## 2 Materials and Methods
### Surface Forces  
The theory of surface forces has been utilized to characterize the inter-surface interaction between NP-mineral and fine-mineral based on the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. It describes the forces between charged surfaces interacting in a medium. The DLVO theory combines the effect of attraction due to van der Waals interaction and the electrostatic repulsion due to the double layer of counter ions around charged surfaces in a medium. Silica NP and fines have sizes in the order of 100-500 nm and 0.1-5 \(\upmu\)m respectively. These are much smaller than the size the pore throats of the porous medium. Due to this size difference, the curvature of the mineral surfaces may be neglected and the interactions can be modelled as Sphere-Plate collector geometry [8; 20; 25; 26; 27]. The forces acting on a particle approaching a mineral surface are the sum of van der Waals attraction, electric double layer repulsion and Born repulsion. \[V_{t}(h)=V_{LVA}(h)+V_{EDLR}(h)+V_{BR}(h) \tag{1}\]  
where \(V\) is the potential of interaction as a function of separation distance (\(h\)) between the particle and the collector surface. The subscripts \(t\), \(LVA\), \(BR\), \(EDLR\) denote total, London-van der Waal interaction, electric double layer interaction and Born Repulsion, respectively. The sign of the total interaction potential indicates attractive potential and repulsive potential for negative and positive signs respectively. The interaction potential can be represented in non-dimensional (\(ND\)) form as follows:  
\[V_{t,ND}(h)=\frac{V_{t}(h)}{k_{B}\times T} \tag{2}\]  
where \(k_{B}\) is the Boltzmann constant (1.38 \(\times\) 10\({}^{-23}\) J\(\cdot\)K\({}^{-1}\)) and \(T\) is temperature. In this study, all the experiments are conducted at room temperature hence \(T\) = 297 K. The contributions due to the different types of interactions in Equation (1) can be calculated as follows [8; 25; 26]:  
\[V_{LVA}(h)=-\frac{A_{132}}{6}\bigg{[}\frac{2(1+H)}{H(2+H)}+ln\bigg{(}\frac{H}{ 2+H}\bigg{)}\bigg{]} \tag{3}\]  
\[V_{EDLR}(h)=\bigg{(}\frac{\epsilon_{0}D_{e}a_{p}}{4}\bigg{)}\bigg{[}2\zeta_{p} \zeta_{s}\ ln\frac{1+exp(-\kappa h)}{1-exp(-\kappa h)}+\bigg{(}\zeta_{p}^{2}+ \zeta_{p}^{2}\bigg{)}\ ln(1-exp(-2\kappa h))\bigg{]} \tag{4}\]  
\[V_{BR}(h)=\frac{A_{132}}{7560}\bigg{(}\frac{\sigma}{a_{p}}\bigg{)}^{6}\bigg{[} \frac{8+H}{\big{(}2+H\big{)}^{7}}+\frac{6-H}{H^{7}}\bigg{]} \tag{5}\]  
where  
\[H=\frac{h}{a_{p}} \tag{6}\]  
And, \(a_{p}\) is the particle radius (m). \(A_{132}\) is the [PERSON]'s constant between the sphere and plate collector which is typically in the range of 10\({}^{-19}\) J. This value of the Hamaker's constant is based on the assumption that the van der Waals interactions occurs in vacuum and is not influenced by the presence of surrounding particles. Hence, to account of the intervening fluid and the surrounding particles, the Hamaker's constant must be modified based on the Lifshitz theory [28]. Based on previous works which are in turn based on the expression for modified Hamaker's constant developed by [PERSON], the [PERSON]'s constant in this study is taken to be equal to 10\({}^{-21}\) J [8]. Also, \(\epsilon_{0}\) is the permittivity of free space (8.854 \(\times\) 10\({}^{-12}\) C\({}^{2}\) J\({}^{-1}\) m\({}^{-1}\)) and \(D_{e}\) is the dielectric constant of water equal to 78 [8; 26]. \(\kappa\) is the inverse Debye length. For pure water used in this study, the inverse Debye length is equal to (9.6 \(\times\) 10\({}^{-7}\))\({}^{-1}\) m\({}^{-1}\)[8]. \(\zeta_{p}\) and \(\zeta_{s}\) are the surface potentials of the particles and the surface respectively which can be replaced by the zeta potential [26]. In Equation (5), \(\sigma\) is the atomic collision diameter and is equal to 0.5 nm [26]. The born repulsive potentials are formed when the particle approaches point of contact with the mineral resulting in overlap of electron clouds. Hence it is a short range interaction and thus calculated only when the distance of separation is less than 1 nm.