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Imbibition into Highly Porous Layers of Aggregated Particles

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Abstract

The imbibition process into the pore structure of highly porous nanoparticle layers is investigated. The layers are fabricated with the Flame-Spray-Pyrolysis process and consist of aggregated \(\hbox {TiO}_{2}\) nanoparticles. Measurements of the pore structure reveal that these layers have a very high porosity and a broad pore size distribution. Imbibition experiments show a deviation of the imbibition process from the classical capillary rising theory (Bell, Cameron, Lucas and Washburn). Therefore, a new capillary rising model is developed that accounts for the complex pore structure within layers of aggregated particles through the implementation of the fractal dimension of tortuosity. Comparison of the new model and the performed imbibition experiments enable the determination of the equivalent pore diameter for the imbibition process. The resulting pore diameter agrees well with the mean pore diameter of the nitrogen physisorption measurements. Hence, the new model enables the prediction of the imbibition process into highly porous layers of aggregated particles.

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Abbreviations

a :

Factor of the power law fitting (\(\hbox {m}^{1+{b}}\,\hbox {s}^{-1}\))

A :

Cross-sectional area of the porous medium (\(\hbox {m}^{2}\))

\({A}_\mathrm{p}\) :

Cross-sectional area of a pore (\(\hbox {m}^{2}\))

b :

Exponent of the power law fitting (–)

BET:

Brunauer–Emmett–Teller

BJH:

Barret–Joyner–Halenda

BCLW:

Bell, Cameron, Lucas and Washburn

Bo :

Bond number (–)

\({c}_{\mathrm{KC}}\) :

Kozeny–Carman constant (–)

\({c}_{\mathrm{Ar}}\) :

Archie constant (–)

\({c}_{0}\) :

Factor of the capillary rising model (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

\({d}_\mathrm{p}\) :

Pore diameter (m)

\(\Delta {d}_\mathrm{p}\) :

Pore diameter interval (m)

\({d}_{\mathrm{p,eq}}\) :

Equivalent pore diameter for the imbibition process (m)

\({\bar{d}}_{\mathrm{p,2}}\) :

Weighted mean pore diameter of the \({q}_{2}\) (\({d}_\mathrm{p}\)) distribution (m)

\({d}_{\mathrm{32,pa}}\) :

Sauter mean diameter of the primary particles (m)

\({D}_\mathrm{t}\) :

Fractal dimension of tortuosity (–)

FSP:

Flame-Spray-Pyrolysis

g :

Gravitational constant (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

\({L}_{0}\) :

Straight-line flow length (m)

\({L}_\mathrm{p}\) :

Flow length inside the pore (m)

\({F}_{\mathrm{fr}}\) :

Friction force (N)

\({F}_{\mathrm{cap}}\) :

Capillary force (N)

\({F}_{\mathrm{gra}}\) :

Gravitational force (N)

m :

Time exponent (–)

\({N}_\mathrm{p}\) :

Number of pores (–)

p :

Pressure (Pa)

\({p}_{\mathrm{cap}}\) :

Capillary pressure (Pa)

\({p}_{\mathrm{gra}}\) :

Gravitational pressure (Pa)

\({p}_{\mathrm{lam}}\) :

Lamination pressure (Pa)

\({q}_{0}(d_\mathrm{p})\) :

Differential pore number distribution (\(\hbox {m}^{-1}\))

\({q}_{2}(d_\mathrm{p})\) :

Differential pore area distribution (\(\hbox {m}^{-1}\))

s :

Standard deviation (a.u.)

\({S}_\mathrm{p}\) :

Surface area of the pore (\(\hbox {m}^{2}\))

\({S}_{\mathrm{pa}}\) :

Surface area of the particles (\(\hbox {m}^{2}\))

SEM:

Scanning-electron-microscope

\(\hbox {SSA}_{\mathrm{fil}}\) :

Specific surface area of the filter paper (\(\hbox {m}^{2}\,\hbox {g}^{-1}\))

\(\hbox {SSA}_{\mathrm{sam}}\) :

Specific surface area of the sample (\(\hbox {m}^{2}\,\hbox {g}^{-1}\))

\(\hbox {SSA}_{\mathrm{pa}}\) :

Specific surface area of the nanoparticles (\(\hbox {m}^{2}\,\hbox {g}^{-1}\))

\({t}_{n}\) :

Time step n (s)

\({v}_{0}\) :

Straight-line imbibition velocity (\(\hbox {m}\,\hbox {s}^{-1}\))

\(v_\mathrm{p}\) :

Specific pore volume (\(\hbox {m}^{3}\,\hbox {g}^{-1}\))

\(\Delta v_\mathrm{p}\) :

Differential specific pore volume (\(\hbox {m}^{3}\,\hbox {g}^{-1}\))

\({V}_\mathrm{p}\) :

Pore volume (\(\hbox {m}^{3}\))

\({V}_{\mathrm{pa}}\) :

Particle volume (\(\hbox {m}^{3}\))

\({\dot{V}}\) :

Volume flow rate (\(\hbox {m}^{3}\,\hbox {s}^{-1}\))

\(\dot{V}_{i}\) :

Volume flow rate in a pore (\(\hbox {m}^{3}\,\hbox {s}^{-1}\))

\({w}_{\mathrm{fil}}\) :

Mass fraction of the filter paper (–)

XRD:

X-ray diffraction

\(\alpha \) :

Statistical significance level (–)

\(\beta \) :

Geometry correction factor for the pore diameter (–)

\(\varepsilon \) :

Longest non-tortuous flow length (m)

\(\eta _\mathrm{l}\) :

Viscosity of the liquid (Pa s)

\(\theta _{\mathrm{adv}}\) :

Advancing contact angle (\({^{\circ }}\))

\(\kappa \) :

Electrical conductivity (\(\hbox {S}\,\hbox {m}^{-1}\))

\(\rho _\mathrm{l}\) :

Density of the liquid (\(\hbox {kg}\,\hbox {m}^{-3}\))

\(\sigma _\mathrm{l}\) :

Surface tension of the liquid (\(\hbox {N}\,\hbox {m}^{-1}\))

\(\tau \) :

Tortuosity (–)

\(\phi \) :

Porosity (–)

\(\psi \) :

Orientation angle to a horizontal plane (\({^{\circ }}\))

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Acknowledgements

We thank the Deutsche Forschungsgemeinschaft (DFG) for founding this work through the project “Experimental and computational analysis of the forces acting on highly porous nanoparticle scaffolds/layers during liquid imbibition” (MA3333/10-1, HA2420/16-1). We also thank the group of Prof. M. Dreyer from the Center of Applied Space Technology and Microgravity (ZARM) at the University of Bremen, especially P. Prengel, for the surface tension measurements.

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Authors and Affiliations

  1. Foundation Institute of Materials Science (IWT), Department of Production Engineering, University of Bremen, Badgasteiner Str. 3, 28359, Bremen, Germany

    Sven O. Schopf, Udo Fritsching & Lutz Mädler

  2. Frauenhofer Institute for Manufacturing Technology and Advanced Materials (IFAM), Wiener Str. 12, 28359, Bremen, Germany

    Andreas Hartwig

  3. MAPEX Center for Materials and Processes, University of Bremen, 28359, Bremen, Germany

    Andreas Hartwig, Udo Fritsching & Lutz Mädler

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  1. Sven O. Schopf
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  2. Andreas Hartwig
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  4. Lutz Mädler
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Correspondence to Lutz Mädler.

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Schopf, S.O., Hartwig, A., Fritsching, U. et al. Imbibition into Highly Porous Layers of Aggregated Particles. Transp Porous Med 119, 119–141 (2017). https://doi.org/10.1007/s11242-017-0876-2

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