Understanding Interactions Driving the Template-Directed Self-Assembly of Colloidal Nanoparticles at Surfaces

Controlled deposition of colloidal nanoparticles using self-assembly is a promising technique for, for example, manufacturing of miniaturized electronics, and it bridges the gap between top-down and bottom-up methods. However, selecting materials and geometry of the target surface for optimal deposition results presents a significant challenge. Here, we describe a predictive framework based on the Derjaguin–Landau–Verwey–Overbeek theory that allows rational design of colloidal nanoparticle deposition setups. The framework is demonstrated for a model system consisting of gold nanoparticles stabilized by trisodium citrate that are directed toward prefabricated sub-100 nm features on a silicon substrate. Experimental results for the model system are presented in conjunction with theoretical analysis to assess its reliability. It is shown that three-dimensional, nickel-coated structures are well suited for attracting gold nanoparticles and that optimization of the feature geometry based on the proposed framework leads to a systematic improvement in the number of successfully deposited particles.


DLVO theory and Surface Element Integration
In the surface element integration (SEI) approach, 1 the interaction energy of a particle with a flat surface f is calculated as the sum of contributions coming from area elements dA of the particle's surface A interacting with f . Given the closest distance d between an area element and f , the contribution is taken as the per unit area interaction energy that two infinite flat surface would have at the same separation, u f (d), multiplied by the projected arean · dA, wheren is the normal of the flat surface. The total interaction free energy is where D is the distance of closest approach between the particle and f . According to DLVO theory, the integrand u f is given by the sum of a vdW contribution and an EDL contribution The (non-retarded) vdW interaction per unit area is readily given by 2 The EDL interaction u edl f is determined by the electrostatic potential Ψ, which, under equilibrium and assuming an electrolyte consisting of ions with charge q j and bulk concentration c ∞ j , obeys the non-linear Poisson-Boltzmann (PB) equation Here, r is the dielectric constant of the electrolyte and x a spatial coordinate normal to the plates. The next step in obtaining the EDL contribution to the interaction energy is to calculate the disjoining pressure of the two plates, given by the sum of the osmotic and electrostatic pressures In this equation Ψ d denotes the electrostatic potential obtained from solving Eq. (S5) for two flat surfaces at separation d. Furthermore, we note that the terms on the right-hand side with a spatial dependence may be evaluated at any x since Π is spatially uniform in equilibrium. The total interaction energy per unit area can subsequently be obtained by integrating In the case of constant, but unequal surfaces potentials φ 1 and φ 2 on the plates, there is no available closed-form, analytic solution PB equation, even in the one-dimensional case. A common approach is therefore to linearize, yielding where κ is the inverse of the Debye length and provides a measure of the effective range of the EDL interaction. From Eq. (S9) and assuming constant surface potentials φ 1 and φ 2 , an analytic expression for u edl f can be ob- this is the well-known Hogg-Healy-Fuerstenau formula. 3 It is crucial to note, however that the error incurred in computing the potential from the linearized PB equation for a symmetric electrolyte, while for an asymmetric electrolyte such as citrate, the error is

Interaction Energy Profiles
Fig. S1 shows a schematic interaction energy curve for a NP with potential Ψ p interacting with a flat plate with a potential Ψ f of the same sign. In this case, the EDL interaction is repulsive, and its competition with the attractive vdW interaction will lead to an energy barrier E B the particle needs to overcome before it can become bound to the surface. To determine if deposition is probable it is then natural to compare this barrier to the average thermal energy k B T . Note that this work neglects any contributions to the free energy resulting from ligand layer confinement effects. Incorporating such contributions would require knowledge of the structure of the ligand layer and the resulting change in the free energy is typically negligible beyond the physical boundary of the layer. Fig. S2 is similar to the schematic energy profile, but shows actual data for an AuNP over an SiO 2 surface using realistic parameters from the main paper. Here, we can see that the resulting energy barrier servation in the main paper that no deposition takes place on bare SiO 2 . Furthermore, the barrier is located 2.16 nm away from the surface and is thus not affected by steric repulsion from the citrate ligand layer. Figure S1: Schematic representation of deposition energetics. A repulsive electric double layer interaction between an incoming particle and a surface typically dominates the van der Waals interaction at intermediate separations. The result is a free energy barrier towards deposition of the particle. As the particle descends into the shaded region the additional contributions to the energy will arise from the ligand layer, but these do not significantly alter the barrier.  Figure S2: Numerical results for the free energy of an AuNP over a SiO 2 surface using realistic parameters reported in the main paper. A large energy barrier of 37k B T prevents the particle from depositing. The barrier is located 2.16 nm from the surface i.e. beyond the citrate ligand layer.

Implementation
For the implementation of the SEI method, consider a spherical particle with radius a and center of mass displacement x c relative to the cell origin, the interaction energy with each plate with normaln (incuding the substrate) can then be numerically calculated from Eqs. (S1) and (S2) using the following steps: 1. Create a mesh of the plate with nodes x i , i = 1, . . . , I and area ∆A i .
2. Calculate the projection of the particle position onto the plate P || ( where P ⊥ (x c ) = (n · x c )n.

For each mesh element that satisfies
to the interaction energy as Note that this algorithm assumes a partitioning of the flat surface into a mesh instead of the particle as described for Eq. (S1). Indeed, these two approaches are equivalent and which one is more convenient depends on whether one expects to treat spatial or material heterogeneities for the surface or the particle. The full source code can be found at https: //gitlab.com/joalof/seipy.

Optimization of the Forklift Angle
To complete the set of geometry optimization's for the forklift geometry we present an optimization of the forklift angle, here defined as β = 1 − α, where α is the opening angle as illustrated in the main text. We note that this result is primarily a consequence of simple geometrical considerations. Here, c salt denotes the electrolyte concentrations, which in this case is trisodium citrate. We find that β = 45 • minimizes the integrated free energy, and thus the optimal opening angle is α = 135 • . Figure S3: Optimization of the angle of a forklift geometry.

Experimental
The SEM micrographs that is used to calculate the percentage of successfully depositions in fig. 6c can be seen in Fig. S4-Fig. S10.