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Adhesive penetration of hardwood: a generic penetration model

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Abstract

An analytical model to predict the penetration of adhesives into hardwood is proposed. Penetration into hardwood is dominated by the vessel network which prohibits porous medium approximations. The model considers two scales: (1) a one dimensional capillary fluid transport of a hardening adhesive through a single, straight vessel with diffusion of solvent through the walls of the vessel; and (2) a mesoscopic scale based on topological characteristics of the vessel network. Given an initial amount of adhesive and applied bonding pressure, the portion of the filled structure could be calculated. The model was applied to beech samples joined with three different types of adhesive (PUR, UF, PVAc) under various growth ring angles as described by Hass et al. (2011). The model contains one free parameter that can be adjusted in order to fit the experimental data.

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Abbreviations

η g :

Viscosity of the fluid without any hardening process (mPa s)

γ :

Viscosity parameter fit with experiments (no units)

α :

Viscosity parameter fit with experiments (s−1)

β :

Viscosity parameter fit with experiments ([C]−1)

η :

Viscosity of the adhesive (mPa s)

a 1 :

Viscosity parameter fit with experiments (s−1)

b 2 :

Viscosity parameter fit with experiments (no units)

c 1 :

Viscosity parameter fit with experiments (s)

d 1 :

Viscosity parameter fit with experiments (mPa s2)

σ :

Surface tension (N/m)

θ c :

Contact angle between adhesive and cell walls (°)

R :

Radius of the vessels (m)

K :

Wave number for the single wavy vessels (m−1)

N = a/4:

Amplitude of the wavy vessel (m)

λ :

Wavelength for the single wavy vessels (m)

A :

Vessel cross section (m2)

V u :

Volume of a single unit cell (m3)

x T,R,L :

Tangential, radial, and longitudinal coordinates (m)

s b :

Path along one radial wave (m)

s total :

Total path along the network radial wave (m)

L T,R,L :

Tangential, radial, and longitudinal lengths of the sample (m)

N uL,uR,uT :

Number of unit cells along the longitudinal, radial, and tangential directions (no units)

N uLf,uRf,uTf :

Number of unit cells on the RT, LT, and RL planes (no units)

N v :

Number of vessels in the plane R–T (no units)

a :

Maximum distance between two consecutive vessels (m)

b :

Wavelength of the vessels (m)

c :

Maximum distance between two consecutive radial waviness (m)

d :

Wavelength of the radial waviness (m)

g T,R :

1 + (a 2 π 2/16 b 2), 1 + (c 2 π 2 /8d 2) (no units)

θ :

Deviation angle of the vessels respect to the long. axis (°)

ψ :

Growth ring angle (°)

t :

Time (s)

x, y, z :

Local coordinate system for a single vessel description (m)

R, θ, z :

Cylindrical coordinate system for a single vessel description (m, °, m)

X, Y, Z :

Global coordinate system (m)

L, R, T :

Body fixed coordinate system (m)

l :

Penetrated distance inside a single cylindrical vessel (m)

l v :

Height of the single wavy vessels (m)

δ L,R,T :

Maximum penetration depths in the longitudinal, radial, and tangential directions (m)

δ Lh,Rh,Th :

Maximum penetration depths in the longitudinal, radial, and tangential directions with hardening process (m)

δ V :

Maximum penetration depth for volume limiting factor (m)

δ h :

Maximum penetration depth for hardening process limiting factor (m)

\( \underline{\delta } \) :

Penetration vector in the system (R–T–L) (m)

Δ L,R,T :

Tangential, longitudinal, and radial components of the penetration vector (m)

V = V p :

Volume of adhesive inside a single vessel (m3)

V T :

Total volume of adhesive inside the network (m3)

C, C 0 :

Concentration, initial concentration of the solvent [C]

N 0 :

Initial amount of solvent (m3)

D :

Diffusivity of the solvent through cell walls (m2/s)

P A :

Applied pressure to the adhesive (Pa)

\( \underline{{\underline{M} }} \) :

Rotation matrix (no units)

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Acknowledgments

The authors are grateful for the financial support of the Swiss National Science Foundation (SNF) under Grant No. 116052, and SNF SINERGIA under Grant No. CRSI22_125184.

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Authors and Affiliations

  1. Institute for Building Materials, ETH Zurich, Schafmattsr. 6, 8093, Zurich, Switzerland

    Miller Mendoza, Philipp Hass, Falk K. Wittel, Peter Niemz & Hans J. Herrmann

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  1. Miller Mendoza
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  2. Philipp Hass
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  3. Falk K. Wittel
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  4. Peter Niemz
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  5. Hans J. Herrmann
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Correspondence to Miller Mendoza.

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Mendoza, M., Hass, P., Wittel, F.K. et al. Adhesive penetration of hardwood: a generic penetration model. Wood Sci Technol 46, 529–549 (2012). https://doi.org/10.1007/s00226-011-0422-2

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