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Phase Uncertainty in Digital Holographic Microscopy Measurements in the Presence of Solution Flow Conditions.

1. Introduction

Digital holographic microscopy (DHM) is a novel surface topography technique that has recently become commercially available. It is a quantitative phase microscopy technique capable of being configured for reflection or transmission microscopy, and it has been used for a variety of applications [1], including static and dynamic surface metrology [2-7], particle tracking [8,9], tracking and monitoring of live biological cells [3,10-12], and monitoring surface dissolution [13,14] or growth [15] kinetics. DHM and related quantitative phase imaging technologies have been extensively developed for the study of biological specimens [16-18]. The DHM operates by separating and recombining object and reference beams at a single wavelength to generate the interferometric data as a hologram, which is then numerically reconstructed to yield amplitude and phase information [19-21], examples of which are shown in Fig. 1. The primary advantage of DHM is that full-field two-dimensional (2D) holograms are collected at tens of frames per second and can be collected in situ.

The numerically reconstructed phase is directly proportional to height, therefore yielding three-dimensional (3D) surface topography (see Fig. 1e), with sub-nanometer vertical accuracy being reported [22]. However, one complication of the technique is the presence of inherent (intrinsic and extrinsic) noise [23-29], which has not undergone a rigorous analysis to relate it to uncertainty in the measured phase (and therefore height) data. Furthermore, recent studies have utilized reflection DHM (1) to track in situ surface topography changes of dissolving mineral phases in static and flowing water [13,14], but it is inconclusive from those studies how the presence of water (or any solution) or the use of flowing water conditions affects the noise and uncertainty in the measurement. Therefore, the objective of the present study is to further quantify the uncertainty in such measurements in order to better understand the limitations of the instrument, sampling statistics, and minimum measurable surface topography changes.

2. Principles of DHM

Interferometric principles provide the basis for DHM. A coherent, monochromatic light source is split into object and reference beams. In a reflection mode DHM, the object beam passes through an objective lens, interacts with the sample surface, and is reflected back through the objective lens, where it is recombined with the reference beam (Fig. 2). The interferogram produced by the recombined object and reference beams is recorded as a 2D hologram on a charge coupled device (CCD) camera [19,20]. Numerical reconstruction of the hologram produces a reconstruction wavefront in an observation plane as a function of the recorded hologram intensity. This reconstruction wavefront consists of real and imaginary parts, from which a 2D amplitude image (similar to what would be observed in a conventional light microscope) and a 2D phase image are generated [19]. Additional processing accounts for aberrations and the shape (e.g., tilt) of the sample surface [30,31]. The phase, [phi], at a given pixel location ([xi], [eta]) in the image can be converted to height, h, as a function of the known wavelength, [lambda], and the index of refraction, n:

h([xi], [eta]) = [lambda]/4[pi]n [phi]([xi], [eta]). (1)

The index of refraction used in Eq. (1) is that of the medium in which the sample surface is being measured, which in this study is air (n = 1.0) and water (n = 1.33).

The use of two wavelengths simultaneously is possible in DHM [32]. Assuming a dispersionless system and stable wavelengths [32,33], at two wavelengths, [[lambda].sub.i] and [[lambda].sub.j], the difference in phase (i.e., [[phi].sub.i] - [[phi].sub.j]) yields a "synthetic" or "beat" wavelength, [[LAMBDA].sup.ij] [32,34]:

[[LAMBDA].sub.ij] = [absolute value of [[lambda].sub.i][[lambda].sub.j]/[[lambda].sub.i] - [[lambda].sub.j]]. (2)

This allows for greater height differences to be measured on a given surface, albeit with greater uncertainty. In addition, since the data are recorded simultaneously [32] at each of the two wavelengths, two data sets are generated for a given sample, so the noise contribution at each wavelength should be evaluated.

One type of intrinsic noise present in a DHM has been attributed to shot noise [23-26,28,29], which is related to the intrinsic variability in photons incident on the CCD camera [35]. Shot noise can be modelled by a Poisson distribution [35]. Temporal averaging of multiple reconstructed phase images can be used to reduce the effects of shot noise [22,24].

So-called "Gaussian noise" is also present in DHM measurements (i.e., in the reconstructed wavefront) and includes the combined effects of numerous noise sources, such as readout, thermal, and quantization noise, but it can also be attributable to extrinsic sources, such as dust or dirt on the lenses or spurious reflections [25]. Spatial averaging of phase images obtained at two different wavelengths in a dual-wavelength DHM configuration has been shown to reduce noise in the measurement [22], and noise reduction has also been demonstrated by averaging holograms collected at multiple wavelengths [36] or with varying incident beam intensities [37].

The DHM utilized in this study was a Lyncee Tec Model R-2203 (Lausanne, Switzerland). (2) The instrument is equipped to produce three different wavelengths ([[lambda].sub.1] = 665.5651 nm, [[lambda].sub.2] = 793.2365 nm, or [[lambda].sub.3] = 681.0068 nm) and can be operated in a single-wavelength ([[lambda].sub.1] only) or dual-wavelength ([[lambda].sub.1] and [[lambda].sub.2] or [[lambda].sub.1] and [[lambda].sub.3]) mode. The resultant synthetic wavelengths in dual mode are [[LAMBDA].sub.12] = 4135 nm and [[LAMBDA].sub.13] = 29.35 [micro]m. In manual collection mode, hologram acquisition can be started, paused, and stopped at will by the operator, and the CCD camera (Model ECO285MVGE, SVS-Vistek, Seefeld, Germany) can collect holograms at frame rates of up to 12.5 [s.sup.-1], but a continuous collection "video" mode can collect holograms at a frame rate of 25 s-1. The objective lenses used in the DHM configuration for this study are summarized in Table 1, including the pixel size and wavelength-dependent depth of field for each objective lens. The DHM is equipped with objective lenses for use in air as well as water immersion lenses. The DHM is configured for reflection only.

A reaction cell was constructed to contain a sample and allow continuous flow of water (or other solution). The cell was composed of polyether ether ketone (PEEK), which is chemically inert in most aqueous environments. Samples were affixed to a titanium stub, which was then screwed into the reaction cell, as shown in Fig. 3. The same cell was used in solution flow experiments and can be configured to use an immersion lens (Fig. 3 a) or use a glass window with an objective lens in air (Fig. 3b). For each experimental configuration, the optical path length of the reference beam was adjusted to account for the object beam traveling through multiple media (e.g., air, glass, and water) to optimize coherence.

3. Uncertainty in DHM Phase Measurements

The uncertainty was quantified in the measured phase (and therefore height) of a nominally flat glass slide with a thin film of chromium. The thickness of the chromium was measured by DHM at various locations on the sample to be about 80 nm to 100 nm; this was performed by examining sections of the surface that had been masked and did not have chromium. At this thickness, it is possible that the chromium layer is semitransparent, which would affect the results, because the reflected wavefront can be generated from both the top and the bottom of the chromium layer [39,40]. Under the assumption that the chromium is transparent, the reflection, R, from the top of the chromium surface would be a function of the Fresnel reflection coefficients from the top (air-chromium interface) and bottom (chromium-glass interface) surfaces, [r.sub.01] and [r.sub.12], respectively, and the phase change, [beta],

R = [r.sub.01] + [r.sub.12] exp(-i2[beta])/1 + [r.sub.01][r.sub.12] exp(-i2[beta])' (3)

where [beta] is a function of the cosine of the incident beam angle, wavelength, layer index of refraction, and layer thickness [40]. The Fresnel reflection coefficient, [r.sub.pq], at the interface of two media, p and q, for perpendicularly incident light is a function of the refractive indices of the two media, [n.sub.p] and [n.sub.q]:

[r.sub.pq] = [[absolute value of [n.sub.p] - [n.sub.q]/[n.sub.p] + [n.sub.q]].sup.2]. (4)

At wavelength A1, the refractive indices of chromium and glass are around 3.5 [41] and 1.5, respectively, so R is approximately 78 %, which suggests that even if the chromium is transparent, the reflectivity is relatively high. For very thin layers, it has been shown that chromium does demonstrate transparency (e.g., about 50 % transmittance with a wavelength around [[lambda].sub.1] for an 18 nm chromium layer [42]), but thicker coatings of chromium have demonstrated zero transmittance. For instance, Rauf et al. [43] found zero transmittance in the wavelength range of 300 nm to 700 nm for a chromium layer thickness of 200 nm, and Wang et al. [44] found zero transmittance at a wavelength of 550 nm for chromium layer thicknesses [greater than or equal to] 100 nm. Therefore, it is assumed that the chromium layer thickness in the present study is thick enough to exhibit minimal effects, if any, of semitransparency.

Actual quantification of the uncertainty was performed by evaluating the temporal standard deviation per pixel. As each phase map consists of 650 pixels by 650 pixels, two nominally defect-free regions were selected: One region was selected as the reference offset height, while another was selected as the region-of-interest (ROI) from which the samples were collected (Fig. 4). The temporal standard deviation, s, was then defined per pixel along the time dimension for N number of holograms, where [x.sub.i] is the phase at a given pixel location in the phase map from the ith hologram, and [bar.x] is the mean phase at a given pixel location across N phase maps:

s = [square root of (1/N - 1 [[summation].sup.N.sub.i=1][([x.sub.i] - [bar.x]).sup.2])] . (5)

Type A uncertainty [45] is defined using the temporal standard deviation. The effects of three variables were considered: (1) magnification, which was changed using various objective lenses and which also influences the lateral resolution (see Table 1); (2) medium, which was air or water; and (3) water flow rate. Experiments under flowing conditions were separated into two categories: (1) measurements in flowing water using an immersion objective lens (see Fig. 3 a configuration); and (2) measurements in flowing water using an objective lens in air and viewing through a glass window (see Fig. 3b configuration). All experiments were conducted at ambient laboratory conditions at (23 [+ or -] 1) [degrees]C.

The uncertainty was quantified by determining the temporal standard deviation of a given pixel from a given number of reconstructed phase images. For each experiment, unless otherwise noted, 100 holograms were collected at a frame rate of 25 [s.sup.-1], although the effects of sample size and hologram acquisition rate were also explored. The effect of sample size was investigated by randomly sampling a sequential subset of the 100-hologram data set. Subsets of 5, 10, 20, 30, and 50 holograms were taken.

As the uncertainty in the phase data was found to be a non-normal distribution, several metrics were used to describe the data set, including the mean, median, interquartile range (IQR), skewness, and kurtosis [46,47]. The IQR is the difference between the 75th and 25th percentiles of the data, which is the middle 50 % of the data. The IQR is an indication of the data variability and provides a metric of the spread of the data and helps to identify outliers. Skewness is a representation of the symmetry (or asymmetry) of a distribution. A normal distribution has zero skewness. Positive skewness indicates that the distribution has more values to the right of the mean, while a negative skewness suggests the opposite. Kurtosis can be used to evaluate the normality of a distribution, where a kurtosis of 3 represents a normal distribution. Distributions with a kurtosis greater than 3 are more prone to outliers than a normal distribution or have heavy tails, while those with a kurtosis less than 3 are less prone to outliers than a normal distribution or have light tails.

Analysis and comparison of the uncertainty distributions across multiple configurations were performed using quantile-quantile plots and a two-sample Kolmogorov-Smirnov test. A quantile-quantile plot compares the quantiles from one data set to the quantiles of a second data set, which helps to assess if the two data sets are from a similar distribution. A reference line is shown in a quantile-quantile plot along with the data; if the two data sets are from a similar distribution, the plotted data points will fall along the reference line. The reference line represents a linear fit through the second and third quantiles of the two data sets (i.e., the middle 50 % of the data), which is then extrapolated for the first and fourth quantiles. A one-sample quantile-quantile plot can also be generated, which is used to compare the sample distribution to a known distribution (e.g., normal, Poisson, etc.) by plotting the quantiles of the sample against quantiles of an ideal distribution. One-sample quantile-quantile plots will also have a reference line that represents the linear fit through the second and third quantiles of the data. The two-sample Kolmogorov-Smirnov test is a nonparametric test to evaluate if two data sets come from a common distribution. To make inferences between two distributions, multiple iterations of the Kolmogorov-Smirnov test were performed with random samplings of 100 data points from the large data set populations in this study.

3.1 Measurements in Air

The uncertainty over 100 reconstructed phase maps collected at 25 [s.sup.-1] was evaluated for each objective lens in air for [[lambda].sub.1], and the resulting histograms are shown in Fig. 5. The mean uncertainty and median uncertainty were less than 0.5 nm for all objective lenses (Table 2 and Table 3). These values are similar to the value of 0.4 nm reported by Charriere et al. [2] from phase data collected from 4500 holograms with a 20x objective lens and a wavelength of 635 nm.

In Table 2 and Table 3, the uncertainty appears to decrease with increasing magnification, considering the 50- and 100-hologram data sets, with the exception of the data at 5x and at 40x. In addition, the distribution, mean, and median uncertainty values at 100x appear to be significantly different from the other magnifications. However, when considering the resolution of the objective lens (i.e., 0.61[lambda][NA.sup.-1], where NA is the numerical aperture) using [[lambda].sub.1], the ratio of the resolution to the pixel size (Table 1) at a given magnification is 2.0, 2.4, 2.0, 3.0, 4.0, and 6.5 for the 2.5x, 5x, 10x, 20x, 40x, and 100x objective lenses, respectively. The ratio is greatest at 100x, which also yielded the lowest uncertainty. The uncertainty at 5x appears to be typically less than the uncertainty at 2.5x or 10x, which also correlates to the ratio of the resolution to pixel size. The ratio for the 40x objective lens is greater than the 20x objective lens, and the uncertainty is less for 40x compared to 20x magnification when considering 30 holograms or fewer. The resolving power and numerical aperture can be considered intrinsic factors affecting the uncertainty, although the data collected also include extrinsic factors, which influence the variability in uncertainty in Table 2 and Table 3.

The effect of sample size on the standard deviation histogram is shown in Fig. 6, and, as expected, the histogram shape was altered as a function of sample size. As can be seen in Table 2, the sample size of holograms minimally affected the mean uncertainty for the majority of objective lenses. As a function of sample size, the uncertainty IQR shows that the middle 50 % of the uncertainty lies within 0.1 nm for sample sizes of 10 holograms or more and within 0.2 nm for sample sizes of 5 holograms (Table 4).

The general shapes of these histograms are right-skewed and non-normal. For data collected from 100 holograms, the uncertainty is less skewed and more normal at higher magnifications (40x and 100x), as shown in Table 5. The uncertainty distribution approaches normality at smaller sample sizes (e.g., phase data from 20 holograms or fewer), as shown in Table 6. The skewness of the data is also evident in a quantile-quantile plot of the data (Fig. 7), which shows that, relative to a normal distribution, the standard deviation data are skewed at the tails, particularly the right tail, which agrees with the findings from the analysis of the skewness value (Table 5 and Table 6).

3.1.1 Measurements at Different Acquisition Rates

Using the 10x objective lens in air with A1, 100 holograms were collected at two different acquisition rates of 25 [s.sup.-1] and 12.5 [s.sup.-1]. The same area of the sample was examined as the acquisition rate was changed. The uncertainties based on mean and median values were comparable between the two acquisition rates, being on the order of 0.4 nm to 0.5 nm, depending on the number of holograms (see Table 7 and Fig. 8). In general, the skewness and kurtosis values were slightly greater for data acquired at 12.5 [s.sup.-1] relative to the 25 [s.sup.-1] acquisition rate, but the IQR values were relatively unaffected.

Quantile-quantile plots comparing the phase uncertainty at the two acquisition rates are shown in Fig. 9, which suggest that the uncertainty distributions of the two acquisition rates may be similar. Therefore, a two-sample Kolmogorov-Smirnov test was used. To evaluate, 1000 iterations of the Kolmogorov-Smirnov test were conducted by randomly sampling 100 data points from the two data sets with different acquisition rates. The results suggest that the data sets are likely from similar continuous distributions, since, of the 1000 iterations, with 95 % confidence, 89 % of the Kolmogorov-Smirnov test results indicated that the distributions were the same between the 25 [s.sup.-1] and 12.5 [s.sup.-1] acquisition rates for the 100-hologram data sets. Similarly, for the 50-, 30-, 20-, 10-, and 5-hologram data sets, 77 %, 84 %, 94 %, 88 %, and 84 %, respectively, of the Kolmogorov-Smirnov test results indicated that the distributions were the same between the 25 [s.sup.-1] and 12.5 [s.sup.-1] acquisition rates.

3.1.2 Measurements in Dual-Wavelength Mode

Using the 10x objective lens in air in dual-wavelength mode, 100 holograms were collected at an acquisition rate of 25 [s.sup.-1]. The holograms were collected using the [[lambda].sub.1]-[[lambda].sub.2] and [[lambda].sub.1]-[[lambda].sub.3] configurations. The uncertainty at each individual wavelength and the combined synthetic wavelength are summarized in Table 8 and Table 9. For direct comparability, the same area of the sample was examined as the single [[lambda].sub.1] experiment in Sec. 3.1. As seen in Fig. 10, the synthetic wavelength increases the uncertainty, which is expected since the noise is additive [32, 34, 48-50]. Kuhn et al. [32] reported an amplification of [square root of (2)] in the synthetic-wavelength phase noise. Since the noise is additive, the sum of the standard deviations of the individual wavelengths multiplied by 1/[square root of (2)] should yield the standard deviation of the synthetic- wavelength phase; the mean and median uncertainty values in Table 8 and Table 9 agree very well with this amplification value of [square root of (2)]. (3)

For phase data collected from the [[lambda].sub.1]-[[lambda].sub.2] dual-wavelength mode, the phase uncertainty is on the order of 0.9 nm for the [[lambda].sub.1] data, 1.3 nm for the [[lambda].sub.2] data, and as high as 8.7 nm for the [[LAMBDA].sub.12] data (Table 8). The distributions are still non-normal and skewed right, but the [[lambda].sub.2] and [[LAMBDA].sub.12] data tend towards normality, particularly at smaller sample sizes, at least when compared to the phase data from Sec. 3.1. The IQR values for the single wavelengths are larger in dual-wavelength mode than in single-wavelength mode, with the [[LAMBDA].sub.12] synthetic-wavelength phase uncertainty IQR being an order of magnitude greater than the single- wavelength phase uncertainty.

For phase data collected from the [[lambda].sub.1]-[[lambda].sub.3] dual-wavelength mode, the phase uncertainty is on the order of 1.7 nm for the [[lambda].sub.1] data, 1.1 nm for the [[lambda].sub.3] data, and as high as 91 nm for the [[LAMBDA].sub.13] data (Table 9). The distribution characteristics are similar to the [[lambda].sub.1]-[[lambda].sub.2] data in that the [[lambda].sub.1]- [[lambda].sub.3] data are non-normal and skewed right, but they tend toward normality more so than the single-wavelength data in Sec. 3.1. For a similar dual-wavelength configuration as [[lambda].sub.1]-[[lambda].sub.3] in this study, Kuhn et al. [22] reported the temporal standard deviation of one pixel over a 15 s acquisition time (the number of holograms, though, was not reported) as 1.3 nm ([lambda] = 657 nm) and 1.0 nm ([lambda] = 680 nm), which agrees very well with the findings shown in Table 9. The [[LAMBDA].sub.13] synthetic-wavelength phase uncertainty IQR can be upwards of two orders of magnitude greater than the individual wavelength phase uncertainty IQR obtained in dual-wavelength mode.

For comparison of phase data from single- and dual-wavelength modes, Fig. 11 shows the distributions for [[lambda].sub.1] obtained from both single- and dual-wavelength modes, which clearly demonstrate that the uncertainty is greater in phase data from dual-wavelength mode compared to single-wavelength mode. Table 10 indicates that the uncertainty in the [[lambda].sub.1] phase is increased by a factor of 2.0 to 3.3 when dual- wavelength mode is used compared to single-wavelength mode. Quantile-quantile plots comparing the [[lambda].sub.1] phase uncertainty from single- and dual-wavelength modes (Fig. 12) show that the plots deviate, suggesting that they likely do not come from the same distribution. The two-sample Kolmogorov-Smirnov test using 1000 iterations indicated, with 95 % confidence, that the [[lambda].sub.1] phase uncertainty from the single-wavelength mode is not from the same distribution as either of the [[lambda].sub.1] phase uncertainty data sets collected in dual- wavelength mode.

3.2 Measurements in Flowing Water (with an Immersion Objective)

Using the 20x and 40x immersion lenses in distilled water, the uncertainty was determined over 100 reconstructed phase maps collected at 25 [s.sup.-1]. The uncertainty was evaluated at water flow rates of 0 mL [min.sup.-1], 15 mL [min.sup.-1], 33 mL [min.sup.-1], and 62 mL [min.sup.-1]. These flow rates were selected because 15 mL [min.sup.-1] was the slowest repeatable flow rate that the pump could output, and 62 mL [min.sup.-1] was the fastest flow rate that the reaction cell could handle with the immersion lens. The same area of the sample was examined as the flow rate was changed. For the 20x (Table 11 and Table 12) and 40x (Table 13 and Table 14) immersion lenses, the mean and median uncertainty increased with flow rate (Fig. 13), although the uncertainty was similar between flow rates of 15 mL [min.sup.-1] and 33 mL [min.sup.-1]. With static water (0 mL [min.sup.-1]), the uncertainty was around 0.4 nm at both 20x and 40x. At flow rates of 15 mL [min.sup.-1] and 33 mL [min.sup.-1], the uncertainty was less than 0.5 nm at both 20x and 40x, while at a flow rate of 62 mL [min.sup.-1], the uncertainty was at most 0.6 nm at 20x and 0.5 nm at 40x. Figure 14 demonstrates the shift in the uncertainty distribution as a function of flow rate, while Fig. 15 compares the 0 mL [min.sup.-1] and 62 mL [min.sup.-1] uncertainty distributions.

With regard to the normality of the data as evaluated by skewness and kurtosis, similar trends to Sec. 3.1 can be seen for the immersion lens data, namely, with regard to the distributions approaching normality with increasing magnification and decreasing sample size. With increasing flowing conditions, however, the skewness and kurtosis values both increase, indicating more positive skewness and a greater number of outliers (Table 15). Therefore, while the mean or median uncertainty value may not be greatly affected by slower flow rates (e.g., 15 mL [min.sup.-1] and 33 mL [min.sup.-1]), the distribution skews to a greater number of outliers as the flow rate increases, particularly at high flow rates (e.g., 62 mL [min.sup.-1]), suggesting greater potential for uncertainty in the phase data as flow rate increases. The IQR also increases as the flow rate increases (Table 15), with the IQR from 62 mL [min.sup.-1] potentially being twice the IQR value in static water. At 62 mL [min.sup.-1], the IQR is [less than or equal to] 0.2 nm, while the IQR is [less than or equal to] 0.15 nm for the 0 mL [min.sup.-1], 15 mL [min.sup.-1], and 33 mL [min.sup.-1] conditions. The magnitudes of the IQR values are comparable to the IQR for measurements in air from Sec. 3.1, although the IQR may be greater for the immersion lens data compared to measurements in air at a given number of holograms; for example, with 10 holograms at 20x magnification, the IQR is 0.15 nm for static conditions with the immersion lens, compared to 0.11 nm for measurements in air.

A quantile-quantile plot also demonstrates that the uncertainties at flow rates of 0 mL [min.sup.-1] and 62 mL [min.sup.-1] do not come from the same distribution (Fig. 16). Additionally, quantile-quantile plots comparing other flow rates suggest that the 0 mL [min.sup.-1] and 15 mL [min.sup.-1] uncertainties come from similar distributions (Fig. 17a), while the 15 mL [min.sup.-1] and 33 mL [min.sup.-1] uncertainties may not (Fig. 17b). A plot of the empirical cumulative distribution function (ECDF) suggests similarities in the distributions for 0 mL [min.sup.-1], 15 mL [min.sup.-1], and 33 mL [min.sup.-1] (Fig. 18). To verify this, 1000 iterations of a two-sample Kolmogorov-Smirnov test were performed by randomly sampling 100 data points from each flow rate data set; results suggest that the 0 mL [min.sup.-1], 15 mL [min.sup.-1], and 33 mL [min.sup.-1] data sets were from the same distribution, while the 62 mL [min.sup.-1] data set was not from the same distribution as any of the other flow rate data sets. (4)

In a comparison of the uncertainty from similar magnifications in air to the immersion lens data in water under static conditions (Table 16), the mean phase uncertainty (rad) is greater for the immersion lens data, but the mean height uncertainty (nm) is comparable (0.4 nm to 0.5 nm). This is because the phase-to-height conversion, shown previously as Eq. (1), includes the index of refraction of the medium, so in air, the height is directly proportional to the phase, while in water, the height is directly proportional to the phase divided by 1.33. The two-sample Kolmogorov-Smirnov test suggested that the phase uncertainties are not from the same distribution when comparing the 20x objective lens in air to the 20x immersion objective lens in water and the 40x objective lens in air to the 40x immersion objective lens in water (i.e., all phase data comparison results rejected the null hypothesis with 95 % confidence), although comparing the height uncertainty, the Kolmogorov-Smirnov test suggests that, at lower hologram sample sizes, the height uncertainty may be from similar continuous distributions, (5) evidence of which can also be noted in a quantile-quantile plot of the 40x data (Fig. 19).

3.3 Measurements in Flowing Water (through a Glass Window)

For measurements through a window, only objective lenses up to 20x could be used, because of the limitations of shorter free working distances for the higher magnification objectives (see Table 1). The window used was float glass with a thickness of about 1.0 mm, and it had an anti-reflective coating. Greater flow rates were possible in this configuration, since the water is forced through the outflow (with the immersion lens configuration, an "open-channel" condition permits the water to back up and spill over the top of the reaction cell at high flow rates). Water flow rates of 0 mL [min.sup.-1], 15 mL [min.sup.-1], 33 mL [min.sup.-1], 70 mL [min.sup.-1], and 109 mL [min.sup.-1] were evaluated. For the phase measurements through a glass window, it is evident that the uncertainty increases from static to flowing water conditions (Table 17, Table 18, Table 19, and Fig. 20). In static water conditions, the uncertainty is on the order of [less than or equal to] 0.7 nm at 5x and [less than or equal to] 0.5 nm at 10x and 20x magnification. At slow flow conditions (up to 15 mL [min.sup.-1]), the uncertainty is less than 1 nm for all magnifications, but at faster flow conditions (up to 109 mL [min.sup.-1]), the uncertainty is as high as 1.8 nm at 5x, 1.0 nm at 10x, and 2.4 nm at 20x magnification. The sudden increase in mean uncertainty at 20x from 70 mL [min.sup.-1] to 109 mL [min.sup.-1] is unexpected, considering that the mean uncertainty decreased at 5x and 10x with the same change in flow rate (Fig. 20); this is likely caused by some additional extrinsic factors (e.g., turbulence, vibration).

For this enclosed configuration, the uncertainty distributions for the 5x (Fig. 21) and 10 x (Fig. 22) objective lenses suggest that uncertainty increases from 0 mL [min.sup.-1] to 33 mL [min.sup.-1] and decreases from 33 mL [min.sup.-1] or 70 mL [min.sup.-1] to 109 mL [min.sup.-1]. This may perhaps be indicative of transitions from turbulent to more laminar flow conditions within the fluid cell. Turbulence in the water will affect interferometric measurements, with greater magnitudes of turbulence inducing more noise in the phase measurement [51]. This explains why there is more noise in the phase measurement when flowing conditions are present, and this suggests that, at least for the through-window configuration, the most turbulent conditions in the fluid cell occur at the 33 mL [min.sup.-1] and 70 mL [min.sup.-1] flow rates. As seen in the Table 20 comparison, there is more noise at a given flow rate for the through-window configuration, and the noise does not change substantially from 0 mL [min.sup.-1] to 33 mL [min.sup.-1] for the immersion lens configuration, which suggests that these turbulent conditions occur in the through-window configuration and are not as prevalent in the immersion lens configuration.

In a comparison of measurements in flowing conditions by an immersion lens to those through a glass window (Fig. 23 and Table 20), the glass window measurements yield greater uncertainty. In static conditions, the uncertainty is comparable: 0.4 nm for the immersion lens compared to 0.5 nm for the through-window measurements. With flowing conditions, the uncertainties are on the order of 0.5 nm and 0.8 nm at 15 mL [min.sup.-1] for the immersion lens and through-window measurements, respectively, and 0.5 nm and 1.1 nm at 33 mL [min.sup.-1] for the immersion lens and through-window measurements, respectively. A quantile-quantile plot comparing the immersion lens and through-window measurements suggests that the uncertainties are from dissimilar distributions (Fig. 24), which was confirmed by a two sample Kolmogorov-Smirnov test. (6)

One possibility in the flowing condition is that the flow could affect the spatial distribution of uncertainty per pixel. That is to say, if the "downstream" pixels represent a greater uncertainty than the "upstream" pixels, such as by some effect of water-induced vibration, turbulence of the water, etc. Ultimately, no evidence of such a flow direction bias was found. Plotting uncertainty values per pixel as a 3D surface (Fig. 25), it can be seen that, aside from a few scattered peaks, the spatial distribution of values is relatively uniform at 0 mL [min.sup.-1], 15 mL [min.sup.-1], and 109 mL [min.sup.-1], with a slightly greater spatial distribution for the data at 70 mL [min.sup.-1]. This agrees with Fig. 22, where the distribution of uncertainty is greater at 70 mL [min.sup.-1], and it further suggests that perhaps the flow is more turbulent at 70 mL [min.sup.-1] compared to 0 mL [min.sup.-1], 15 mL [min.sup.-1], and 109 mL [min.sup.-1].

4. Discussion

4.1 How Many Holograms are Needed?

Kuhn et al. [22] argued that temporal averaging over a 25-hologram sequence (with an acquisition rate of 25 [s.sup.-1]) can reduce the effects of shot noise in the DHM. However, the findings from this study suggest that the number of holograms required for a given measurement is dependent on the experimental configuration, the sensitivity of the value to be measured, and the user-defined allowable uncertainty tolerance in the measurement. For measurements in air (Sec. 3.1), the mean and median uncertainties were minimally affected by the number of holograms, where the values were around 0.4 nm to 0.5 nm at magnifications from 2.5x to 40x. The skewness and kurtosis increased while the IQR decreased with increasing sample size. The IQR was [less than or equal to] 0.1 nm for sample sizes of 10 holograms or more and [less than or equal to] 0.2 nm for 5 holograms, which suggests that a minimum of 10 holograms may be sufficient when a 25 [s.sup.-1] acquisition rate is used, depending on the allowable desired measurement tolerance. Similar behavior was noted at a lower acquisition rate of 12.5 [s.sup.-1], which suggests that 10 holograms may also be sufficient at that acquisition rate. With an immersion lens, the IQR is slightly greater compared to measurements in air, suggesting that sample sizes greater than 10 holograms may be beneficial, and similar arguments can be made for measurements through a glass window. Additionally, with flowing water conditions, it may be advisable to collect additional holograms to further reduce the effects of noise and uncertainty in the measurement. The effects of the number of holograms and allowable sensitivity are further discussed through hypothetical measurements in Sec. 5.

4.2 Sources of Noise

As discussed in Sec. 2, the primary sources of noise in the DHM are shot noise, which is related to the intrinsic variability in photons incident on the CCD camera, and Gaussian noise, which is the combined effects from numerous intrinsic and extrinsic sources [23-29]. Given that the findings in this study evaluated the global uncertainty in measurements, it is not possible to reliably state which source (or sources) of noise is the primary cause of the quantified uncertainty. However, for a given experimental configuration, certain noise sources may be more likely to be present, which will be the main discussion of this section.

4.2.1 Effect of Acquisition Rate

As was demonstrated in Sec. 3.1.1, there was not a significant effect of acquisition rate (25 [s.sup.-1] compared to 12.5 [s.sup.-1]) on the uncertainty, and results from the Kolmogorov-Smirnov test suggested that the uncertainties were from similar distributions. The effects of shot noise should be identical at these two acquisition rates, since the CCD camera settings remained unchanged. Therefore, any change in the uncertainty at different acquisition rates would be attributable to extrinsic factors, such as if there were some drifting or vibrations, but that was not evident in the experiments conducted in this study.

4.2.2 Dual-Wavelength Mode

The additive nature of noise in multiple-wavelength configurations relative to a single-wavelength configuration has been discussed in the literature [32,34,48-50], particularly with regard to how 2[pi] phase ambiguities can be removed. While the additive nature of noise in this configuration is known, additional potential sources of noise will be discussed in this section.

In dual-wavelength mode, there is the possibility of crosstalk between the signals for two wavelengths, which results in additional uncertainty. One experiment with a dual-wavelength DHM configuration (wavelengths of 632.8 nm and 532.8 nm) predicted a 5 % crosstalk [52]. (7) It is therefore possible that crosstalk is occurring in the [[lambda].sub.1]-[[lambda].sub.2] and [[lambda].sub.1]-[[lambda].sub.3] dual-wavelength modes, which could be one source of increased noise when comparing the single- to the dual-wavelength modes.

The CCD camera settings are different between the single- and dual-wavelength modes, and this affects the shot noise contribution. In dual-wavelength mode, the camera shutter speed is reduced relative to the single-wavelength mode to account for the additional photons from the second wavelength. Because of the longer exposure for the single-wavelength mode, it can be expected that the shot noise is greater. However, the longer exposure for single-wavelength mode also results in a higher signal-to-noise ratio (SNR), therefore reducing the relative effect of shot noise and yielding less uncertainty for [[lambda].sub.1] phase data collected from single-wavelength mode compared to the [[lambda].sub.1] phase data collected from dual-wavelength mode.

To further evaluate the effects of noise in the single- and dual-wavelength modes, a brief experiment was conducted. For the [[lambda].sub.1]-[[lambda].sub.2] dual-wavelength mode, using the 10x objective lens in air and an acquisition rate of 25 [s.sup.-1] to collect 100 holograms, an optimized hologram was generated, and the [[lambda].sub.1] phase data were extracted (Test D1 in Table 21). Then, using the same optimized camera settings, the [[lambda].sub.2] laser source was switched off, and 100 holograms were collected to extract the [[lambda].sub.1] phase data (Test D2). These two data sets from the dual-wavelength mode were then compared to two configurations for the [[lambda].sub.1] single-wavelength mode: one with an optimized hologram and optimized camera settings (Test S1), and another with the camera settings set to the dual-wavelength mode optimized hologram (Test S2). The same ROI was evaluated across all experiments. The results are summarized in Table 21, which demonstrates that once the possible crosstalk and fluctuations from [[lambda].sub.2] are removed, the uncertainty is reduced, and the uncertainty characteristics are similar to those obtained in the single-wavelength mode with reduced camera settings (i.e., compare Test D2 to Test S2). To compare the distributions, a quantile-quantile plot (Fig. 26) confirms, along with the two-sample Kolmogorov-Smirnov test, that Test D2 and Test S2 come from similar distributions. Of the 1000 iterations, 94 % of the Kolmogorov-Smirnov tests returned (with 95 % confidence) that Test D2 and Test S2 were from the same distribution. The results also demonstrate that the reduced shutter speed yields an increase in the uncertainty because of the reduction in SNR (e.g., compare Test S2 to Test S1). This confirms that that main causes of increased uncertainty in the dual-wavelength mode are attributable to: crosstalk and fluctuations from the second wavelength and the reduced shutter speed.

4.2.3 Flowing Solution

Relative to static conditions (0 mL [min.sup.-1]), the presence of flowing water in either the immersion lens or the through-window experimental configuration was shown to increase the uncertainty. This can primarily be attributed to the greater noise induced by turbulent flow conditions [51], which is therefore an extrinsic noise contribution that can only be minimized as a function of the flow rate magnitude.

The through-window configuration was found to yield greater uncertainties relative to the immersion lens configuration. This is likely attributable to changes in the SNR. In the hologram, the maximum SNR of a given pixel is a function of the object and reference wave intensities, [I.sup.O] and [I.sub.R], respectively, where the numerator represents the maximum signal that could be obtained, while the denominator represents the shot noise [23]:

[SNR.sub.max] = 2[square root of ([I.sub.O][I.sub.R])]/[square root of ([I.sub.O] + [I.sub.R])]. (6)

In the through-window configuration, some of the signal is lost due to reflections at the window, and multiple reflections inside the window can affect the reflected signal [53]. For perpendicularly incident light, the Fresnel reflection coefficient was previously defined in Eq. (4). Considering the index of refraction for various media, [[eta].sub.air] = 1.0, [[eta].sub.water] = 1.33, and [[eta].sub.glass] [approximately equal to] 1.5, the Fresnel reflection coefficient for incident light perpendicular to the surface is 4 % for the air-glass interface and 0.4 % for the glass-water interface. In the through-window configuration, the signal lost from reflection at interfaces occurs at the air-glass interface of the objective lens, air-glass interface of the window, and the water-glass interface of the window, thus diminishing [I.sub.O] and resulting in a potential decrease in the hologram SNR. There is less uncertainty in the immersion lens configuration, because signal lost from these reflections only occurs at the water-glass interface of the immersion lens, and therefore the immersion lens configuration has potentially greater SNR than the through-window configuration. However, it is theoretically possible that, for a constant [I.sub.R], a decrease in [I.sub.O] may yield an increase in the phase image SNR, at least when considering SNR relative to shot noise only [23]. Considering that the present study considered the global intrinsic and extrinsic noise effects (and not just shot noise), it is likely that this argument is still feasible considering SNR relative to all noise factors.

Another loss in signal can occur at the interface of the sample surface. As was shown in Sec. 3.2 (Table 16), the phase uncertainty in static water is greater than the phase uncertainty in static air at the same magnification. Consider measurements of the surface of the glass without a chromium film. The Fresnel reflection coefficient is lower in water (0.4 %) compared to in air (4 %), so there is less reflected signal when measurements are conducted in water. Because of the relationship in Eq. (4), measurements of a given sample in water will likely have greater phase uncertainty than measurements in air when considering the combined effects of all intrinsic and extrinsic factors.

4.2.4 Sample Surface

As a general comment, the sample type, condition, and preparation are all critical to the phase uncertainty. As already established, because of Eq. (4), measurements in any media where n > 1.0 will likely yield greater phase uncertainty than measurements in air as a result of the reduction in SNR.

Similarly, considering any measurement in any given medium, the index of refraction of the sample surface is also important. For example, measurements of a selenite (gypsum) surface (n [approximately equal to] 1.52) will contain more uncertainty than measurements of a calcite surface (n [approximately equal to] 1.66). The opaqueness of the sample as well as the intensity of incident light (e.g., single wavelength vs. dual wavelength) are also important factors when considering the experimental configuration and potential magnitude of uncertainty.

5. Effect of Measurement Uncertainty on Measured Dissolution Fluxes

Recent research at NIST is utilizing the DHM to quantify mineral dissolution fluxes by observing in situ changes in surface topography over time [13,14], similar to what is performed in geochemical studies by measurements with vertical scanning interferometry (e.g., [54-56]). Through these measurements, the surface-normal dissolution flux, [k.sub.s], is computed based on the surface-normal dissolution velocity, [v.sub.s], which is simply the change in height over time of a pixel or set of pixels, [DELTA]h/[DELTA]t, and the molar volume, [V.sub.m] [54]:

[k.sub.s] = [v.sub.s]/[V.sub.m] = [DELTA]h/[DELTA]t 1/[V.sub.m]. (7)

Though discussed for dissolution (i.e., when [v.sub.s] < 0 and [k.sub.s] < 0), since that was the primary focus of the recent studies [13, 14], Eq. (7) applies equally to precipitation or growth studies (i.e., when [v.sub.s] > 0 and [k.sub.s] > 0). Note that in these experiments, an inert reference mask was partially applied to the surface to serve as a reference height offset, so that relative measurements were determined. In experiments where only one reaction is occurring (i.e., only dissolution, precipitation, or growth), spatial averaging can be performed, which can provide details on the spatial variability of fluxes [13,14] and can also reduce the effects of noise [22,32]. Spatial averaging is conducted by averaging over an ROI and evaluating the mean change in height over time, so [v.sub.s] essentially becomes a mean height change over time rather than a height change per pixel over time.

To evaluate the effect of uncertainty, assume, for example, a hypothetical perfectly flat mineral surface with known values [V.sub.m] = 3.7 x [10.sup.-5] [m.sup.3] [mol.sup.-1] and [k.sub.s] = -0.1 [micro]mol [m.sup.-2] [s.sup.-1], which are approximate values for calcite [13]. Consider an experiment in which DHM is used with a 20x immersion lens and [[lambda].sub.1] single- wavelength mode with a flow rate of 15 mL [min.sup.-1]. Holograms are collected at an acquisition rate of 25 [s.sup.-1], and a set of 10 holograms is collected every minute for an experiment duration of 1 h. The phase map is 300 pixels by 300 pixels. The data for a perfectly dissolving surface with these values are shown in Fig. 27, which indicates a dissolution flux of exactly -0.1 [micro]mol [m.sup.-2] [s.sup.-1]. Using the uncertainty from Sec. 3.2 for this experimental configuration, for each pixel in the phase map at each time step, a randomly (8) selected value from the uncertainty distribution is added or subtracted at random from the given pixel value, yielding the hypothetically "measured" dissolution data in Fig. 27, which indicate a flux of (-0.100 [+ or -] 0.001) [micro]mol [m.sup.-2] [s.sup.-1]. The data in Fig. 27 are a representation of the spatially averaged mean height over the entire phase map per time step. Therefore, it can be concluded that, at least for calcite, the dissolution flux is large enough (i.e., [DELTA]h is large enough per [DELTA]t time step in this experiment) such that the effect of phase measurement uncertainty is relatively negligible (i.e., the difference is 0.4 %). (9)

If the computed calcite dissolution flux is relatively unaffected by DHM phase measurement uncertainty, what is the limiting dissolution flux such that the measured changes in height are indistinguishable from the uncertainty? Since minerals can have the same [k.sub.s] but different [V.sub.m], the product of these parameters (i.e., dissolution velocity, [v.sub.s]) will be the independent variable. Assuming the same experiment configuration as for calcite, the perfect linear dissolving mineral surface is compared to the "measured" height changes per pixel with added uncertainty, as shown in Table 22. Some of the "known" [v.sub.s] values are hypothetical, while others are derived from the literature. Table 22 shows that any mineral with a surface-normal dissolution velocity greater than about -[10.sup.-12] m [s.sup.-1] can be safely measured with the DHM (for the assumed experimental configuration) without significant detriment to the measurement from phase uncertainty. All "measured" [v.sub.s] values in Table 22 had a standard error of regression on the order of [10.sup.-13.3] m [s.sup.-1], which also appears to be the threshold where the known and "measured" [v.sub.s] values begin to differ by [greater than or equal to] 100 %. Also, at very low dissolution [v.sub.s], such as -[10.sup.-15] m [s.sup.-1], uncertainty in the measurement may even result in a measured precipitation or growth velocity because the uncertainty is greater than the actual dissolution rate.

Considering a different experimental configuration (and therefore different uncertainty), Table 23 demonstrates that a configuration with greater uncertainty (such as using a glass window and a faster flow rate) results in a larger difference between the known and "measured" [v.sub.s], as expected. While the dissolution of gypsum and calcite, as measured by the DHM, were still fast enough to not be affected by the uncertainty, the limit appears to be around -[10.sup.-11.8] m [s.sup.-1] for the lowest dissolution velocity measured by DHM in this configuration before uncertainty significantly affects the measurement. As with the immersion lens configuration, at low dissolution velocities (e.g., [less than or equal to] -[10.sup.-13.7] m [s.sup.-1]), the uncertainty is significant enough for a precipitation or growth to be measured rather than dissolution. As expected, since the uncertainty is greater for the configuration in Table 23, the error of regression is greater than the immersion lens configuration in Table 22, and the typical error of regression also appears to be related to the velocity at which the difference between known and measured values is [greater than or equal to] 100 %.

Based on the results in Table 22 and Table 23, the limiting dissolution velocity for a number of experimental configurations with water using an immersion lens and through-window measurements is summarized in Table 24. The limiting velocity was selected based on a percent difference threshold of 5 %. This limiting velocity is applicable to dissolution or precipitation measurements by accounting for a change in sign (e.g., dissolution would be a negative velocity). In general, the number of holograms did not greatly impact the limiting surface-normal velocity, so for each experiment configuration and flow rate in Table 24, a recommended limit is suggested based on the most conservative case. When using a 20x or 40x immersion lens, regardless of the flow rate, a surface-normal velocity limit of [10.sup.-11.7] m [s.sup.-1] can be assumed. When using a glass window, there is greater uncertainty, especially when flowing water is used, so a surface-normal velocity limit of [10.sup.-11.7] m [s.sup.-1] can be assumed with static water (0 mL [min.sup.-1]), a surface-normal velocity limit of [10.sup.-11.4] m [s.sup.-1] can be assumed with slow flow (15 mL [min.sup.-1]), and a surface-normal velocity limit of [10.sup.-11.0] m [s.sup.-1] can be assumed with fast flow (up to 109 mL [min.sup.-1]). Based on these recommended limits, it is evident that the dissolution fluxes measured for calcite [13] and gypsum [14] are valid. These findings also demonstrate that, in general, the extrinsic noise factors with a flowing solution (e.g., turbulence, vibration), as a direct result of increasing the uncertainty, reduce the reliable confidence with which dissolution fluxes can be measured (i.e., a velocity as low as [10.sup.-11.7] m [s.sup.-1] could be reliably measured in static water through a glass window, but such a low velocity cannot be reliably measured at higher flow rates, since the velocity limits are [10.sup.-11.4] m [s.sup.-1] and [10.sup.-11.0] m [s.sup.-1] at flow rates of 15 mL [min.sup.-1] and 109 mL [min.sup.-1], respectively).

The discussion of these simulated data has so far been concerned with temporally and spatially averaged data. The temporal averaging was the result of the assumed averaging of 10 holograms, and the spatial averaging occurred as the mean surface height per time step (e.g., Fig. 27). However, these analyses are only applicable to experiments with only one reaction occurring (e.g., only dissolution, precipitation, or growth). Some mineralogical systems react through a coupled dissolution-precipitation reaction [59], which means that the temporal phase data from the DHM would indicate that certain pixels or local groups of pixels would have a negative [v.sub.s] (i.e., dissolution), while others would have a positive [v.sub.s] (i.e., precipitation). For an example, consider an experiment on a mineral surface experiencing coupled precipitation (known [v.sub.s,precipitation] = [10.sup.-11.9]) and dissolution (known [v.sub.s,dissolution] is 1.5 times the precipitation velocity), where a 20x immersion lens is used with a 15 mL [min.sup.-1] flow rate, and 10 holograms are collected every minute. Applying the uncertainty as before, the "measured" [v.sub.s] values are determined on a per pixel basis (i.e., for every pixel, the slope of height over time is determined), which actually yields a distribution of values, as shown in Fig. 28. The mean surface-normal precipitation and dissolution velocities are ([10.sup.-11.91] [+ or -] [10.sup.-14.64]) m [s.sup.-1] and (-[10.sup.-10.73] [+ or -] [10.sup.-14.46]) m [s.sup.-1], respectively, both of which are 2.3 % different from the known velocity. Therefore, the mean values are comparable (10) between the simulated minerals experiencing pure dissolution, precipitation, or growth and those experiencing coupled dissolution-precipitation, so the surface-normal velocity limits in Table 24 are valid for coupled dissolution-precipitation experiments as well.

6. Summary and Conclusions

Type A uncertainty was quantified in this study for measurements conducted with a digital holographic microscope considering various experimental configurations, including objective lens magnification, objective lens type (air objective and immersion objective), measurement medium (air and water), and flowing water conditions. The findings suggest that the uncertainty has a non-normal distribution of values, with mean values of [less than or equal to] 0.5 nm up to 40x magnification for measurements in air and in static water (immersion lens). With increasing water flow rates, the mean uncertainty was found to be [less than or equal to] 0.6 nm up to 40x magnification with an immersion lens. For measurements conducted through a glass window up to 20x magnification, the mean uncertainty was [less than or equal to] 0.7 nm in static water conditions, [less than or equal to] 1.0 nm in slow-flowing water, and [less than or equal to] 2.4 nm in fast-flowing water. The acquisition rate did not significantly impact the uncertainty when varied from 25 [s.sup.-1] to 12.5 [s.sup.-1]. Collecting holograms in single-wavelength versus dual-wavelength modes did impact the uncertainty, where mean uncertainty at 10x magnification was [less than or equal to] 0.5 nm from the single-wavelength mode compared to [less than or equal to] 1.5 nm from the dual-wavelength mode.

Based on the uncertainties quantified in this study, limitations are posited for allowable average changes in surface topography in a given time step that can be confidently measured. For example, for a hypothetical dissolving (or growing) mineral surface examined with an immersion lens in flowing water conditions, surface topography changes need to be [greater than or equal to] [10.sup.-11.7] m [s.sup.-1] in order to be confident that the measured height changes are not significantly (i.e., differ by more than 5 %) affected by the inherent uncertainty. Similarly, in a configuration with a glass window, the surface topography changes need to be [greater than or equal to] [10.sup.-11.7] m [s.sup.-1], [greater than or equal to] [10.sup.-11.4] m [s.sup.-1], and [greater than or equal to] [10.sup.-11.0] m [s.sup.-1] for conditions with static (0 mL [min.sup.-1]), slow ([less than or equal to] 15 mL [min.sup.-1]), and fast ([less than or equal to] 109 mL [min.sup.-1]) water flow conditions, respectively.

The uncertainty characterized in this study can be used to further develop experimental designs and methodologies. For example, based on the expected dissolution rate of a given mineral, the experimental conditions of the DHM can be modified such that the effects of uncertainty in the measurement are minimized. In addition, these uncertainty data are needed in order to have confidence in the phase output provided by the DHM and in the data produced (such as computed dissolution fluxes) in the postprocessing analyses.

Acknowledgments

The author would like to acknowledge Adam Pintar (NIST) for his helpful discussion of the applicability of various statistical tests, Jeffrey W. Bullard (NIST) and Pan Feng (Southeast University) for their helpful discussions, Tristan Colomb (Lyncee Tec) and additional anonymous reviewers for their helpful comments and recommendations, and Jay Brandenburg (NIST) for fabrication of the reaction cell. The author would like to acknowledge the National Research Council for support through the NRC Postdoctoral Research Associateship Program at NIST.

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About the author: Alexander S. Brand, Ph.D., is in the Materials and Structural Systems Division of the Engineering Laboratory at NIST, where he started in January 2016 as a National Research Council Postdoctoral Research Associate. He received his Ph.D. in civil engineering from the University of Illinois at Urbana-Champaign in December 2015. His research interests focus on the materials science, microstructure development, and reaction kinetics of cementitious materials

The National Institute of Standards and Technology is an agency of the U.S. Department of Commerce

Alexander S. Brand

National Institute of Standards and Technology Gaithersburg, MD 20899, USA

alexander.brand@nist.gov

Accepted: March 20, 2017

Published: March 27, 2017

https://doi.org/10.6028/jres.122.022

(1) Several studies have used DHM or related quantitative phase imaging techniques in a transmission configuration to study biological specimens under flowing conditions [60-64]. While one study assigned uncertainty to their measured and computed values [60], it is unclear how the effects of flow conditions, such as flow rate, influenced this uncertainty. Other studies demonstrated how flow rate influences image quality when particles (e.g., cells, colloids) are in solution [61,62].

(2) Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.

(3) Though not investigated in the present study, it is worth noting that techniques have been developed [34,48] to reduce phase ambiguities by combining the synthetic-wavelength and single-wavelength data, which may reduce the effects of noise and uncertainty.

(4) Of the 1000 iterations conducted, 82 % of the Kolmogorov-Smirnov tests returned (with 95 % confidence) that the 0 mL [min.sup.-1] and 15 mL [min.sup.-1] data sets were from the same distribution. Similarly, 84 %, 94 %, 0 %, and 0 % of the tests returned that the 0 mL [min.sup.-1] and 33 mL [min.sup.-1] data sets, 15 mL [min.sup.-1] and 33 mL [min.sup.-1] data sets, 0 mL [min.sup.-1] and 62 mL [min.sup.-1] data sets, and 33 mL [min.sup.-1] and 62 mL [min.sup.-1] data sets were from the same distribution, respectively.

(5) Of the 1000 iterations conducted, 73 % and 65 % of the Kolmogorov-Smirnov tests returned (with 95 % confidence) that the height data sets for the 20x objective lens in air and 20x immersion lens in water were from the same distribution for the 10- and 5-hologram sample sizes, respectively, and 80 %, 79 %, 94 %, and 87 % of the Kolmogorov-Smirnov tests returned (with 95 % confidence) that the height data sets for the 40* objective lens in air and 40* immersion lens in water were from the same distribution for the 30-, 20-, 10-, and 5-hologram sample sizes, respectively.

Caption: 6 Over 1000 iterations, comparing the 20x immersion lens to the 20x objective lens through a window, 50 % of the Kolmogorov-Smirnov tests returned (with 95 % confidence) that the static flow (0 mL [min.sup.-1]) datasets were from the same distribution when data from 20 holograms were considered. Similarly, when data from 10 and 5 holograms were used, 88 % and 67 % of the Kolmogorov-Smirnov tests returned (with 95 % confidence), respectively, that the static flow (0 mL [min.sup.-1]) datasets were from the same distribution. This suggests that, for the 10-hologram dataset, the uncertainties under static flow conditions may be from the same distribution. For the same comparison under flow conditions (15 mL [min.sup.-1] and 33 mL [min.sup.-1]), the Kolmogorov- Smirnov test results suggest that the datasets are not from the same distribution, regardless of the number of holograms considered.

(7) Note that the DHM configuration and sensor in Ref. [52] is different than that of the present study. Therefore, the 5 % crosstalk estimated by Ref. [52] is not representative of the potential crosstalk in this study and is included here only as an illustrative discussion point.

(8) By "random selection," a uniform sampling was used. Values from the uncertainty data set in Sec. 3.2 were sampled uniformly at random. The decision to add or subtract the uncertainty value to or from the given pixel height value was also determined by a uniform random distribution.

(9) This is the result of one simulated experiment. In 10 replicate simulated experiments, the mean percent difference was found to be 1.4 %, with values ranging from 0.3 % to 3.4 % difference.

(10) It can also be noted that the standard error is lower for these data compared to Table 22, which is expected given the larger number of samples examined in this simulation.

Caption: Fig. 1. DHM observations of microlenses, indicating (a) a zoomed-in view of the collected hologram showing the fringe interference pattern and the entire reconstructed (b) amplitude and (c) phase maps. The (b,c) images measure 443 [micro]m by 443 [micro]m. The output phase data are "wrapped" (d) and confined to the interval [-[pi], [pi]], which can be numerically unwrapped and converted to height (e).

Caption: Fig. 2. Lyncee Tec R2200 Series reflection DHM, indicating the object (O) and reference (R) beams [38]. Reproduced with permission from Lyncee Tec SA.

Caption: Fig. 3. Fluid reaction cell constructed for the DHM experiments, indicating the configurations for (a) immersion lens use and (b) glass window use, as well as (c) a cross-section schematic. Note: The stage is lowered, and no water is present in (a,b) to more clearly show the sample and cell setup.

Caption: Fig. 4. The specified reference height offset (red box) and ROI (blue box) of the phase image for the 20x objective lens in air analyses. The image measures 223 [micro]m by 223 [micro]m.

Caption: Fig. 5. Uncertainty over 100 reconstructed phase maps for a glass sample in air with a hologram acquisition rate of 25 [s.sup.-1].

Caption: Fig. 6. Histograms of the standard deviation per pixel over 5, 10, 20, 30, 50, and 100 reconstructed phase maps for a glass sample in air with the 20x objective lens and a hologram acquisition rate of 25 [s.sup.-1].

Caption: Fig. 7. Quantile-quantile plot of the uncertainty over 30 reconstructed phase maps for a glass sample in air with the 20x objective lens compared to quantiles for a normal distribution. A reference line (linear fit of the second and third quantiles) is shown in red. The plot indicates that the body of the distribution is normal, and the tails (particularly the right tail) are non-normal, as they deviate from the red line.

Caption: Fig. 8. Uncertainty over 10, 30, and 100 reconstructed phase maps for a glass sample in air with the 10x objective lens comparing the effect of hologram acquisition rates at 25 [s.sup.-1] and 12.5 [s.sup.-1].

Caption: Fig. 9. Quantile-quantile plots of the phase uncertainty (blue data) over (a) 100 and (b) 30 reconstructed phase maps for a glass sample in air with the 10x objective lens comparing the hologram acquisition rates at 25 [s.sup.-1] and 12.5 [s.sup.-1]. A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 10. Uncertainty over 100 reconstructed phase maps for a glass sample in air with the 10x objective lens comparing the data from [[lambda].sub.1]-[[lambda].sub.2] dual-wavelength mode with the phase data obtained from the individual [[lambda].sub.1] and [[lambda].sub.2] wavelengths and the synthetic [[LAMBDA].sub.12] wavelength.

Caption: Fig. 11. Uncertainty over 30 reconstructed phase maps for a glass sample in air with the 10x objective lens comparing phase data for [[lambda].sub.1] obtained from single-wavelength mode, [[lambda].sub.1]-[[lambda].sub.2] dual-wavelength mode, and [[lambda].sub.1]-[[lambda].sub.3] dual-wavelength mode.

Caption: Fig. 12. Quantile-quantile plot of the phase uncertainty over 30 reconstructed phase maps for a glass sample in air with the 10x objective lens comparing (a) phase data for X 1 obtained from single-wavelength mode to [[lambda].sub.1] obtained from [[lambda].sub.1]-[[lambda].sub.2] dual-wavelength mode and (b) phase data for [[lambda].sub.1] obtained from single-wavelength mode to [[lambda].sub.1] obtained from [[lambda].sub.1]-[[lambda].sub.3] dual-wavelength mode. A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 13. Effect of flow rate on the mean and median uncertainty over 50 reconstructed phase maps for the 20x and 40x immersion lenses in water.

Cpaption: Fig. 14. Uncertainty over 100 reconstructed phase maps for a glass sample in water examined by a 40x immersion lens at various water flow rates and hologram acquisition rate at 25 [s.sup.-1].

Caption: Fig. 15. Uncertainty over 30 reconstructed phase maps for a glass sample in water examined by 20x and 40x immersion lenses at two water flow rates and hologram acquisition rate at 25 [s.sup.-1].

Caption: Fig. 16. Quantile-quantile plot of the phase uncertainty over 100 reconstructed phase maps for a glass sample in water examined by a 40x immersion lens, comparing uncertainty at flow rates of 0 mL [min.sup.-1] and 62 mL [min.sup.-1]. Hologram acquisition rate at 25 [s.sup.-1]. A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 17. Quantile-quantile plot of the phase uncertainty over 50 reconstructed phase maps for a glass sample in water examined by a 40x immersion lens, comparing uncertainty at flow rates of (a) 0 mL [min.sup.-1] and 15 mL [min.sup.-1] and (b) 15 mL [min.sup.-1] and 33 mL [min.sup.-1]. A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 18. ECDF plot of the phase uncertainty over 50 reconstructed phase maps for a glass sample in water examined by a 40x immersion lens for each flow rate.

Caption: Fig. 19. Quantile-quantile plot of the phase uncertainty over (a) 100 and (b) 10 reconstructed phase maps for a glass sample examined by a 40x lens, comparing measurements in air and water. A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 20. Effect of water flow rate on the median uncertainty over 50 reconstructed phase maps for the 5x, 10x, and 20x objective lenses through a glass window.

Caption: Fig. 21. Uncertainty over 5 reconstructed phase maps for a glass sample in water at various water flow rates examined by 5x objective lens through a glass window.

Caption: Fig. 22. Uncertainty over 20 reconstructed phase maps for a glass sample in water at various water flow rates examined by 10x objective lens through a glass window.

Caption: Fig. 23. Uncertainty over 100 reconstructed phase maps for a glass sample in water examined by 20x lenses comparing immersion lens data (solid lines) to an air objective lens through a glass window (dashed lines) at flow rates of 0 mL [min.sup.- 1], 15 mL [min.sup.-1], and 33 mL [min.sup.-1].

Caption: Fig. 24. Quantile-quantile plot of the phase uncertainty from 100 reconstructed phase maps examined by a 20x objective lens, comparing the immersion lens and through-window measurements for the (a) static (0 mL [min.sup.-1]) and (b) 33 mL [min.sup.-1] flow conditions. A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 25. Phase uncertainty from 5 holograms plotted as a function of spatial distribution for measurements through a glass window with 10x magnification and a water flow rates of 0 mL [min.sup.-1], 15 mL [min.sup.-1], 70 mL [min.sup.-1], and 109 mL [min.sup.-1]. The scale on the color bar represents height uncertainty (nm).

Caption: Fig. 26. A quantile-quantile plot comparing the uncertainty results from a dual-wavelength configuration with the second wavelength turned off (Test D2) to the uncertainty results from a single-wavelength configuration with the same shutter speed as the dual-wavelength configuration (Test S2). A reference line (linear fit of the second and third quantiles) is shown in red.

Caption: Fig. 27. Surface-normal dissolution velocity data for a perfectly linear dissolving surface and considering the uncertainty in the phase measurement. Linear regression through the data suggests surface-normal dissolution velocities of exactly -0.2220 nm [min.sup.-1] (dissolution flux of exactly -0.1 [micro]mol [m.sup.-2] [s.sup.-1]) for perfect dissolution and (-0.2230 [+ or -] 0.0032) nm [min.sup.-1], which is a dissolution flux of (-0.1004 [+ or -] 0.0014) [micro]mol [m.sup.-2] [s.sup.-1], when uncertainty is added.

Caption: Fig. 28. Simulated surface-normal velocity results for a mineral experiencing a coupled dissolution- precipitation reaction. Note: all values are nonzero, despite the histogram bins overlapping zero.
Table 1. DHM objective lens details.

Mag.   Description            Lens Type   Numerical    Pixel
                                          Aperture     Size

2.5x   Leica N Plan           Air         0.07         2.84
       2.5x/0.07                                       [micro]m
5x     Leica N Plan EPI       Air         0.12         1.39
       5x/0.12                                         [micro]m
10x    Leica HC PL Fluotar    Air         0.30         683 nm
       10x/0.30
20x    Leica HC PL Fluotar    Air         0.40         344 nm
       20x/0.40 Corr
20x    Leica HCX APO          Immersion   0.50         343 nm
       L20x/0.50 W U-V-I
40x    Leica HC PL Fluotar    Air         0.60         171 nm
       40x/0.60 Corr
40x    Leica HCX APO          Immersion   0.80         170 nm
       L40x/0.80 W U-V-I
100x   Leica HC PL Fluotar    Air         0.90         68.9 nm
       100x/0.90

Mag.   Free Working       Depth of Field ([micro]m) (a)
       Distance (mm)

                       [[lambda].sub.1]   [[lambda].sub.2]

2.5x   11.2                  140                 160

5x     14.0                   46                 55

10x    11.0                  7.4                 8.8

20x    6.9                   4.2                 5.0

20x    3.5                   3.5                 4.2

40x    3.3                   1.8                 2.2

40x    3.3                   1.4                 1.6

100x   1.0                   0.8                 1.0

Mag.   Depth of Field ([micro]m) (a)

         [[lambda].sub.3]

2.5x         140

5x            47

10x          7.6

20x          4.3

20x          3.6

40x          1.9

40x          1.4

100x         0.8

(a) Varies as a function of wavelength: [[lambda].sub.1] = 665.5651
nm, [[lambda].sub.2] = 793.2365 nm, [[lambda].sub.3] = 681.0068 nm

Table 2. Mean uncertainty in the phase measurement per objective lens
in air.

                                            Objective Lens
                  Uncertainty
                                 2.5x       5x      10x      20x

Collected from    rad           0.0089    0.0080   0.0086   0.0086
100 Holograms     nm             0.47      0.42     0.46     0.46
Collected from    rad           0.0089    0.0080   0.0085   0.0087
50 Holograms      nm             0.47      0.42     0.45     0.46
Collected from    rad           0.0089    0.0080   0.0085   0.0086
30 Holograms      nm             0.47      0.42     0.45     0.46
Collected from    rad           0.0087    0.0079   0.0087   0.0084
20 Holograms      nm             0.46      0.42     0.46     0.44
Collected from    rad           0.0089    0.0078   0.0086   0.0082
10 Holograms      nm             0.47      0.41     0.46     0.43
Collected from    rad           0.0083    0.0077   0.0078   0.0084
5 Holograms       nm             0.44      0.41     0.41     0.44

                  Objective Lens

                   40x      100x

Collected from    0.0093   0.0049
100 Holograms      0.49     0.26
Collected from    0.0088   0.005
50 Holograms       0.47     0.26
Collected from    0.0083   0.0049
30 Holograms       0.44     0.26
Collected from    0.0080   0.005
20 Holograms       0.42     0.26
Collected from    0.0075   0.0047
10 Holograms       0.40     0.25
Collected from    0.0075   0.0048
5 Holograms        0.40     0.25

Table 3. Median uncertainty in the phase measurement per objective lens
in air.

                                             Objective Lens
                 Uncertainty
                                2.5x       5x      10x      20x

Collected from   rad           0.0086    0.0079   0.0085   0.0085
100 Holograms    nm             0.46      0.42     0.45     0.45
Collected from   rad           0.0086    0.0079   0.0084   0.0086
50 Holograms     nm             0.46      0.42     0.44     0.46
Collected from   rad           0.0086    0.0079   0.0083   0.0085
30 Holograms     nm             0.46      0.42     0.44     0.45
Collected from   rad           0.0085    0.0078   0.0085   0.0083
20 Holograms     nm             0.45      0.41     0.45     0.44
Collected from   rad           0.0087    0.0076   0.0084   0.0081
10 Holograms     nm             0.46      0.40     0.44     0.43
Collected from   rad           0.0080    0.0074   0.0076   0.0081
5 Holograms      nm             0.42      0.39     0.40     0.43

                  Objective Lens

                  40x      100x

Collected from   0.0090   0.0049
100 Holograms     0.48     0.26
Collected from   0.0086   0.0049
50 Holograms      0.46     0.26
Collected from   0.0081   0.0048
30 Holograms      0.43     0.25
Collected from   0.0079   0.0049
20 Holograms      0.42     0.26
Collected from   0.0074   0.0046
10 Holograms      0.39     0.24
Collected from   0.0073   0.0047
5 Holograms       0.39     0.25

Table 4. Uncertainty IQR in the phase measurement per objective lens
in air.

                                            Objective Lens
                 Uncertainty
                                2.5x      5x      10x      20x

Collected from   rad           0.0017   0.0012   0.0014   0.0011
100 Holograms    nm             0.07     0.05     0.06     0.04
Collected from   rad           0.0020   0.0015   0.0015   0.0014
50 Holograms     nm             0.08     0.06     0.06     0.06
Collected from   rad           0.0023   0.0017   0.0018   0.0017
30 Holograms     nm             0.09     0.07     0.07     0.07
Collected from   rad           0.0023   0.0020   0.0022   0.0019
20 Holograms     nm             0.09     0.08     0.09     0.08
Collected from   rad           0.0033   0.0027   0.0030   0.0027
10 Holograms     nm             0.13     0.11     0.12     0.11
Collected from   rad           0.0043   0.0039   0.0039   0.0042
5 Holograms      nm             0.17     0.16     0.16     0.17

                   Objective Lens

                   40x      100x

Collected from   0.0020   0.0007
100 Holograms     0.08     0.03
Collected from   0.0019   0.0009
50 Holograms      0.08     0.04
Collected from   0.0019   0.0010
30 Holograms      0.08     0.04
Collected from   0.0020   0.0012
20 Holograms      0.08     0.05
Collected from   0.0026   0.0016
10 Holograms      0.10     0.06
Collected from   0.0038   0.0025
5 Holograms       0.15     0.10

Table 5. Skewness and kurtosis of data in Fig. 5.

Objective Lens   2.5x    5x    10x    20x    40x   100x

Skewness         2.1    2.3    3.6    2.4    1.0   1.0
Kurtosis         17.2   30.1   59.3   35.8   4.5   5.2

Table 6. Skewness and kurtosis of data in Fig. 6.

No. of Holograms    5    10    20    30     50    100

Skewness           0.5   0.3   0.4   0.8   1.7    2.4
Kurtosis           3.3   3.2   4.9   7.6   21.7   35.8

Table 7. Effect of acquisition rate on the uncertainty characteristics.

Acquisition       Data Value                 No. of Holograms
Rate
                                   5        10       20       30

                  Mean (rad)     0.0078   0.0086   0.0087   0.0085
                  Mean (nm)       0.41     0.46     0.46     0.45
                  Median (rad)   0.0076   0.0084   0.0085   0.0083
25 [s.sup.-1]     Median (nm)     0.40     0.44     0.45     0.44
                  IQR (rad)      0.0039   0.0030   0.0022   0.0018
                  IQR (nm)        0.16     0.12     0.09     0.07
                  Skewness        0.5      1.2      2.1      1.4
                  Kurtosis        3.1      11.5     27.0     16.0

                  Mean (rad)     0.0083   0.0083   0.0085   0.0087
                  Mean (nm)       0.44     0.44     0.45     0.46
                  Median (rad)   0.0080   0.0081   0.0084   0.0086
12.5 [s.sup.-1]   Median (nm)     0.42     0.43     0.44     0.46
                  IQR (rad)      0.0042   0.0028   0.0022   0.0019
                  IQR (nm)        0.17     0.11     0.09     0.08
                  Skewness        0.8      0.6      1.3      2.5
                  Kurtosis        5.5      4.8      15.6     38.0

Acquisition       Data Value
Rate
                                    50       100

                  Mean (rad)      0.0085   0.0086
                  Mean (nm)        0.45     0.46
                  Median (rad)    0.0084   0.0085
25 [s.sup.-1]     Median (nm)      0.44     0.45
                  IQR (rad)       0.0015   0.0014
                  IQR (nm)         0.06     0.06
                  Skewness         1.7       3.6
                  Kurtosis         20.6     59.3

                  Mean (rad)      0.0087   0.0088
                  Mean (nm)        0.46     0.47
                  Median (rad)    0.0086   0.0086
12.5 [s.sup.-1]   Median (nm)      0.46     0.46
                  IQR (rad)       0.0016   0.0014
                  IQR (nm)         0.06     0.06
                  Skewness         3.5       4.8
                  Kurtosis         58.0     91.3

Table 8. Uncertainty characteristics for phase data from
[[lambda].sub.1]-[[lambda].sub.2] dual-wavelength mode.

Wavelength                                   No. of Holograms
                    Data Value
                                      5        10       20       30

                    Mean (rad)     0.0162    0.0169   0.0169   0.0171
                    Mean (nm)       0.86      0.90     0.90     0.91
                    Median (rad)   0.0156    0.0165   0.0166   0.0168
[[lambda].sub.1]    Median (nm)     0.83      0.87     0.88     0.89
                    IQR (rad)      0.0083    0.0060   0.0044   0.0040
                    IQR (nm)        0.44      0.32     0.23     0.21
                    Skewness         0.6      0.7      0.7      1.0
                    Kurtosis         3.7      5.8      6.0      10.1

                    Mean (rad)     0.0188    0.0197   0.0197   0.0199
                    Mean (nm)       1.19      1.24     1.24     1.26
                    Median (rad)   0.0182    0.0194   0.0195   0.0197
[[lambda].sub.2]    Median (nm)     1.15      1.22     1.23     1.24
                    IQR (rad)      0.0095    0.0067   0.0049   0.0042
                    IQR (nm)        0.60      0.42     0.31     0.27
                    Skewness         0.5      0.4      0.4      0.4
                    Kurtosis         3.3      3.3      3.3      3.5

                    Mean (rad)     0.0247    0.0259   0.0259   0.0262
                    Mean (nm)       8.13      8.52     8.52     8.62
                    Median (rad)   0.0240    0.0255   0.0256   0.0260
[[LAMBDA].sub.12]   Median (nm)     7.90      8.39     8.42     8.56
                    IQR (rad)      0.0125    0.0087   0.0062   0.0054
                    IQR (nm)         4.1      2.9      2.0      1.8
                    Skewness         0.5      0.4      0.4      0.5
                    Kurtosis         3.2      3.5      3.5      4.3

Wavelength
                    Data Value     No. of Holograms

                                     50       100

                    Mean (rad)      0.0172   0.0171
                    Mean (nm)        0.91     0.91
                    Median (rad)    0.0169   0.0169
[[lambda].sub.1]    Median (nm)      0.90     0.90
                    IQR (rad)       0.0035   0.0031
                    IQR (nm)         0.19     0.16
                    Skewness         1.4       1.5
                    Kurtosis         15.3     19.0

                    Mean (rad)      0.0200   0.0200
                    Mean (nm)        1.26     1.26
                    Median (rad)    0.0198   0.0198
[[lambda].sub.2]    Median (nm)      1.25     1.25
                    IQR (rad)       0.0036   0.0030
                    IQR (nm)         0.23     0.19
                    Skewness         0.6       0.6
                    Kurtosis         3.9       4.3

                    Mean (rad)      0.0264   0.0263
                    Mean (nm)        8.69     8.65
                    Median (rad)    0.0261   0.0261
[[LAMBDA].sub.12]   Median (nm)      8.59     8.59
                    IQR (rad)       0.0046   0.0037
                    IQR (nm)         1.5       1.2
                    Skewness         0.6       0.8
                    Kurtosis         5.4       6.7

Table 9. Uncertainty characteristics for phase data from
[[lambda].sub.1]-[[lambda].sub.3] dual-wavelength mode.

Wavelength                                    No. of Holograms
                   Data Value
                                     5        10       20       30

                   Mean (rad)     0.0227    0.0232   0.0257   0.0284
                   Mean (nm)       1.20      1.23     1.36     1.50
                   Median (rad)   0.0219    0.0227   0.0249   0.0271
[[lambda].sub.1]   Median (nm)     1.16      1.20     1.32     1.44
                   IQR (rad)      0.0117    0.0082   0.0075   0.0083
                   IQR (nm)        0.62      0.43     0.40     0.44
                   Skewness         0.6      0.5      1.1      1.6
                   Kurtosis         3.4      3.4      6.8      9.8

                   Mean (rad)     0.0187    0.0190   0.0199   0.0201
                   Mean (nm)       1.01      1.03     1.08     1.09
                   Median (rad)   0.0181    0.0187   0.0196   0.0199
[[lambda].sub.3]   Median (nm)     0.98      1.01     1.06     1.08
                   IQR (rad)      0.0093    0.0065   0.0050   0.0043
                   IQR (nm)        0.50      0.35     0.27     0.23
                   Skewness         0.5      0.4      0.4      0.5
                   Kurtosis         3.4      3.2      3.5      3.6

                   Mean (rad)     0.0294    0.0299   0.0325   0.0349
                   Mean (nm)       68.7      69.8     75.9     81.5
                   Median (rad)   0.0285    0.0294   0.0318   0.0339
[[LAMBDA].sub.13]  Median (nm)     66.6      68.7     74.3     79.2
                   IQR (rad)      0.0149    0.0103   0.0087   0.0089
                   IQR (nm)        34.8      24.1     20.3     20.8
                   Skewness         0.5      0.4      0.7      1.2
                   Kurtosis         3.3      3.3      4.4      7.8

Wavelength                         No. of Holograms
                    Data Value
                                     50       100

                    Mean (rad)     0.0316   0.0329
                    Mean (nm)       1.67     1.74
                    Median (rad)   0.0297   0.0309
[[lambda].sub.1]    Median (nm)     1.57     1.64
                    IQR (rad)      0.0103   0.0112
                    IQR (nm)        0.55     0.59
                    Skewness        1.7       1.6
                    Kurtosis        9.2       8.4

                    Mean (rad)     0.0203   0.0202
                    Mean (nm)       1.10     1.09
                    Median (rad)   0.0200   0.0200
[[lambda].sub.3]    Median (nm)     1.08     1.08
                    IQR (rad)      0.0037   0.0031
                    IQR (nm)        0.20     0.17
                    Skewness        0.5       0.6
                    Kurtosis        3.7       3.9

                    Mean (rad)     0.0376   0.0388
                    Mean (nm)       87.8     90.6
                    Median (rad)   0.0361   0.0370
[[LAMBDA].sub.13]   Median (nm)     84.3     86.4
                    IQR (rad)      0.0098   0.0102
                    IQR (nm)        22.9     23.8
                    Skewness        1.6       1.6
                    Kurtosis        9.8       9.0

Table 10. Uncertainty characteristics for phase data in Fig. 11.

              [[lambda].sub.1]      [[lambda].sub.1]
               (Single Mode)     (Dual [[lambda].sub.1]-
                                 [[lambda].sub.2] Mode)

Mean (rad)         0.0085                0.0171
Mean (nm)           0.45                  0.91
Median (rad)       0.0083                0.0168
Median (nm)         0.44                  0.89
IQR (rad)          0.0018                0.0040
IQR (nm)            0.07                  0.21
Skewness            1.4                    1.0
Kurtosis            16.0                  10.1

                  [[lambda].sub.1]
               (Dual [[lambda].sub.1]-
                [[lambda].sub.3] Mode)

Mean (rad)            0.0284
Mean (nm)              1.50
Median (rad)          0.0271
Median (nm)            1.44
IQR (rad)             0.0083
IQR (nm)               0.44
Skewness                1.6
Kurtosis                9.8

Table 11. Mean uncertainty in the phase measurement with the 20x
immersion lens in water.

                                       Water Flow Rate (mL
                 Uncertainty              [min.sup.-1])

                                  0        15       33       62

Collected from   rad           0.0104    0.0116   0.0115   0.0150
100 Holograms    nm             0.41      0.46     0.46     0.60
Collected from   rad           0.0100    0.0117   0.0116   0.0153
50 Holograms     nm             0.40      0.47     0.46     0.61
Collected from   rad           0.0099    0.0116   0.0111   0.0125
30 Holograms     nm             0.39      0.46     0.44     0.50
Collected from   rad           0.0106    0.0115   0.0102   0.0146
20 Holograms     nm             0.42      0.46     0.41     0.58
Collected from   rad           0.0105    0.0123   0.0099   0.0124
10 Holograms     nm             0.42      0.49     0.39     0.49
Collected from   rad           0.0102    0.0114   0.0093   0.0118
5 Holograms      nm             0.41      0.45     0.37     0.47

Table 12. Median uncertainty in the phase measurement with the 20x
immersion lens in water.

                                          Water Flow Rate
                 Uncertainty             (mL [min.sup.-1])

                                 0        15       33       62

Collected from   rad           0.0101   0.0112   0.0111   0.0143
100 Holograms    nm             0.40     0.45     0.44     0.57
Collected from   rad           0.0098   0.0113   0.0111   0.0142
50 Holograms     nm             0.39     0.45     0.44     0.57
Collected from   rad           0.0098   0.0112   0.0107   0.0121
30 Holograms     nm             0.39     0.45     0.43     0.48
Collected from   rad           0.0103   0.0112   0.0099   0.0140
20 Holograms     nm             0.41     0.45     0.39     0.56
Collected from   rad           0.0102   0.012    0.0096   0.0119
10 Holograms     nm             0.41     0.48     0.38     0.47
Collected from   rad           0.0098   0.0108   0.0090   0.0113
5 Holograms      nm             0.39     0.43     0.36     0.45

Table 13. Mean uncertainty in the phase measurement with the 40x
immersion lens in water.

                  Uncertainty           Water Flow Rate
                                       (mL [min.sup.-1])

                                  0        15       33       62

Collected from    rad           0.0102   0.0112   0.0112   0.0134
100 Holograms     nm             0.41     0.45     0.45     0.53
Collected from    rad           0.0104   0.0109   0.0107   0.0133
50 Holograms      nm             0.41     0.43     0.43     0.53
Collected from    rad           0.0107   0.0113   0.0106   0.0127
30 Holograms      nm             0.43     0.45     0.42     0.51
Collected from    rad           0.0103   0.0106   0.0106   0.0135
20 Holograms      nm             0.41     0.42     0.42     0.54
Collected from    rad           0.0099   0.0109   0.0103   0.0119
10 Holograms      nm             0.39     0.43     0.41     0.47
Collected from    rad           0.0106   0.0113   0.0105   0.0118
5 Holograms       nm             0.42     0.45     0.42     0.47

Table 14. Median uncertainty in the phase measurement with the 40x
immersion lens in water.

                                         Water Flow Rate
                  Uncertainty           (mL [min.sup.-1])

                                   0        15       33       62

Collected from    rad           0.0106    0.011    0.0109   0.0129
100 Holograms     nm             0.42      0.44     0.43     0.51
Collected from    rad           0.0102    0.0108   0.0105   0.0127
50 Holograms      nm             0.41      0.43     0.42     0.51
Collected from    rad           0.0105    0.0111   0.0104   0.0123
30 Holograms      nm             0.42      0.44     0.41     0.49
Collected from    rad           0.0101    0.0104   0.0104   0.0130
20 Holograms      nm             0.40      0.41     0.41     0.52
Collected from    rad           0.0098    0.0107   0.0101   0.0114
10 Holograms      nm             0.39      0.43     0.40     0.45
Collected from    rad           0.0102    0.0109   0.0101   0.0112
5 Holograms       nm             0.41      0.43     0.40     0.45

Table 15. IQR, skewness, and kurtosis of data for flowing water with
an immersion lens.

Immersion   No. of      Metric      Water Flow Rate (mL [min.sup.-1)
Lens        Holograms
                                       0        15       33       62

                        IQR (rad)   0.0017    0.0021   0.0026   0.0035
            100         IQR (nm)     0.07      0.08     0.10     0.14
                        Skewness      7.8      8.2      9.8      4.2
                        Kurtosis     176.6    213.5    270.1     66.1
                        IQR (rad)   0.0018    0.0025   0.0028   0.0038
            50          IQR (nm)     0.07      0.10     0.11     0.15
                        Skewness      6.7      6.9      8.3      3.8
                        Kurtosis     154.1    167.5    201.7     32.4
                        IQR (rad)   0.0021    0.0029   0.0028   0.0033
            30          IQR (nm)     0.08      0.12     0.11     0.13
                        Skewness      5.7      6.8      7.5      5.5
20x                     Kurtosis     127.9    174.3    162.3    162.8
                        IQR (rad)   0.0029    0.0030   0.0026   0.0047
            20          IQR (nm)     0.12      0.12     0.10     0.19
                        Skewness      4.7      4.3      7.6      2.9
                        Kurtosis     85.8      92.5    168.6     45.8
                        IQR (rad)   0.0038    0.0042   0.0034   0.0050
            10          IQR (nm)     0.15      0.17     0.14     0.20
                        Skewness      2.4      2.1      2.7      3.8
                        Kurtosis     31.2      28.9     35.4     98.7
                        IQR (rad)   0.0053    0.0060   0.0047   0.0063
            5           IQR (nm)     0.21      0.24     0.19     0.25
                        Skewness      1.2      1.0      1.5      2.6
                        Kurtosis     10.2      6.4      14.4     51.9

                        IQR (rad)   0.0017    0.0018   0.0020   0.0027
            100         IQR (nm)     0.07      0.07     0.08     0.11
                        Skewness      3.1      6.4      17.9     18.4
                        Kurtosis     46.7     175.7     1042    496.5
                        IQR (rad)   0.0018    0.0019   0.0020   0.0028
            50          IQR (nm)     0.07      0.08     0.08     0.11
                        Skewness      2.5      4.3      19.8     18.3
                        Kurtosis     36.6     109.8     1181    487.3
                        IQR (rad)   0.0024    0.0025   0.0024   0.0030
            30          IQR (nm)     0.10      0.10     0.10     0.12
                        Skewness      1.8      2.7      11.3     14.4
40x                     Kurtosis     20.7      47.6    553.8    381.4
                        IQR (rad)   0.0026    0.0026   0.0027   0.0036
            20          IQR (nm)     0.10      0.10     0.11     0.14
                        Skewness      2.1      2.3      11.5     16.4
                        Kurtosis     29.6      47.4    573.0    427.5
                        IQR (rad)   0.0033    0.0038   0.0035   0.0042
            10          IQR (nm)     0.13      0.15     0.14     0.17
                        Skewness      0.5      0.9      7.8      12.8
                        Kurtosis      4.6      9.2     355.1    331.1
                        IQR (rad)   0.0055    0.0056   0.0054   0.0060
            5           IQR (nm)     0.22      0.22     0.22     0.24
                        Skewness      0.7      0.6      2.5      9.5
                        Kurtosis      4.0      4.2      56.8    219.8

Table 16. Mean uncertainty comparing 20x and 40x objective lenses in
air and in water (static conditions).

                 Mean                   Objective Lens and Medium
                 Uncertainty
                               20x in Air   20x in Water   40x in Air

Collected from   rad             0.0086        0.0104        0.0093
100 Holograms    nm               0.46          0.41          0.49
Collected from   rad             0.0087        0.0100        0.0088
50 Holograms     nm               0.46          0.40          0.47
Collected from   rad             0.0086        0.0099        0.0083
30 Holograms     nm               0.46          0.39          0.44
Collected from   rad             0.0084        0.0106        0.0080
20 Holograms     nm               0.44          0.42          0.42
Collected from   rad             0.0082        0.0105        0.0075
10 Holograms     nm               0.43          0.42          0.40
Collected from   rad             0.0084        0.0102        0.0075
5 Holograms      nm               0.44          0.41          0.40

                 Objective Lens and Medium

                   40x in Water

Collected from       0.0102
100 Holograms         0.41
Collected from       0.0104
50 Holograms          0.41
Collected from       0.0107
30 Holograms          0.43
Collected from       0.0103
20 Holograms          0.41
Collected from       0.0099
10 Holograms          0.39
Collected from       0.0106
5 Holograms           0.42

Table 17. Phase measurement uncertainty characteristics with a 5x lens
through a glass window.

Flow Rate                                  No. of Holograms
(mL [min.sup.-1])   Data Value
                                      5        10       20       30

                    Mean (rad)     0.0115    0.0108   0.0126   0.0126
                    Mean (nm)       0.46      0.43     0.50     0.50
                    Median (rad)   0.0111    0.0105   0.0123   0.0123
0                   Median (nm)     0.44      0.42     0.49     0.49
                    IQR (rad)      0.0060    0.0040   0.0032   0.0032
                    IQR (nm)        0.24      0.16     0.13     0.13
                    Skewness         0.5      0.5      0.7      0.8
                    Kurtosis         3.3      3.4      3.7      3.9

                    Mean (rad)     0.0214    0.0166   0.0213   0.0247
                    Mean (nm)       0.85      0.66     0.85     0.98
                    Median (rad)   0.0193    0.0150   0.0185   0.0214
15                  Median (nm)     0.77      0.60     0.74     0.85
                    IQR (rad)      0.0157    0.0088   0.0152   0.0183
                    IQR (nm)        0.62      0.35     0.60     0.73
                    Skewness         0.7      2.5      0.9      0.8
                    Kurtosis         3.0      16.8     3.1      2.9

                    Mean (rad)     0.0448    0.0555   0.0373   0.0417
                    Mean (nm)       1.78      2.21     1.48     1.66
                    Median (rad)   0.0386    0.0479   0.0330   0.0365
33                  Median (nm)     1.53      1.90     1.31     1.45
                    IQR (rad)      0.0357    0.0454   0.0228   0.0277
                    IQR (nm)        1.42      1.80     0.91     1.10
                    Skewness         1.0      0.9      1.0      0.9
                    Kurtosis         3.8      3.1       4       3.3

                    Mean (rad)     0.0248    0.0527   0.0407   0.0379
                    Mean (nm)       0.99      2.09     1.62     1.51
                    Median (rad)   0.0217    0.0433   0.0334   0.0311
70                  Median (nm)     0.86      1.72     1.33     1.24
                    IQR (rad)      0.0208    0.0570   0.0420   0.0378
                    IQR (nm)        0.83      2.26     1.67     1.50
                    Skewness         0.6      0.5      0.5      0.6
                    Kurtosis         2.3      2.0      2.0      2.0

                    Mean (rad)     0.0123    0.0244   0.0250   0.0313
                    Mean (nm)       0.49      0.97     0.99     1.24
                    Median (rad)   0.0113    0.0209   0.0211   0.0259
109                 Median (nm)     0.45      0.83     0.84     1.03
                    IQR (rad)      0.0071    0.0204   0.0206   0.0294
                    IQR (nm)        0.28      0.81     0.82     1.17
                    Skewness         1.0      0.6      0.6      0.6
                    Kurtosis         4.2      2.3      2.2      2.1

Flow Rate                          No. of Holograms
(mL [min.sup.-1])   Data Value
                                     50       100

                    Mean (rad)      0.0130   0.0184
                    Mean (nm)        0.52     0.73
                    Median (rad)    0.0126   0.0167
0                   Median (nm)      0.50     0.66
                    IQR (rad)       0.0035   0.0080
                    IQR (nm)         0.14     0.32
                    Skewness         0.9       1.0
                    Kurtosis         3.9       3.3

                    Mean (rad)      0.0245   0.0230
                    Mean (nm)        0.97     0.91
                    Median (rad)    0.0213   0.0200
15                  Median (nm)      0.85     0.79
                    IQR (rad)       0.0183   0.0164
                    IQR (nm)         0.73     0.65
                    Skewness         0.8       0.8
                    Kurtosis         2.8       2.9

                    Mean (rad)      0.0435   0.0446
                    Mean (nm)        1.73     1.77
                    Median (rad)    0.0379   0.0387
33                  Median (nm)      1.51     1.54
                    IQR (rad)       0.0296   0.0309
                    IQR (nm)         1.18     1.23
                    Skewness         0.9       0.9
                    Kurtosis         3.3       3.3

                    Mean (rad)      0.0357   0.0428
                    Mean (nm)        1.42     1.70
                    Median (rad)    0.0292   0.0349
70                  Median (nm)      1.16     1.39
                    IQR (rad)       0.0350   0.0443
                    IQR (nm)         1.39     1.76
                    Skewness         0.6       0.5
                    Kurtosis         2.0       2.0

                    Mean (rad)      0.0286   0.0312
                    Mean (nm)        1.14     1.24
                    Median (rad)    0.0238   0.0258
109                 Median (nm)      0.95     1.03
                    IQR (rad)       0.0255   0.0289
                    IQR (nm)         1.01     1.15
                    Skewness         0.6       0.6
                    Kurtosis         2.1       2.1

Table 18. Phase measurement uncertainty characteristics with a 10x
lens through a glass window.

Flow Rate                                   No. of Holograms
(mL [min.sup.-1])   Data Value

                                      5        10        20        30

                    Mean (rad)     0.0103    0.0103    0.0110    0.0106
                    Mean (nm)       0.41      0.41      0.44      0.42
                    Median (rad)   0.0099    0.0101    0.0108    0.0104
0                   Median (nm)     0.39      0.40      0.43      0.41
                    IQR (rad)      0.0053    0.0036    0.0030    0.0024
                    IQR (nm)        0.21      0.14      0.12      0.10
                    Skewness         0.6       0.5       0.6       0.6
                    Kurtosis         3.6       3.4       3.8       3.7
                    Mean (rad)     0.0108    0.0165    0.0180    0.0172
                    Mean (nm)       0.43      0.66      0.72      0.68
                    Median (rad)   0.0103    0.0152    0.0168    0.0160
15                  Median (nm)     0.41      0.60      0.67      0.64
                    IQR (rad)      0.0057    0.0080    0.0083    0.0077
                    IQR (nm)        0.23      0.32      0.33      0.31
                    Skewness         0.9       1.1       0.9       1.1
                    Kurtosis         5.8       6.1       4.5       5.5
                    Mean (rad)     0.0267    0.0250    0.0221    0.0233
                    Mean (nm)       1.06      0.99      0.88      0.93
                    Median (rad)   0.0232    0.0221    0.0200    0.0210
33                  Median (nm)     0.92      0.88      0.79      0.83
                    IQR (rad)      0.0216    0.0171    0.0128    0.0138
                    IQR (nm)        0.86      0.68      0.51      0.55
                    Skewness         0.9       0.9       0.9       0.9
                    Kurtosis         3.2       3.4       3.5       3.4
                    Mean (rad)     0.0197    0.0223    0.0219    0.0237
                    Mean (nm)       0.78      0.89      0.87      0.94
                    Median (rad)   0.0184    0.0212    0.0209    0.0227
70                  Median (nm)     0.73      0.84      0.83      0.90
                    IQR (rad)      0.0121    0.0132    0.0123    0.0143
                    IQR (nm)        0.48      0.52      0.49      0.57
                    Skewness         0.9       0.5       0.6       0.4
                    Kurtosis         4.5        3        3.3       2.7
                    Mean (rad)     0.0131    0.0135    0.0132    0.0134
                    Mean (nm)       0.52      0.54      0.52      0.53
                    Median (rad)   0.0122    0.0128    0.0127    0.0128
109                 Median (nm)     0.48      0.51      0.50      0.51
                    IQR (rad)      0.0075    0.0056    0.0045    0.0043
                    IQR (nm)        0.30      0.22      0.18      0.17
                    Skewness         1.2       1.6       1.9       2.2
                    Kurtosis         7.7      13.9      19.8      22.2

Flow Rate                           No. of Holograms
(mL [min.sup.-1])   Data Value

                                     50        100

                    Mean (rad)     0.0111    0.0115
                    Mean (nm)       0.44      0.46
                    Median (rad)   0.0109    0.0112
0                   Median (nm)     0.43      0.45
                    IQR (rad)      0.0024    0.0022
                    IQR (nm)        0.10      0.09
                    Skewness         1.0       1.2
                    Kurtosis         5.1       6.1
                    Mean (rad)     0.0186    0.0196
                    Mean (nm)       0.74      0.78
                    Median (rad)   0.0173    0.0179
15                  Median (nm)     0.69      0.71
                    IQR (rad)      0.0087    0.0095
                    IQR (nm)        0.35      0.38
                    Skewness         1.0       1.0
                    Kurtosis         4.6       4.2
                    Mean (rad)     0.0227    0.0233
                    Mean (nm)       0.90      0.93
                    Median (rad)   0.0205    0.0210
33                  Median (nm)     0.81      0.83
                    IQR (rad)      0.0130    0.0138
                    IQR (nm)        0.52      0.55
                    Skewness         0.9       0.9
                    Kurtosis         3.4       3.4
                    Mean (rad)     0.0245    0.0233
                    Mean (nm)       0.97      0.93
                    Median (rad)   0.0235    0.0223
70                  Median (nm)     0.93      0.89
                    IQR (rad)      0.0153    0.0138
                    IQR (nm)        0.61      0.55
                    Skewness         0.4       0.4
                    Kurtosis         2.5       2.7
                    Mean (rad)     0.0133    0.0132
                    Mean (nm)       0.53      0.52
                    Median (rad)   0.0126    0.0126
109                 Median (nm)     0.50      0.50
                    IQR (rad)      0.0040    0.0038
                    IQR (nm)        0.16      0.15
                    Skewness         2.4       2.5
                    Kurtosis        25.0      27.6

Table 19. Phase measurement uncertainty characteristics with a 20x lens
through a glass window.

Flow Rate                                    No. of Holograms
(mL [min.sup.-1])   Data Value
                                      5        10       20       30

                    Mean (rad)     0.0112    0.0108   0.0112   0.0112
                    Mean (nm)       0.45      0.43     0.45     0.45
                    Median (rad)   0.0108    0.0105   0.0110   0.0111
0                   Median (nm)     0.43      0.42     0.44     0.44
                    IQR (rad)      0.0058    0.0038   0.0030   0.0025
                    IQR (nm)        0.23      0.15     0.12     0.10
                    Skewness         0.6      0.5      0.8      0.7
                    Kurtosis         3.4      3.4      4.9      4.2

                    Mean (rad)     0.0193    0.0204   0.0201   0.0190
                    Mean (nm)       0.77      0.81     0.80     0.75
                    Median (rad)   0.0188    0.0196   0.0195   0.0184
15                  Median (nm)     0.75      0.78     0.77     0.73
                    IQR (rad)      0.0098    0.0099   0.0082   0.0072
                    IQR (nm)        0.39      0.39     0.33     0.29
                    Skewness         0.3      0.6      0.5      0.5
                    Kurtosis         2.9      3.0      2.9      2.8

                    Mean (rad)     0.0216    0.0265   0.028    0.0259
                    Mean (nm)       0.86      1.05     1.11     1.03
                    Median (rad)   0.0208    0.0261   0.0274   0.0253
33                  Median (nm)     0.83      1.04     1.09     1.01
                    IQR (rad)      0.0120    0.0121   0.0132   0.0114
                    IQR (nm)        0.48      0.48     0.52     0.45
                    Skewness         0.5      0.2      0.3      0.3
                    Kurtosis         3.0      2.5      2.3      2.3

                    Mean (rad)     0.0224    0.0179   0.0245   0.0199
                    Mean (nm)       0.89      0.71     0.97     0.79
                    Median (rad)   0.0214    0.0175   0.0238   0.0194
70                  Median (nm)     0.85      0.70     0.95     0.77
                    IQR (rad)      0.0128    0.0070   0.0119   0.0075
                    IQR (nm)        0.51      0.28     0.47     0.30
                    Skewness         0.5      0.4      0.3      0.4
                    Kurtosis         3.1      3.0      2.4      2.6

                    Mean (rad)     0.0493    0.0347   0.0595   0.0520
                    Mean (nm)       1.96      1.38     2.36     2.07
                    Median (rad)   0.0488    0.0337   0.0583   0.0510
109                 Median (nm)     1.94      1.34     2.32     2.03
                    IQR (rad)      0.0343    0.0237   0.0465   0.0382
                    IQR (nm)        1.36      0.94     1.85     1.52
                    Skewness         0.1      0.2      0.1      0.1
                    Kurtosis         2.1      2.1      1.9      1.9

Flow Rate                          No. of Holograms
(mL [min.sup.-1])   Data Value
                                      50       100

                    Mean (rad)      0.0115   0.0118
                    Mean (nm)        0.46     0.47
                    Median (rad)    0.0112   0.0115
0                   Median (nm)      0.45     0.46
                    IQR (rad)       0.0023   0.0021
                    IQR (nm)         0.09     0.08
                    Skewness         1.3       1.3
                    Kurtosis         8.2       6.2

                    Mean (rad)      0.0200   0.0206
                    Mean (nm)        0.79     0.82
                    Median (rad)    0.0194   0.0200
15                  Median (nm)      0.77     0.79
                    IQR (rad)       0.0076   0.0080
                    IQR (nm)         0.30     0.32
                    Skewness         0.4       0.4
                    Kurtosis         2.5       2.4

                    Mean (rad)      0.0259   0.0276
                    Mean (nm)        1.03     1.10
                    Median (rad)    0.0253   0.0268
33                  Median (nm)      1.01     1.06
                    IQR (rad)       0.0112   0.0133
                    IQR (nm)         0.45     0.53
                    Skewness         0.3       0.3
                    Kurtosis         2.3       2.2

                    Mean (rad)      0.0202   0.0232
                    Mean (nm)        0.80     0.92
                    Median (rad)    0.0197   0.0224
70                  Median (nm)      0.78     0.89
                    IQR (rad)       0.0076   0.0102
                    IQR (nm)         0.30     0.41
                    Skewness         0.4       0.3
                    Kurtosis         2.6       2.3

                    Mean (rad)      0.0551   0.0557
                    Mean (nm)        2.19     2.21
                    Median (rad)    0.0539   0.0544
109                 Median (nm)      2.14     2.16
                    IQR (rad)       0.0415   0.0419
                    IQR (nm)         1.65     1.66
                    Skewness         0.1       0.3
                    Kurtosis         1.9       1.9

Table 20. Uncertainty in water comparing 20x immersion lens and
through-window measurements.

No. of      Mean                   Water Flow Rate (mL [min.sup.-1])
Holograms   Uncertainty
                                   0                    15

                          Immersion   Window   Immersion   Window

100         rad            0.0104     0.0112    0.0116     0.0193
            nm              0.41       0.45      0.46       0.77
50          rad            0.0100     0.0108    0.0117     0.0204
            nm              0.40       0.43      0.47       0.81
30          rad            0.0099     0.0112    0.0116     0.0201
            nm              0.39       0.45      0.46       0.80
20          rad            0.0106     0.0112    0.0115     0.0190
            nm              0.42       0.45      0.46       0.75
10          rad            0.0105     0.0015    0.0123     0.0200
            nm              0.42       0.46      0.49       0.79
5           rad            0.0102     0.0118    0.0114     0.206
            nm              0.41       0.47      0.45       0.82

No. of      Mean             Water Flow Rate
Holograms   Uncertainty     (mL [min.sup.-1])

                                   33

                          Immersion   Window

100         rad            0.0115     0.0216
            nm              0.46       0.86
50          rad            0.0116     0.0265
            nm              0.46       1.05
30          rad            0.0111     0.0280
            nm              0.44       1.11
20          rad            0.0102     0.0259
            nm              0.41       1.03
10          rad            0.0099     0.0259
            nm              0.39       1.03
5           rad            0.0093     0.0276
            nm              0.37       1.10

Table 21. Comparison of noise in different configurations of single-
and dual-wavelength mode.

                                  [[lambda].sub.1]-[[lambda].sub.2]
                                        Dual-Wavelength Mode

                                     Test D1               Test D2

Experiment                     [[lambda].sub.1] and   [[lambda].sub.2]
Description                     [[lambda].sub.2]      source turned off
                                sources turned on

Shutter Speed                          492                   492
([micro]s)
                Mean (rad)            0.0148               0.0116
                Mean (nm)              0.78                 0.61
                Median (rad)          0.0148               0.0115
Uncertainty     Median (nm)            0.78                 0.61
                IQR (rad)             0.0017               0.0014
                IQR (nm)               0.09                 0.07
                Skewness               0.5                   1.8
                Kurtosis               4.3                  23.5

                                       [[lambda].sub.1]
                                    Single-Wavelength Mode

                                   Test S1             Test S2

Experiment                        Optimized        Camera settings
Description                       hologram    set to optimized hologram
                                               in dual-wavelength mode

Shutter Speed                        872                 492
([micro]s)
                Mean (rad)         0.0084               0.0115
                Mean (nm)           0.44                 0.61
                Median (rad)       0.0084               0.0114
Uncertainty     Median (nm)         0.44                 0.60
                IQR (rad)          0.0011               0.0013
                IQR (nm)            0.06                 0.07
                Skewness             2.3                 1.3
                Kurtosis            29.2                 13.4

Table 22. Effect of uncertainty on hypothetical dissolving mineral
surfaces of known [v.sub.s] (Experiment 1). (a)

Known [v.sub.s]   "Measured" [v.sub.s]         Percent
(m [s.sup.-1])    (m [s.sup.-1])               Difference

-[10.sup.-9.65]   -[10.sup.-9.65] [+ or -]     Negligible
                  [10.sup.-13.29]

-[10.sup.-11.43]  -[10.sup.-11.43] [+ or -]    Negligible
                  [10.sup.-13.27]

-[10.sup.-12.0]   -[10.sup.-12.01] [+ or -]    1.6 %
                  [10.sup.-13.29]
-[10.sup.-12.2]   -[10.sup.-12.27] [+ or -]    14.9 %
                  [10.sup.-13.26]
-[10.sup.-13.0]   -[10.sup.-13.59] [+ or -]    74.3 %
                  [10.sup.-13.17]

-[10.sup.-13.2]   -[10.sup.-14.17] [+ or -]    89.3 %
                  [10.sup.-13.21]
-[10.sup.-13.3]   -[10.sup.-14.95] [+ or -]    97.8 %
                  [10.sup.-13.23]

-[10.sup.-13.7]   -[10.sup.-13.29] [+ or -]    157 %
                  [10.sup.-13.22]
-[10.sup.-14.0]   -[10.sup.-13.31] [+ or -]    390 %
                  [10.sup.-13.21]
-[10.sup.-15.0]   [10.sup.-1318 [+ or -]       6510 %
                  [10.sup.-13.18]

Known [v.sub.s]   Example Mineral Dissolution
(m [s.sup.-1])

-[10.sup.-9.65]   Gypsum in water ([k.sub.s] =
                  -3.0 [micro]mol [m.sup.-2]
                  [s.sup.-1]; [V.sub.m] =
                  7.45 x [10.sup.-5] [m.sup.3]
                  [mol.sup.-1]) [14]

-[10.sup.-11.43]  Calcite in water ([k.sub.s] =
                  -0.1 [micro]mol [m.sup.-2]
                  [s.sup.-1]; [V.sub.m] =
                  3.7 x [10.sup.-5] [m.sup.3]
                  [mol.sup.-1]) [13]
-[10.sup.-12.0]

-[10.sup.-12.2]   Anorthite dissolution at pH 3.0
                  ([k.sub.s] = -5.7 x [10.sup.-9]
                  mol [m.sup.-2] [s.sup.-1]) [54];
                  [V.sub.m] assumed 1.05 x [10.sup.-4]
                  [m.sup.3] [mol.sup.-1]

-[10.sup.-13.0]

-[10.sup.-13.2]   Pyrite dissolution at pH 1.0
                  ([k.sub.s] = -2.8 x [10.sup.-9]
                   mol [m.sup.-2] [s.sup.-1]; [V.sub.m]
                  = 2.4 x [10.sup.-5] [m.sup.3]
                  [mol.sup.-1]) [57]

-[10.sup.-13.3]

-[10.sup.-13.7]   Muscovite dissolution at pH 9.4
                  and 155 [degrees]C ([k.sub.s] = -1.4 x
                  [10.sup.-10] mol [m.sup.-2] [s.sup.-1])
                  [58]; [V.sub.m] assumed 1.4 x [10.sup.-4]
                  [m.sup.3] [mol.sup.-1]

-[10.sup.-14.0]

-[10.sup.-15.0]

(a) Assuming 20x immersion lens, 10-hologram collection,
[[lambda].sub.1] single-wavelength mode, 25 [s.sup.-1] acquisition
rate, and 15 mL [min.sup.-1] water flow rate

Table 23. Effect of uncertainty on hypothetical dissolving mineral
surfaces of known [v.sub.s] (Experiment 2). (a)

Known [v.sub.s]    "Measured" [v.sub.s]        Percent
(m [s.sup.-1])     (m [s.sup.-1])              Difference

-[10.sup.-9.65]    -[10.sup.-9.65 [+ or -]     Negligible
                   [10.sup.-12.91]

-[10.sup.-11.43]   -[10.sup.-11.43 [+ or -]    Negligible
                   [10.sup.-12.92]

-[10.sup.-11.7]    -[10.sup.-11.70 [+ or -]    Negligible
                   [10.sup.-12.94]
-[10.sup.-11.8]    -[10.sup.-11.79 [+ or -]    2.2 %
                   [10.sup.-12.90]
-[10.sup.-12.0]    -[10.sup.-11.95 [+ or -]    12.2 %
                   [10.sup.-12.84]
-[10.sup.-12.2]    -[10.sup.-12.15 [+ or -]    12.2 %
                   [10.sup.-12.92]

-[10.sup.-12.9]    -[10.sup.-12.67 [+ or -]    68.1 %
                   [10.sup.-12.92]
-[10.sup.-13.0]    -[10.sup.-12.64 [+ or -]    129 %
                   [10.sup.-13.01]
-[10.sup.-13.2]    -[10.sup.-12.56 [+ or -]    337 %
                   [10.sup.-12.87]

-[10.sup.-13.7]    [10.sup.-13.05] [+ or -]    547 %
                   [10.sup.-12.99]

-[10.sup.-14.0]    [10-.sup.12.45] [+ or -]    3650 %
                   [10.sup.-12.83]
-[10.sup.-15.0]    [10.sup.-12.96] [+ or -]    11 100 %
                   [10.sup.-12.84]

Known [v.sub.s]    Example Mineral Dissolution
(m [s.sup.-1])

-[10.sup.-9.65]    Gypsum in water ([k.sub.s] =
                   -3.0 [micro]mol [m.sup.-2]
                   [s.sup.-1]; [V.sub.m]
                   7.45 x [10.sup.-5] [m.sup.3]
                   [mol.sup.-1]) [14]

-[10.sup.-11.43]   Calcite in water ([k.sub.s] =
                   -0.1 [micro]mol [m.sup.-2]
                   [s.sup.-1]; [V.sub.m] =
                   3.7 x [10.sup.-5] [m.sup.3]
                   [mol.sup.-1]) [13]

-[10.sup.-11.7]

-[10.sup.-11.8]

-[10.sup.-12.0]

-[10.sup.-12.2]    Anorthite dissolution at pH 3.0
                   ([k.sub.s] = -5.7 x [10.sup.-9]
                   mol [m.sup.-2] [s.sup.-1]) [54];
                   [V.sub.m] assumed 1.05 x [10.sup.-4]
                   [m.sup.3] [mol.sup.-1]

-[10.sup.-12.9]

-[10.sup.-13.0]

-[10.sup.-13.2]    Pyrite dissolution at pH 1.0
                   ([k.sub.s] = -2.8 x [10.sup.-9]
                   mol [m.sup.-2] [s.sup.-1]; [V.sub.m]
                   = 2.4 x [10.sup.-5] [m.sup.3]
                   [mol.sup.-1]) [57]

-[10.sup.-13.7]    Muscovite dissolution at pH 9.4
                   and 155 [degrees]C ([k.sub.s] = -1.4 x
                   [10.sup.-10] mol [m.sup.-2] [s.sup.-1])
                   [58]; [V.sub.m] assumed 1.4 x [10.sup.-4]
                   [m.sup.3] [mol.sup.-1]

-[10.sup.-14.0]

-[10.sup.-15.0]

(a) Assuming 20x immersion lens, 10-hologram collection,
[[lambda].sub.1] single-wavelength mode, 25 [s.sup.-1] acquisition
rate, and 33 mL [min.sup.-1] water flow rate

Table 24. Limiting surface-normal velocity (m [s.sup.-1]) for
conditions with flowing water.

Objective    Flow Rate                   No. of Holograms
Lens         (mL [min.sup.-1])
                                       5              10

             0                   [10.sup-11.9]   [10.sup-11.8]
20x          15                  [10.sup-12.0]   [10.sup-12.0]
Immersion    33                  [10.sup-11.9]   [10.sup-12.3]
             62                  [10.sup-11.9]   [10.sup-12.1]

             0                   [10.sup-12.0]   [10.sup-12.3]
40x          15                  [10.sup-12.0]   [10.sup-12.2]
Immersion    33                  [10.sup-11.9]   [10.sup-11.8]
             62                  [10.sup-11.9]   [10.sup-12.1]

             0                   [10.sup-12.0]   [10.sup-12.1]
             15                  [10.sup-11.5]   [10.sup-12.1]
5x Window    33                  [10.sup-11.2]   [10.sup-11.2]
             70                  [10.sup-11.5]   [10.sup-11.4]
             109                 [10.sup-11.7]   [10.sup-11.7]

             0                   [10.sup-11.8]   [10.sup-12.0]
             15                  [10.sup-11.6]   [10.sup-11.5]
10x Window   33                  [10.sup-11.5]   [10.sup-11.6]
             70                  [10.sup-11.5]   [10.sup-11.5]
             109                 [10.sup-11.8]   [10.sup-11.5]

             0                   [10.sup-11.9]   [10.sup-12.0]
             15                  [10.sup-11.6]   [10.sup-11.7]
20x Window   33                  [10.sup-11.5]   [10.sup-11.8]
             70                  [10.sup-11.6]   [10.sup-11.6]
             109                 [10.sup-11.3]   [10.sup-11.1]

Objective    Flow Rate                  No. of Holograms
Lens         (mL [min.sup.-1])
                                      20              30

             0                   [10.sup-11.9]   [10.sup-12.1]
20x          15                  [10.sup-12.1]   [10.sup-12.1]
Immersion    33                  [10.sup-12.1]   [10.sup-11.9]
             62                  [10.sup-12.1]   [10.sup-11.8]

             0                   [10.sup-11.9]   [10.sup-12.1]
40x          15                  [10.sup-11.7]   [10.sup-12.0]
Immersion    33                  [10.sup-12.0]   [10.sup-11.9]
             62                  [10.sup-11.9]   [10.sup-12.2]

             0                   [10.sup-11.8]   [10.sup-11.7]
             15                  [10.sup-11.5]   [10.sup-11.4]
5x Window    33                  [10.sup-11.1]   [10.sup-11.3]
             70                  [10.sup-11.4]   [10.sup-11.5]
             109                 [10.sup-11.4]   [10.sup-11.8]

             0                   [10.sup-11.9]   [10.sup-12.1]
             15                  [10.sup-11.4]   [10.sup-11.6]
10x Window   33                  [10.sup-11.5]   [10.sup-11.6]
             70                  [10.sup-11.7]   [10.sup-11.6]
             109                 [10.sup-11.7]   [10.sup-11.8]

             0                   [10.sup-12.1]   [10.sup-12.0]
             15                  [10.sup-11.1]   [10.sup-11.6]
20x Window   33                  [10.sup-11.7]   [10.sup-11.7]
             70                  [10.sup-11.5]   [10.sup-11.8]
             109                 [10.sup-11.4]   [10.sup-11.0]

Objective    Flow Rate                   No. of Holograms
Lens         (mL [min.sup.-1])
                                      50              100

             0                   [10.sup-11.7]   [10.sup-11.9]
20x          15                  [10.sup-11.9]   [10.sup-11.9]
Immersion    33                  [10.sup-11.9]   [10.sup-11.9]
             62                  [10.sup-11.7]   [10.sup-11.8]

             0                   [10.sup-11.8]   [10.sup-12.3]
40x          15                  [10.sup-11.8]   [10.sup-11.9]
Immersion    33                  [10.sup-12.4]   [10.sup-12.0]
             62                  [10.sup-12.0]   [10.sup-12.2]

             0                   [10.sup-11.7]   [10.sup-11.7]
             15                  [10.sup-11.9]   [10.sup-11.5]
5x Window    33                  [10.sup-11.5]   [10.sup-11.1]
             70                  [10.sup-11.6]   [10.sup-11.4]
             109                 [10.sup-11.6]   [10.sup-11.4]

             0                   [10.sup-11.9]   [10.sup-12.0]
             15                  [10.sup-11.8]   [10.sup-11.6]
10x Window   33                  [10.sup-11.8]   [10.sup-11.8]
             70                  [10.sup-11.8]   [10.sup-11.7]
             109                 [10.sup-11.7]   [10.sup-11.9]

             0                   [10.sup-12.2]   [10.sup-11.9]
             15                  [10.sup-11.6]   [10.sup-11.7]
20x Window   33                  [10.sup-11.7]   [10.sup-11.2]
             70                  [10.sup-11.4]   [10.sup-11.5]
             109                 [10.sup-11.2]   [10.sup-11.2]

Objective    Flow Rate
Lens         (mL [min.sup.-1])    Recommended
                                   Limit (a)

             0                   [10.sup-11.7]
20x          15                  [10.sup-11.9]
Immersion    33                  [10.sup-11.9]
             62                  [10.sup-11.7]

             0                   [10.sup-11.8]
40x          15                  [10.sup-11.7]
Immersion    33                  [10.sup-11.8]
             62                  [10.sup-11.9]

             0                   [10.sup-11.7]
             15                  [10.sup-11.4]
5x Window    33                  [10.sup-11.1]
             70                  [10.sup-11.4]
             109                 [10.sup-11.4]

             0                   [10.sup-11.8]
             15                  [10.sup-11.4]
10x Window   33                  [10.sup-11.5]
             70                  [10.sup-11.5]
             109                 [10.sup-11.5]

             0                   [10.sup-11.9]
             15                  [10.sup-11.6]
20x Window   33                  [10.sup-11.2]
             70                  [10.sup-11.4]
             109                 [10.sup-11.0]
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Author:Brand, Alexander S.
Publication:Journal of Research of the National Institute of Standards and Technology
Date:Jan 1, 2017
Words:21586
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