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List of symbols
List of symbols Fundamental constants g k Na R
acceleration due to gravity Boltzmann constant Avogadro's number gas constant
Abbreviations L MPE P S TL V 2D(3D)
liquid multiphase equilibrium reaction product solid triple line vapour two (three) dimensions
Common symbols ai
b C d D e
E E' Eb Ec Ediss Eg Ep Ev c~.LV EiO)
thermodynamic activity of component i radius of curvature at sessile drop apex concentration width of a parallel sided gap or joint diffusion coefficient thickness of reaction product at the solid/liquid interface potential energy of a bulk atom potential energy of a surface atom energy barrier cohesion energy dissipated energy gap energy potential energy of a spreading drop vibrational energy adsorption energy at liquid/vapour surface of solute i at infinite dilution in solvent j
405
406
Wettability at high temperatures
oo,SL
Ei(j) f
fd
fi. fv F Fb Fs G AG~
adsorption energy at solid/liquid interface of solute i at infinite dilution in solvent j force compacity factor driving force surface fraction of phase i inertial force viscous force free energy free energy of bulk phase surface free energy Gibbs energy standard Gibbs energy of formation of compound i
h H AHf(o AH~0)
partial excess Gibbs energy of mixing at infinite dilution of a solute i in a solvent j height of wetting ridge or depth of grain boundary groove sessile drop height enthalpy of formation of compound i standard enthalpy of formation of compound i
AHi0)
partial enthalpy of mixing of component i in component j
(-JIG)
(X)
AH~0)
partial enthalpy of mixing at infinite dilution of a solute i in a solvent j
AHm Kl
enthalpy of mixing of an alloy constant first used in equation (2.4) constant in equation (2.20) equilibrium constant of a dissolution reaction equilibrium constant of reaction of formation of a new phase distance capillary length equilibrium rise or depression of a liquid in a parallel sided gap energy or heat of evaporation length of a facet on a sawtooth surface energy or heat of melting energy or heat of sublimation for a bulk atom, fraction of nearest neighbour lying in an adjacent plane for a bulk atom, fraction of nearest neighbour lying in the same plane mass of a drop power term in equation (1.33) coordination number of an oxygen atom dissolved in a metallic phase (equation (6.20))
K2
KD Kp l le Le Lf Lm
Ls ml
m2 md n n
List of symbols
Ph Pi
number of moles of component i perimeter pressure capillary pressure hydrostatic pressure partial pressure of species i
Po E
nitrogen partial pressure for dissociation of a nitride
Po2
oxygen partial pressure
Pf2
oxygen partial pressure in the furnace atmosphere
P~2
equivalent oxygen partial pressure at a solid/liquid interface
ni
P P Pc
407
oxygen partial pressure for decomposition of oxide MO Po2(M) Psat
r rd
R R0 RI, R2 Ra RF S Sr
Sj Sv Ss ". ~ X S , ~
A~itj)
t to tF
T TF U
U Uo Ur V Vm W
oxygen partial pressure for oxidation of metal M partial pressure at saturation of a species radius radius of a spherical drop base radius of a sessile drop initial sessile drop base radius principal radii of curvature of a surface average height of asperities on a surface final or stationary drop base radius length of line on a liquid surface roughness parameter of Wenzel (equation (1.35)) joint strength yield strength surface excess entropy partial excess entropy of mixing of solute i at infinite dilution in solvent j time time corresponding to the initial contact angle 0o time needed to reach a constant drop base radius RF temperature melting temperature fluid velocity triple line velocity initial triple line velocity rate of reaction volume molar volume length of a parallel sided joint
408
Wettability at high temperatures
Xi
weight of a meniscus work of adhesion work of adhesion in the absence of adsorption on the solid work of cohesion work of capillary forces (equation (2.2)) work of immersion work of spreading horizontal coordinate equatorial radius of a sessile drop molar fraction of component i
•
molar fraction of component i in phase j
Wm
Wa
wo W~ Wd Wi W~ x X9o
xl
molar fraction of component i at the solid/liquid interface q
xl Yi Yi Z z90 z* z*max Zb Ze
Z
~ij
molar fraction of component i for a reaction at equilibrium molar fraction of component i at point J corresponding to a three phase equilibrium (figure 7.2) molar fraction of oxygen at the liquid surface molar fraction of component i adsorbed at surface or interface monolayer molar fraction of component i around oxygen atoms dissolved in an alloy i-j (equation (6.19)) vertical coordinate equator-apex height of a sessile drop meniscus height maximum (or equilibrium) meniscus height depth to which the base of a solid is immersed in a liquid bath equilibrium rise or depression of a liquid in a circular cross section capillary coordination number i.e. number of nearest neighbour atoms angle reduction of the metallic bond energy due to oxygen (equation (6.20)) evaporation coefficient slope of surface asperities inclination of a surface to the vertical parameter in equation (3.7) elastic strain bond energy between i and j atoms Wagner interaction parameter between oxygen and solute i joining angle of a meniscus on a solid surface dihedral angle of grain boundary groove at solid/liquid interface
List of ,~ymbols (I)i ~o Yi YLV ~SV Fi
rl q q~ K ;t 2i_j 2a
A P 0 00 a
0app 0c 0eq 0equiv OF 0i
0i 0M 0p 0~ 0w 0v P O'LV ?
409
dihedral angles at a triple line (i = 1,2, 3) flow of oxygen activity coefficient of species i activity coefficient of species i at infinite dilution in a solvent liquid / vapour surface tension solid/vapour surface tension (equation (1.3)) absolute adsorption of species i relative adsorption of species i with respect to species j dynamic viscosity angle between the normal to the liquid surface and the axis of rotation of a sessile drop curvature regular solution parameter regular solution parameter of a binary i-j solution average wavelength of asperities on a surface Gibbs-Thomson parameter chemical potential contact angle initial contact angle or contact angle in the absence of reaction (equation (2.14)) advancing contact angle apparent contact angle Cassie contact angle on a chemically heterogeneous surface equilibrium contact angle equivalent contact angle of a small solid drop final or stationary contact angle contact angle on solid phase i contact angle of liquid phase i macroscopic contact angle equilibrium contact angle on a reaction product receding contact angle Wenzel contact angle on a rough surface Young contact angle mass density liquid / vapour surface energy
O'LV
temperature coefficient of liquid/vapour surface energy
O'SV
solid / vapour surface energy
~r~v cr~v
van der Waals component of solid/vapour surface energy solid/vapour surface energy in the absence of adsorption or reaction
410
Wettability at high temperatures
O'SL
solid/liquid interfacial energy
~~L
solid/liquid interfacial energy in the absence of reaction grain boundary energy specific excess free energy of triple line atomic surface area surface area base area of solid in wetting balance test molar surface area dihedral angle of grain boundary groove at solid/vapour surface dihedral angle of grain boundary groove at adsorption free solid / vapour surface
O'gb T fD
f~ f~ f~m
0
acceleration due to gravity Boltzmann constant Avogadro's number gas constant
Abbreviations L MPE P S TL V 2D(3D)
liquid multiphase equilibrium reaction product solid triple line vapour two (three) dimensions
Common symbols ai
b C d D e
E E' Eb Ec Ediss Eg Ep Ev c~.LV EiO)
thermodynamic activity of component i radius of curvature at sessile drop apex concentration width of a parallel sided gap or joint diffusion coefficient thickness of reaction product at the solid/liquid interface potential energy of a bulk atom potential energy of a surface atom energy barrier cohesion energy dissipated energy gap energy potential energy of a spreading drop vibrational energy adsorption energy at liquid/vapour surface of solute i at infinite dilution in solvent j
405
406
Wettability at high temperatures
oo,SL
Ei(j) f
fd
fi. fv F Fb Fs G AG~
adsorption energy at solid/liquid interface of solute i at infinite dilution in solvent j force compacity factor driving force surface fraction of phase i inertial force viscous force free energy free energy of bulk phase surface free energy Gibbs energy standard Gibbs energy of formation of compound i
h H AHf(o AH~0)
partial excess Gibbs energy of mixing at infinite dilution of a solute i in a solvent j height of wetting ridge or depth of grain boundary groove sessile drop height enthalpy of formation of compound i standard enthalpy of formation of compound i
AHi0)
partial enthalpy of mixing of component i in component j
(-JIG)
(X)
AH~0)
partial enthalpy of mixing at infinite dilution of a solute i in a solvent j
AHm Kl
enthalpy of mixing of an alloy constant first used in equation (2.4) constant in equation (2.20) equilibrium constant of a dissolution reaction equilibrium constant of reaction of formation of a new phase distance capillary length equilibrium rise or depression of a liquid in a parallel sided gap energy or heat of evaporation length of a facet on a sawtooth surface energy or heat of melting energy or heat of sublimation for a bulk atom, fraction of nearest neighbour lying in an adjacent plane for a bulk atom, fraction of nearest neighbour lying in the same plane mass of a drop power term in equation (1.33) coordination number of an oxygen atom dissolved in a metallic phase (equation (6.20))
K2
KD Kp l le Le Lf Lm
Ls ml
m2 md n n
List of symbols
Ph Pi
number of moles of component i perimeter pressure capillary pressure hydrostatic pressure partial pressure of species i
Po E
nitrogen partial pressure for dissociation of a nitride
Po2
oxygen partial pressure
Pf2
oxygen partial pressure in the furnace atmosphere
P~2
equivalent oxygen partial pressure at a solid/liquid interface
ni
P P Pc
407
oxygen partial pressure for decomposition of oxide MO Po2(M) Psat
r rd
R R0 RI, R2 Ra RF S Sr
Sj Sv Ss ". ~ X S , ~
A~itj)
t to tF
T TF U
U Uo Ur V Vm W
oxygen partial pressure for oxidation of metal M partial pressure at saturation of a species radius radius of a spherical drop base radius of a sessile drop initial sessile drop base radius principal radii of curvature of a surface average height of asperities on a surface final or stationary drop base radius length of line on a liquid surface roughness parameter of Wenzel (equation (1.35)) joint strength yield strength surface excess entropy partial excess entropy of mixing of solute i at infinite dilution in solvent j time time corresponding to the initial contact angle 0o time needed to reach a constant drop base radius RF temperature melting temperature fluid velocity triple line velocity initial triple line velocity rate of reaction volume molar volume length of a parallel sided joint
408
Wettability at high temperatures
Xi
weight of a meniscus work of adhesion work of adhesion in the absence of adsorption on the solid work of cohesion work of capillary forces (equation (2.2)) work of immersion work of spreading horizontal coordinate equatorial radius of a sessile drop molar fraction of component i
•
molar fraction of component i in phase j
Wm
Wa
wo W~ Wd Wi W~ x X9o
xl
molar fraction of component i at the solid/liquid interface q
xl Yi Yi Z z90 z* z*max Zb Ze
Z
~ij
molar fraction of component i for a reaction at equilibrium molar fraction of component i at point J corresponding to a three phase equilibrium (figure 7.2) molar fraction of oxygen at the liquid surface molar fraction of component i adsorbed at surface or interface monolayer molar fraction of component i around oxygen atoms dissolved in an alloy i-j (equation (6.19)) vertical coordinate equator-apex height of a sessile drop meniscus height maximum (or equilibrium) meniscus height depth to which the base of a solid is immersed in a liquid bath equilibrium rise or depression of a liquid in a circular cross section capillary coordination number i.e. number of nearest neighbour atoms angle reduction of the metallic bond energy due to oxygen (equation (6.20)) evaporation coefficient slope of surface asperities inclination of a surface to the vertical parameter in equation (3.7) elastic strain bond energy between i and j atoms Wagner interaction parameter between oxygen and solute i joining angle of a meniscus on a solid surface dihedral angle of grain boundary groove at solid/liquid interface
List of ,~ymbols (I)i ~o Yi YLV ~SV Fi
rl q q~ K ;t 2i_j 2a
A P 0 00 a
0app 0c 0eq 0equiv OF 0i
0i 0M 0p 0~ 0w 0v P O'LV ?
409
dihedral angles at a triple line (i = 1,2, 3) flow of oxygen activity coefficient of species i activity coefficient of species i at infinite dilution in a solvent liquid / vapour surface tension solid/vapour surface tension (equation (1.3)) absolute adsorption of species i relative adsorption of species i with respect to species j dynamic viscosity angle between the normal to the liquid surface and the axis of rotation of a sessile drop curvature regular solution parameter regular solution parameter of a binary i-j solution average wavelength of asperities on a surface Gibbs-Thomson parameter chemical potential contact angle initial contact angle or contact angle in the absence of reaction (equation (2.14)) advancing contact angle apparent contact angle Cassie contact angle on a chemically heterogeneous surface equilibrium contact angle equivalent contact angle of a small solid drop final or stationary contact angle contact angle on solid phase i contact angle of liquid phase i macroscopic contact angle equilibrium contact angle on a reaction product receding contact angle Wenzel contact angle on a rough surface Young contact angle mass density liquid / vapour surface energy
O'LV
temperature coefficient of liquid/vapour surface energy
O'SV
solid / vapour surface energy
~r~v cr~v
van der Waals component of solid/vapour surface energy solid/vapour surface energy in the absence of adsorption or reaction
410
Wettability at high temperatures
O'SL
solid/liquid interfacial energy
~~L
solid/liquid interfacial energy in the absence of reaction grain boundary energy specific excess free energy of triple line atomic surface area surface area base area of solid in wetting balance test molar surface area dihedral angle of grain boundary groove at solid/vapour surface dihedral angle of grain boundary groove at adsorption free solid / vapour surface
O'gb T fD
f~ f~ f~m
0