Influence of the Processes Parameters on the Properties of The ...

Chapter 3.
Analytical Methods and Designs <strong>of</strong> Experiments
where γ 12 is <strong>the</strong> surface energy <strong>of</strong> contact region between phase 1 and 2, γ 1 (γ 2 ) is <strong>the</strong> surface
energy <strong>of</strong> <strong>the</strong> phase 1 (phase 2) and <strong>the</strong> exponents γ d (γ p ) correspond to <strong>the</strong> dispersive fraction and <strong>the</strong> polar
fraction <strong>of</strong> <strong>the</strong> surface energy.
The force needed to extend <strong>the</strong> contact region between two immiscible phases relies on <strong>the</strong> force
needed to extend each phase separately minus <strong>the</strong> interaction that <strong>the</strong>y have with each o<strong>the</strong>r.
Thermodynamically this equation can be interpretated as follows: each substance is seeking for <strong>the</strong> lowest
energy possible. Since a surface has a higher energy state, each phase tries to reduce its surface area. When
two immiscible liquids are combined, a contact region is formed between both. To extend this region, energy
is needed. This energy depends on <strong>the</strong> energy needed to break <strong>the</strong> bindings <strong>of</strong> each phase separately minus
<strong>the</strong> interactions <strong>the</strong>y have with each o<strong>the</strong>r. This negative sign is explained since, by creating two new
surfaces, new interactions between <strong>the</strong> different phases will occur, causing a lower energy and <strong>the</strong>refore a
more favourable state. Considering equation 3.19, <strong>the</strong> following equation can be set up in order to describe
<strong>the</strong> interaction <strong>of</strong> a solid sample with a liquid:
SL
d d 1/
2 p p 1/ 2
SV LV 2[(
SV
.
LV ) (
SV
.
LV ) ]
(3.20)
where γ SL = solid-liquid surface energy, γ SV = solid-vapour surface energy, γ LV = liquid-vapour
surface tension, γ d = dispersive fraction and γ P = polar fraction <strong>of</strong> <strong>the</strong> surface energy.
This equation is, next to <strong>the</strong> Young-Dupré’ equation, <strong>the</strong> second equation needed for calculating
<strong>the</strong> surface free energy <strong>of</strong> a solid sample [Rudawska and Jacniacka, 2008; Owens and Wendt, 1969; Fowkes,
1962]. By combining equations 3.20 and 3.17, a final ma<strong>the</strong>matical statement can be made in order to
calculate <strong>the</strong> surface energy <strong>of</strong> a solid sample (γ SV ).[Rudawska and Jacniacka, 2008; Mykhaylyk et al.,
2003]:
SV
SV
SL LV Cos
SV LV SL LV ( Cos
1)
(3.21)
d d 1/
2 p p 1/ 2
LV
SL 2[(
SV
.
LV ) (
SV
.
LV ) ]
(3.22)
By substituting value from equation 3.21 into equation 3.22, we get:
d d 1/
2 p p 1/ 2
LV ( Cos
1) 2[(
SV
.
LV ) (
SV
.
LV ) ]
(3.23)
where γ LV = liquid-vapour surface energy, θ = contact angle, γ SV = solid-vapour surface energy, γ d =
dispersive fraction and γ p = polar fraction <strong>of</strong> <strong>the</strong> surface energy.
By using a liquid that only interacts on a dispersive level (diiodomethane or -bromonaphtalene)
with o<strong>the</strong>r phases, equation 3.23 can be simplified by excluding <strong>the</strong> polar interactions. The liquid vapor
surface energy and <strong>the</strong> contact angle can be directly measured. The dispersive tension <strong>of</strong> <strong>the</strong> liquid vapour
surface energy (γ d LV) <strong>of</strong> <strong>the</strong> diiodomethane is equal to <strong>the</strong> total liquid vapour surface energy (γ LV ) since <strong>the</strong>
polar interactions are null (cf. Table 3.1). Consequently, <strong>the</strong> only unknown parameter (γ d SV) can be
calculated.
Extrapolating <strong>the</strong> γ d SV value and using a second liquid having dispersive and polar interactions
with <strong>the</strong> solid sample, <strong>the</strong> only unknown parameter (γ p SV) can be determined. The sum <strong>of</strong> both calculated
fractions <strong>of</strong> <strong>the</strong> solid material (γ p SV and γ d SV) gives <strong>the</strong> total surface free energy <strong>of</strong> a solid sample [Wu, 2001;
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