Charge-Regulated Interactions: The Case of a Nanoparticle and a Sphere of Arbitrary Dielectric Constants.

Surfaces of objects immersed in liquid develop an electric charge density that depends on the types and concentrations of dissolved ions. The strength and spatial distribution of this charge density controls a myriad of processes, from biological to industrial processes. In addition, the lack of a full understanding of the charge density precludes a complete foundational interpretation of liquid-mediated many-body interactions. This understanding is especially obscured by charge regulation, whereby the charge on an object hinges, in addition, on the charges and locations of all other charged objects in the liquid. Here, we present a rigorous mathematical approach based on the Poisson-Boltzmann Equation, with field-dependent boundary conditions, and apply it to obtain the liquid-mediated interaction energy between a charged dielectric sphere and a charged particle. The framework that we develop in this article should be of use beyond the limits of the example application considered here: it should be useful as a conceptual and technical starting point to obtain charge-regulated many-body interactions in liquids.

Charge Calibration Standard for Atomic Force Microscope Tips in Liquids.

An electric charge standard with nanoscale resolution is created using the known charge distribution of a single tobacco mosaic virus coat protein combined with the known packing of these proteins in the virus capsid. This advances the ability to measure charge on nanometric samples. Experimental atomic force microscope (AFM) force-distance curves are collected under aqueous conditions with controlled pH and ion concentration. A mathematical model that considers a polarizable dielectric tip immersed in an electrolyte is used to obtain charge density from the AFM measurements. Interactions between the tip and the sample are modeled using theory that includes monopolar electrostatic interactions, dipolar interactions, screening from both the dielectric nature of ambient water and solvated ions as described by the linear Poisson-Boltzmann equation, and hard-core repulsion. It is found that the tip charge density changes on a timescale of hours requiring recalibration of the tip for experiments lasting more than an hour. As an example of how a charge-calibrated tip may be used, the surface charge densities on 20 individual carboxylate-modified polystyrene (PS) beads are measured. The average of these AFM-measured bead charge densities is compared with the value obtained from conventional titration combined with electron microscopy. The two values are found to agree within 20%. While the comparison demonstrates similarity of the two charge measurements, hypotheses are put forward as to why the two techniques might be expected not to provide identical mean charge densities. The considerations used to build these hypotheses thus underscore the relevance of the method performed here if charge information is required on individual nanoparticles.

Method to Quantify Nanoscale Surface Charge in Liquid with Atomic Force Microscopy.

A theory is presented to obtain surface charge density on nanoscale objects from data in the snap-to-contact portion of an atomic force microscope force-separation curve. The mathematical model takes into account the tip's dielectric constant using the Self-Consistent Sum of Dipoles theory which includes the charge-charge interaction and the charge-dipole interaction with electrolyte-induced exponentially decaying screening, Debye and London dipolar force, and fluid viscosity including confined fluid layers to account for energy dissipation. Using previously published experimental data, the mathematical model is applied to measure the surface charge density on an individual nanoscale amine-modified polystyrene bead immobilized on the basal plane of highly oriented pyrolytic graphite in buffered aqueous solution. Within the experimental uncertainty, the magnitude of the charge density on a single bead obtained using the new method falls within the distribution of values determined by the manufacturer using titration and electron microscopy.

Euler-Bernoulli theory accurately predicts atomic force microscope cantilever shape during non-equilibrium snap-to-contact motion.

We prove that the Euler-Bernoulli elastic beam theory can be reliably used to describe the dynamics of an atomic force microscope cantilever during the far from equilibrium snap-to-contact event. In conventional atomic force microscope operation, force-separation curves are obtained by post-processing voltage versus time traces produced by measuring one point on the cantilever close to the hanging end. In this article, we assess the validity of the Euler-Bernoulli equation during the snap-to-contact event. The assessment is based on a direct comparison between experiment and theory. The experiment uses Doppler vibrometry to measure displacement versus time for many points along the long axis of the cantilever. The theoretical algorithm is based on a solution of the Euler-Bernoulli equation to obtain the full shape of the cantilever as a function of time. The algorithm uses as boundary conditions, experimentally obtained information only near the hanging end of the cantilever. The solution is obtained in a manner that takes into account non-equilibrium motion. Within experimental error, the theory agrees with experiment indicating that the Euler-Bernoulli theory is appropriate to predict the cantilever kinematics during snap-to-contact. Since forces on the tip can be obtained from the instantaneous shape of the cantilever, this work should allow for computation of tip-sample forces during the snap-to-contact event from a conventional force-distance measured input.

Electrostatic force curves in finite-size-ion electrolytes.

We obtain analytical expressions for electrostatic forces between an atomic force microscope tip and a sample immersed in an electrolyte. These simple expressions relate force to tip-sample separation explicitly incorporating tip size, solvent ion size, and solvent ion concentration. If the ions are much smaller than the tip-sample gap, the force decays monotonically, a consequence of the corresponding monotonic decays of the correlation function in the Debye-Hückel context. If the ions are of size comparable to the tip-sample gap, then oscillations appear superimposed on the overall decay, a consequence of the geometric mismatch between ion cluster size and the gap size.

Obtaining charge distributions on geometrically generic nanostructures using scanning force microscopy.

We develop the self-consistent sum of dipoles (SCSD) theory for the purpose of recovering charge densities present on nanostructures using scanning force microscope (SFM) force-separation experiments. The dielectric probe is discretized into volume elements characterized by their atomic polarizabilities. Magnitudes of the induced dipole in each element are calculated based on discrete charges placed on the surfaces, dipole-dipole interactions, and dielectric and ionic properties of the surrounding medium. We perform two model-model comparisons, one with a macroscopic dielectric sphere and one with a nanocluster of silicon atoms. In both cases, using a single adjustable parameter, our SCSD theory agrees with the accepted theories to better than 99%. Force-separation curves between a silicon nitride probe and the basal plane of highly oriented pyrolytic graphite in nine ionic concentration and pH combinations were fit with a root-mean-square error of 3.6 pN, an improvement over the 12 pN error obtained using the Derjaguin approximation. These results suggest that the SCSD will be useful in modeling SFM force-separation data to obtain spatially varying charge densities on surfaces with complex geometries.

Signal distortion in atomic force microscopy photodetector.

The frequency-dependent complex impedance of an atomic force microscope photodetector is measured. The inverse problem is solved obtaining the voltage that would have been collected with a hypothetical, perfectly flat-frequency-response photodetector from the experimentally available voltage. This information is used to study the distortion that the true input signal undergoes as it passes through the photodetector on the way to becoming the experimentally measured output signal. It is found that signals with features of interest shorter than 10 µs render noticeable differences between the true and measured raw voltages and forces. Signals with features shorter than 1 µs produce experimentally measured force curves that deviate substantially from the true force curves. A method is proposed for correcting the measured raw voltage signal.

Dry friction avalanches: experiment and theory.

Experimental evidence and theoretical models are presented supporting the conjecture that dry friction stick-slip is described by self-organized criticality. We use the data, obtained with a pin-on-disk tribometer set to measure lateral force, to examine the variation of the friction force as a function of time. We study nominally flat surfaces of matching aluminum and steel. The probability distribution of force drops follows a negative power law with exponents mu in the range 3.2-3.5. The frequency power spectrum follows a 1/f alpha pattern with alpha in the range 1-1.8. We first compare these experimental results with the well-known Robin Hood model of self-organized criticality. We find good agreement between theory and experiment for the force-drop distribution but not for the power spectrum. We explain this on a physical basis and propose a model which takes explicitly into account the stiffness and inertia of the tribometer. Specifically, we numerically solve the equation of motion of a block on a friction surface pulled by a spring and show that for certain spring constants the motion is characterized by the same power law spectrum as in experiments. We propose a physical picture relating the fluctuations of the force drops to the microscopic geometry of the surface.

Spectroscopy of atomic chains with tunneling microscopy.

Recently, it has been possible to create metallic chains down to seven-atom long. This has two implications. Experimentally it is necessary to have algorithms that allow converting measured data into microscopic information as, for example, processing a current-voltage curve to obtain electronic density of states (DOS). From a theoretical standpoint, we have a new framework to test extant theories and to further develop them. With this in mind, we present a new theoretical solution to the problem of mapping scanning tunneling microscopy current-voltage curves into DOS-energy curves. Our model is based on a self-consistent solution to the quantum problem of electrons in the presence of an array of attractive centers. The problem, being simpler than its three-dimensional counterparts is solved exactly for DOS and for the current-voltage curves. The main results are that the current-voltage peaks at the position of the energies of the electron in the chain and, when an impurity is present in the chain, its chemical information can be extracted from the value of analytical curves.

Scanning tunneling microscope spectroscopy of polymers.

This paper presents theoretical results on the relationship between density of states (DOS) and scanning tunneling microscope current-voltage curves in polymers. We considered samples of linear hydrocarbons electrically grounded at one of their extremes. The other extreme is electrically connected to the microscope tip via electron tunneling through vacuum. When a voltage, V, is applied to the tip, electric current, I, flows in the tip-sample circuit. This current varies as the voltage varies and depends on the DOS to the extent that no current would flow if no electron states exist at a certain energy (or voltage). The detailed relationship between DOS and the current-voltage (I-V) curve is not known a priori. We solve the corresponding quantum problem in the context of tight binding and find that I-V reproduces accurately the resonant energy peaks of the DOS. We apply the results to 100 atom-long alkane and alkene chains and found that there is a significant voltage shift in the corresponding curves as to discriminate one structure from the other.