## 3 Model Development
### Prediction of the Induction Time without AS (t0A)  
At supersaturation, the critical number of molecules in the clusters is determined by Equation (1) using free energy arguments [1,19]. \[\mathrm{n_{c}}=32\pi\mathrm{V_{m}}^{2}\ \gamma^{2}/3(\mathrm{kTInS_{a}})^{3} \tag{1}\]  
where Vm is the molar volume of a target scallant and \(\gamma\) is the cluster surface energy density. A value of \(\gamma=9.25\ \mathrm{mJ/m^{2}}\) was used in this study [24]; k is the Boltzmann constant, T is the absolute temperature, and Sa is the supersaturation ratio, defined in Equation (2). \[\mathrm{S_{a}}=\mathrm{C_{f}}/\mathrm{C_{e}} \tag{2}\]  
where Cf is the target scallant feed concentration (mM) and Ce is the saturation level (mM). In this model, the induction time t0A refers to the period during which a cluster grows until it reaches the critical size [27,28]. Theoretical derivation of the t0A can commence with the consideration of the rate of cluster growth. A diffusion-reaction approach was employed to account for the cluster's growth, involving two distinct steps: a diffusion process, wherein solute molecules are transported to the surface of the cluster, and a first-order reaction, where the solute molecules arrange themselves within the lattice of the cluster [29,30]. This approach is presented as follows:  
\[\mathrm{dm/dt}=\mathrm{k_{c}A(C_{f}-C_{e})^{z}} \tag{3}\]  
where dm/dt is the rate of mass m deposited over time t, kc is the cluster growth coefficient [kg/(s m2 mM^2)], A is the cluster surface area, and z is the order of the overall cluster growth process. When z = 1, the deposition process is primarily controlled by diffusion, whereas at z > 1, the process is dominated by the surface reaction. The cluster size is determined based on a balance between the adhesive forces and hydrodynamic or shear forces that act at the solution-cluster interface. The shear forces are induced by the velocity difference \(\mathrm{u_{d}}=\mathrm{u_{f}}-\mathrm{u_{c}}\) (between the feed u_f and cluster u_c velocities) at the cluster surface, the viscosity of the solution, and the cluster's surface roughness. Removal from the cluster's surface occurs through mechanisms, such as dissolution, erosion, and spalling [20,29]. Spacer-filled RO channels create local shear stresses resulting in the detachment of deposited particles on a cluster [15]. The present analytical model is based on the assumption that the overall shear forces Fs, acting on a cluster surface, slow down the rate of the cluster growth. This assumption leads to the modification of Equations (3) and (4). \[\mathrm{dm/dt}=\mathrm{k_{c}A(C_{f}-C_{e})^{z}-k_{s}F_{s}} \tag{4}\]  
where ks is a shear rate constant (s/m), which depends on the cluster's surface shape and roughness, the adhesion strength of its molecules, the local velocity profile in the feed solution, and the concentrations Cf and Ce (M). The local velocity profile may be affected by factors, such as the presence or absence of membrane spacers or the types of spacers used [31]. The flow around a cluster surface is influenced by the velocity difference ud. Due to the small magnitudes of ud (<u_f < 1 m/s [32]) and the cluster size (<10\({}^{-8}\) m [33]), the Reynolds number is very low (Re << 1). In this case, the Stokes analytical solution of the velocity field around a sphere [34] is used to derive the shear stress (\(\tau\)) of the flow on the cluster surface. Integration of \(\tau\) over the cluster surface area A yields the resulting total shear force (Fs) acting on a cluster's surface:  
\[\mathrm{F_{s}}=\int_{0}^{\pi}\tau\mathrm{dA}-3/2\pi^{2}\mathrm{nRu_{d}} \tag{5}\]where R is the cluster radius and \(\eta\) is the dynamic viscosity of the solution. The velocity difference \(u_{d}\) can be derived from the force balance between Stokes' drag force and the inertial force resulting from the change in the cluster mass over time (Equation (6)). \[6\pi\eta Ru_{d}=(u_{f}-u_{d})dm/dt \tag{6}\]  
where \(u_{f}\) is the flow velocity of the feed solution.