1. Introduction

Techniques to measure forces between discrete entities in fluids, ambient air, or vacuum have been vital in advancing our understanding of condensed matter. From the invention of the surface force apparatus (SFA) that permitted early measurements of interactions between macroscopic surfaces [1], followed by atomic force microscopy (AFM) which carried this principle over to micro and nanoscale entities [2], to the more recent use of optically and magnetically trapped microspheres as indirect sensors of forces in macromolecules [3], the development of tools for interaction potential measurements continues to be a vibrant area of research. While the methods listed above rely on the mechanical motion (e.g. bending or displacement) of a sensing element in response to the action of an intermolecular force, optical microscopy has emerged as a sensitive, non-invasive, and often calibration-free tool, that offers a thermodynamic method to directly measure interaction energies [4]. In particular, it facilitates highly parallel realizations of the measurement. The technique has been successfully used to study a variety of interactions in soft matter: electrostatic interactions of charged colloids, polymers and lipid membranes [5–7] as well as van der Waals [8], gravitational [9], depletion [10] and critical-Casimir [11] forces to name a few.

These measurements are founded in the Boltzmann principle for a system in thermodynamic equilibrium, where the potential ascribed to a state depends exclusively on the probability of occupancy of that state. Thus in colloidal systems, spatially varying interaction potentials may be measured by acquiring a large number of snapshots of a probe particle thermally sampling its surroundings. Normalized probability density distributions of the particle’s instantaneous location, $P(r)$ measured in the experiment can be converted to local potentials via the Boltzmann relation, $U(r)=-\ln P(r)$.

Although AFM and SFA provide angstrom spatial resolution and may be ubiquitously applied, optical imaging could deliver better sensitivity in particular experimental situations. In optical imaging, the particle localization accuracy determines the spatial resolution with which the interaction can be measured and this is generally on the order of a nanometer [12]. However since the range of the measurement, especially in the lateral dimension, can be on the order of micrometers, the technique is particularly well suited to measuring potentials that change monotonically and gradually with distance, thus putting fN force measurement within its reach.

Provided that snapshots of the particle can be acquired with a high signal-to-noise ratio (SNR), optical imaging is an excellent calibration-free method for direct mapping of potential landscapes of arbitrary shape and large range. Although quadrant photodiodes (QPDs) offer outstanding temporal resolution, and have been successfully used to calibrate optical traps [13], their position sensitivity is linear over a limited range of displacements (<1 μm), and they cannot be used to track more than one particle at a time. Direct imaging, therefore, offers distinct advantages in parallelized measurements of potentials over an extended spatial range, and can provide full three dimensional (3D) information on an interaction in a single run.

Most imaging-based interaction potential measurements deal with micron-scale objects, which typically permit very high SNR imaging, and with few exceptions [14] concentrate on the measurement of 2D potentials in the imaging plane. Studies that measure potentials in the axial dimension using nanoscale probes are less common and mainly use evanescent excitation at the fluid-glass interface (TIRM) [15]. TIRM is intrinsically constrained by the fact that the local excitation field falls off exponentially from the interface and therefore limits measurements to within ~100 nm of the interface. Other 3D particle tracking methods such as holographic microscopy [16, 17], point-spread function engineering [18, 19], off-focus imaging [20, 21], and Fresnel particle tracking [22] offer the ability to track particle motion over an extended range of axial displacements, however face challenges in supporting nanometer scale positional accuracy combined with >kHz imaging rates, which are required when using nanoscale objects as probes to directly map arbitrarily shaped potential landscapes.

Apart from the SNR, a further subtlety in performing accurate position measurements, on small particles (<100 nm in diameter) in water, relates to the exposure time of the measurement, . A particle trapped in a harmonic potential well of spring constant, k has a relaxation time given by $\tau =\gamma /k$, where $\gamma$ is the viscous drag experienced by the particle. When $\sigma$ is comparable to the relaxation time $\tau$, the timescale over which the particle experiences net motion towards the bottom of the well, the measured position of the particle is weighted towards the bottom of the potential, making the trap appear stiffer than it really is [23, 24]. Measurement and correction of “motion blur” has been discussed in detail for harmonic confining potentials that arise in the context of optical trapping [24]. Thus the need arises for an optical imaging technique delivering 3D particle tracking with nanometer spatial and submillisecond temporal resolution.

Here we describe the details of an optical technique, “interferometric scattering” (iSCAT) imaging [25–27] that delivers accurate particle localization in 3D and can in principle be used to reconstruct spatial interaction energy landscapes for nanometer scale particles in 3D. Our experimental measurement addresses the interactions of charged gold particles thermally sampling an electrostatic potential landscape generated in a fluidic slit. In contact with water, the walls of the slit acquire a surface charge comparable to that of the particle. The electrostatic potential near an isolated glass or SiO₂ surface, $\psi$ decays exponentially away from the surface value, ${\psi }_0$ as: $\psi (z)={\psi }_0{e}^{-\kappa z}$, where ${\kappa }^{-1}$ is the “Debye length” and for a monovalent electrolyte may be given by $0.304/\sqrt{C}$, where C is the salt concentration in solution expressed in mol/L [28]. Two charged planes separated by a gap 2h in a fluid thus give rise to an electrostatic potential minimum midway between them. For low values of ${\psi }_0$, the potential at the midplane due to both surfaces can be taken to be simply additive and is given by ${\psi }_{\text{m}}=2{\psi }_{\text{0}}{e}^{-\kappa h}$ (Fig. 1(c) ). A local increase in the gap width of d would give a local potential minimum of ${\psi }_{\text{m}}=2{\psi }_{\text{0}}{e}^{-\kappa (h+d/2)}$. So a point charge q, traversing a width modulation in the gap would experience a change in electrostatic energy given by $\Delta U=2q{\psi }_{\text{0}}{e}^{-\kappa h}(1-e^{-\kappa d/2})$. When $d\to 0$, i.e., the width modulation vanishes, the slit consists of two flat parallel walls facing each other, and $\Delta U\to 0$. For large $\kappa d$, on the other hand, $\Delta U=2q{\psi }_{\text{0}}{e}^{-\kappa h}$. While these simple considerations - based on the linearization of the governing equations valid for low surface potentials or far away from surfaces - are not quantitatively exact, they give a physical picture of how geometrical modulation of a gap can translate to a modulation of the electrostatic potential in a fluid. They also furnish key insight into optimal design of electrostatic landscapes to trap and manipulate single charged particles in fluids. Accordingly, systems with small values of $\kappa h$ and walls with a high surface potential, ${\psi }_{\text{0}}$ would be expected to work best in creating deep local potential wells, capable of retaining a charged object for a long time. Furthermore under a given set of conditions, i.e. ionic strength and slit depth, the shape and depth of each local potential well can be tailored using the geometry of the surface indentation. We have demonstrated stable trapping of charged nanospheres as small as 20 nm polystyrene and 50 nm aqueous lipid vesicles in water [29] and have recently described how spatial mapping of potential wells can be used in conjunction with free energy calculations to directly measure the charge of single trapped objects in a highly parallel fashion [30]. In this work we consider the electrostatic interaction between charged gold particles, 80 nm in diameter, and confining walls, composed of glass and SiO₂ with etched cylindrical indentations, or “pockets” on the top surface, and discuss details on the use of iSCAT imaging to probe in 3D the morphology of potential wells generated by different pocket geometries.

 

Fig. 1 (a) Atomic force micrograph of an array of 500 nm pockets separated by 3 μm. (b) Line plot of the topography of one pocket indicated in (a). (c) Schematic of the electrostatic potential between two parallel walls separated by a smaller (left) and larger (right) gap.

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In our scanning focused beam setup [29–31], a collimated laser beam is continuously swept by a two-axis acousto-optic deflector (AOD). In such a system the deflection angle from the propagation axis, z corresponds to an xy position of the focused diffraction-limited spot in the vicinity of trapped particles. The combination of large scanning bandwidth of each AOD (between 50 and 100 kHz), which results in the rapid scanning of the diffraction-limited spot, and a fast CMOS camera, ensures a uniform illumination area in each frame, acquired over an exposure time of $\sigma =1\text{}\text{ms}$. While faster acquisition is possible using more rapidly scanning AODs, we achieve exposure times down to 200 µs in the current configuration [30], however with a greatly reduced field of view. 1 ms exposure times produce a stable uniform wide-field illumination that we use to study particles trapped by pockets of various geometries. In such a coherent detection scheme, the fields that produce the image at the camera predominantly originate from the field strongly scattered by the trapped nanoparticle at the beam focus and the highly reflecting SiO₂/Si interface. The scanned illumination eliminates undesired interference fringes originating from the light reflected from distant locations and interfaces in the multilayer device.

We further note that in our scanning beam illumination, the particle is sampled approximately 10 times during an exposure time of 1 ms. This may be compared with imaging and tracking the center of mass of a particle during the same exposure time, but with continuous illumination in a static beam. In the latter, light from the particle continuously accumulates on the camera and the fit to the final accumulated intensity distribution gives its average spatial location during the illumination. The use of a static beam that continuously illuminates the particle, or a scanning beam that stroboscopically samples the particle multiple times during in the exposure, primarily changes the accumulated intensity in the image. The “scanned” beam case naturally contains fewer samples than the “static” beam case, but since we acquire ~10 samples with scanning, the average inferred position approximates the continuously illuminated value reasonably well. Finite sampling could however be expected to reduce the effective exposure time compared to the camera exposure time. This difference may be estimated to be $\sigma /s$, where s is the number of samples, so that as s gets larger the values converge. In our experiments we estimate that the sampling rate results in an effective exposure time of 0.9 ms for a $\sigma =1\text{ms}$ camera exposure. For smaller exposure times, use of a greatly reduced field of view ensures that the illumination as seen by the particle tends to the continuously illuminated case.

2. Results and discussion

A schematic of our nanofluidic trapping device is shown in Fig. 2(a) . We study the effect of three different trap geometries created by cylindrical indentations of diameter D = 100, 200 and 500 nm, of constant depth, d = 100 nm (Fig. 1(b)). The pockets in the SiO₂ were etched in arrays with a spacing of 3 μm (Fig. 1(a)), which facilitated parallel imaging and tracking of several trapped particles. Gold nanoparticles 80 nm in diameter suspended in water at low ionic strength solution, ~0.05mM, were used as test objects, with zeta potential measurements providing an average charge estimate of ~-80e per particle.

 

Fig. 2 (a) Schematic of electromagnetic fields involved in forming an iSCAT image in the trapping device. (b) iSCAT image of a trapped 80 nm gold nanoparticle (left) and a D = 500 nm pocket (right). The intensity profile along the dashed line in each image is plotted in the graph below it.

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iSCAT imaging is based on coherent detection of the interference between elastically scattered light from an object and a reference beam [25, 26, 29]. Figure 2(a) illustrates the optical fields that create an iSCAT image of a trapped object, illuminated by a scanning focused laser beam. In this configuration, the full 3D information on the particle locus is embedded in the interferometric dip in the image due to the beam reflected from the Si/SiO₂ interface, $E_{\text{ref}}$ and the field scattered by the trapped particle, $E_{\text{bd}}$. The 3D locus of the particle can be extracted from this signal by fitting a 2D Gaussian function to the spatial intensity distribution in the image. The peak position of the Gaussian fit yields the lateral coordinates of the particle, x and y, while the amplitude of the fit function can be directly related to the axial location, z via a calibration function. The channel depth, 2h in the range of 200 to 220nm and SiO₂ thickness, l~200 nm can be used to manipulate the optical path length (OPL) difference between the detected waves and optimize the axial calibration function as described later. We present details of the axial calibration procedure after reviewing the particle trapping results.

As explained earlier, the shape of a potential well may be reconstructed from the measured spatial probability density distribution of a trapped particle, thermally sampling the landscape. Figure 3(a) illustrates scatter plots in the xy plane of representative particles confined in potential wells with different geometries, where each point represents the particle’s position at one specific frame. Normalized radial probability density distributions, $P(r)$ from such scatter plots can be used to make quantitative comparisons of experiment with theory [30]. The average radial probability distributions of particles trapped by the three geometries, obtained by tracking an ensemble of objects, is presented in Fig. 3(b). As displayed, the pocket width, D significantly influences both the axial and lateral range of motion of a confined particle. The measured $P(r)$ distribution for a particle trapped by a D = 500 nm pocket reveals a square-well potential with soft walls. Furthermore, the 3D scatter plot of a representative particle (Fig. 3(d)) reveals that not only are particles confined to a much shorter range in the axial dimension (~50 nm) compared to the radial, but that the scatter plot exhibits a conical shape, reflecting the presence of the indentation overhead. Decreasing D results in stronger spatial confinement in all dimensions. As depicted in Figs. 3(a) and 3(b), the lateral confinement of particles trapped by D = 200nm decreases substantially in comparison with D = 500nm. In particular $P(r)$ in this case has a profile, which is well approximated by a Gaussian function, indicating that the particle is trapped in a harmonic potential; the stiffness of this potential has been shown to strongly depend on the charge of the particle [30].

 

Fig. 3 (a) Scatter plots of lateral motion for representative 80 nm gold particles trapped by D = 500 nm (red), 200 nm (blue), and 100 nm (green) pockets acquired with $\sigma =1\text{ms}$. (b) Averaged radial and (c) axial probability density distributions obtained by tracking an ensemble of particles trapped by the three pocket geometries. (d), (e), (f) 3D scatter plots of representative particles trapped in the three geometries and (g), (h), (i) their overlay on the corresponding calculated electrostatic potential distribution. The panels under (h) and (i) represent the same plots magnified 3 and 8 times respectively. The electrostatic potential is presented on a color scale going from high (black) to low energy (yellow) for a unit negative charge. For emphasis, only the minimum of the well is shown.

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In order to make controlled comparisons between our experimental findings and theoretical predictions, we performed numerical calculations of the potential distribution in the trapping nanostructure using COMSOL Multiphysics [29]. Figures 3(g)-3(i) depict the projection of 3D scatter plots of Figs. 3(d)-3(f) in the xz plane, overlaid on the calculated potential distribution. In all three cases, the measured most probable location of a particle coincides with the location of the potential minimum as expected from Boltzmann statistics. As depicted in Fig. 3(c), increasing D not only increases the range of levitation, but the trapped particles also levitate about higher mean axial positions. The midplane of levitation, $z_{\text{m}}$, defined as the average z value over the entire measurement, could thus be used to compare the experimental and theoretical results. Table 1 displays the experimentally measured and calculated values which are in good agreement for all three geometries considered.

Table 1. Geometrical parameters of the devices and predicted and ensemble-averaged measured midplanes of levitation. All units are in nanometers.

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3. Method of 3D iSCAT

In the present coherent imaging scheme, the signal at the detector, $I_{\text{d}}$ consists of the interference of fields scattered by the trapped bead, $E_{\text{bd}}$, the trapping nanostructure or pocket overhead, $E_{\text{pok}}$, and the field reflected from the Si/SiO₂ interface, $E_{\text{ref}}$ (Fig. 2(a)). In analyzing the overall interference, the vectorial nature of the fields is ignored while the phase is taken into account. In our experiments $E_{\text{bd}}$ and $E_{\text{pok}}$ have the same order of magnitude and are small in comparison to $E_{\text{ref}}$, due to the strong reflection of the SiO₂/Si interface near the beam focus. As a result, the pure scattering terms, ${\left|E_{\text{bd}}\right|}^2$ and ${\left|E_{\text{pok}}\right|}^2$ can be omitted and the detected signal consists of a constant background, ${\left|E_{\text{ref}}\right|}^2$ and the interference of $E_{\text{bd}}$ and $E_{\text{pok}}$ with $E_{\text{ref}}$. $I_{\text{d}}$ could therefore be written as:

Full 3D motion of the particle is obtained by fitting a 2D Gaussian function to the intensity distribution in each frame. The axial location of the particle, z directly correlates with the amplitude of the fit, $I(z)$. The magnitude of $I(z)$ results from the far-field interference of $E_{\text{ref}}$ and $E_{\text{bd}}$, which varies with a changing OPL difference between these waves. Consider the propagation of the two waves (Fig. 2(a)) based on the device geometry. For a particle levitating at a height z the measured intensity may be written as follows:

 

Fig. 4 (a) Calculated electromagnetic field intensity (in logarithmic scale) of a focused Gaussian beam propagating in water and reflecting off a Si surface 170 nm away from the focus. The intensity scale bar indicates the intensity in logarithmic scale, the solid lines show the geometry of the D = 200nm trapping pocket in a slit of depth d = 215 nm and the points highlighted by the arrow represent the scatter plot displayed in Fig. 3(h). (b) The black line is the axial profile of the total intensity in (a) along x = 0 indicated by the dashed line. $z^{\prime }$is the axial distance from the beam focus. The yellow line is the intensity in the absence of the reflected field. (c) Schematic drawing of the particles attached to the SiO₂/H₂O interface used to determine the calibration function for axial motion. (d) Curves used to calibrate the motion of particles trapped by D = 500 nm (red), 200 nm (blue), and 100 nm (green) pockets. The solid segments of each curve represent the axial range sampled by corresponding levitating particles. The black line represents the calibration curve for trapping by the D = 200 nm pocket, when not taking the standing wave effect into account.

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We point out that while the highly reflective Si/SiO₂ interface facilitates iSCAT measurements on strongly scattering particles, the intense reflection also results in an axially non-uniform excitation field in the vicinity of the particle, having the form of a partial standing wave. As a result, $I_0$ in Eq. (2) is not a constant but itself depends on z, altering the effective calibration function for particle contrast. We account for this modulation by mathematically modelling a focused beam reflected at a water/Si interface by first, decomposing $E_{\text{inc}}$into plane waves propagating in all directions within the angular spectrum [34]. The field reflected from the interface, $E_{\text{ref}}$ is consequently found by multiplying the amplitude of each incident plane wave by the corresponding Fresnel reflection coefficient [35]. The total local field experienced by the illuminated particle is obtained by the superposition of $E_{\text{inc}}$ and $E_{\text{ref}}$. To model $E_{\text{inc}}$we considered the experimental parameters that were used, namely, a 1.4NA oil-immersion objective with an underfilling fraction of ~70%, associated with the incident Gaussian beam partially covering the back aperture of the microscope objective [36]. The maximum focusing angle was set by the total internal reflection of the beam at the glass/water interface (62°). We emphasize that although the calculation of the local electric field assumes that the beam propagates through water, the SiO₂ layer is taken into consideration in the OPL differences in Eq. (2). Figure 4(a) shows a 2D contour plot of the total illumination intensity in the vicinity of the Si/SiO₂ interface on a logarithmic scale. In Fig. 4(b) the normalized on-axis variation of the total intensity can be compared in the presence and absence of the Si/SiO₂ interface reflection. Clearly, the reflection produces a significant axial modulation in the field experienced by the particle, similar to the standing wave excitation in Fluorescence Interference Contrast (FLIC) microscopy [37].

The axial excitation profile further depends on the distance $z_{\text{f}}$ of the beam focus from the Si/SiO₂ interface and was calculated for different values of $z_{\text{f}}$; we denote the calculated profile as $I_0(z,z_{\text{f}})$. Since we do not have a direct measurement of $z_{\text{f}}$, we determined it in an indirect fashion as follows. We used the same focusing criterion in all experiments: the axial location of the device relative to the microscope objective was adjusted using a piezo-mounted stage to ensure that a D = 500 nm pocket displayed maximum contrast. We performed trapping experiments on particles using pockets of different diameters in devices with different values of l and 2h, and found the maximum experimentally obtainable value of particle contrast, $I_{\text{max}}$. We then used the intensity and position data of the calibration particles $I(z_1)$ and $I(z_2)$ and a function representing the theoretical axial excitation profile $I_0(z,z_{\text{f}})$ in Eq. (2) to determine the maximum expected particle contrast $I_{\text{max}}$ and system phase, ${\phi }_{\text{sys}}$ for a given device configuration. Agreement between $I_{\text{max}}$ and $I_0(z,z_{\text{f}})$ was taken as a criterion to determine the value of $z_{\text{f}}$ and hence the calibration curve for that device. Small changes in $z_{\text{f}}$ from the determined value in each case gave $I_0(z,z_{\text{f}})$ magnitudes larger or smaller than the experimentally observed $I_{\text{max}}$. For example for the case shown in Fig. 3(e), an increase in from 170 to 180 nm would increase $I_0(z,z_{\text{f}})$ by 17% and also shift the measured mean levitation plane by + 5 nm. $z_{\text{f}}$ is thus a fit parameter in the axial calibration and its value for various device configurations lies in a fairly tight range between 170 and 200 nm. Figure 4(d) shows the axial calibration curves obtained for the three different device configurations that were used in the experiments. The solid sections of the curves show the range of axial displacements sampled by the particles. Comparing calibration curves for the D = 200nm case including the reflection (blue) and excluding it (black) demonstrates that the standing wave mainly alters the amplitude of the expected particle contrast, but also slightly shifts the location of this peak. Accurate measurements of the axial motion can be obtained over half a period of Eq. (2), where the signal varies from maximally bright to dark, which corresponds to ~100nm in these experiments. In the trap geometries and imaging conditions we explored, particles levitate over a range <50nm, a distance over which the contrast is expected to change from 0 to $I_{\text{max}}$, thus permitting accurate axial localization. In such an experiment the OPL in Eq. (2) may be optimized by choosing values of l and 2h to ensure that I depends monotonically on z. As the slit depth 2h is, in fact, a trap parameter, altering it also influences the levitation midplane as well as the range of motion of the trapped particle. An increase in slit depth of $\delta l$for example, effectively shifts $z_{\text{m}}$ by $\sim \delta l/2$. Manipulating these thicknesses, therefore, enables us to shift the calibration function and levitation plane of the particle relative to each other in order to ensure that the range of particle’s axial motion is contained within a monotonic segment of the calibration function.

In our iSCAT imaging scheme, the high SNR and sensitivity to the axial motion enables very accurate particle tracking. Figure 5 illustrates the transverse ($\delta x$) and axial ($\delta z$) localization precisions as a function of the particle contrast. $\delta x$ monotonically decreases as the signal increases [12]. In the case of particles trapped by D = 500 nm pockets, the contrast varies from nearly zero to almost $I_{\text{max}}$, which accordingly corresponds to an average 2.5 nm lateral localization precision and an axial localization precision in the range 0.18 - 1.61 nm, with an average of 0.32nm. $\delta z$ is obtained by considering the uncertainty in determining the amplitude of the Gaussian fit function ($\delta I$) according to $\delta z=\left|\partial z/\partial I\right|\delta I$. As shown in Fig. 5, the localization precision at low contrasts is subnanometric and it monotonically increases with contrast. Although $\delta I$ is smaller for images with higher contrast, when $I\to I_0$ the derivative of the above equation rapidly increases. In summary, for a given range of axial motion the OPLs may be adjusted to optimize the axial resolution over the entire measurement.

 

Fig. 5 Measured lateral (red) and axial (blue) localization precision dependence on the contrast of 80nm particles trapped by a D = 500 nm pocket.

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We note that the two calibration intensities $I(z_1)$ and $I(z_2)$ were obtained by averaging the intensities of several particles at the SiO₂/H₂O interface. As a result, Eq. (2) applies for particles of the canonical average size. Since the amount of light scattered by a particle depends strongly on its size [26], the measured average values of $I(z)$ on single particles, denoted as $I_{\text{m}}$, should reflect the size dispersion in the sample. Further, the mean of these $I_{\text{m}}$ values for an ensemble of trapped particles corresponds to the contrast of the canonical trapped particle with the average size. Thus for each trapped particle, the axial calibration was carried out by normalizing the two values of $I(z_1)$ and $I(z_2)$ by $m_{\text{c}}$, the ratio of $I_{\text{m}}$ of a given particle to the mean contrast of all trapped particles ($m_{\text{c}}=1$ corresponds to an 80nm particle), and then proceeding with the axial calibration procedure as described previously.

Interestingly, as illustrated in Fig. 6 , the measured values of $I_{\text{m}}$ may be further used to estimate the size of the individual trapped particles [38]. The first step in this regard is to use the generalized Mie theory for a spherical particle in a focused beam [39] in order to calculate $E_{\text{inc}}$ and $E_{\text{bd}}$, and consequently $m_{\text{s}}^2$, the ratio of the total scattered power ($P_{\text{bd}}$) to the incident power ($P_{\text{inc}}$), as a function of the particle size. Since both $I_{\text{m}}$ and $m_{\text{s}}$ are proportional to the iSCAT signal, the calculated functional dependence of $m_{\text{s}}$ may be used to directly relate, $I_{\text{m}}$ obtained from the measured optical signal to the particle size. To do this, we set the calculated value of $m_{\text{s}}$ for an 80nm particle to correspond to the average experimentally measured contrast of all trapped particles i.e., to the particle with $m_{\text{c}}=1$. The estimated sizes of all other trapped particles can then be read off the graph based on their $m_{\text{c}}$ values. Based on this analysis we estimate a particle size dispersion of 9% in the sample, which is in good agreement with the value of 8% quoted by the manufacturer.

 

Fig. 6 Calculated dependence of $m_{\text{s}}$ on particle size (black curve) and size estimates of 18 gold nanoparticles trapped by D = 200 nm pockets.

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4. Conclusion

In our experiment, for a gold bead of diameter 80 nm trapped in a D = 200 nm pocket, calculations show that the bottom of the trap ($r<60\text{nm}$) is well described by a harmonic potential over a wide range of particle charge. A spring constant of confinement of say 7.5 × 10⁻³ pN/nm, easily achieved for particles under consideration, corresponds to a relaxation time, of around 100 μs. Using the attainment of saturation in the plateau values of the mean square displacement with decreasing exposure time as a criterion [40], it is apparent that $\sigma \ll \tau$ is required in order to directly map a potential landscape in a measurement with a single exposure time. In shot-noise limited imaging, a reduction in exposure time by a factor n leads to a localization accuracy poorer by $\sqrt{n}$ [12]. Since a technique like iSCAT is based on elastic scattering of the incident field, a reduction in exposure time can be simply compensated by an increase in excitation power, thereby maintaining the same SNR ratio in each image and in theory permitting access to arbitrarily small exposure times. Our combination of device structure and optical set-up currently permits reliable imaging and tracking using iSCAT down to exposure times of $\sigma =200$μs.

Since the average measured position of the particle in a symmetric potential well should coincide with the bottom of the well regardless of the exposure time, and the bottom of the trap in our experiments is harmonic in the axial dimension in all cases (Fig. 3(c)), we find indeed that the axial potential minimum $z_{\text{m}}$ may be nonetheless located accurately independent of exposure time considerations. Furthermore, while we have shown that for harmonic interaction potentials it is indeed possible to reconstruct the true free energy landscape using $\sigma \ge \tau$ [30], rapid (>10 kHz), high SNR imaging techniques would foster direct measurements of arbitrarily-shaped interaction potentials using nanoscale probes.