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.

Introduction

Guided assembly of nanosized particles (NPs) and clusters[1] onto surfaces is of interest for the continued miniaturization of circuits using molecular electronics and other applications, for example, in plasmonic-based sensors,[2−4] stimulation of cell adhesion at nanostructured interfaces[5] and the engineering of neuronal cell function.[6]

Early investigations revealed that a Coulomb blockade[7,8] can be measured through a quantum dot or an NP when positioned between two electrodes. Bar-Joseph et al. subsequently demonstrated that a circuit could be constructed with NP dimers interlinked by molecules and electrostatically captured between electrodes.[9,10] This method, however, only allows for the capture of a single NP or a dimerized NP pair. In order to compete with conventional semiconductor integrated circuits, methods capable of simultaneously constructing nanofeatures and nanogaps in parallel[11,12] and assembling multiple nanoparticles onto specific sites on a sample or device are required.[13]

Previously, methods for parallel NP assembly have been proposed based on, for example, meniscus flow,[14−16] direct NP growth onto the sites of interest,[17] selective chemical activation and passivization of a surface,[18,19] or polymer-templated self-assembly.[20,21] Recently, a facile method for parallel delivery of NPs to selected parts of a sample was put forward by Eklöf et al.[22] based on wet chemical deposition. The key idea is to leverage colloidal interactions between suspended nanoparticles and prefabricated nanosized features (nanofeatures) on a substrate to guide particles toward target areas on the sample (Figure ). The technique has been demonstrated for both single and dimerized gold nanoparticles stabilized by trisodium citrate using patches of metal coating and three-dimensional structures.[23,24]

Figure 1

Schematic of the depositing setup. A chip with a variety of prefabricated structures is positioned on a three-dimensional printed scaffold placed in a beaker of water. A drop containing the NP dispersion is placed on top of the chip, and everything is sealed with a glass lid in order to prevent evaporation of the drop.

While colloidal deposition has many advantages for guided nanoparticle assembly, including good scalability and no need for chemical pretreatments of the surface, it also presents many challenges. In particular, the combination of a complex chemical environment with the stochastic nature of deposition events renders an uncertainty regarding the positioning of NPs. To remedy this situation, we here present a framework for rational design of guided NP deposition based on the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory. As a model system, we consider the deposition of citrate-stabilized AuNPs on metal-coated nanofeatures due to its key role in the development of the experimental methodology. Particular emphasis is put on design aspects such as the choice of the metal nanofeatures and the selection of optimal shapes. To keep the model system realistic for applications in small-scale electronics and to restrict the design space, the nanofeatures considered generally consist of two identical parts placed in a mirrored configuration to represent electrodes. Complementary to the modeling, experimental depositions have been carried out for the model system in order to gauge the predictive power of the theoretical analysis.

Two common simplifications found in the literature regarding the DLVO theory are avoided in the implementation of our framework. First, instead of the Derjaguin approximation,[25,26] the surface element integration (SEI) method[27,28] is employed. This enables the description of particle–surface interactions in the presence of both chemical[29−31] and spatial heterogeneities[32−34] and has the additional advantage of being applicable even when the characteristic range of the interactions is on the order of the diameter of the particle. Second, we do not linearize the Poisson–Boltzmann (PB) equation, and the solution of which is required when calculating the electrostatic double layer (EDL) contribution to the interaction energy. This is motivated in part by the fact that the experimental electrolyte consists mainly of dissolved citrate, an asymmetric 3:1 salt. From a broader perspective, however, linearization of the PB equation is not valid when the surface potential is on the order of the average thermal energy, which is frequently the case in the type of deposition setups of interest here.[24]

Experimental Section

Citrate-stabilized NPs (Sigma-Aldrich, 742015) with a diameter of 60 nm were used in the following experiments. Nanofeatures were fabricated using a double-layer resist system consisting of a 100 nm-thick lift of resist (MCC NANO Copolymer EL6, Microlithography Chemicals Corp.) and a 50 nm-thick e-beam resist (6200.13:Anisol 1:2; supplier, Allresist GmBH), all spin-coated onto a Si(100) wafer with a 400 nm-thick thermally grown SiO2 layer. The wafer was exposed to an electron beam lithography system (JEOL JBX-9300FS, operating at 100 kV) according to the predesigned pattern described below. A 70 nm-thick metal layer of Ni was evaporated after development followed by a lift-off in acetone. The NP concentration in the system was increased over two centrifugation cycles (2400g, 10 min), where excess solution was replaced with deionized water after each cycle.

2-Propanol was then mixed into the particle dispersion. A droplet of the dispersion was placed on a chip supported by a homebuilt setup with controlled humidity in order to reduce evaporation of the droplet.[22,24] The droplet was rinsed away after the deposition with a mixture of 2-propanol and deionized water (in the same ratio as the dispersion), and the chip was rinsed with deionized water and blow-dried under a stream of N2. The schematic of the setup is shown in Figure . A preliminary investigation of the samples was carried out with an optical dark-field microscope (Carl Zeiss Microscopy GmbH: Axio) where larger parts of the sample were inspected for deposited particles. A more thorough scanning electron microscopy (SEM) investigation was then performed at 7 × 10–7 mbar using the 30 μm aperture of a Zeiss Supra 60 VP equipped with an in-lens detector with an accelerating voltage of 12 kV.

Deposition Energetics

Due to the characteristic length and timescales of the systems involved, atomistic treatment of colloidal deposition problems is prohibitively expensive. Hence, studies are typically conducted on a continuum level within the framework of the DLVO theory,[35] where the interaction free energy between two objects is taken as the sum of a van der Waals (vdW) contribution and a contribution from electrical double layer (EDL) interactions.

The key quantity that must be obtained is the interaction free energy between a depositing particle and a flat surface. This is accomplished using the SEI approach, which has shown to yield vdW energies that match the exact analytic solution for sphere–surface interactions and EDL energies derived from finite element solutions of the PB equation.[27] In SEI, the interaction energy U is calculated as the sum of contributions coming from area elements of the particle surface interacting with a flat surface f. In particular, for spherical particles, U only depends on the distance of the closest approach, D, between the particle and flat surface. The vdW contribution to U can be evaluated analytically,[36] but the EDL contribution requires a numerical solution of the nonlinear PB equation[37] due to the presence of citrate, an asymmetric 3:1 salt. A more detailed account of the relevant parts of the DLVO theory and SEI and how to efficiently evaluate U can be found in the Supporting Information.

Once U is known as a function of D, we can determine if a surface is amenable to deposition of particles. For an NP with potential ψp interacting with a flat surface with potential ψ, a repulsive EDL interaction will lead to a free-energy barrier EB that the particle has to overcome before it can attach to the surface. This barrier arises from the competition between the EDL interaction and the attractive vdW interaction (Figure S1). Steric interactions arising due to the stabilized layer of citrate formed around the NPs are neglected in this work. Their contribution to the free energy can in principle be evaluated,[38,39] given sufficient knowledge of the structure and properties of the ligand layer,[40−42] and affect the location and depth of the free-energy minima corresponding to deposition. Here, we adopt a pragmatic modeling approach where we are primarily concerned with determining if NPs will deposit at all, as determined by the free-energy barrier. For realistic parameter choices, this barrier occurs beyond the extent of the citrate layer (Figure S2) and hence a short-ranged repulsive force is not included.

In general, deposition of NPs onto a surface f is probable if the energy barrier EB is on the order of kBT or smaller. In this context, we adopt the notation Ψ◦[EB] for the surface potential required to yield a barrier of height EB and refer to this as the deposition potential corresponding to EB. As we shall see, the deposition potential is a useful quantity when selecting materials for a deposition setup.

The interaction of a depositing particle with a nanofeature can be obtained by regarding the surrounding walls of the feature as a set of flat surfaces and summing up the contributions U coming from each wall:Here, r is the position of the particle center of mass, and D(r) is the distance of the closest approach from r to the surface f. Since the nanofeatures considered in this work consist of two identical parts placed in a two-fold axis configuration, they enclose a region that we refer as the nanogap. The nanogap volume is bound by the inner walls of the nanofeature and the plane defined by its top surface. A scalar measure of the attractiveness of a general nanogap can then be obtained by integrating U(r) over VV, the partial nanogap volume that is accessible to the particleWhile I[U] clearly cannot encode the full complexity underlying a deposition event, it possesses several key characteristics expected of an ideal measure. These include scaling with the volume of the nanogap and the fact that a more negative I[U] indicates a larger chance of deposition occurring and vice versa.

As a complement to the theoretical description of the model, we have made the source code available as a Python package1.

Selection of Materials

We first discuss the importance of the choice of materials in achieving selective deposition of NPs, with an emphasis on the underlying interactions and the guidance offered by the DLVO theory.

As a consequence of the negative charges present on both the citrate-stabilized AuNPs and an unmodified SiO2 substrate and the relatively low Hamaker constant (eq S3) of the latter, the energy barrier for deposition is effectively insurmountable.[22] Targeted deposition on a SiO2 substrate can then be achieved by introducing patches or structures coated with a material that attracts NPs.[23,24] For this purpose, it is natural to consider metals as candidate materials as they generally have higher Hamaker constants than non-metals;[43] hence, the attractive vdW force between a transition-metal surface and an NP is correspondingly stronger.

The number of candidate materials can be narrowed down by calculating the theoretical deposition potential of the particle–surface system under different conditions. To illustrate the process, we analyzed the deposition potential Ψ◦[kBT] as a function of the particle surface potential for our citrate-stabilized AuNP model system (Figure ). The use of a target energy barrier EB = kBT in these calculations implies that the calculated deposition potentials correspond to a system where NP deposition occurs rapidly since a large fraction of particles will have a kinetic energy in the range of the barrier. Other values for the target barrier are possible; for instance, to study the limit where deposition becomes improbable, one can consider a larger value, for example, Ψ◦[10kBT]. Two surfaces were included in the analysis with Hamaker constants AH[SiO2] = 7.2 × 10–20 J[44] and AH[M] = 40 × 10–20 J. The latter value is typical for the Hamaker constant of a late transition metal M.[43] Given AH[Au] = 45 × 10–20 J and AH[H2O] = 4.8 × 10–20 J,[43,45] effective Hamaker constants for all vdW interactions in the system could subsequently be determined using the standard combining rule (eq S4). Another important consideration is the citrate concentration for which we note that the centrifugation and partial replacement of solvent with deionized water leads to an estimated range on the order of 0.1–1 mM, that is, lower than what is typically used for synthesis of citrate-stabilized AuNPs.

Figure 2

(a) Material selection for targeted NP deposition can be simplified by a deposition potential analysis. Particle potentials are mapped onto the surface potential that a candidate material with a fixed Hamaker constant would need in order to achieve a given deposition energy barrier EB. Illustrated here is the EB = kBT case, corresponding to rapid NP deposition. (b) SEM micrographs of trisodium citrate-stabilized NPs after deposition on SiO2 at the top and NiO at the bottom. The arrows point toward corresponding plots in (a).

If the measured ζ potential of −34 mV[22] of the AuNPs is taken as an approximation of the AuNP potential, we can conclude, on the basis of Figure a, that rapid NP deposition can be achieved on a metal surface even if the surface potential is negative (Figure a). In this case, the EDL interaction is still repulsive but deposition can occur since the vdW interaction is strong enough to overcome it over distances on the order of the system’s Debye length (eq S9). At c∞ = 1 mM citrate concentration, a metal surface potential of −40 mV would, in fact, still yield a barrier height of EB = kBT. For a surface with a smaller Hamaker constant than SiO2, rapid NP deposition is still possible for negative surface potentials not exceeding approximately −10 mV in magnitude. In this case, however, concentration dependence is less pronounced. These results thus underscore the sensitivity of the outcome of a deposition experiment to the Hamaker constants of the chosen materials.

The usefulness of a theoretical analysis in terms of deposition potentials depends upon whether the surface potentials of the candidate materials can be estimated. In ref (24), Ni was found to be highly amenable to deposition of citrate-stabilized AuNPs dispersed in 1 mM KCl,[24] in which case the measured surface potential was ψ = 61.9 mV. Since AH[Ni] ≈ 45 × 10–20 J,[43] this is consistent with Figure . Indeed, for positive surface potentials, the EDL interaction with citrate-stabilized AuNP is attractive; hence, in isolation from other surfaces and NPs, there can be no barrier toward deposition. The same study also found that no deposition took place on SiO2, for which a surface potential ψ = −57.3 mV was measured. This is again consistent with Figure a, as −57.3 mV is a significantly more negative potential than the −10 mV that would be required to obtain a barrier height of EB = kBT.

Deposition of AuNPs on Ni and SiO2 surfaces carried out as part of the present study confirms this picture: AuNPs are only found on the Ni surface (Figure b). No new surface potential measurements have been made in this case, however, due to the significant effort and difficulty they entail. Since surface potentials are not transferable when different electrolytes are considered, we can, however, only expect to gain qualitative understanding from the DLVO theory in the present case.

Fully realizing the potential of the DLVO theory as a tool for analyzing and predicting the outcome of deposition problems would require a multiscale approach where atomistic methods are used to calculate the prerequisite surface potentials.[46,47]

Geometry Optimization

In addition to the choice of NPs and sample materials, the impact of the nanofeature geometry on the outcome of the deposition process was investigated. The primary design goal, according to which different nanofeatures were selected for evaluation, was their conduciveness to NPs depositing inside the nanogap. Theoretically, we quantify the attractiveness of a nanofeature using the integrated free energy, I[U], introduced in eq . Changes in I[U] are then studied as a function of the design parameters of selected nanofeatures. Typically, the final application of the deposited nanoparticles imposes some restrictions on the maximum allowed area for the nanogaps, which is taken into account when comparing different designs.

The simplest nanofeature design considered consists of two metal bars positioned in parallel to partially enclose a nanogap. If the dimension of each bar is fixed, the only adjustable design parameter of this feature is the spacing between the bars, that is, the nanogap width. Experimentally, deposition of citrate-stabilized AuNPs was observed on parallel Ni bar features for a range of nanogap widths. As apparent from SEM micrographs (Figure a,b), deposition of NPs occurs both inside the nanogap and on the outside walls of the Ni bars. Proximity to a single surface of Ni is thus enough to overcome the repulsion from the Si substrate, consistent with the assignment of a positive surface potential to the Ni surface, as discussed in the previous section. A careful examination of the SEM images reveals that no particles are found in the middle of the nanogaps, indicating that the decay of the EDL interactions over the length of the nanogap is rapid enough for a deposition close to one of the Ni surface to be more favorable than deposition in between the two surfaces. The parallel bar geometry is thus not optimal for deposition since a particle cannot attain sufficient proximity to more than one of the nanofeature walls. Furthermore, since increasing the nanogap width from 50 to 125 nm results in a slight decrease in the number of successfully deposited particles (Figure c), the nanogap width as a design parameter does not offer much control over the deposition results. Further insight into the parallel bar geometry can be obtained from the DLVO theory. For this purpose, the interaction free energy between a AuNP and the surrounding sample was calculated over a vertical plane spanning the nanogap between the bars (Figure a, top). In these calculations, the surface potentials were set to representative values ψ = – ψp = – ψs = 50 mV and a citrate concentration of 1.0 mM was assumed. From the resulting free-energy maps, it can be seen that the most favorable positions for the particle are, as expected from the experimental results, those closest to a Ni surface (Figure a, middle) for both narrow 80 nm nanogaps and wider 125 nm gaps.

Figure 3

Cross-sectional attraction maps for two parallel Ni bars with a spacing of (a) 85 and (b) 125 nm. The x axis indicates the distance from an NP to the wall of one bar. Only the left wall is shown in the map. Each map has a corresponding schematic above; the cross section of which can be seen in (b). (c) Integrating the free energy throughout planar cuts made perpendicular to the parallel Ni bar geometry gives a measure of the overall ability of the nanogap to attract nanoparticles. This measure attains a maximum when the nanogap is slightly wider than the particle diameter, although the effect is less pronounced at higher salt concentrations.

In addition to the free-energy maps, the integrated free energy was obtained for the parallel bar geometry as a function of the nanogap width under different concentrations (Figure c).2 Intuitively, one might expect an increase in I[U] with nanogap width since a wider gap exposes more of the repulsive substrate, which can also be seen by comparing the energy maps for the narrow and wide gaps (Figure a,b). Starting from narrow 70 nm gaps, however, I[U] initially becomes more negative and the expected increase is only exhibited for gaps wider than 80 nm, depending on the concentration. As a result, a minimum exists corresponding to the optimal nanogap width most likely to capture an NP. This minimum occurs at 98 nm when c∞ = 1 mM and at around 82 nm for concentrations between 0.5–1.0 mM. The overall trend is that, for lower concentrations c∞ ≪ 1 mM, the minima are both deeper and more pronounced, indicating that the nanogap is more attractive but its capture efficiency is also more sensitive to concentration.

While the parallel bar design is good at capturing particles, it does not allow lateral confinement of the deposited particles, which becomes a problem when high-precision positioning is necessary. It is possible to simultaneously remedy this issue and improve the probability for successful particle deposition by considering geometries where the nanogap is enclosed by Ni walls to a larger extent. For this purpose, geometries where the two nanofeature parts resemble opposing forklifts were constructed. If the maximum nanogap area is restricted, geometries belonging to the forklift family can achieve more negative I[U] values than parallel bar geometries since, in the simplest case, a rectangular forklift geometry can be constructed by extending the parallel bar setup with walls that partially enclose those sides of the nanogap previously left uncovered (Figure a,b). Further improvement of the rectangular forklift design can be achieved by cutting away the 90° corners at an angle α and introducing a new corner wall along the direction of the cut (Figure b,c). Symmetry dictates that the optimal opening angle α of the corner wall must be close to 135°, which is also confirmed by calculations (Figure S3).

Figure 4

Schematic representation of successive improvements to a parallel bar design. (a) Attractive walls are added to the sides to better enclose the target deposition region. (b) Sharp inner corners are cut away and replaced by corner walls with an optimal opening angle of 135°. Finally, the length of the corner wall can be tuned. If the opening angle remains fixed, extending the length of the corner wall beyond a certain point leads to a decrease in the nanogap area available for particle deposition; hence, the optimal choice of the corner length is not immediately clear.

The remaining design parameter to optimize is the length of the corner wall, denoted as L. From Figure b, it can be seen that the integrated free energy exhibits a nontrivial dependence on this length, with two distinct minima at 26 and 36, the latter being the global minimum. The existence of these two minima closely reflects changes in the nanogap area available to a depositing particle. More precisely, the first minima occurs when the corner wall reaches its maximally allowed value under the constraint that the available area does not decrease. Further increase in L initially leads to an increase in I[U] due to the associated decrease in available area, but this trend is reversed once regions that were previously far removed from the corner come within the range of its attractive interaction. A second minimum consequently appears when the corner wall has become sufficiently long such that the regions closest to the side walls become unavailable if L is increased further. Contrary to what might be expected, the attractive power of a nanogap can thus, in certain situations, be increased to an extent by reducing the area available for depositing particles in favor of greater proximity to attractive feature surfaces.

Experimentally, a set of four nanofeatures designed for both lateral and vertical confinement, that is, two-dimensional control of depositing NPs, was considered, including the forklift design (Figure a–d). Counting the number of successful NP depositions on each array yielded a success rate of 31% for the forklift geometry and lower rates ranging from 14–18% for the other candidates (Figure ). In terms of our theoretical model, this result can be attributed to the fact that the forklift nanogap is enclosed to a higher degree by the attractive Ni walls than the other geometries, which decreases the integrated free energy as seen in Figure and Figure S3.

Figure 5

(a–d) Partial SEM micrograph over four different arrays of 70 nm-thick Ni features. Each array corresponds to a different candidate geometry that offers both lateral and vertical control of the position of depositing NPs. (e) Deposition success rates after a sample submersion period of 72 h.

Figure 6

SEM micrographs over two arrays of Ni bars with a height of 70 nm on a Si/SiO2 substrate. The arrays seen in this figure are parts of a greater set of arrays. The parallel bars are spaced with (a) 85 nm and (b) 125 nm. (c) Percentage of successful depositions of a single NP. NPs were allowed to deposit over a period of 72 h. The full SEM micrographs used in this analysis can be found in the Supporting Information, Figures S4–S10.

The forklift geometry was also compared to the parallel bar geometries with varying nanogap widths (Figure ). Here, the percentage of successfully deposited single NPs is ∼40% for a 50 nm-wide gap and 37% for a 60 nm-wide gap. The latter number is suitable for comparison with the forklift geometry where the vertical dimension of the nanogap is also 60 nm. This leads to the conclusion that the double bar geometry has a slightly higher success rate for single NP deposits, which is explained theoretically by the increase in available area close to attractive Ni walls. It must be stressed, however, that the increase in success rate of deposited NPs in this case comes with a complete loss of lateral control over the NPs positioning, rendering the double bar geometry unsuitable for many applications.

Comparisons between different nanofeature geometries are thus most meaningful when the total nanogap area is constrained from above by the final application, in which case design parameter optimizations of the kind illustrated in Figures c,4 and Figure S3 are straightforward to apply.

As a caveat to the theoretical predictions presented above, it should be noted that the optimal ranges of certain design parameters are always wider in experimental settings since particle sizes are not fixed, rather they follow a distribution with a given mean.

Chemical Environment

In addition to the choice of sample materials and nanofeature geometry, several aspects of the chemical deposition environment merit further discussion.

It was observed that the addition of 2-propanol to the citrate-covered AuNP solution is required to successfully achieve NP deposition. From Figure S11, it can be seen that 2-propanol greatly improves the number of NPs successfully deposited on the prefabricated nanostructures. Optimal results were obtained with a 50% mixture between AuNP solution and 2-propanol, and further addition results in significant NP agglomeration. Intuitively, this could be attributed to the decrease in the Debye length caused by the presence of 2-propanol, which implies greater screening of the repulsive electrostatic interactions from the SiO2 substrate. The situation is complicated, however, by the fact that the addition of propanol to the solution can also affect the local chemical environment of the surfaces of Ni and SiO2 and the AuNPs.[48−50] A more thorough investigation of the impact of 2-propanol on NP deposition is therefore an important topic for future work. As a consequence, the DLVO-based modeling was restricted to the case of pure water to maintain a consistent set of simulation parameters. This, in turn, implies that our current calculations do not capture the full chemical complexity of the experiments. We emphasize, however, that in spite of this deficiency, the important general trends of the system are captured, as evident from the comparison with experimental data. Furthermore, given a suitable set of parameters for samples where the solvent is a water–propanol mixture, the theoretical analysis can be repeated in much the same fashion.

Another important consideration in designing deposition setups is the role of pH, which influences, for example, the surface potentials and speciation of citrate. In the present work, measurements performed after dilution with propanol yielded a pH value of 6.9. Under such conditions, the majority of citrate ions in the solution are trivalent,[51] and hence no other valencies were considered in the DLVO model.

Rinsing and drying of the sample after the liquid-phase deposition is also a point of concern as it can lead to removal of NPs, in particular, if they are weakly bound to the surface.[52] To investigate whether any significant removal of particles was occurring on our samples, chips were subjected to repeated rinsing and drying cycles with intermediate SEM imaging. The typical result can be seen in Figure S12 where no observable change in the number of deposited NPs is found when comparing the same area of a single chip that has undergone one and three cleaning cycles, respectively.

Conclusions

We have presented a predictive framework for analyzing guided deposition of colloidal nanoparticles based on a combination of the DLVO theory and empirical experiments. The framework was illustrated for a system consisting of citrate-stabilized NPs in aqueous solution above sub-100 nm metal/metal oxide nanosized features on a Si/SiO2 substrate.

With regard to materials, it was found that Ni is particularly suitable for the deposition of AuNPs, which can be attributed to its surface charge being of opposite sign of the NPs and its high Hamaker constant. We further explored, using our model system of citrate-stabilized AuNPs, how the concept of deposition potentials defined with respect to a fixed value deposition barrier can aid in the process of selecting materials for a deposition setup. In particular, this analysis revealed that the range of surface potentials over which a material supports rapid deposition of NPs is highly sensitive to the Hamaker constant, with larger allowed ranges for metals with high Hamaker constants such as Ni. It was also noted that while the current modeling framework based on DLVO can describe much of the chemical environment, it requires knowledge of parameters that are difficult to measure experimentally such as surface potentials. Here, atomistic simulations could provide a solution for calculating properties that are otherwise difficult to assess.

The selection of optimal geometries for three-dimensional nanofeatures was also investigated. Starting from a basic design consisting of two parallel, bar-shaped nanofeatures, we demonstrated that the design could be systematically improved in terms of the number of successfully deposited particles. More precisely, by extending the bars into angled structures resembling forklifts, the number of successfully deposited nanoparticles could be increased. This type of design also allows a more selective deposition process where the number of nanoparticles captured between opposing structures can be controlled, which is important, for example, for avoiding short circuits in electronics applications.

For more complex nanofeature geometries, additional design parameters are introduced in the form of angles and structure dimensions. Here, the proposed modeling framework offers an inexpensive solution for selecting optimized parameter values using the integrated interaction free energy as the objective function, as demonstrated for the forklift geometries.

The model is not only limited toward the deposition of spherical particles; other more complex shapes such as cubes, stars or rods could also be used with some modifications made to the model. One could as well think of more complex structures used to attract the particles, all depending on the application. Only flat surfaces have been considered in this work excluding parameters such as roughness and local defects. Possible continuation for this article could be the deposition inside nanosized trenches or tubes used in, for example, microfluidics.