text: fig1 ( a ) illustrates the setup of a single electrostatic trap 1 , and fig1 ( b ) illustrates a possible arrangement of traps in an array . an array of such traps was fabricated by lithographically patterning the surface of a − 400 nm deep silicon dioxide layer 12 on a p - type silicon substrate 13 and subsequent wet - etching the silicon dioxide layer 12 to a depth of 2 h ≈ 200 nm in buffered hf ( ammonium fluoride - hf mixture , sigma - aldrich ). the floors of these trenches were then patterned with submicron - scale features such as cylindrical pockets 14 having a pocket diameter d using electron beam lithography and subsequent reactive ion etching of the silicon dioxide to a pocket depth of d = 100 nm . a fluidic device with fully functional fluidic slits having a depth 2 h of approximately 200 nm and a width of 20 micrometers was obtained by irreversibly bonding the processed silicon dioxide - silicon substrates with a glass substrate 11 compatible with high - na microscopy ( planoptik , ag ) using field - assisted bonding . for details , reference is made to reference [ 1 ], “ methods ” section . gold nanospheres 15 having a diameter of 80 nm ( british biocell international ) were centrifuged and resuspended in deionized h 2 o ( 18 mωcm − 1 ) twice to remove traces of salt or other contaminants . nanoslits loaded with an aqueous suspension of the nanometric object of interest ( number density ca . 10 10 p / ml for gold particles ) by the capillary effect , were allowed to equilibrate at room temperature for 1 - 2 h before commencing with optical measurements . high - speed interferometric scattering detection ( iscat ) was used to image the 3d motion of individual particles trapped in harmonic potential wells created by pockets 14 of diameter d = 200 nm or 500 nm and depth d = 100 nm in a fluidic slit of depth 2 h = 215 nm , using an imaging device 2 in the form of a laser scanning microscope set - up shown in fig2 . the gaussian output beam of a 30 mw diode - pumped solid state laser 21 ( tecgl - 30 , wstech ) at λ = 532 nm was expanded by a 4 × telescope lens system 22 and passed through a half - wave plate 23 for polarization adjustment , followed by a two - axis acousto - optic deflector ( aod ) 24 ( dtsxy , aa opto - electronic ). the deflected beam was delivered via a telecentric system 25 and a beamsplitter cube 26 to the back focal plane of a microscope objective 27 ( 1 . 4 na , 100 × uplasapo - olympus ) mounted on an inverted microscope equipped with a three - dimensional piezoelectric translation stage 17 ( pt1 , thorlabs ). the fluidic device comprising the traps 1 was positioned using the translation stage 17 such that the scanned beam illuminated the area of interest . the scanning rates of the aods were between 50 khz and 100 khz and were adjusted to achieve a uniform wide field of illumination for a given exposure time . light scattered by the particle and reflected by the device was collected by the same microscope objective 27 and imaged via a tube lens 28 onto a cmos camera 29 ( mv - d1024e - 160 - cl - 12 , photonfocus ). image data from the camera 29 was transferred to an analyzing device 3 in the form of an appropriately programmed general - purpose computer , which carried out the subsequent analysis of the image data as described in more detail below . when imaging a trapped particle using interferometric scattering detection ( iscat ), the detected signal results from the interference of the electromagnetic field scattered by the particle , the background pocket , and the beam reflected from the sio 2 / si interface . the interference allows determining complete information on the 3d location of the particle , not only along the lateral plane , but also along the axial direction . for details , reference is made to reference [ 1 ], “ method ” section ( see above ). the gold nanospheres gave a signal - to - noise ratio ( snr ) of ˜ 100 suitable for particle tracking with a localization precision of ˜ 2 nm . they also carried a substantial amount of charge , of the order of − 100 e , making them amenable to trapping for long periods of time ( several minutes to hours ) and therefore convenient to study . the observed 3d motion of single trapped particles is presented in fig3 and 4 . as apparent from fig3 , the particles sample the confining potential both in the radial ( lateral ) direction and in the axial direction with a near - gaussian probability distribution . the spatial sampling of an electrostatic potential well by a particle strongly depends on the charge it carries : the higher the charge on the particle , the greater the expected stiffness of its confinement , which manifests in the experiment as a smaller r . m . s . spatial displacement . importantly , any increase or decrease in stiffness arising from particle charge would be expected to appear in all spatial dimensions . a scatter plot of the r . m . s . displacement of each particle in the axial ( s z ) vs . radial dimension ( s r ) is shown in fig3 . this plot convincingly demonstrates just this correlation . the fact that s z & lt ; s r confirms higher trap stiffness in the axial compared to the radial dimension . the solution ionic strength in these measurements was 0 . 04 mm . measuring the trap stiffness thus presents a simple and rapid route to measure the relative charge dispersion in a sample at the single object level . in order to measure the absolute net charge carried by any given particle , the characteristics of the confining potential , as determined by the measurements , can be compared with a potential model obtained from numerical calculations . in particular , if the probability density distribution of particle displacement p ( r ) has been measured , the boltzmann relation , u ( r )/ k b t =− ln p ( r ), yields the spatially dependent potential u ( r ). once this potential is known , the charge of the particle can be directly obtained by comparison with a model obtained from free energy calculations . comsol multiphysics was used to create such a potential model by calculating the spatial distribution of electrostatic potential in the trapping nanostructure by numerically solving the non - linear poisson - boltzmann equation in 3d . as shown in fig5 , the model system consisted of a sphere of a fixed surface charge density embedded in an electrolyte , which in turn is bounded by surfaces of a given charged density representing the walls of the trapping nanostructure . the inputs to the calculation were the wall charge density , the solution ionic strength and the size and surface charge density of the object . the background electrolyte ionic strength and an estimate of the wall charge density were obtained from conductivity and electroosmotic flow measurements , respectively . thus , for a particle of a given diameter , its charge remained the only free parameter in the calculation . the free energy of the system as a function of particle position was calculated by summing the electrostatic field energies and entropies over all charges in the system . fig6 shows a series of calculated radial free energy curves as a function of particle charge q for the conditions of the measurements in fig1 - 4 . the free energy curves were fitted to a function u ( r )= ½kr 2 for r & lt ; 50 nm to obtain the spring constant of confinement k in each case . the relationship between k and q derived from such comparisons is shown in fig7 . this figure enables a direct readout of the charge of a particle once its spring constant is known . further , the linearity of the relation for q ← 100 e and the low uncertainty in the single fit coefficient (˜ 0 . 5 %) implies that if the particle &# 39 ; s spring constant is measured with a comparable accuracy , the measurements could be very close to single - charge resolved ( fig7 inset ). this raises prospects both for fundamental studies on ( dis -) charging processes on matter in solution as well as for ultrasensitive single - nanoparticle based molecular binding sensors . the procedure outlined above requires that snapshots of the particle can be acquired with a high signal - to - noise ratio and at exposure times which are much shorter than the relaxation times of the particles . provided that these conditions are fulfilled , optical imaging is an excellent calibration - free method for direct mapping of potential landscapes of arbitrary shape and large range , and offers distinct advantages in high - throughput analysis of a dense array of trapped objects . unfortunately , high snr imaging with an exposure time much smaller than the relaxation time of the particle can be challenging . in a harmonic confining potential , however , the spring constant of confinement k can be obtained even for exposure times which are comparable or even larger than the relaxation time t of the particle . this will explained in the following . the spring constant and the relaxation time are related via the drag coefficient γ as k = γ / τ ( equation 1 ), where γ = 3πηα for a sphere of diameter a in a solution of viscosity η . a spring constant of k = 7 . 5 × 10 − 3 pn / nm for example , easily achieved for the particles under consideration , corresponds to a relaxation time of around 100 μs , assuming η = 1 × 10 − 3 kg / ms , the viscosity of water in free solution . if the drag coefficient is known , a measurement of the relaxation time thus yields the spring constant . the motion of a single particle can be investigated using different exposure times , σ & gt ; τ and the corresponding apparent mean squared displacements ( msd ) [ δx ( t , σ )] 2 can be evaluated as a function of lag time t . rather than continuously increasing as a function of t , the msd of a trapped particle eventually saturates at a value for a harmonic trapping potential . in this case , the plateau value of the msd , [ δx ( σ )] p 2 measured using an exposure time , σ approaches the true value asymptotically as a function of the σ / τ ratio : this is illustrated in fig1 for a selected particle trapped in a 200 nm pocket , imaged at exposure times of 0 . 2 ms and 1 ms , respectively . using the values of [ δx ( σ )] p 2 from the msd measurements , equation 2 gives the relaxation time and , through equation 1 , the spring constant of confinement for the particle under consideration . a relaxation time of τ = 85 ± 5 μs was obtained for the selected particle . note that the same result may be obtained from an analysis which uses the apparent variance s x 2 of the measured probability distribution p ( x ) in x ( or y ), as shown in fig1 , in place of [ δx ( σ )] p 2 , and which uses the true variance a value of τ = 85 μs corresponds to a spring constant of confinement k = 8 . 8 × 10 − 3 pn / nm , depicted by the lines in fig1 and 13 . a similar analysis can be carried out for other potential shapes . having experimentally deduced the true spring constant of confinement , the measurement can be compared with the calculation to obtain the charge of the particle . the squares in fig1 and 13 represent the free energy as a function of radial distance from the trap center r calculated for a particle 80 nm in diameter carrying a total surface charge of − 62 e ; the wall charge density and background electrolyte concentration were − 0 . 01 e / nm 2 and 0 . 03 mm respectively . the uncertainty in the measured relaxation time implies that the charge on a single particle can be determined to within ± 10 % ( fig1 inset ). the precision in the relaxation time can be enhanced with measurements at additional exposure times , albeit at the possible expense of time resolution in the overall charge measurement . further , the charge on the particle deduced from its lateral motion can be independently confirmed by a similar analysis of its motion in the axial dimension . experimental relaxation times and deduced net charges for four different particles are presented in the following table . the solution ionic strength for particles ( i - iii ) was 0 . 03 mm , while that for particle ( iv ) was 0 . 04 mm . the size of the particles can be measured independently . one way to do this is via an analysis of the msd of the trapped particle . the data series in fig9 shows msd data for a particle trapped in a well created by a d = 500 nm pocket , which may be roughly approximated by a square well . fitting the linear portion of the data , a translational diffusion coefficient of 1 . 62 ± 0 . 17 μm 2 / s was obtained , averaged over 10 different cases . the measured diffusivity , of the particle yields its hydrodynamic diameter which can then serve as an input to the free energy calculation . an alternative route is to leave the quantity in eq . 2 as a free parameter and obtain both γ and τ from the fit to measurements at multiple exposure times . interestingly , the free energy calculations suggest that for a given particle charge , the size of the particle starts to contribute more strongly to the shape of the trap at longer range , say r & gt ; 80 nm , than it does closer to the center ( r & lt ; 50 nm ). an accurate long - range spatial map of the potential could therefore further provide an independent measurement of the particle diameter . the present invention thus shows that a few seconds worth of high spatio - temporal resolution imaging of electrostatically trapped objects can yield both size and charge information on thousands of individual entities trapped in parallel in high - density arrays . this equilibrium measurement directly addresses the surface of a single nano - object , raising prospects for measuring charge fluctuations in matter , monitoring the progress of chemical reactions in real time and fostering the elucidation of fundamental phenomena at the poorly understood solid - liquid interface . the method may be extended to higher solution ionic strengths by employing smaller slit depths although iscat was used in the above example , a variety of other imaging techniques that deliver sufficient spatial and temporal resolution , e . g . wide - field or laser scanning fluorescence , or dark field microscopy may be used to the same end . given further advances in high - speed , high - sensitivity imaging technology , weakly scattering but labeled entities that are only transiently trapped (& lt ; 1 s )— such as small and / or weakly charged matter , or biological macromolecules in solutions of higher ionic strength — could be studied by this technique . progress in imaging based on scattering or absorption would go a long way in fostering label - free measurements of this nature on nanoscopic entities . it goes without saying that the present invention is not limited to the above - described example . in particular , differently shaped traps may be employed , depending on the nature of the objects under investigation . for example , the traps may be elongated for elongated objects such as nanofibers . while in the above - described example electrostatic traps were obtained by surface patterning in a fluid slit according to ref . [ 1 ], other types of electrostatic traps may be employed .