3. FACTORS AFFECTING PICKERING STABILIZATION OF THE INTERFACE
3.5. Intermolecular and surface forces
The stability of the fluid-fluid interface is governed by intermolecular forces (IMFs) and forces between surfaces and fluid. The stability of the interface is governed by the colloidal stability of particles. It should be noted that the interactions listed are not exhaustive and it is outside the scope of the current study to discuss each of these interactions in detail. This article will only discuss these forces very briefly in a qualitative manner and serve to provide a brief introduction to the subject. IMF and surface forces acting on colloidal systems can be broadly categories into
1. DLVO interactions
a. Van der Waals forces
b. Electrostatic double-layer forces 2. Non-DLVO interactions
a. Hydration forces
b. Hydrophobic interactions c. Depletion forces
3. Capillary forces
14 3.5.1. DLVO interactions
DLVO interactions have been used to successfully explain the properties of many colloidal systems, emulsions, and foams.[71,72] Van der Waals interactions are short-range attractive forces that arise due to the oscillation of electron clouds. The strength of van der Waal’s attraction is affected by the dielectric properties of colloidal particles and background medium. [72–75] Electrostatic double-layer interactions occur due to the formation of non-homogenous charge distribution around the surface of colloidal particles that have a surface charge. The layer of charge is often called double layer and the width of the layer (Debye length) depends on the concentration of electrolytes in the medium. Debye length (λD) is measured using equation (16) and is dependent on the electric permittivity of medium (ε), the concentration of an ith component in bulk solution (Ci) and valency of ith electrolytes present in the solution (Zi) [72–
78]
𝜆𝐷 = √( 𝑒2
𝜀𝜀0𝑘𝐵𝑇∑ 𝐶𝑖 𝑖𝑍𝑖2) (16)
Where e is the charge of an electron, ε0 is dielectric constant, kB is Boltzmann constant, and T is the temperature. Van der Waals forces and electrostatic double-layer forces are often also reported as DLVO forces. Hence, DLVO theory can be used to explain the stability of aqueous colloidal suspension as an interaction between van der Waals force and repulsive electrostatic double-layer force.[72] Electrostatic double-layer pressure between two charged surface, wherein there are no interactions between counterions and the surface, can be estimated using contact value theorem expressed in equation (17), where P(D) is double repulsive pressure at distance D, k is Boltzmann constant, T is temperature ρs and ρm are ionic densities at surface and midplane respectively.
𝑃(𝐷) = 𝑘𝑇[𝜌𝑠(𝐷) − 𝜌𝑠(∞)] = 𝑘𝑇[𝜌𝑚(𝐷) − 𝜌𝑚(∞)] (17)
15 3.5.2. Non-DLVO interactions
As the proximity between two surfaces reduces to the nanometer range, the theory of DLVO forces often fails to describe the interactions between 2 surfaces. At such proximity, another set of short-range forces dominates DLVO forces. These short-range forces are often described as non-DLVO forces and can be attractive, repulsive, or oscillatory. Solvation of solutes/ particles in the aqueous phase can be affected by properties of both solvent and surfaces, at short range solvation (hydration if water) interaction arises between particles and extended surfaces. Equation 10 can be extended to estimate repulsive solvation pressure is given by equation (18). [79]
𝑃(𝐷 → 0) = −𝜌(∞)𝑘𝑇 (18)
While the solvation force discussed earlier can be oscillatory depending on the
separation, solvation forces can also be monotonically repulsive or attractive depending on the properties of solvent and surfaces. If the solvent in consideration is water hydration forces can act as repulsive forces whereas hydrophobic interactions can act as attractive forces.[80] Yet another non-DLVO force that could play an important role in colloidal stabilization of systems consisting of particles and polymer/surfactants is depletion forces. These forces arise when large macromolecules are suspended in a dilute solution of non-adsorbing polymers/micelles due to the exclusion of small non-adsorbed species (depletants) from the space between two large particles when the particles are close enough. Depletion forces are equal to osmotic pressure outside the gap between two particles and can be either attractive (promotes flocculation) or repulsive (prevents flocculation) in nature depending on the distance between the particles, concentration of depletant.[81–84] Derjaguin approximation (Equation 19) is often used to quantify depletion forces (fs) between particles of radius R, separated by the distance h and have
16 osmotic pressure .
𝑓𝑠(ℎ) = −𝜋𝑅 ∫ Π(ℎ∞ℎ ′)𝑑ℎ′ (19) 3.5.3. Capillary forces
Capillary forces are caused by the formation of the thin film at the curved interface between fluids. The curvature at the interface generated due to surface/interfacial tension of the fluids creates capillary pressure that is used as a measure of the stability of the film which in turn is correlated stability of the interface (emulsions/foams). Maximum capillary pressure (𝑃𝑐𝑚𝑎𝑥) provides the measure of the maximum pressure the thin film between the fluids can withstand before it ruptures resulting in droplet/bubble coalescence. Denkov (1992) developed a model to calculate (𝑃𝑐𝑚𝑎𝑥) for the thin film between two fluids with surface tension () stabilized by a monolayer of spherical particles of radius R, they found that the maximum capillary pressure can be calculated using equation 20.
𝑃𝑐𝑚𝑎𝑥 = 𝑝∗ 2𝜎
𝑅 (20)
Where p* is a positive dimensionless parameter that is a function of interface coverage and packing parameter of the particles at the interface.[85,86] This equation was further extended to develop equation 14 which can be used to calculate maximum capillary pressure in the film stabilized by a double layer of particles that take hexagonal packing formation.
𝑃𝑐𝑚𝑎𝑥 = 𝑝∗ 2𝜎
𝑅 (cos 𝜃 + 𝑧) (21)
In equation 21, is the contact angle between particle and fluids and z is a constant dependent on the arrangement of fluids in the film.[57,87] The films when initially formed are in swollen state and drain until the equilibrium is reached, the kinetics of film thinning (lamella drainage) can be characterized by an equation known as Reynolds equation (Equation 22).
17
−𝑑𝐻
𝑑𝑡 =2𝐻3∆𝑃
3𝜇𝑅𝑓2 (22)
Where H is the film thickness, t is drainage time, P is the pressure drop causing the film thinning, is bulk viscosity and Rf is the film radius.[88,89]