 1

q 0=

=∑j

cN

[1] Chris H. Rycroft, Voro++: A three-dimensional Voronoi cell library in C++, Chaos 19, 041111 (2009)

• Not all the volume is available for diffusion due to the porosity

• Capture and immobilize part of the liquid,

• Increasing in volume (60 times) modify the porosity of the medium

• Swelling kinetics affects the history of the flow

( ) ( )

( )

22

b 0sphere t t 0

r, t r, tD r for 0 r R(t) and t 0t r r r

and∂ =

∂χ ∂χ ∂= < < > ∂ ∂ ∂

χ = χ χ = χ

( )bR

dRD 1r dt

∂χ= −χ

∂

Diffusion into a spherical particle of radius R is described by the following set of equations:

Volume balance due to swelling:

where is the volume fraction of liquid. ( )tχ

PRELIMINARY RESULTS FOR IMBIBITIONS