Understanding How Charged Nanoparticles Electrostatically

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Understanding how Charged Nanoparticles Electrostatically Assemble and Distribute in 1-D Keith M. Carroll, Heiko Wolf, Armin W Knoll, Jennifer E Curtis, Yadong Zhang, Seth R. Marder, Elisa Riedo, and Urs Duerig Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03471 • Publication Date (Web): 30 Nov 2016 Downloaded from http://pubs.acs.org on December 2, 2016

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Understanding how Charged Nanoparticles Electrostatically Assemble and Distribute in 1-D 1

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Keith M. Carroll , Heiko Wolf , Armin Knoll , Jennifer E. Curtis , Yadong Zhang , Seth R. Marder , Elisa 4-6

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Riedo , Urs Duerig 1

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IBM Research – Zurich; School of Physics, Georgia Institute of Technology; School of Chemistry and 4

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Biochemistry, Georgia Institute of Technology; CUNY-Advanced Science Research Center; Department 6

of Physics, CUNY-City College of New York; Physics Program, CUNY-The Graduate Center. Abstract The effects of increasing the driving forces for a 1-D assembly of nanoparticles onto a surface is investigated with experimental results and models. Modifications, which take into account not only the particle-particle interactions, but also particle-surface interactions, to previously established extended Random Sequential Adsorption simulations are tested and verified. Both data and model are compared against the heterogeneous Random Sequential Adsorption simulations, and finally a connection between the two models is suggested. The experiments and models show that increasing the particle-surface interaction leads to narrower particle distribution; this narrowing is attributed to the surface interactions compensating against the particle-particle interactions. The long term advantage of this work is that the assembly of nanoparticles in solution is now understood as controlled not only by particle-particle interactions, but also by particle-surface interactions.

Both particle-particle and particle-surface

interactions can be used to tune how nanoparticles distribute themselves on a surface. Introduction The surface assembly of microparticles and nanoparticles (NPs) is of particular importance for a 1

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myriad of fields such as: photonics , anti-reflection coatings , cellular studies , superhydrophobic 4

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surfaces , micro- and nano-fabrication , and more . Most of these applications require site-specific and controlled assembly of colloids or nanoparticles onto a substrate, and there is plenty of experimental, theoretical and computational work dedicated to understand and predict the placement of a single particle onto a surface at a specific location

7–9

. Part of the attraction, in addition to the applications, to working

with colloidal particles and in particular NPs is the tractability and computational accessibility of the 10

interactions acting on the particle at these scales .

For the most part, particle assembly can be

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decomposed into two main components: the particle-particle forces, which describe how particles interact amongst themselves in solution, and particle-surface forces, which describe the driving force to bring the particles down to the surface. 14

physical , and/or electrostatic

15,16

Examples of these interactions are intra-molecular

11,12

13

, chemical ,

. In particular, electrostatic driving forces are important beyond their

applications to colloidal and NP assembly because their principles can, in part, be applied to charged molecules. Charged molecules include, for example, bio-molecules, such as proteins, DNA, and cellular membranes, all of whose charges derive from hydrogen-ion dissociation or protonation of side groups and charged backbones

17,18

. While the assembly process for small molecules and macro-molecules is

more complicated and can not be attributed to a single force or interaction such as the electrostatic interaction, by making the assumption, such as that made in DLVO theory, that interaction forces can be decomposed into different, orthogonal components, we start to resolve the assembly process component by component. Studying electrostatic interaction in scenarios, where there are little or no other forms of interactions, provides us with a controlled method to appreciate and faithfully approximate different electrostatic effects and parameters without external interference; these isolated interaction scenarios are difficult to accomplish with molecular assemblies, but are attainable with colloidal assembly. Studies and results from these “simpler” colloidal experiments may contribute significant input for the understanding of molecular assembly, while leaving the other driving forces, such as van der Waals or chemical interactions to be de-coupled with separate experiments and studies. Moreover, measuring the dynamics of single molecule assembly processes is, at best, difficult and tedious; therefore, we also benefit from less intricate methods to measure the assembly process. With this as an underlying motivation, we aim to understand and quantify how the assembly of charged particles, more specifically of nano-particles, evolves as a function of different parameters. For example, previous studies and models have focused on explaining and measuring how particles assemble onto a surface by accounting for only the particleparticle interactions, which is a valid approach as long as the particle-particle interaction is large compared to the particle-surface interaction and that the assembly driving forces are uniform. But these 19

models, with few exceptions , fail to account for changes in the assembly process when the particlesurface interaction become important; in particular, one exception is the work

20

by Ma et al, which

examines how charged particles are electrostatically funneled down onto a surface patterned with


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oppositely charged stripes. Here, we specifically study particle assembly in the presence of a significant particle-surface interaction. Electrostatic potentials in solutions have been described by several models and combination of 21

models, such as the Helmholtz and Gouy Chapman-theories . As part of the evolution of understanding electrostatic potentials and forces in solution, the Poisson Boltzmann equation (PBE) developed as a method to approximate and compute the interactions and forces between charged species in solution. The results from the PBE have been applied to understand the assembly process of charged colloids 21,20

onto a surface

, only accounting for the particle-particle interaction. The computational accessibility of

the electrostatic driving forces have allowed researchers to generalize the assembly process with physics based as well as abstract techniques; two such techniques are the Random Sequential Adsorption (RSA) 21,22

method and the extended Random Sequential Adsorption model

. Here, we refer to the extended

Random Sequential Adsorption model as the Monto Carlo (MC) or the Monte Carlo Random Sequential Adsorption (MC-RSA).

The final concentration, in the MC-RSA models, is controlled in part by the

amount of time the simulation runs and the strength of the particle-particle interaction

14,17

, and not

explicitly on the particle-surface interaction. Intuitively, we expect the final concentration to relate not only to the particle-particle interactions, but also to the particle-surface driving force, and there is experimental evidence

23–25

that this is indeed true, but the connection between the particle-surface interaction and the

final concentration is elusive. RSA simulations remove the complications of explicit knowledge of the particle-particle and particle-surface interactions, but do so by obscuring the underlying physics. Succinctly stated, the RSA methods are robust simulations, which compute the adsorption process without the complications of knowing all details of the particle-particle and particle-substrate interactions. RSA simulations are simple to implement; however, there is disconnect between the physics describing the assembly process and the parameters that can be extracted from the simulations. One additional challenge, therefore, is to link the RSA results to the physics that explain how different driving forces effect the assembly process. Techniques such as the RSA or MC methods are often used to predict the distribution of charged particles on a 2-D surface, and for the most part they succeed, under the assumption the particle-surface interaction is locally constant (i.e. not changing as a function of position). What is needed, however, for


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the MC and RSA models is a way to account for variable particle-surface interactions and while many studies benefit from understanding the 2-D distribution, we work with the simpler 1-D case, which is just 26

as valuable and insightful . The beauty of 1-D studies is several-fold. First, the physics reduces down to a single dimension, removing many of the complications caused by having a second - or third dimension. This reduction highlights the underlying physics; moreover, because of this reduction in complexity, the assembly process and description become less complex and more intuitive to changes in parameters, such as enhancing the driving forces. Finally, in 1-D, simulation time is greatly reduced, allowing one to span over a large space of parameters in a shorter time. The goal of this work is to provide a detailed, experimentally verified description of how charged nano-particles assemble along 1-D lines, which have different charge densities. To do this, we use RSA and MC style techniques and modify them to account for different assembly driving forces. Furthermore, we apply our modifications to make a connection between the description of the assembly physics and the abstract RSA techniques. Experimental Details Since we are studying 1-D assembly, we need to fabricate thin, nano-scale charged lines. One of the simplest ways to create charged lines is to modify the substrate’s chemistry with functional chemical groups, such as with carboxylic acid groups

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, which can dissociate hydrogen ions to create negative

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areas, or with amine functional groups , which form positive patches upon protonation. We rely on thermal probe techniques such as thermal Scanning Probe Lithography (tSPL) and Thermo-Chemical Nanolithography (TCNL).

tSPL/TCNL are similar nano-lithographic techniques which use semi-

conducting, thermal probes to site-specifically alter a surface’s physical or chemical features through a thermally activated transformation

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. The thermal probe heats up when biased with a voltage, and

when brought into contact with a surface, the heated probe induces a localized temperature profile into the substrate. resulting elsewhere

in

The site-specific increase in temperature causes the surface to undergo a reaction, a

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chemical

or

physical

change

(Figure

1).

The

details

are

well-described

.These techniques have been used reliably to fabricate patterns on the nano-scale

which makes them an attractive candidate to study charged nano-particle assembly in 1-D.


30,31

,

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Figure 1. a) Schematic showing thermal probe lithography techniques working on a bi-layered material composed of a cinnamate polymer (red) and PPA (blue). b) Chemical structure and thermally induced reaction of the cinnamate polymer. The exposed amine groups in solution protonate to create positive charges. c) Schematic showing the assembly of negative particles (green) on top of positive lines (red) against a background of negative charge (blue).

In order for us to study the assembly process in an effective manner, we use silica NPs for the assembly process.

Silica nanoparticles are attractive because under neutral pH-levels they are

negatively charged from the dissociation of hydrogen ion from the external hydroxyl groups (i.e. their charge is native, and not introduced by labile groups or chains); moreover, because of the high pKa for the silanol group’s protonation, we have isolated our experiments to mono-charged particles, removing complications that could be introduced by dipole interactions which may exist in zwitterionic molecules or with appropriately designed janus particles. Solutions of silica particles are stable without any additives, making the solvent clean from any potential contaminants. In addition, the mono-dispersity of the silica


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NPs can be well controlled, removing complexity caused by size variations. For this study, we use 50 ± 5 nm diameter silica NPs (Nanocomposix); and for the first part of this work, we work with de-ionized (Millipore) water to create a long Debye length, which also means that our pH cannot be accurately measured since there is no added buffer. Full details are provided in the methods sub-section. To assemble the negatively charged NPs, we need a surface which upon modification, contains a positive charge; furthermore, the rest of the surface needs to be passivated against un-specific binding of the particles. For these purposes, we have layered two polymer materials as shown in Figure 1a. The first, top layer is a 7 nm layer of polyphthaldehyde (PPA). While the mechanism remains elusive, PPA 35

picks up a negative charge in aqueous solutions ; we believe this charge derives from the preferential adsorption of anions from solution. The PPA acts as a passivating agent against negatively charged particles because of the electrostatic repulsion. The positive lines on the surface come from an activated form of the bottom layer of polymer material. This second layer is composed of ~50 nm of cinnamate polymer

(poly((tetrahydropyran-2-yl

N-(2-methacryloxyethyl)carbamate)-co-(methyl

4-(3-

methacryloyloxypropoxy)cinnamate))); when activated this polymer has an exposed primary amine group. Once in solution, the primary amine group protonates, creating positive charges. Figure 1b shows the chemical structure and the reaction the cinnamate polymer undergoes with thermal treatment: the tetrahydropyran side group cleaves, leaving behind an exposed, active amine group on the surface (hereto we refer to this dissociation process as de-protection). To de-protect and expose the underlying cinnamate polymer, we use tSPL to pattern the substrate before placing the patterns into solution. tSPL induces the thermal reactions in both PPA and the cinnamate polymer.

When heated, PPA decomposes from its polymer structure into volatile

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monomeric units . In tSPL, the thermal probes localize this unzipping mechanism and, by layering the materials in the manner depicted in Figure 1a, we can remove the top layer of the PPA and simultaneously de-protect the cinnamate polymer. When the patterned substrate is submerged into a solution with negatively charged particles, the particles assemble only on the patterned lines, as depicted in the schematic Figure 1c, where the exposed amine groups are shown in red, the areas covered with PPA are depicted in the blue, and the particles are shown in green. The driving force acting on the particles originates from the electrostatic


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attraction between the negatively charged particles and the protonated, exposed amine areas written into the cinnamate polymer. We can control this driving force by varying the level of de-protection that takes place in the cinnamate polymer. The extent of the thermal probe induced surface transformations can be tuned and controlled through the writing parameters such as thermal probe’s heater temperature, the duration of writing time (writing speed), or the load applied by the probe

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.

Methods and Materials Polymer surfaces were fabricated in a similar way as previously described

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. Briefly, silica

wafer were cut into inch by inch wafers. They were cleaned and sonicated for 30 min with separate solutions of 5% Triton-X, de-ionized water, and finally ethanol. The samples were rinsed with Ethanol and plasma cleaned for 2 min. Cinnamate polymer solution at concentrations of 20 mg/mL in cyclohexanone was spun down at 1000 RPM for 2 min, and UV cross-linked for 5 minutes. After crosslinking, a PPA layer of 7 nm was spun down on top (5798 RPM for 80 seconds). The PPA solution was prepared at concentrations of 0.5% by weight dissolved in cyclohexanone. Particle solution were obtained from Nanocomposix; they were diluted down from 1% down to .001% concentrations by weight. The diluted solution was treated with ion-exchange resins. Prior to the experiments the particles were further diluted down to .0001% with either de-ionized water or with salty -4

-5

-6

water (NaCl 10 , 10 , 10 M solutions). All patterning was done on in lab developed instruments that have been previously described

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After patterning, the substrates were placed in de-ionized water or salt water for about 5-10 minutes to allow for electrostatic equilibrium to take place. The solution of particles were then proportionally added (final concentration of .00005% by weight). After about 15 minutes, the samples were removed with water by either diluting the solution 20 times and adding ~1% Ethanol to fix the particles or by directly removing the water. To remove complications from external contamination, we isolate our samples in a vessel with limited atmospheric exchange (i.e. a sealed compartment), and use 2

water with an initial conductivity of 18.2 MΩ/cm or water that has been treated with mixed bed deionizing resin particles for several days. All SEM measurements were carried out on a LEO1550 SEM. The resulting images were processed and tracked with ImageJ software.


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Results and Discussion In a typical experiment, we pattern sets of parallel lines (each set is composed of 10 lines spaced by ~1 µm, with widths measured as ~70 nm, lengths of 10 µm); each set is written at a writing speeds of ~350 µm/s and at a unique, fixed temperature in a range of 650-900 ºC. Examples of the patterned lines are provided in the SI, but briefly we highlight that the roughness (RMS) of the bottom of the lines is about 1-1.2 nm. We adjust the writing force to reach fixed depths of 15-20 nm to ensure the writing contact area is approximately the same. This guarantees the thermal contact resistance of the silicon tip to the polymer to be similar for each writing temperature. The heater temperature is the temperature measured across the thermal probe’s heating element. Because of losses through the silicon tip the ratio between 38

the surface temperature and the heater temperature with respect to room temperature is ~0.3-0.6 . By ensuring a constant thermal resistance across the tip-surface contact temperature ratio which is an unknown parameter is fixed. By writing at different temperatures, we tune how much de-protection the cinnamate polymer undergoes, acting as a control parameter for the lines’ charge density: lines written with higher temperatures have more charge, whereas the lines written at lower temperatures have a lower charge. The strength of thermal probe lithographic techniques is that we can employ these tuning parameters to probe the particle-line interactions. To the best of our knowledge, most models and simulations and, with few exceptions, experimental techniques have only looked at tuning the particle-particle interactions, leaving the particle-substrate interaction alone. There has been limited experimental work to demonstrate that different charge concentrations lead to different concentrations of particles; examples have shown that the particle concentration varied as a function of a chemical gradient over distances much greater than the particle-particle interaction

23,24

. Moreover, there has been attempts to create model parameters 19

which account for the surface interaction , but these studies in particular could not verify these parameters with a variable surface parameter interaction. The results of one typical experiment are shown in Figure 2, where we have varied the writer temperature from 650ºC up to 900ºC. After exposing the sample with 50 nm silica NPs (see methods section for full details), we can measure the particle distributions with the scanning electron microscope


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(SEM). Figure 2a, b show the corresponding SEM images for two writing temperatures: 700 ºC and 900 ºC. Examples for all writing temperatures (650-950 ºC) are provided in the supporting information. The silica particles are seen to bind to the patterned lines with virtually no particles bound elsewhere. Figure 2 demonstrates how the distribution changes as a function of different substrate charges. The lines written at 900ºC are more densely coated with NPs than the lower temperature lines. Intuitively, we expect that as we increase the writing temperature and thus the surface charge of the lines, the magnitude of the particle-line attraction will increase. Consequently, the particles will be able to pack at higher densities because the attraction to the line compensates against the particle-particle repulsion. The particle density as a function of writer temperature is shown in the inset of Figure 2c, and we see that as the temperatures increases, the density also increases up to some saturation point. We interpret this levelling of the particle density as derived from the saturation of the polymer’s de-protection reaction: above a certain temperature the polymer is fully converted, preventing any further increase in the surface charge.

Figure 2. SEM measurement of silica NPs bound to lines written at a) 700 °C and b) 900 °C. Higher written temperatures lead to more activation of the cinnamate polymer, which in turn leads to an


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enhanced particle-line interaction.

c) Plots of the neighboring particle distribution function as a function

of distance. Experimental results for writing temperatures ranging from 650 to 900 °C are shown as points, solid lines represent simulations from the RSA method, and the dashed lines are fits from the modified MC-RSA method we introduce. There is excellent agreement between our two models and the experimental measurements. Inset is a plot of the density of particles per µm.

Our focus for this work is to understand how charged particles distribute along a charged line. It is known and intuitively understood that particle density will increase with increasing surface charge. To quantitatively understand how particles interact with a charged surface, we need to consider and study the statistics of the particle distribution.

For these purposes, scientists typically rely on the radial

distribution function. The radial distribution function measures how the particle density, on average, fluctuates as a function of the radial distance from a particle’s center. The setup of our experiments makes it extremely difficult to collect a large amount of statistics to faithfully measure the radial distribution function. Instead, we suggest measuring and analyzing the neighboring particle distribution. The neighboring particle distribution, succinctly, is the probability distribution function for the center-tocenter particle distances separating two particles next to one another. We expect a few things to be true of the neighboring particle distributions: 1) since there is a hard wall caused by steric hindrance between two particles, the distribution should be zero up to twice the particle radius; 2) since the particle-particle electrostatic repulsion enhances as the particles approach, the distribution should start low and increase up to some peak; 3) when the particles are sufficiently far apart such that their interactions are negligible, the distribution must decay at a rate proportional to the density, i.e. a Poisson distribution for noninteracting particles. These properties for the neighboring particle distribution are expected as long as there is an attractive driving force to cause assembly to the surface. The details of the distribution (such as the peak value) are subject to the strength of the line’s driving force. Examples of the neighboring particle distribution measured from SEM images are shown in Figure 2c. The experimental results are shown as data points, and the distribution curves qualitatively match our predictions about the particle-particle distribution.

The fits, shown as solid and dashed lines, come from the RSA and the MC models


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respectively; both models are described in the next section. Examining the distribution for the writer temperature at 700ºC, Figure 2c shows that initially there are no particles closer than twice the radius as expected from the hard wall repulsion. Past the hard wall distance, the distribution function increases until it peaks around 250 nm.

Similar curves are found for the other writing temperatures with the peaks

varying from ~150 nm (800-900 ºC) up to ~500 nm (650ºC). The distributions for 800-900 ºC converge onto a single curve; this coalescence of the high temperature distribution curves comes from the saturation of the cinnamate polymer’s de-protection and is related to the saturation of the particle density seen in the inset. The general trend observed in Figure 2c is that as the temperature increases the distribution function shifts its peak value to lower values and the distribution narrows. The evolution of the particles’ distribution from low temperatures to high temperatures can be understood through the driving force acting to assemble the particles versus the inter-particle repulsive forces. Because of the greater surface charge density, patterns written at higher temperatures have a larger driving force, leading to a more densely packed particle ensemble. A higher density shifts the peak of the distribution function towards lower values and narrows the distribution’s width because as the particles can pack in tighter and closer, the mean-spacing between particles decreases and the distribution must decay asymptotically at a rate proportional to the density. These effects result in a narrower, leftward shifted distribution profile. For low temperature patterns, we expect a lower surface charge density; a smaller surface charge density corresponds to a weaker attractive force, which does not compensate as well against the particle-particle repulsion. With stronger particle-particle repulsion, the particles space out further, manifesting in a wide, rightward shift of the neighboring particle distribution. While the plots show how the distribution function changes as a function of writing conditions, we need to further analyze the data in conjunction with models to extract some quantitative information. Models and Analysis To explain and fully model the assembly process, we need to account for interaction from particles in the suspension, the attraction of the NPs to the surface, and repulsion from particles already bound to the surface. The interactions from the particles still in the suspension originate primarily from the change in the chemical potential when a particle binds to the line. We can ignore these effects from the particle solution by working at concentrations diluted enough that we can ignore particle-particle


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interactions from particles in solution, but concentrated enough that any change in entropy is negligible (i.e. at concentrations where the chemical potential of the solution is approximately constant as particles are added to the lines). The remaining two interactions, namely the interactions between the assembled particles and the particle-line interaction, cannot be ignored and we describe and detail how we can model the assembly process with two distinct methods. The first approach consists of a modified Monte Carlo (MC) simulation with certain, typical assumptions to account not only for the particle-particle interactions, but also for the tunable particle-surface interaction from the surface’s charge density. The second method relies on a modified version of the well-described and reported Random Sequential Adsorption (RSA) for particles adsorbing to a surface; this technique is easy to understand from a simulation perspective, but more abstract with respect to the physics. In order for us to compare between the experimental results and the models, we have assumed that the particles once down, do not move. This sticky particle assumption is rather difficult to prove explicitly; however, results from other experiments indicate that when particle deposit interacting with a surface they show little movement when 39

the water is removed . This sticky particle assumption is also fairly typical for RSA and extended RSA 20

simulations, and has been used in 2-D scenarios to measure the radial distribution function . We begin with a description about the MC method, then briefly review the RSA techniques employed, and finally we conclude by drawing a connection between the two methods. The MC method requires an approximate form for the particle-particle and particle-line interactions.

These two interactions can be further decomposed into different components: van der

Waals forces, electrostatic forces, and osmotic pressures from the co-/counter-ions in the surrounding solution.

We ignore the van der Waals forces, since silica is known to have small van der Waals 40

contributions and because we work with small particles . Electrostatic potentials in solution are well described by the self-consistent Poisson-Boltzmann Equation (PBE), which successfully estimates how charged, dissociated ions re-arrange and distribute near a charged particle/surface or a particle/surface 21

held at a constant potential . In low ionic strength solutions, knowledge of the electrostatic potential translates into the knowledge of the ionic distribution through a Boltzmann factor; the ionic distribution can 41

then be used to compute the osmotic pressure .

With the help of electrostatic force tensors and

pressure-stress tensors, one can compute the total force and consequently the total interaction between


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two entities in solution (such as particle-particle or particle-line). Because of the coupling between the electrostatic potential and the ionic distribution, computations of interaction energies are dependent upon the correct computation of the electrostatic potential from the PBE. The PBE, however, is a non-linear equation for which very few analytic solutions exist presently; instead, most analytic solutions come from a linearized theory known as the Debye-Hückel (DH) approximation. The DH approximation is valid reasonably far from the surface and for potentials less than kT, but it has also been shown that it provides surprisingly high agreement for the charge distributions near the particle/wall surface upon substitution of effective charges and potentials

21,42

.

The characteristic length scale, which can be derived from either the few solutions that exist to the PBE or directly from the DH approximations, is the Debye length. The Debye length represents the length scale over which the electrostatic potential decays; physically, the decay is caused and understood as the counterbalancing effects from the cloud of co-/counter-ions in the solution which rearrange as a result of the electrostatic potential. The Debye length is determined by a number of factors including the dielectric coefficient, temperature, ionic valency, and the ionic concentration. Both the PBE and the DH approximation only estimate the electrostatic potential in solution caused by a charged particle/surface, but they do not directly quantify the interaction between two entities. The total interaction energies involve a combination of electrostatic and osmotic forces acting on 21

the particle/surface components .

Further complicating the process is how the co-/counter-ions

rearrange themselves when two components are placed near one another. The Linear Superposition Approximation (LSA) helps to overcome this obstacle. Typically, the LSA operates under the assumption that the particles/surface entities are sufficiently separated such that the ionic re-arrangement caused by the presence of the second component is negligible. However, it has been shown theoretically that the 21

LSA is a decent approximation even at short distances, for which the assumption is no longer valid . It has also been measured and shown experimentally with optical tweezers that the LSA and DH 43

approximations are valid in regimes in which the Debye length is large and both assumptions fail . In order for us to advance our description and understanding of the particle assembly physics, we work under the assumption that both the DH approximation and LSA are valid for the particle-particle and


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particle-surface interactions in our experiments. For these approximations, excluding van der Waals forces, we can write the particle-particle interaction in two equivalent ways as: =

ψ = 4 0 4 1 + [1]

where Z is an effective valency number, q is the electron charge, εr is the dielectric constant of the solvent -1

(for water ~80), ε0 is the permittivity of free space, κ is the Debye length, r is the radial distance from the particle’s center, a is the particle radius, and ψo is the effective potential of the nanoparticles. We write this down in these two ways to be commensurate with literature and because both are useful for comparison and estimations. The functional form in Equation 1 is composed of two distinct, distance dependent factors. The first factor is the 1/r component, which we can understand from point charges in electrostatics; the second is an exponentially decaying factor with the decay rate determined by the Debye length. The term which dominates Equation 1 depends upon the circumstances: for short distances and radii smaller than the -1

Debye length, the 1/r term dominates; very far away (r>> κ ), the exponentially decaying term dominates. In order for us to enhance effects from the particle-substrate interaction, we needed to work with a particle-particle interaction which decays at the slowest rate possible. Consequently, we designed our experiments using Debye lengths which are large compared to the particle radius.

For long Debye

lengths, the decay of the particle-particle interaction is mostly attributed to the 1/r term in the regions of interest (i.e. Uint > 0.5 kT). It is known that if the water resistivity measures 18.2 MΩ·cm, the water has a 44

pH extraordinarily close to pH 7 and the water is essentially pure . Pure water, with a pH of 7 and no added salt, would have a theoretical Debye length of approximately 950 nm. Most water, however, is contaminated by either the container or just by the ambient environment (i.e. through the absorption of CO2). As long as the solution is fresh or pre-treated with ionic exchange resins and all containers are 40

cleaned, the contamination of the water primarily originates from the atmospheric CO2 . When the CO2 concentration in water has reached equilibrium, we calculate the Debye length to be ~250 nm (238 nm). We work under the assumption that we have a long Debye length (>250 nm). For the simulations shown in Figures 2 and 3, we assume a Debye length of ~900 nm; this is approximately the Debye length in pure H2O.

This approach is justified by the fact that we found the simulated distributions to have little


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dependence on the salt concentrations over Debye lengths ranging from ~250-900 nm, though the exact values of interaction parameters slightly change. See SI for these results. Equation 1 requires an effective valency charge (Z) to be assigned to the silica NPs, and as a 45

2

good estimate, we extract a value found in the literature . Assuming the effective charge scales as r , we estimate the effective charge as 18-28 electrons per silica particle. When we implemented our simulations and matched to experiment, we found a similar range in the valency: 17-35 electron per silica particle. This value is smaller than expected from other calculations; as has been previously pointed out, we attribute this discrepancy to charge regulation

45,46

. This value also gives potentials that are close to zeta 47

potential measurements of ~20 mV at neutral pH , which would be a decent approximation for the effective potential. Now that we have an approximate equation describing the particle-particle interaction, we focus on the more complex particle-line interaction. To the best of our knowledge, there does not exist a closed form solution for the particle-line interaction. Instead, we rely on a more generic approach here. We start by examining the known particle-surface interaction, for a uniform plane surface, which assuming the DH and LSA hold, the interaction can be written as: / ℎ = 4 !

" ψ $ = 4 ψ0 ψS $ = % $ 1 + S [2]

where ψS is the effective surface potential, h is the gap between the surface and the particle, and Uo is an effective term defined when h=0 as = 4 ψ0 ψS . In equation 2, we also ignore van der Waals forces since they are known to be small for silica. Similarly, from Equation 1, we learn that when the gap between two particles is 0 (r=2a), the particle-particle interaction is also fixed. It is reasonable to assume that the same could be said of a particle-line interaction, and we reduce the particle-line interaction to: /& ℎ = %

'ℎ '0 [3]

Where h now represents the gap between the particle and the line and f(h) is the function describing how the particle-line interaction changes as a function of h. We have no way of measuring f(h) with our current experimental setup because we only measure our results for h=0. As a result, we can simplify the


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complex particle-line interaction by replacing the lines effect with a constant, Uo; this is the parameter that we plug into our simulations and extract from our experimental data. We refer to this Uo parameter as the particle-line interaction, and we note it is also proportional to the surface’s potential, so we use the concepts interchangeably. The Uo parameter is equivalent to the discussed Uc parameter in previous 19

literature . Physically, Uo can be interpreted as the interaction energy to keep a particle bound to a line. It is not necessarily the binding energy which would involve a combination of factors such as the van der Waals forces, electrostatic forces, changes in entropy, etc. Because forces like van der Waals are short ranged (~ 1nm), Uo should mostly be attributed to the longer ranged electrostatic forces and instead van der Waals are likely to be responsible for locking the particles in place while electrostatics guides the assembly process. Uo also accounts for the total electrostatic force imposed by the attractive line in combination with the surrounding repelling background. The strength of this approach is that it requires no explicit knowledge of either the surrounding surface’s repulsive manner or the strength of the repulsive background, and instead relies only on the fact that the final state of the particle’s energy must be attractive to the patterned line. The form for the finite line should look different from the particle-line interaction, but the effects from the finite length should be negligible when the particles are very close to the line. It may be that the ends of the lines also have a different form, but we ignore this complication since the bulk of our statistics come from the central part of the line. Figure 3a shows how we can envision and model the particle-particle interactions coupled with the particle-line interaction. We make the assumption here that the pairwise superposition of particle interactions holds

48,49

; we justify this assumption because the perturbations caused by the small size of

the particles as well as their small potentials should have a negligible effect on the total interaction. Our intent in these simulations is to also highlight that it is not just the particle-particle interaction but also the particle-surface interaction that must be taken into account, and so for simplicity, we ignore the many body effects. Now that we have developed a simple model to describe the assembly process, we can perform a MC simulation under different particle-line interaction conditions, with a fixed valency charge (28 e), and with a fixed Debye length (here we start with 900 nm, which is the approximate Debye length in pure water); this last restriction is akin to our experimental setup since the sets of lines concurrently


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undergo the binding process with the same Debye length. A complete description and algorithm flow chart are provided in the supporting information, but we briefly highlight the procedures here. We start by allowing a single particle to assemble anywhere along a line of fixed length (10 µm), and we compute the average energy per particle (Uo for the first particle); a second particle assembles weighted by the interaction of the first particle and the surface through a Boltzmann factor. In our simulations, we do not let the particles vary in the direction transverse to the line; we justify this because our experimental results which show little variation from the center of the lines. This alignment of particles with the line’s center 20

could be due to the funneling effect discussed in the literature . The rejection conditions are two-fold: 1) the hard wall interaction must hold (i.e. no two particles can overlap), and 2) the total energy along the line is less than -0.5 kT. With each new particle assembled, we re-compute the energy along the line and particles are discretely added until the second condition fails. We assume that once a particle goes down, 4

it does not move. For each potential, we run 10 simulations, and we neglect interactions between the lines since they are sufficiently spaced apart (~1 µm). This simulation is a similar prescription offered for MC simulations of particle assembly except that we have modified it to account for a changing particleline interaction from the charged lines.


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Figure 3. a) Schematic showing the parameters in the MC-RSA simulations for particles assembling onto a line. b) MC-RSA simulations of neighboring particle distribution for a fixed Debye length (900 nm) and fixed charge (28 e per particle) but with varying particle-line interaction values as indicated in the figure. The simulations are shown as points, while the line is a fit from a gamma distribution, added only to aid the eye. As the magnitude of the particle-line interaction increases the distribution curves shift to the left and their widths narrow.

This behavior can be understood as the particle-line interaction balancing

against the particle-particle repulsion.

Results for some simulations are shown in Figure 3b where the particle-line interaction is varied from ~-1 kT down to -5.5 kT with a fixed Debye length of 900 nm. The simulated distribution functions for the neighboring particles show a similar trend to what we measure experimentally. As the particle-line interaction increases, there is a narrowing and a leftward shift of the neighboring particle distribution. Although the relationship between the particle-line interaction and surface charge are not necessarily linear from complications caused by ionic distribution, we understand that an increase in the magnitude of the particle-line interaction is akin to an increase in the magnitude of surface charge (for example Grahame’s equation

46

assuming a fixed Debye length). The evolution of the distribution function can be

interpreted in the same way as for charge: the increase in the magnitude of the line’s interaction compensates against the repulsive behavior of the particle-particle interaction. We compare our experimental results in Figure 2c with fits from a series of MC simulations which are done with different particle-line interaction (~-.5 kT down to -7 kT) and different effective charges per NP (we used a range of 1-25 electron per NP).

The MC simulation, shown as dashed lines, compare

well against the experimental data in Figure 2c. Table 1 provides the fit parameters for the experimental data; the fit parameters (charge and particle-line interaction) are extracted from the simulations which showed the best fit to the data (least squares comparison of the curves). In Figure 4, we plot the magnitude of the particle-line interaction, Uo, measured from the simulations. The particle-line interaction starts at approximately -1 kT for the lowest writing temperature and monotonically decreases down to -3.5 to -4 kT, where it then plateaus for higher temperatures (800-900 ºC). We interpret the increase in the magnitude of the particle-line interaction as a measure of the increase in the magnitude of the surface


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charge density. The results in Figure 4 are also in agreement with previous works describing how this surface reaction is tuned with temperature and they are easily understood as being part of the “S” shaped curve typical for this reaction mechanism. As the temperature goes up, so too does the surface charge and the particle-line interaction. The plateauing is related to the saturation of the cinnamate polymer’s chemical reaction that we discussed with the density of particles in Figure 2. Because the simulations and experiments agree well and because the values extracted from the simulation are in agreement with previous results, we believe our modified MC model describes quite well the underlying physics.

Figure 4. Plot of the extracted particle-line interaction versus the writing temperature. The curve shows that the magnitude of the particle-line interaction increases with writing temperature up to some saturation point. This effect is understood to be caused by the saturation of the underlying, cinnamate polymer’s deprotection reaction.

Inset: Plot of the RSA patch size versus writer temperature; the trends for the

particle-line interaction and the patch size are similar.

The second method, against which we compare our experimental results, is the RSA method. The RSA method is a simulation technique developed to model and understand the dynamics of particle assembly. We focus on comparing to the RSA for two reasons. The first reason is the RSA method is blind to the physics; there is no requirement of prior knowledge of the exact interactions between species. Our interest in comparing against the RSA stems from the fact that the RSA was shown to do well in scenarios in which particle-particle interactions were the dominating factor in describing the assembly process, but never against data in which more than one interaction could tune the distribution.


By

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comparing against experiments in which there are two interactions controlling the assembly process, we provide some evidence that the RSA is generic enough to be applied to even more complicated situations with a multitude of interactions; the multitude of interactions could, for example, be interactions which are variable as a function of density or a fluctuating external parameter. The second reason we focus on the RSA relates to our desire to make a connection between the physics based MC method and the RSA technique. By mapping from one technique to the other, we can use the parameters from the RSA to extract information about the underlying physics. Typically, the RSA methods are used for 2-D assemblies, but can easily be converted to 1-D scenarios as well. In RSA simulations, particles are allowed to bind to a surface subject only to the condition that there is no overlap between the particles (i.e. only a hard wall repulsion is accounted for). To account for possible electrostatic repulsions, the hard wall radius can be replaced with an effective radius

21,50,51

, aeff, which can be computed with: ()) =

/ 1 4 + , -1 − 012 3 2

[4] where u(r) is the particle-particle interaction. In previous studies, RSA simulations were done under the assumption of homogeneous surfaces where a particle can bind anywhere without discretion; later modifications of the model accounted for a more realistic, heterogeneous surface, in which pre-simulated patches or active regions of the surface 16

were allowed to accept particles binding to the surface . The adsorption conditions are limited to cases in which a particle was simulated to land on a patch and must not overlap with other particles (hard wall). The three parameters that can be varied for the heterogeneous model are the particle size, the active 22

patch size, and the patch density . We will work with a modified version of the heterogeneous RSA technique; in this case the exact definition of a patch is not well-defined, but rather is an abstraction of the total particle-particle interaction coupled with the particle-line interaction. The modification that we impose is that a single particle is allowed to land on a patch, as opposed to multiple particles. The exact procedures and schematic are provided in the supporting information, and we briefly discuss the significance of the three parameters


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(particle size, patch size, patch density) with our new modification. The particle size is understood as the hard wall approximation; it prevents two particles from overlapping, and introduces a lower bound on the neighboring particle distribution to twice the particle radius. The patch size and density are the two components, which control how particles will distribute themselves onto a surface, but do so in different ways. More specifically, the patch density controls the long distance spacing, whereas the patch size controls intermediate and short distance separations. We can understand the patch size as relating to the effective size of the particles including the interactions from the line: the more the surface attracts the particles, the less the particles repel one another. A stronger attraction to the surface manifests as a smaller patch size. Figure 2c shows the modified-RSA model also compares well against our experimental results; the RSA model is shown as the solid lines. The fits are made by comparing the experimental data against a series of RSA simulations which vary both the patch size (50-400 nm) and the patch density (variable because of the different patch sizes and the packing limit). The parameter of interest to us is the patch size as we can use it to compare against the MC results; we extract the value from the least squares best fit of the simulations to the data. Both the modified MC and the modified RSA models sufficiently describe the distribution of particle assembly. Both models correctly account for the long tails/wide distributions seen in the low temperatures, and for the short tails/narrow distributions for the higher temperatures. We also plot, in the inset of Figure 4, how the patch size varies as a function of writer temperature. There is a similar trend for the patch size as we observed for the Uo parameter in our modified MC simulations: as the writing temperature increases, the patch size decreases until it hits a lower bound of ~80 nm. This lower bound on the patch size represents the saturation of the surface charge from the de-protected cinnamate polymer. We now want to compare our MC results with our RSA results; to do this, we work with a modified version of Equation 4: we can insert the effect from the particle-line interaction by limiting our integration to distances over which there is repulsive behavior between the particles and the surface (i.e. {r | u(r) |Uo|>0}). Assuming just a single particle-particle interaction (a first order approximation), table 1 shows the results, ()) ,

against the RSA patch size values.

The results are not great for low or high


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temperatures; suggesting that the approximation made in Equation 4 is not comprehensive enough. Equation 4 only takes into account the effect of interactions resulting from the neighboring particles distribution; it neglects to account for next neighbor and next-next neighbor interactions. We can modify Equation 4 by computing effective interactions (U1 and U2) for these fellow neighboring particles, and add the interactions to the particle-particle interaction. Approximate contributions from the next neighbor and the next, next neighbor can be computed by convolving the distribution functions with the particle-particle interactions (see supporting information for more details). We account for the effect of the surfaceparticle interaction in the same way as we did for the first order approximation (by limiting the integration range). The results are shown in the table as 5()) and ()) . We see that as we add more and more

contribution, the RSA parameters and the parameters extracted from the MC models start to compare well against one another. With this understanding, we interpret the patch size as the effective size resulting from the total particle-particle interactions coupled with the surface-particle interactions. The patch size can in addition be viewed as the average length of activated cinnamate polymer needed to counter-act the total particle-particle repulsion. This suggests why not having prior knowledge of the exact particle-particle interaction still works; we just need the interactions to be close enough that they approximate the total interaction acting on an assembling particle. RSA

MC-RSA

Heater Temperature (ºC)

Patch Size (RSA) (nm)

()) (nm)

5()) (nm)

()) (nm)

Charge (e)

ψ0 (mV)

U0 (kT)

650

260

237

274

286

25

17.5

-0.9

700

170

130

155

166

25

17.5

-2.2

750

130

89

108

117

21

14.7

-2.4

800

80

50

56

60

18

12.6

-4

850

80

52

61

67

18

12.6

-3.5

900

90

51

58

63

18

12.6

-3.8

Table 1: Comparison of the two methods for analyzing Particle absorption. N.B. the step size for the RSA simulations were 10 nm.

Interestingly, from the parameters derived from the fits, we note that the potential of the particle decreases with increasing background surface charge. We do not know the exact mechanism behind this


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change, but we believe it may have to do with increased charge regulation and the neutralization of the charged particle by the background potential. For the experiments seen in figure 2c, we work with de-ionized water to create a long Debye length to enhance the effects from the particle-line interaction. We show in the supporting information that we can do the exact same analysis with different Debye lengths (220 nm-900 nm) to within reason. The fits measure essentially the same results for the surface potential and the particle charged down to Debye lengths of 250/300 nm; the key to our assumption here, however, is not so much having the exact, correct value of the Debye length, but that most of the particle-particle interaction is buried in the 1/r term and not lost through the decay imposed by the counter-/co-ions in solution. This long Debye length assumption will clearly breakdown if we impose shorter Debye lengths by adding salt. In order for us to demonstrate this in an effective manner, we want to enhance the change in the distribution as much as possible by working with a very low particle-line interaction, fabricated by writing at low temperatures.

This study is complicated by the fact that direct comparisons of the

distribution and U0 are difficult since we expect this value to change as a function of Debye length. Qualitatively, we expect that as we shorten the particle-particle interaction, the distribution will narrow. -6

Figure 5 shows this effect at three different salt concentrations: 10 M NaCl (Debye length: ~300 nm), 10 5

M NaCl (Debye length: ~100 nm), and 10

-4

-

M NaCl (Debye length: ~30 nm). Investigations beyond

these salt regimes showed that the packing was too dense for a proper 1-D analysis (example shown in supporting information). We also note that for Debye lengths of ~3 nm, the particles did not stick to the positive patches; we believe this is because the particle-surface interactions become small enough that they cannot compete with particle’s entropy. We investigated this in 2-D as well and saw a similar effects.


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Figure 5. Particle distribution for three different Debye lengths (300 nm, 100 nm, and 30 nm). The Debye lengths are approximations made from the amount of salt added, and do not necessarily reflect the actual values. As the Debye length decreases, there is a noticeable shift in the distribution curve towards smaller separation distances. This shift can be attributed to the truncated particle-particle interaction from the short Debye lengths. Dashed lines are MC simulations, solid lines are RSA simulations, and the experimental measurements are the data points.

Figure 5 shows that as we vary the particle-particle interactions, the particles are able to pack in more densely.

With shorter Debye lengths, we truncate the particle-particle interaction at shorter

distances and consequently the particles pack at higher densities. We can see this trend for the three Debye lengths supplied; qualitatively, we can apply our two models to analyze these data sets, and again we find the agreement between both is quite good. As the Debye length decreases, we find that the data becomes increasingly difficult to analyze; our analysis works best for well separated particles because we remove complications caused by particles too close to each by neglecting the aggregated particles (see methods section). It is also true that we neglect other particle interactions such as the van der Waals forces, which may become appreciable when the particle repulsion is truncated; the data and fit still demonstrate the expected trends. Conclusions


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In this work, we presented systematic experimental results to understand the colloidal assembly process on differently charged lines. We observed that as we increased the surface charge, we could tune the 1-D distribution to a more densely packed array. We measured the density as a function of different surface charges and we saw that the density packing saturated when we reached the limit of the surface interaction. We also verified that by tuning the particle-particle interaction with the Debye length, we could also tune the assembly distribution as well. To understand the results, we applied two models to understand the distribution of charged particles onto a charged 1-D pattern, and we introduced modifications to both models to account for different driving forces. To the best of our knowledge, this is the first attempt to describe and modify these simulations to account for different attractive forces from the surface, and we believe we have not only successfully captured the added complexity of different particlesurface interactions, but we were also able to map the results between the full accounting MC method and the abstract RSA method. One of the long term advantages to this work on NPs assembly is that since we have worked with scenarios in which the dominating forces for assembly are electrostatic, we have, in an uncontaminated fashion, furthered our understanding of how electrostatic interaction tune and control the assembly process; future studies which isolate other forms of interactions, such as van der Waals, could in principle be applied in conjunction with the work presented here to understand the more complicated, multi-component molecular assembly processes. Acknowledgments We acknowledge the support and help provided by Ute Drechsler, Martin Spieser, and Colin Rawlings. This publications is supported by the Swiss National Science Foundation as part of the NCCR Molecular Systems Engineering. We thank the support of the Office of Basic Energy Sciences of the Department of Energy (DE 0000224414) and the Army Research Office. Supporting Information Detailed descriptions of the models, analysis, and algorithms; a sample set of data to understand how the distribution functions are extracted from SEM measurements; additional fits for other long Debye lengths; and an example profile of the patterned lines. References


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Understanding How Charged Nanoparticles Electrostatically

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