Measuring Boltzmann's constant through holographic video microscopy of a single colloidal sphere

Bhaskar Jyoti Krishnatreya
Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003
Arielle Colen-Landy
Packer Collegiate Institute, 170 Joralemon Street, Brooklyn, NY 11201
Paige Hasebe
Department of Physics, NYU Abu Dhabi, United Arab Emirates
Breanna A. Bell
Jasmine R. Jones
Anderson Sunda-Meya
Department of Physics, Xavier University of Louisiana, New Orleans, LA 70125
David G. Grier
Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003

The trajectory of a colloidal sphere diffusing in water records a history of the random forces exerted on the sphere by thermally-driven fluctuations in the suspending fluid. The trajectory therefore can be used to characterize the spectrum of thermal fluctuations, and thus to obtain an estimate for Boltzmann's constant. We demonstrate how to use holographic video microscopy to track a colloidal's sphere's three-dimensional motions with nanometer precision while simultaneously measuring its radius to within a few nanometers. The combination of tracking and characterization data reliably yields Boltzmann's constant to within two percent, and also provides the basis for many other useful and interesting measurements in statistical physics, physical chemistry, and materials science.

§ I. Introduction

Early in his annus mirabilis in 1905, Albert Einstein published an explanation for the constant random motion that small objects undergo when suspended in fluids (1). He proposed that these visible motions are caused by invisibly small fluid molecules colliding at random with the object's surface. The statistical properties of the object's position and velocity fluctuations therefore should reflect those of the underlying thermal bath of molecules. Specifically, Einstein predicted that the particle's mean-squared displacement would be proportional to the thermal energy scale, k_{B}T, where T is the fluid's absolute temperature and k_{B} is Boltzmann's constant. These predictions were borne out by Jean Perrin's direct microscopic observations (2), first published in 1912, which yielded a value for k_{B} that was consistent with values that previously had been obtained from tests of the ideal gas law, and from measurements of black-body radiation. This consistency constituted the first experimental demonstration of the physical reality of atoms, and molecules and earned Perrin the Nobel Prize in Physics for 1926.

Perrin carried out his experiments entirely manually, measuring the positions of micrometer-scale colloidal spheres through the eyepiece of optical microscope and recording the results on graph paper. This process has since been automated with the introduction of digital video microscopy (3) and the introduction of automated numerical methods (4) to identify individual spheres in digitized video images and to link their positions in consecutive video frames into time-resolved trajectories for analysis. Quantitative video microscopy of colloidal spheres has been used not only to reproduce Perrin's experiment (3); (4); (5); (6); (7); (8), but also to measure colloidal particles' interactions with each other (9); (10); (11), to probe the statistical physics of particles moving through structured force fields (12); (13); (14); (15); (16), and as the basis for particle-tracking microrheology (17), which is used to characterize the viscoelastic properties of complex fluids and biological materials.

Conventional video microscopy is most useful for measuring colloidal particles' in-plane positions, and provides limited information about their motion along the microscope's optical axis (4); (18). Because diffraction blurs small objects' images, moreover, little information can be gleaned from conventional images regarding the size, shape or composition of individual colloidal particles. These limitations diminish the utility and reliability of results obtained by conventional particle tracking. Perrin's own result for Boltzmann's constant, for example, differed from the accepted value by more than 20 percent (2).

This Article describes how holographic video microscopy can be used to track micrometer-scale colloidal spheres with nanometer precision in all three dimensions while simultaneously measuring each sphere's radius and refractive index with part-per-thousand precision. Implemented with low-cost equipment, holographic video microscopy yields such a wealth of high-resolution information that a five-minute video of a single diffusing sphere can yield a measurement of Boltzmann's constant to a few parts per thousand. The same information also can be used to weigh an individual colloidal sphere, to measure its porosity, and to perform self-consistent experimental tests of the fluctuation-dissipation theorem.

§ II. Colloidal samples

The experiment consists of tracking a single micrometer-scale colloidal sphere as it moves unhindered through quiescent water at a fixed temperature. The particles used for this study are chemically synthesized by emulsion polymerization. This ensures that they are spherical to within a percent and have well characterized and consistent chemical and optical properties (19). Before the sample can be mounted in a microscope, it must be sealed in a suitable container. Our container consists of the small rectangular volume made by gluing the edges of a glass microscope cover slip (Corning Life Sciences, 22~\mathrm{m}\mathrm{m}\times 22~\mathrm{m}\mathrm{m}, #1.5) to the face of a standard glass microscope slide (Fisher Scientific, Fisherbrand Economy Plain). Even this simple apparatus requires some care in its preparation.

Small colloidal spheres tend to stick to surfaces irreversibly because of van der Waals attraction (19). This can be avoided if the spheres and the walls of their container all carry a high enough electrostatic charge to prevent deposition. Many commercially available brands of colloidal spheres are synthesized with large numbers of ionizable surface groups specifically for this purpose. The surface groups dissociate in water, leaving behind bound charges. Our samples consist of polystyrene sulfonate spheres (Duke Scientific, catalog 4016A, nominal diameter 1.587\pm 0.018~\mathrm{\upmu}\mathrm{m}) that develop large negative surface charges. Clean glass surfaces, similarly develop negative surface charges in contact with water (20), which will tend to repel the negatively charged particles.

As delivered, microscope slides and cover slips often are coated with lubricants to prevent them from sticking to each other. These lubricants tend to suppress the surface charge, and thus tend to allow colloidal particles to stick. We therefore thoroughly clean the glass surfaces before assembling our samples.

First, we use a soap solution and a soft bristled brush to clean the slide and cover slip, and then rinse them clean with deionized water. To prevent water-borne residue from contaminating the surfaces, we dry them with a stream of nitrogen gas, taking care to blow excess liquid off the corners. Then we place the slide and cover slip in an oven at 70{}^{{\circ}}\mathrm{C} for 10 minutes until it is thoroughly dry. To minimize contamination, the glass pieces rest on their edges during drying. Once the glass is dry, we rinse it first with acetone to remove remaining organic residue, then isopropanol, and finally ethanol to facilitate drying. Once the pieces are dried under streaming nitrogen, they are clean enough to assemble a sample that will remain viable for a day or two.

Creating samples that are stable over longer periods requires surfaces with higher and more uniform surface charge. We achieve this by further cleaning the glass pieces in a 100 \mathrm{W} oxygen plasma etcher (SPI Plasma Prep II) for 30 \mathrm{min}. This removes any remaining organic residue and oxygenates the silica surface so it develops a higher surface charge when it comes into contact with water. The entire cleaning process including plasma etching takes two hours.

The sample container is assembled as shown in Fig. 1 by placing the coverslip over the microscope slide with thin glass spacers to set the separation. We used two slivers of glass cut from a clean cover to set the spacing at roughly 120 \mathrm{\upmu}\mathrm{m}. The spacers are aligned with the edge of the cover slip, and then glued in place with optical grade adhesive (Norland Industries, uv-cured adhesive No. 68). The two remaining edges of the cover slip are left open to form a channel into which the colloidal dispersion is introduced.

The goal in preparing the colloidal sample is to have no more than one or two particles in a typical field of view. Substantially fewer particles would make it difficult to find a particle to study. Substantially more would interfere with particle tracking. Assuming a field of view of 50~\mathrm{\upmu}\mathrm{m}\times 50~\mathrm{\upmu}\mathrm{m}\times 120~\mathrm{\upmu}\mathrm{m} and a single-particle volume of 2 \mathrm{\mathrm{\upmu}\mathrm{m}}^{{3}}, we prepare samples with a volume fraction of roughly 10^{{-5}} by diluting the stock solution of spheres from the manufacturer with deionized water. A little more than 80 \mathrm{\upmu}\mathrm{L} of the diluted suspension fills the glass channel. We then seal the ends of the channel with more optical adhesive, taking care to avoid bubbles, and cure the glue thoroughly under an ultraviolet light. This last step requires an additional half hour.

§ III. Holographic video microscopy

Figure 1. (color online) Holographic video microscope. The microscope slide containing the sample is mounted on a three-axis positioner, which is not shown for clarity.

§ III.1. Hardware

A holographic microscope suitable for tracking and characterizing micrometer-scale colloidal spheres may be created from a conventional optical microscope by replacing the illuminator with a collimated laser source (21); (22); (23). Our custom-made implementation is shown schematically in Fig. 1. The collimated light from a fiber-coupled laser passes through the sample and is collected by an objective lens before being relayed by a tube lens to a video camera.

Laser light scattered by the sample interferes with the rest of the illumination in the microscope's focal plane. The camera records the intensity of the interference pattern. Each snapshot in the resulting video stream is a hologram of the sample, which encodes comprehensive information on the sample's position, size and optical properties. A sequence of holographic snapshots constitutes a time-resolved record of the sample's behavior and characteristics.

An ideal laser illuminator provides a circular, linearly polarized beam in a single Gaussian mode with no longitudinal or transverse mode hopping. Many low-cost diode lasers have these characteristics. The beam should be collimated to within 1 \mathrm{m}\mathrm{rad}, and should be aligned with the microscope's optical axis to within 10 \mathrm{m}\mathrm{rad} (24). The beam's diameter should be large enough to fully illuminate the field of view with roughly uniform intensity, and the beam should be bright enough to use the camera's full dynamic range without saturation. In our implementation, the 3 \mathrm{m}\mathrm{m}-diameter beam from a 25 \mathrm{m}\mathrm{W} diode laser (Coherent Cube, fiber coupled, operating at a vacuum wavelength of \lambda=447~\mathrm{n}\mathrm{m}) provides more than enough light. Stable operation is important for this application. Some laser pointers, for example, suffer from an instability known as mode hopping that renders them unsuitable.

The magnification and axial range of the holographic microscope are determined by the combination of the objective lens, the tube lens and the video camera. We use a Nikon 100× oil-immersion objective with a numerical aperture of 1.4, which also provides a good basis for optical trapping applications (25). Any oil-immersion objective lens with well-corrected geometric aberrations should perform well for holographic microscopy. Long working distance is not necessary because the lens gathers light scattered along the entire length of the illuminating laser beam. Short depth of focus is desirable because it reduces blurring. This consideration favors lenses with higher numerical apertures.

When combined with a 200 \mathrm{m}\mathrm{m} tube lens, a 100× objective yields a magnification of 135 \mathrm{n}\mathrm{m}\mathrm{/}pixel on the half-inch detector of a 640×480 pixel video camera. We confirmed this figure by measuring the spacing of a grid calibration standard. This overall magnification factor is the only instrumental calibration constant required for the measurement.

The magnification establishes the resolution with which interference fringes can be measured. These fringes become more closely spaced as an object approaches the focal plane. The magnification thus establishes the lower limit of the accessible axial range. Our system can reliably measure the position and properties of colloidal spheres that are more than 5 \mathrm{\upmu}\mathrm{m} above the focal plane.

The interference pattern spreads out and loses contrast as the particle moves away from the focal plane. The limited field of view and the sensitivity of the camera together establish the upper limit of the axial range. We are able to track and characterize micrometer-diameter colloidal spheres up to 100 \mathrm{\upmu}\mathrm{m} above the focal plane.

The video camera that records the holograms ultimately determines the quality of the recorded data and the time resolution with which colloidal particles may be tracked and characterized. Remarkably, even a relatively low-cost surveillance camera suffices for high-precision holographic tracking and characterization, both because commodity video sensors have evolved to high standards of performance, and also because the hologram of a sphere can subtend a large number of pixels. Our instrument uses an NEC TI-324A grayscale camera, whose analog video output is recorded as an uncompressed digital video stream at a nominal intensity sensitivity of 8 bits per pixel using a Panasonic EH50S digital video recorder.

§ III.2. Lorenz-Mie analysis

Figure 2. Video holography of a colloidal polystyrene sphere in water. (a) Measured hologram, I(\mathbf{r}) of a colloidal polystyrene sphere in water. (b) Background intensity, I_{0}(\mathbf{r}), estimated from the running median of 10,000 video frames. (c) Normalized hologram, b(\mathbf{r}) obtained by dividing the hologram in (a) by the background in (b). (d) Fit of (c) to the prediction of Lorenz-Mie theory from Eq. (7).

The intensity I(\mathbf{r}) recorded at position \mathbf{r} in the image plane results from the interference of the illuminating wave with light scattered by the particle. We describe the system in Cartesian coordinates \mathbf{r}=(x,y,z), with the origin located at the center of the microscope's focal plane. The \hat{x} and \hat{y} coordinates lie in the focal plane and \hat{z} is directed along the optical axis. We model the electric field of the illumination as a monochromatic plane wave polarized in the \hat{x} direction and propagating in the -\hat{z} direction (downward):

\mathbf{E}_{0}(\mathbf{r},t)=u(\mathbf{r})\, e^{{-ikz}}\, e^{{-i\omega t}}\,\hat{x}. (1)

The wavenumber k of the light is related to its frequency \omega through the dispersion relation k=n_{m}\omega/c, where c is the speed of light in vacuum and n_{m} is the refractive index of the medium. Most lasers are characterized by the vacuum wavelength of the light they emit, rather than the frequency. These are related by \lambda=2\pi c/\omega. Ideally, the amplitude profile of the beam u(\mathbf{r}) would be a smoothly varying function such as a Gaussian. In practice, as we shall see, light scattered by dust and other imperfections in the optical train can make u(\mathbf{r}) quite complicated.

A small particle centered at \mathbf{r}_{p} scatters a small amount of the incident light from \mathbf{r}_{p} to position \mathbf{r} in the imaging plane:

\mathbf{E}_{s}(\mathbf{r},t)=E_{0}(\mathbf{r}_{p},t)\,\mathbf{f}_{s}\left(k\left(\mathbf{r}-\mathbf{r}_{p}\right)\right). (2)

The scattering function \mathbf{f}_{s}(k\mathbf{r}) describes how light of wavenumber k that is initially polarized in the \hat{x} direction is scattered by a particle with specified properties located at the origin of the coordinate system. In Eq. (2), \mathbf{r}-\mathbf{r}_{p} is the displacement from the particle's center to the point of observation.

Calculating the scattering function is notoriously difficult, even for the simplest cases. Light scattering of a plane wave by an isotropic homogeneous dielectric sphere is described by the Lorenz-Mie theory (26); (27) and is parameterized by the radius of the sphere, a_{p}, and its complex refractive index, n_{p}.

The intensity of the interference pattern due to a sphere of radius a_{p} and refractive index n_{p} located at position \mathbf{r}_{p} relative to the center of the microscope's focal plane is

I(\mathbf{r})=\left|\mathbf{E}_{0}(\mathbf{r},t)+\mathbf{E}_{s}(\mathbf{r},t)\right|^{2}. (3)

A recording of this intensity distribution is a hologram of the sphere, and contains information about the sphere's position, \mathbf{r}_{p}, its radius, a_{p}, and its refractive index, n_{p}. Because a sphere's scattering pattern is largely symmetric about the axis passing through its center, such a hologram should resemble concentric rings on a featureless background. In practice, the expected ringlike pattern usually is obscured by a rugged landscape of interference fringes due to imperfections in the illumination. These characteristics are apparent in the measured hologram of a colloidal polystyrene sphere reproduced in Fig. 2(a).

The influence of these distracting intensity variations can be minimized by factoring illumination artifacts out of the observed interference pattern:

I(\mathbf{r})=I_{0}(\mathbf{r})\,\left|\hat{x}+\frac{u(\mathbf{r}_{p})}{u(\mathbf{r})}\, e^{{-ikz_{p}}}\,\mathbf{f}_{s}\left(k\left(\mathbf{r}-\mathbf{r}_{p}\right)\right)\right|^{2}. (4)

Here, I_{0}(\mathbf{r})=\left|u(\mathbf{r})\right|^{2} is the intensity distribution of the illumination in the absence of the scatterer. Figure 2(b) shows this background intensity distribution for the hologram in Fig. 2(a). Dividing I(\mathbf{r}) by I_{0}(\mathbf{r}) yields a normalized hologram (22); (23),

b(\mathbf{r})\equiv\frac{I(\mathbf{r})}{I_{0}(\mathbf{r})}, (5)

that is comparatively free of illumination artifacts, as may be seen in Fig. 2(c).

If we assume that the amplitude and phase of the illumination vary little over the size of the recorded scattering pattern, we may approximate u(\mathbf{r}_{p})/u(\mathbf{r}) by a complex-valued parameter \alpha to obtain (23); (28); (29)

\displaystyle b(\mathbf{r}) \displaystyle\approx b(\mathbf{r}|\left\{\mathbf{r}_{p},a_{p},n_{p},\alpha\right\}) (6)
\displaystyle\equiv\left|\hat{x}+\alpha\, e^{{-ikz_{p}}}\,\mathbf{f}_{s}\left(k\left(\mathbf{r}-\mathbf{r}_{p}\right)\right)\right|^{2}. (7)

The Lorenz-Mie model for a sphere's normalized hologram, b(\mathbf{r}|\left\{\mathbf{r}_{p},a_{p},n_{p},\alpha\right\}), is an exceedingly complicated function of the particle's three-dimensional position, \mathbf{r}_{p}, radius, a_{p}, and refractive index n_{p}. It also depends on the adjustable parameter \alpha that accounts for variations in illumination at the position of the particle. Fortunately, open-source software is available (30) for calculating b(\mathbf{r}|\left\{\mathbf{r}_{p},a_{p},n_{p},\alpha\right\}). Normalized holograms therefore can be fit pixel by pixel to Eq. (7) to locate and characterize each colloidal sphere in each holographic snapshot.

Fitting b(\mathbf{r}) to b(\mathbf{r}|\left\{\mathbf{r}_{p},a_{p},n_{p},\alpha\right\}) can be performed with standard numerical packages for nonlinear least-squares fitting. Our implementation uses an open-source Levenberg-Marquardt fitting routine (31). Assuming that Eq. (7) is the correct model for the data and that the reduced chi-squared statistic for the fit is of order unity, the uncertainties in the fit parameters may be interpreted as estimates for the associated measurement errors. Using this interpretation, fits to Eq. (7) routinely yields the in-plane position of a micrometer-diameter sphere to within one or two nanometers, the axial position to within five nanometers, the radius to about a nanometer, and the refractive index to within a part per thousand (23); (28); (32).

Obtaining meaningful estimates for the measurement errors in the position, size and refractive index requires good estimates for the error in the value of b(\mathbf{r}) at each pixel in the normalized hologram. Each video snapshot suffers from additive noise arising from the recording process, which is likely to be influenced by normalization with the background image. We estimate an overall value for the noise in each image using the median-absolute deviation (MAD) estimator (33). For the normalized hologram in Fig. 2(c), this value is \Delta b=0.009, or roughly 1 percent of the average value. This is a reasonable figure for images digitized with 8 bits per pixel.

In addition to random noise, the normalized hologram also suffers from uncorrected background variations. These tend to break the circular symmetry of a sphere's hologram, and therefore reduce the quality of the fits. To estimate this contribution to the measurement error, we azimuthally average a sphere's image about the center of the scattering pattern to obtain the radial intensity profile b(r), and then compute the variance \Delta b(\mathbf{r})=\left|b(\mathbf{r})-b(r)\right| of the intensity at each radius r. These contributions might be added in quadrature. Instead, we estimate the single-pixel measurement error as \max\left(\Delta b(\mathbf{r}),\Delta b\right), treating \Delta b as a simple noise floor. This choice has no substantial influence on the results of the fits.

The range of radii r over which normalized images are fit to Eq. (7) then is limited by the range over which the signal peeks out over the noise and uncorrected background variations. A convenient estimate for consistent particle tracking and characterization is to identify the maxima and minima in the radial profile b(r) and to set the range to the radius of a particular extremum. The images in Fig. 2 were cropped to the radius of the twentieth extremum. With these considerations, the chi-squared parameter for a typical fit is of order 1, suggesting that computed uncertainties in the adjustable parameters are reasonable estimates for the measurement errors in those parameters.

Figure 2(d) shows the result of fitting the normalized hologram to b(\mathbf{r}|\left\{\mathbf{r}_{p},a_{p},n_{p},\alpha\right\}) using n_{m}=1.3397, which is the accepted value (34) for the refractive index of water at \lambda=447~\mathrm{n}\mathrm{m} and T=24.0^{{\circ}}\mathrm{C}. The only other calibration constant for the measurement is the overall magnification of the optical train, which is found to be 0.135 \mathrm{\upmu}\mathrm{m}\mathrm{/}pixel using a standard reticule. With these two calibration constants, the fit yields the particle's in-plane position to within 0.8 \mathrm{n}\mathrm{m} in the plane and 5.7 \mathrm{n}\mathrm{m} axially.

The fit values for the radius and refractive index are a_{p}=805\pm 1~\mathrm{n}\mathrm{m} and n_{p}=1.5730\pm 0.0006. The former is substantially larger than the nominal 780 \mathrm{n}\mathrm{m} radius specified by the manufacturer. The latter is significantly smaller than the refractive index of bulk polystyrene at this temperature and wavelength, which is 1.592. Chemically synthesized polystyrene spheres are not fully dense, however. The discrepancy presumably results from an effective porosity of 1 or 2 percent assuming the pores to be full of water (35).

§ III.3. Background estimation

The background intensity I_{0}(\mathbf{r}) used to normalize each hologram can be obtained by taking a snapshot of an empty field of view. Moving the microscope's stage is not an effective way to find a particle-free area, however, because the background can change from place to place within a sample. Instead, moving the particle out of the field of view with optical tweezers or another non-contact manipulation technique can be effective.

If optical tweezers are not available, the background intensity can be estimated by recording a sequence of images as the particle moves freely across the field of view. The measured intensity deviates from the background intensity only in the vicinity of the particle's scattering pattern. These intensity variations are either brighter or darker than the local background. The background at each pixel therefore can be estimated as the median value of the intensities measured at each pixel as the particle moves across the field of view. A sufficiently long video sequence of a particle diffusing across the field of view therefore can be used to first to estimate the background intensity and then to analyze the particle's diffusion. The background in Fig. 2(b) was estimated as the median of 900 consecutive images obtained over 30 \mathrm{s}  starting with the image in Fig. 2(a).

Figure 3. (color online) Time-resolved measurements of the radius a_{p} and refractive index n_{p} of a single diffusing colloidal sphere. Dashed lines indicate the mean values and standard deviations from the mean.

§ III.4. Video holography

The sequence of snapshots in a holographic video stream can be analyzed individually to obtain time-resolved information on a particle's position and characteristics. The time interval between snapshots is determined by the video camera's shutter interval, t_{0}, which is 1/29.97~\mathrm{s} for cameras conforming to the National Television Systems Committee (NTSC) standard used in the United States.

Most consumer-grade video cameras do not acquire all of the pixels in a frame simultaneously. Rather they first record the odd-numbered scan lines and then the even, the time between these fields' acquisition being half the shutter period. Such images are said to be interlaced. Because a particle may have moved substantially during the interval between recording the even and odd fields, the two fields of each interlaced image should be analyzed separately. The images in Fig. 2 show the even field of an interlaced hologram, which were interpolated back to full resolution for presentation.

The loss of half the spatial resolution in one direction is compensated by the doubling of the measurement's temporal resolution. If results from the two fields are to be combined into a single time sequence, care must be taken to ensure that the two data streams are interleaved in the correct order, and that the geometric offset between the two fields is accounted for correctly.

Figure 4. (color online) Free diffusion of a sedimenting colloidal sphere. (a) Three-dimensional trajectory measured over 160 \mathrm{s}. Points are colored to indicate the passage of time. In-plane components of the motion are projected onto the xy plane. (b) Axial component of the trajectory showing steady sedimentation together with a linear least-squares fit for the sedimentation velocity. (c) Components of the mean-squared displacement of the trajectory from (a).

In addition to providing time-resolved particle-tracking data, holographic video microscopy also yields time series for the particle's radius a_{p} and refractive index n_{p}. These quantities are useful for monitoring the sample's stability, particularly the stability of the temperature. Ideally, a_{p} and n_{p} should not vary over the course of a measurement by more than the uncertainties in their values. Changes in temperature, however, cause changes in the relative refractive index of the medium and the sphere and therefore influence extracted values for a_{p}, n_{p} and the axial position z_{p}.

The 10,000 data points in Fig. 3 were obtained from the even and odd fields of 5,000 consecutive video snapshots of the same sphere presented in Fig. 2. Variations from the mean values for a_{p} and n_{p} are significantly larger than the single-fit uncertainty estimates. Much of this variability is likely to be due to uncorrected illumination artifacts and slow mechanical drifts in the optical train, including flow in the immersion oil. In addition to these variations, the overall mean values drift slightly over time, particularly near the end of the measurement. This trend suggests that more confidence should be attached to the first 100 seconds of this data set.

§ IV. Analyzing a sphere's sedimentation

Figure 4(a) shows the three-dimensional trajectory \mathbf{r}_{p}(t) of the same colloidal sphere. The plot is colored by time, and the data from Fig. 2 contributed the first of 10,000 points. In-plane distances in Fig. 4(a) are measured relative to one corner of the field of view. The axial distance is measured relative to the height of the focal plane. Based on the particle's motion at very long times, we estimate the axial position of the lower glass surface of the sample chamber to be roughly z_{0}=5~\mathrm{\upmu}\mathrm{m} on this scale. Because hydrodynamic coupling can substantially influence a particle's diffusion when it approaches a solid surface (36); (37); (38), we restrict our analysis to z_{p}(t)>15~\mathrm{\upmu}\mathrm{m}. Being able to rule out hydrodynamic coupling as a source of systematic error is one of the noteworthy strengths of using holographic video microscopy to measure Boltzmann's constant.

Although the in-plane diffusion shown in projection in Fig. 4(a) appears to be random, there is a clear downward trend in the axial position plotted in Fig. 4(b). This shows that the sphere is slightly more dense than the surrounding water, and therefore sediments under gravity. Fitting z_{p}(t) to a linear trend yields a mean axial velocity of v_{z}=75.27\pm 0.01~\mathrm{n}\mathrm{m}\mathrm{/}\mathrm{s}. A sphere of radius a_{p} and mass density \rho _{p} should achieve a terminal velocity

v_{z}=\frac{2}{9}a_{p}^{2}\,\frac{(\rho _{p}-\rho _{m})g}{\eta}, (8)

in a medium of mass density \rho _{m} and dynamic viscosity \eta. From this, we may estimate the particle's density. Taking the viscosity of the water to be \eta=0.912~\mathrm{m}\mathrm{Pa}\;\mathrm{s} at T=24.0^{{\circ}}\mathrm{C} and its density to be \rho _{m}=997.3~\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{\mathrm{m}}^{{3}}, and using g=9.80363~\mathrm{m}\mathrm{/}\mathrm{\mathrm{s}}^{{2}} for the acceleration due to gravity in New York City, we obtain for the density of the sphere \rho _{p}=1045.7~\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{\mathrm{m}}^{{3}}. When compared with the density of bulk polystyrene, \rho _{\text{PS}}=1047~\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{\mathrm{m}}^{{3}}, the value for \rho _{p} estimated from sedimentation is consistent with a porosity of 1 or 2 percent, assuming the pores to be filled with water (35). This agrees with the estimate for the sphere's porosity obtained from optical data alone in Sec. III.2.

§ V. Analyzing a sphere's diffusion

A small particle immersed in a fluid responds both to deterministic forces, such as the force of gravity, and also to random thermal forces resulting from collisions with the fluid's molecules. The former influence causes the particle to drift with a constant velocity \mathbf{v}. The latter renders a diffusing particle's trajectory, \mathbf{r}_{p}(t), unpredictable and is a defining characteristic of Brownian motion. For clarity, we will drop the subscript p in the following analysis of the particle's trajectory.

Even though a Brownian particle's position cannot be anticipated, a statistical ensemble of such trajectories nevertheless has predictable properties. In the absence of drift due to external forces, the ensemble average of the particle's displacements over a time interval \Delta t is expected to vanish. This is not surprising because the random forces themselves vanish on average in accord with the second law of thermodynamics. More remarkably, Einstein (1) and Smoluchowski (39) predicted that a diffusing particle's mean-squared displacement should increase linearly with time. For a particle that diffuses as it drifts, the Einstein-Smoluchowski relation predicts

\displaystyle\Delta r^{2}(\Delta t) \displaystyle\equiv\left<\left|\mathbf{r}(t+\Delta t)-\mathbf{r}(t)\right|^{2}\right> (9)
\displaystyle=2d\, D\,\Delta t+(v\,\Delta t)^{2}, (10)

where d is the dimensionality of the trajectory, and where D is a constant called the diffusion coefficient. The angle brackets in Eq. (9) indicate either an average over different but equivalent trajectories or else an average over statistically independent starting times, t, in a single trajectory. In the limit that D/\Delta t is much smaller than v^{2}, Eq. (9) reduces to the familiar result for motion with constant velocity, \mathbf{r}(t+\Delta t)=\mathbf{r}(t)+\mathbf{v}\,\Delta t.

By considering the statistics of molecular motion in the suspending fluid, Einstein showed that the macroscopic particle's diffusion coefficient is proportional to the thermal energy scale of the microscopic molecules:

D=\gamma^{{-1}}\, k_{B}T. (11)

The constant of proportionality is the viscous drag coefficient \gamma that characterizes the force F=\gamma v required to move the particle through the fluid at speed v. The drag coefficient therefore establishes the power P=\gamma v^{2} that is dissipated in moving the particle through the fluid. In this sense, Eq. (11) relates microscopic fluctuations to macroscopic dissipation and so is an example of a fluctuation-dissipation theorem.

For the particular case of a spherical particle of radius a_{p} moving through an unbounded Newtonian fluid of dynamic viscosity \eta, Sir George Stokes calculated the viscous drag coefficient to be \gamma=6\pi\eta a_{p}. The associated Stokes-Einstein diffusion coefficient,

D=\frac{k_{B}T}{6\pi\eta a_{p}}, (12)

together with the Einstein-Smoluchowski equation in Eq. (9) provide a basis for estimating Boltzmann's constant from tracking data.

Holographic particle tracking offers the substantial benefit over conventional video microscopy that it yields a precise estimate for the radius a_{p} of the particular sphere being tracked, including an estimate for the uncertainty in this value (40). Without this information, the probe particle's size typically has been estimated through light-scattering analysis of a representative sample of similar spheres, with an uncertainty no smaller than the sample's polydispersity, which typically is on the order of a few percent.

§ V.1. Analyzing a discretely sampled trajectory

The experimentally measured trajectory is sampled at discrete times t_{n}=nt_{0} that may be indexed by the integer n and are separated by the video camera's shutter interval t_{0}. The trajectory thus consists of a sequence of positions, \mathbf{r}_{n}=\mathbf{r}(t_{n})+\boldsymbol{\epsilon}_{n}, each of which includes a measurement error \boldsymbol{\epsilon}_{n}. Unlike conventional particle tracking (4), holographic tracking yields an estimate for the localization error at each position. This additional information will prove useful in obtaining Boltzmann's constant from a single-particle trajectory.

We assume that the uncertainty in the measurement time, t_{n}, is small enough to be ignored. For a conventional video camera, the inter-frame jitter is typically less than 5 \mathrm{n}\mathrm{s}. Similarly, we will assume that the camera's exposure time is sufficiently short that blurring due to the particle's motion can be ignored (41); (42); (28); (43). In the experiments we report, the camera's exposure time is set to 100 \mathrm{\upmu}\mathrm{s}. Whether this is sufficiently fast can be established once the particle's measured trajectory has been analyzed.

A complete data set consists of N samples from which the mean-squared displacement at lag time \Delta t=st_{0} may be estimated as

\Delta r^{2}_{s}=\frac{1}{N_{s}}\,\sum _{{n=0}}^{{N_{s}-1}}\left|\mathbf{r}_{{(n+1)s}}-\mathbf{r}_{{ns}}\right|^{2}, (13)

where N_{s}=\left\lfloor N/s\right\rfloor is the integer part of N/s. The plots in Fig. 4(c) show the mean-squared displacements for each of the three components of the trajectory in Fig. 4(a) computed with Eq. (13).

The statistical variance in the estimate for \Delta r^{2}_{s} is

\sigma _{s}^{2}=\frac{1}{N_{s}-1}\,\sum _{{n=0}}^{{N_{s}-1}}\left[\Delta r_{s}^{2}-\left|\mathbf{r}_{{(n+1)s}}-\mathbf{r}_{{ns}}\right|^{2}\right]^{2}. (14)

The variance can be computed for each of the Cartesian components of \mathbf{r}(t) independently. The error bars in Fig. 4(c) are computed in this way.

Whereas statistical fluctuations may either increase or decrease the estimated value of \Delta r_{s}^{2} relative to its ideal value, measurement errors only increase the apparent positional fluctuations. To see this, imagine that the particle were truly stationary at the origin of the coordinate system, so that \mathbf{r}_{n}=\boldsymbol{\epsilon}_{n} for all values of s. The true mean-squared displacement vanishes in this thought experiment. The apparent mean-squared displacement, however, is

\displaystyle\Delta r^{2}_{s} \displaystyle=\frac{1}{N_{s}}\,\sum _{{n=0}}^{{N_{s}-1}}\left|\boldsymbol{\epsilon}_{{(n+1)s}}-\boldsymbol{\epsilon}_{{ns}}\right|^{2} (15)
\displaystyle=\frac{1}{N_{s}}\,\sum _{{n=0}}^{{N_{s}-1}}\left[\epsilon _{{(n+1)s}}^{2}+\epsilon _{{ns}}^{2}\right]-\frac{2}{N_{s}}\,\sum _{{n=0}}^{{N_{s}-1}}\boldsymbol{\epsilon}_{{(n+1)s}}\cdot\boldsymbol{\epsilon}_{{ns}}. (16)

If the measurement errors are random, the second term in Eq. (16) vanishes. If we further assume that the measurement errors have the same statistical distribution during the entire experiment, we may define an apparent mean-squared distribution associated specifically with measurement errors

2d\,\epsilon^{2}=\frac{1}{N_{s}}\,\sum _{{n=0}}^{{N_{s}-1}}\left[\epsilon _{{(n+1)s}}^{2}+\epsilon _{{ns}}^{2}\right] (17)

that is independent of the measurement interval s. Measurement errors therefore systematically increase the mean-squared displacement computed from a trajectory.

Rather than incorporating measurement errors into the error bars in \Delta r^{2}_{s}, we instead modify the Einstein-Smoluchowski relation to account for the measurement-related offset:

\Delta r^{2}_{s}=2d\,\epsilon^{2}+2d\, D(st_{0})+v^{2}(st_{0})^{2}. (18)

In this case, only the statistical uncertainties \sigma^{2}_{s} contribute to the error estimate for each value of s.

Fits of the data in Fig. 4(c) to Eq. (18) yield values for the tracking error in each of the coordinates of \epsilon _{x}=3\pm 8~\mathrm{n}\mathrm{m}, \epsilon _{y}=4\pm 8~\mathrm{n}\mathrm{m} and \epsilon _{z}=11\pm 8~\mathrm{n}\mathrm{m}. These values are statistically consistent with the average position errors obtained from fitting to the individual images, which are 0.6~\mathrm{n}\mathrm{m}, 0.6~\mathrm{n}\mathrm{m} and 4~\mathrm{n}\mathrm{m}, respectively. The dynamical values obtained from Eq. (18) might reasonably be expected to be larger because they also incorporate tracking errors due to mechanical vibrations in the apparatus.

§ V.2. Statistical cross-checks

Analyzing motions in each coordinate independently permits useful tests for potential experimental artifacts. Hydrodynamic coupling to the sample container's glass walls, for example, increases hydrodynamic drag by an amount that depends on height above the wall (36) and thereby reduces the observed diffusion coefficient. This was one of the principal sources of error in Perrin's particle-tracking experiments. Because hydrodynamic coupling has a larger effect in the axial direction (36); (44); (38); (45), agreement between in-plane and axial diffusion coefficients may be used to verify that this source of error is negligibly small. Non-uniform drift due to unsteady fluid flow in a poorly sealed sample similarly would appear as inconsistencies among results for the three coordinates.

The single-component values are related to the overall mean-squared displacement by \Delta r^{2}_{s}=\Delta x^{2}_{s}+\Delta y^{2}_{s}+\Delta z^{2}_{s}. If the system is homogeneous and isotropic, therefore, the sphere should diffuse equally freely along each axis: \Delta x^{2}_{s}=\Delta y^{2}_{s}=\Delta z^{2}_{s}=\frac{1}{3}\,\Delta r^{2}_{s}. Statistical uncertainties in these estimates also should agree: \sigma _{{x,s}}^{2}=\sigma _{{y,s}}^{2}=\sigma _{{z,s}}^{2}=3\sigma _{s}^{2}.

For the data set in Fig. 4, fits to Eq. (18) yield D_{x}=0.3015\pm 0.0042~\mathrm{\mathrm{\upmu}\mathrm{m}}^{{2}}\mathrm{/}\mathrm{s}, D_{y}=0.3009\pm 0.0046~\mathrm{\mathrm{\upmu}\mathrm{m}}^{{2}}\mathrm{/}\mathrm{s} and D_{z}=0.2924\pm 0.0044~\mathrm{\mathrm{\upmu}\mathrm{m}}^{{2}}\mathrm{/}\mathrm{s}. As anticipated, the three diffusion coefficients are consistent with each other to within their estimated uncertainties. Similarly, the statistical variances in the mean-squared displacements all are consistent with each other for each measurement interval s.

Agreement among the single-coordinate measurements is consistent with the assumptions that hydrodynamic coupling to the bounding surfaces may be ignored and that any nonuniform drifts were small. Limits on the axial range set in Sec. IV therefore appear to have been chosen appropriately, and the sample cell appears to have been stably sealed and well equilibrated. We are justified, therefore, in estimating the particle's free diffusion coefficient as the average of the single-component values, D=0.2986\pm 0.0025~\mathrm{\mathrm{\upmu}\mathrm{m}}^{{2}}\mathrm{/}\mathrm{s}.

§ V.3. Independent sampling and greedy sampling

Equation (13) yields an estimate for \Delta r^{2}_{s} based on a subset \{\mathbf{r}_{0},\mathbf{r}_{s},\dots,\mathbf{r}_{{sN_{s}}}\} of the measurements in the trajectory starting from the first measurement in the series. An equivalent estimate could be obtained by starting from the j-th measurement

\Delta r^{2}_{{s,j}}=\frac{1}{N_{{s,j}}}\,\sum _{{n=0}}^{{N_{{s,j}}-1}}\left|\mathbf{r}_{{(n+1)s+j}}-\mathbf{r}_{{ns+j}}\right|^{2}, (19)

with j<s and where N_{{s,j}}=\left\lfloor(N-j+1)/s\right\rfloor. Each such estimate is based on a statistically independent sampling of the particle's trajectory distinguished by its starting index j.

It is tempting to treat each value of \Delta r^{2}_{{s,j}} as a separate measurement of the mean-squared displacement and to combine them,

\Delta r^{2}_{s}=\frac{1}{s}\,\sum _{{j=0}}^{{s-1}}\Delta r^{2}_{{s,j}}, (20)

thereby improving statistical sampling by a factor of \sqrt{s}. This choice makes use of all of the available data, and therefore is known as greedy sampling.

The statistically independent samples that go into the greedy estimate for \Delta r^{2}_{s} are not, however, statistically independent of each other. To see this, consider that a trajectory starting at \mathbf{r}_{0} and advancing to \mathbf{r}_{s} after s time intervals shares s-1 time steps of diffusion with the equivalent trajectory starting at \mathbf{r}_{1} and advancing to \mathbf{r}_{{s+1}}. Such correlated estimates nevertheless may be combined according to Eq. (20) provided that particular care is taken to account for their correlations (3). If the underlying trajectory consists of statistically independent steps, as in the case of a freely diffusing sphere, then greedy sampling can beneficially reduce the variance in the estimate for the mean-squared displacement without biasing its value (3). For the present data set, greedy sampling yields D=0.2982\pm 0.0008~\mathrm{\mathrm{\upmu}\mathrm{m}}^{{2}}\mathrm{/}\mathrm{s}.

Greedy sampling may be risky, however. If the underlying trajectory is correlated, then greedy sampling may bias the apparent mean-squared displacement. Such correlations may occur, for example, if the particle's diffusion coefficient depends on its position, or if the particle is subject to a force that varies with position or time. To avoid such biases, we adopt the statistically independent estimator described in Eq. (13), which is equivalent to \Delta r^{2}_{{s,0}} from Eq. (19).

§ VI. Measuring Boltzmann's constant

Given the foregoing results, Boltzmann's constant may be estimated with Eq. (12) as

k_{B}=\frac{6\pi\eta\, a_{p}\, D}{T}. (21)

For the radius of the particle, we use the average a_{p}=0.802\pm 0.011~\mathrm{\upmu}\mathrm{m} of the values obtained in all of the fits. From this and the previously discussed values for \eta, D and T, we obtain k_{B}=(1.387\pm 0.021)\times 10^{{-23}}~\mathrm{J}\mathrm{/}\mathrm{K}, which agrees to within half a percent with the IUPAC accepted value of (1.3806488\pm 0.0000013)\times 10^{{-23}}~\mathrm{J}\mathrm{/}\mathrm{K}.

This result compares favorably with conventional video microscopy measurements (5); (7); (8), which have yielded less precise and less accurate values for k_{B} despite amassing larger data sets. Errors in these earlier studies were dominated by uncertainty in the radius of the diffusing particle, which was addressed by combining results from multiple particles. Holographic microscopy addresses this problem by providing a direct measurement of the sphere's radius. Other potential sources of error, such as non-steady drift and hydrodynamic coupling to the walls, can be ruled out by comparing results for motion along independent directions. Constancy in the monitored value for the particle's refractive index similarly excludes thermal and instrumental drifts as potential sources of error.

The two most important sources of error in the present measurement are statistical uncertainty and variations in the measured value for the particle's radius. The former can be addressed by acquiring more data. The latter could be reduced by improving the illumination's uniformity to minimize uncorrected background variations, and thus to avoid position-dependent variations in the sphere's apparent size. The ultimate limit on this measurement's precision is likely to be set by uncertainty in the sample's temperature through its influence on the materials' refractive indexes and on the water's viscosity.

§ VII. Conclusions

Through holographic video microscopy of a single micrometer-diameter polystyrene sphere diffusing freely in water, we have obtained a value for the Boltzmann constant of k_{B}=(1.387\pm 0.021)\times 10^{{-23}}~\mathrm{J}\mathrm{/}\mathrm{K}. This measurement took advantage of holographic microscopy's ability to track the sphere's motion in three dimensions with high precision and also to measure the sphere's radius. Comparably good results were obtained with repeated measurements on the same sphere and with measurements on other spheres.

Because no effort was made to match the solvent's density to the sphere's, these holographic tracking measurements also revealed the particle's slow sedimentation under gravity. From the sedimentation velocity and the holographic measurement of the sphere's radius we were able to estimate the sphere's density. Comparing this value with the accepted value for bulk polystyrene then suggests that the sphere is porous with a porosity of roughly 1 percent. This value is consistent with the porosity estimated from the sphere's refractive index, which also was obtained from the same holographic microscopy data. Consistency among these measurements lends confidence to the result for the Boltzmann constant, which is indeed consistent with the presently accepted value.

More generally, this measurement highlights the capabilities of holographic video microscopy for particle tracking and particle characterization. Comparatively simple apparatus can be used to track individual colloidal particles over large distances with excellent precision, while simultaneously offering insights into particle-resolved properties that are not available with any other technique. These capabilities already have been applied to measure the microrheological properties of complex fluids (40), to characterize variations in colloidal particles' properties (46); (47) and to perform precision measurements in statistical physics (48); (49). They should be useful for a host of other experiments in soft-matter physics, physical chemistry, and chemical engineering.

§ Acknowledgments

This work was supported in part by a grant from Procter & Gamble and in part by the PREM program of the National Science Foundation through Grant Number DMR-0934111.


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