Method and apparatus for measuring charge and size of single objects in a fluid

Abstract

In a method for determining charge and/or size of an object (15) suspended in a fluid, the object (15) is introduced, together with the fluid, into an electrostatic trap (1) defining an electrostatic confining potential. The thermal motion of the object (15) in the fluid is observed under influence of the confining potential, and charge and/or size are determined from the observed thermal motion. In particular, the viscous drag on the object yields a measure of its size, while the stiffness of its confinement can be compared with a potential model to reveal the total charge it carries. Also disclosed are an apparatus and software for carrying out the method.

Claims

1. A method for determining at least one of charge and size of an object suspended in a fluid, the method comprising: introducing the object, together with the fluid, into an electrostatic trap defining an electrostatic confining potential (U); with an imaging device, observing thermal motion of the object in the fluid under influence of the confining potential (U); and with an analyzing device connected to the imaging device, determining at least one of charge and size from the observed thermal motion, wherein at least one characteristic of the confining potential is determined by the analyzing device from the observed thermal motion and is compared with a potential model of the confining potential, thereby determining at least one of charge and size of the object.

2. The method of claim 1, wherein the characteristic of the confining potential (U) is determined by measuring at least one characteristic of a probability density distribution (P(r)) of object displacement under influence of the confining potential and deriving the characteristic of the confining potential from said characteristic of the probability density distribution (P(r)).

3. The method of claim 2, wherein determining the stiffness parameter comprises: with the imaging device connected to the analyzing device, imaging the thermal motion of the object with at least two different exposure times (); determining, for the at least two different exposure times (), a width parameter of the apparent probability density distribution; deriving a relaxation time () of the object from a comparison of the width parameters at the different exposure times (); and calculating the stiffness parameter (k) from the relaxation time ().

4. The method of claim 3, wherein the drag parameter () is determined from an observation of short-term diffusion behavior of the object.

5. The method of claim 1, wherein the characteristic of the confining potential is a stiffness parameter (k) of the confining potential (U).

6. The method of claim 5, further comprising: with the imaging device and the analyzing device connected to the imaging device, determining a drag parameter () for the object, wherein the stiffness parameter (k) is calculated from the relaxation time () using the drag parameter ().

7. The method of claim 5, wherein both the relaxation time () and a drag parameter () are derived by the analyzing device from a comparison of the width parameters at the different exposure times ().

8. The method of claim 1, further comprising: obtaining with the analyzing device, the potential model by carrying out a calculation of system free energy.

9. The method of claim 1, comprising: with the imaging device and the analyzing device connected to the imaging device, determining a drag parameter () of the object within the confining potential; and calculating the size of the object from the drag parameter ().

10. The method of claim 1, wherein the electrostatic trap comprises two substrates arranged in an opposing configuration so as to form a fluidic slit, the substrates having non-zero surface charge densities inducing an electrostatic potential between the substrates, at least one of the substrates having a surface nanostructure for modulating the electrostatic potential between the substrates so as to generate said electrostatic confining potential.

11. The method of claim 1, wherein a plurality of objects are observed by the imaging device in parallel in an array of traps, and wherein at least one of size and charge is determined for each one of said plurality of objects.

12. An apparatus for determining at least one of size and charge of an object suspended in a fluid, the apparatus comprising: an electrostatic trap for receiving the object together with the fluid and for trapping the object in an electrostatic confining potential; an imaging device for observing thermal motion of the object under influence of the confining potential; and an analyzing device connected to the imaging device, and configured to receive information about the thermal motion from the imaging device and to determine at least one of size and charge of the object from the observed thermal motion, wherein the analyzing device is configured to determine at least one characteristic of the confining potential (U) from the observed thermal motion and to compare the determined characteristic with a potential model to determine at least one of charge and size of the object.

13. The apparatus of claim 12, comprising an array of electrostatic traps for receiving a plurality of objects in parallel, wherein the analyzing device is configured to determine at least one of size and charge of each one of the plurality of objects .

14. The apparatus of claims 12, wherein the electrostatic trap comprises two substrates arranged in an opposing configuration so as to form a fluidic slit, the substrates having non-zero surface charge densities inducing an electrostatic potential between the substrates, at least one of the substrates having a surface nanostructure for modulating the electrostatic potential between the substrates so as to generate said electrostatic confining potential.

15. An analyzing device configured to execute the steps of a computer program code, the steps of the computer program code comprising: receiving information from an imaging device about thermal motion of an object suspended in a fluid under influence of a confining potential in an electrostatic trap ; and determining at least one of charge and size of the object from the received information, by determining at least one characteristic of the confining potential (U) from the received information; and comparing the determined characteristic with a model potential to determine at least one of the charge and size of the object.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Preferred embodiments of the invention are described in the following with reference to the drawings, which are for the purpose of illustrating the present invention and not for the purpose of limiting the same. In the drawings:

(2) FIG. 1 shows, in cross section, the geometry of a single electrostatic fluidic trap (part (a)) and a possible arrangement of traps into an array (part (b));

(3) FIG. 2 shows a diagram illustrating the experimental set-up used for iSCAT imaging;

(4) FIG. 3 shows a 3D scatter plot of positions x, y, z for a representative particle and corresponding histograms for the probability distributions P(r) and P(z) of position in radial coordinate r and axial coordinate z, respectively;

(5) FIG. 4 shows a plot of the r.m.s. values of axial displacement s.sub.z vs. lateral displacement s.sub.r for 18 different particles, numerals adjacent to each data symbol representing particle serial number;

(6) FIG. 5 shows a diagram illustrating the calculated distribution of electrostatic potential in a three-dimensional cylindrical half-space around a particle 80 nm in diameter in a trapping nanostructure with D=200 nm and 2 h=215 nm, the displayed range being truncated at a potential e=2 k.sub.BT;

(7) FIG. 6 shows radial free energy profiles calculated along the contour of the axial energy minimum for particle charges q=40 (open circles), 88 (open squares), 133 (open triangles), 177 (solid circles), 221 (solid squares), 398 (solid triangles) and 1105 e (crosses), for a solution ionic strength 0.04 mM;

(8) FIG. 7 shows a diagram illustrating the variation of the spring constant of confinement k with particle charge q, the inset displaying a linear relationship between k and q for q<100 e (black line);

(9) FIG. 8 shows a diagram illustrating the charge deduced for each particle in FIG. 4 (symbols) based on the k vs. q relationship in FIG. 7 (solid line);

(10) FIG. 9 shows a diagram illustrating the apparent mean square displacement (MSD) in x, [x(t, )].sup.2, as a function of lag-time t for a particle trapped by a D=500 nm pocket imaged with an exposure time =1 ms, together with a linear fit to the data for t<3 ms;

(11) FIG. 10 shows a diagram illustrating apparent MSD in x for a particle trapped by a D=200 nm pocket and imaged using exposure times =0.2 ms (open circles) and 1 ms (solid circles);

(12) FIG. 11 shows radial probability histograms P(r) corresponding to the apparent MSD data of FIG. 10 for exposure times =0.2 ms (grey bars, dotted line) and 1 ms (black bars, dashed line);

(13) FIG. 12 shows a diagram illustrating the apparent local electrostatic potential U(r) derived from the P(r) data shown in FIG. 12 for =0.2 ms (solid circles) and 1 ms (open circles), together with a solid curve representing a harmonic potential of the form U(r)=kr.sup.2 with k=8.810.sup.3 pN/nm inferred from the experimental MSD data, square symbols representing the calculated free energy for a particle of charge 62 e; and

(14) FIG. 13 shows a log-log plot of the experimentally inferred potential and calculated free energy shown in FIG. 12, the inset presenting a zoomed view of the experimental potential (solid line) and free energy calculations (symbols) for particles of charge q=54 e (open circles), 62 e (open squares), and 68 e (solid circles).

DESCRIPTION OF EXEMPLARY EMBODIMENTS

(15) FIG. 1(a) illustrates the setup of a single electrostatic trap 1, and FIG. 1(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 h200 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.

(16) Gold nanospheres 15 having a diameter of 80 nm (British Biocell International) were centrifuged and resuspended in deionized H.sub.2O (18 Mcm.sup.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.sup.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.

(17) 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 FIG. 2. The gaussian output beam of a 30 mW diode-pumped solid state laser 21 (TECGL-30, WSTech) at =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.

(18) 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.sub.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.

(19) The observed 3D motion of single trapped particles is presented in FIGS. 3 and 4. As apparent from FIG. 3, 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.sub.z) vs. radial dimension (s.sub.r) is shown in FIG. 3. This plot convincingly demonstrates just this correlation. The fact that s.sub.z<s.sub.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.

(20) 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.sub.BT=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.

(21) 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 FIG. 5, 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.

(22) FIG. 6 shows a series of calculated radial free energy curves as a function of particle charge q for the conditions of the measurements in FIGS. 1-4. The free energy curves were fitted to a function U(r)=kr.sup.2 for r<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 FIG. 7. 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's spring constant is measured with a comparable accuracy, the measurements could be very close to single-charge resolved (FIG. 7 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.

(23) 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.

(24) 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 of the particle. This will explained in the following.

(25) The spring constant and the relaxation time are related via the drag coefficient as k=/ (Equation 1), where =3a for a sphere of diameter a in a solution of viscosity . A spring constant of k=7.510.sup.3 pN/nm for example, easily achieved for the particles under consideration, corresponds to a relaxation time of around 100 s, assuming =110.sup.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.

(26) The motion of a single particle can be investigated using different exposure times, > and the corresponding apparent mean squared displacements (MSD) [x(t, )].sup.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

(27) $.Math.{\left[x\right]}_p^2.Math.=\frac{2k_{B}T}{k}$
for a harmonic trapping potential. In this case, the plateau value of the MSD, [x()].sub.p.sup.2 measured using an exposure time, approaches the true value asymptotically as a function of the / ratio:

(28) $\begin{array}{cc}\frac{.Math.{\left[x\left(\right)\right]}_p^2.Math.}{.Math.{\left[x\right]}_p^2.Math.}=\frac{2}{/}-\frac{2\left(1-e^{-/}\right)}{{\left(\text{/}\right)}^2}.& \left(\mathrm{Equation}2\right)\end{array}$

(29) This is illustrated in FIG. 10 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 [x()].sub.p.sup.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 =855 s was obtained for the selected particle.

(30) Note that the same result may be obtained from an analysis which uses the apparent variance s.sub.x.sup.2 of the measured probability distribution P(x) in x (or y), as shown in FIG. 11, in place of [x()].sub.p.sup.2, and which uses the true variance

(31) $\frac{k_{B}T}{k}$
in place of [x].sub.p.sup.2.

(32) A value of =85 s corresponds to a spring constant of confinement k=8.810.sup.3 pN/nm, depicted by the lines in FIGS. 12 and 13. A similar analysis can be carried out for other potential shapes.

(33) 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 FIGS. 12 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.sup.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% (FIG. 13 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.

(34) 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.

(35) TABLE-US-00001 particle (s) q (e) i 85 5 54-68 ii 87 13 49-75 iii 85 4 54-68 iv 161 11 26-35

(36) 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 FIG. 9 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 1.620.17 m.sup.2/s was obtained, averaged over 10 different cases. The measured diffusivity,

(37) $\frac{k_{B}T}{}$
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

(38) $.Math.{\left[x\right]}_p^2.Math.=\frac{2k_{B}T}{}$
in Eq. 2 as a free parameter and obtain both and from the fit to measurements at multiple exposure times.

(39) 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>80 nm, than it does closer to the center (r<50 nm). An accurate long-range spatial map of the potential could therefore further provide an independent measurement of the particle diameter.

(40) 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.

(41) The method may be extended to higher solution ionic strengths by employing smaller slit depths

(42) $\left(h~\frac{1}{\sqrt{C}}\right)$
for effective trapping.

(43) 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 (<1 s)such as small and/or weakly charged matter, or biological macromolecules in solutions of higher ionic strengthcould 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.

(44) 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.