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A Sub-Particle Model for Capillary Interaction Between Arbitrarily Shaped Nonmetallic Inclusions With an Undulated Contact Line

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Abstract

A numerical sub-particle model is developed to study the dynamic behavior of nonmetallic inclusions (NMIs) attached to the gas/steel or slag/steel interface due to capillary interaction, where the interface is deformed in a quadrupolar way due to the presence of the NMIs. The model is easy to be implemented since the Young–Laplace equation that describes the meniscus shape does not have to be solved for complex boundary conditions. The parent NMIs/particles are rebuilt in the model by closely packed spherical mono-sized or multi-sized sub-particles, with each sub-particle a small ‘quadrupole.’ The ‘quadrupole’ moment of sub-particles is assumed to be additive, and the net ‘quadrupole’ moment of all sub-particles should be equal to the moment of the parent particle. The capillary interaction between the quadrupolar parent particles is approximated by the net capillary interaction between the constituent sub-particles from one parcel to the other. The conditions for a good representation of the parent particle using a sub-particle parcel were determined, including the appropriate size of the sub-particles and the mass centers’ deviation. The anisotropic meniscus shape around an isolated ellipsoid and the anisotropic interaction energy and force from the side–side and tip–tip configurations are captured by the sub-particle model.

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Abbreviations

a :

Semi-minor axis of an ellipsoid (m)

b :

Semi-major axis of an ellipsoid (m)

e :

Elliptical eccentricity

F QP-QP :

Capillary force between parent ‘quadrupoles’ (N)

f QP- QP :

Capillary force between two quadrupolar sub-particles (N)

g :

Gravity acceleration vector (m/s2)

H Y :

Contact line undulation amplitude of parent particle Y (m)

H m , Y :

Capillary ‘quadrupole’ moment for parent particle Y (m3)

H’ m , Y :

Capillary ‘quadrupole’ moment for sub-particle Y (m)3

I QP-QP, W :

Coefficient for capillary energy

I QP-QP, F :

Coefficient for capillary force

l :

Distance between sub-particle centers (m)

L :

Distance between parent particle centers (m)

q – 1 :

Capillary length (m)

r :

Distance from a particle center (m)

r Y :

Radius of sub-particle Y (m)

r c,Y :

Radius of the contact line for sub-particle Y (m)

R Y :

Radius of parent particle Y (m)

Rc, Y :

Radius of the contact line for parent particle Y (m)

∆W QP - QP :

Capillary interaction energy between quadrupolar parent particles (J)

∆w QP - QP :

Capillary interaction energy between quadrupolar sub-particles (J)

α Y :

Contact angle between the liquid and solid particle Y

α min :

Minimum contact angle due to undulation

α max :

Maximum contact angle due to undulation

Δα :

Angle difference between αmax and αmin

γ :

Surface tension (N/m)

η Y :

Filling fraction of parcel Y

ρ I, ρ II :

Densities of fluid phases (kg/m3)

ρ Y :

Density of particles Y (kg/m3)

ϕ :

Capillary force percentage error

ω :

Capillary energy percentage error

X :

Energy ratio between side–side to tip–tip interaction

\({\sigma }_{\text{m},Y}\) :

Volume density of ‘quadrupole’ moment of parent particle Y

\({\sigma }_{\text{m},Y}^{,}\) :

Volume density of ‘quadrupole’ moment of sub-particle Y

m:

Moment

Y :

Particle index, A or B

W :

Energy

QP:

Capillary ‘quadrupole

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Acknowledgments

The authors thank the China Scholarship Council (CSC) for financial support (File No. 201706080018).

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  1. Department of Materials Engineering, KU Leuven, Kasteelpark Arenberg 44, Box 2450, 3001, Louvain, Belgium

    Zilong Qiu, Annelies Malfliet, Muxing Guo & Bart Blanpain

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Qiu, Z., Malfliet, A., Guo, M. et al. A Sub-Particle Model for Capillary Interaction Between Arbitrarily Shaped Nonmetallic Inclusions With an Undulated Contact Line. Metall Mater Trans B 53, 3442–3458 (2022). https://doi.org/10.1007/s11663-022-02608-0

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