Explaining the Transition from Diffusion Limited to Reaction Limited Surface Assembly of Molecular Species through Spatial Variations.

Surface assembly is often decomposed into two classes: diffusion and reaction limited processes. The transition between the two cases is complex because the dynamics are so different. In this article, we simulate, explain, and experimentally discuss the evolution of the spatial distribution for surface assemblies with diffusion limited and reaction limited processes. Explicitly, we demonstrate that diffusion limited and reaction limited processes show some temporal differences, but more importantly, we show that the spatial arrangements are different enough to discriminate between the two cases. Using fundamental properties, such as the diffusion constant, we calculate the evolution of the spatial profile and derive from physical, heuristic models the assembly rate for reaction and diffusion limited processes based on the individual particle's interactions with the surface. Finally, we confirm the spatial profile differences between diffusion and reaction limited cases by experimentally measuring the surface assembly between two molecules of similar size, but having different assembly routes. Unique to our description is that we have derived and simulated everything through the particle picture in place of ensemble descriptions such as the diffusion equation, and we show the equivalence between our heuristic formulas and those derived from the diffusion equation.

Testing the Equivalence between Spatial Averaging and Temporal Averaging in Highly Dilute Solutions.

Diffusion relates the flux of particles to the local gradient of the particle density in a deterministic way. The question arises as to what happens when the particle density is so low that the local gradient becomes an ill-defined concept. The dilemma was resolved early last century by analyzing the average motion of particles subject to random forces whose magnitude is such that the particles are always in thermal equilibrium with their environment. The diffusion dynamics is now described in terms of the probability density of finding a particle at some position and time and the probabilistic flux density, which is proportional to the gradient of the probability density. In a time average sense, the system thus behaves exactly like the ensemble average. Here, we report on an experimental method and test this fundamental equivalence principle in statistical physics. In the experiment, we study the flux distribution of 20 nm radius polystyrene particles impinging on a circular sink of micrometer dimensions. The particle concentration in the water suspension is approximately 1 particle in a volume element of the dimension of the sink. We demonstrate that the measured flux density is exactly described by the solution of the diffusion equation of an infinite system, and the flux statistics obeys a Poissonian distribution as expected for a Markov process governing the random walk of noninteracting particles. We also rigorously show that a finite system behaves like an infinite system for very long times despite the fact that a finite system converges to a zero flux empty state.

Understanding How Charged Nanoparticles Electrostatically Assemble and Distribute in 1-D.

The effects of increasing the driving forces for a 1-D assembly of nanoparticles onto a surface are 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.

Speed dependence of thermochemical nanolithography for gray-scale patterning.

Thermochemical nanolithography (TCNL) is a high-resolution lithographic technique and, owing to its fast speed, versatility, and unique ability to fabricate arbitrary, gray-scale nanopatterns, this scanning probe technique is relevant both for fundamental scientific research as well as for nanomanufacturing applications. In this work, we study the dependence of the TCNL driven chemical reactions on the translation speed of the thermal cantilever. The experimental data compares well with a model of the chemical kinetics for a first-order reaction. The impact of higher order reactions on the optimization of TCNL is addressed. The reported quantitative description of the speed dependence of TCNL is exploited and illustrated by fabricating controlled gradients of chemical concentration.

Fabricating nanoscale chemical gradients with ThermoChemical NanoLithography.

Production of chemical concentration gradients on the submicrometer scale remains a formidable challenge, despite the broad range of potential applications and their ubiquity throughout nature. We present a strategy to quantitatively prescribe spatial variations in functional group concentration using ThermoChemical NanoLithography (TCNL). The approach uses a heated cantilever to drive a localized nanoscale chemical reaction at an interface, where a reactant is transformed into a product. We show using friction force microscopy that localized gradients in the product concentration have a spatial resolution of ~20 nm where the entire concentration profile is confined to sub-180 nm. To gain quantitative control over the concentration, we introduce a chemical kinetics model of the thermally driven nanoreaction that shows excellent agreement with experiments. The comparison provides a calibration of the nonlinear dependence of product concentration versus temperature, which we use to design two-dimensional temperature maps encoding the prescription for linear and nonlinear gradients. The resultant chemical nanopatterns show high fidelity to the user-defined patterns, including the ability to realize complex chemical patterns with arbitrary variations in peak concentration with a spatial resolution of 180 nm or better. While this work focuses on producing chemical gradients of amine groups, other functionalities are a straightforward modification. We envision that using the basic scheme introduced here, quantitative TCNL will be capable of patterning gradients of other exploitable physical or chemical properties such as fluorescence in conjugated polymers and conductivity in graphene. The access to submicrometer chemical concentration and gradient patterning provides a new dimension of control for nanolithography.

Cartography of high-dimensional flows: a visual guide to sections and slices.

Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a "template." Together these charts provide an atlas of the symmetry-reduced "slice" of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits. Visualizations of these solutions and their unstable manifolds reveal their interrelations and the role they play in organizing turbulence/chaos.

Nonperturbative chemical modification of graphene for protein micropatterning.

Graphene's extraordinary physical properties and its planar geometry make it an ideal candidate for a wide array of applications, many of which require controlled chemical modification and the spatial organization of molecules on its surface. In particular, the ability to functionalize and micropattern graphene with proteins is relevant to bioscience applications such as biomolecular sensors, single-cell sensors, and tissue engineering. We report a general strategy for the noncovalent chemical modification of epitaxial graphene for protein immobilization and micropatterning. We show that bifunctional molecule pyrenebutanoic acid-succinimidyl ester (PYR-NHS), composed of the hydrophobic pyrene and the reactive succinimide ester group, binds to graphene noncovalently but irreversibly. We investigate whether the chemical treatment perturbs the electronic band structure of graphene using X-ray photoemission (XPS) and Raman spectroscopy. Our results show that the sp(2) hybridization remains intact and that the π band maintains its characteristic Lorentzian shape in the Raman spectra. The modified graphene surfaces, which bind specifically to amines in proteins, are micropatterned with arrays of fluorescently labeled proteins that are relevant to glucose sensors (glucose oxidase) and cell sensor and tissue engineering applications (laminin).

Parallelization of thermochemical nanolithography.

One of the most pressing technological challenges in the development of next generation nanoscale devices is the rapid, parallel, precise and robust fabrication of nanostructures. Here, we demonstrate the possibility to parallelize thermochemical nanolithography (TCNL) by employing five nano-tips for the fabrication of conjugated polymer nanostructures and graphene-based nanoribbons.