We propose a technique to evaluate the field of diffusion coefficient for particle dispersion where the Brownian motion is heterogeneous in space and single particle tracking (SPT) analysis is hindered by high concentration of the particles and/or their small size. We realize this “particle image diffusometry” by the principle of the differential dynamic microscopy (DDM). We extend the DDM by introducing the automated objective decision of the scaling regime itself. Label-free evaluation of spatially non-uniform diffusion coefficients without SPT is useful in the diverse applications including crystal nucleation and glass transition where non-invasive observation is desired.

Particle dispersions have long been important in many disciplines of science and technology, ranging from basic studies to industrial developments.1–7 Flow velocity of particles suspended in fluid can be experimentally evaluated by tracking them, and flow velocity field can be evaluated by the particle image velocimetry (PIV) or the particle tracking velocimetry (PTV). PIV can be applied to such systems where particle concentrations are too high to track them individually. The combination with microscopy is often referred to μPIV.8,9

When Brownian motion of colloidal particles or fluorescent labels attached to molecules is significant, the diffusion coefficient can also be evaluated by the single particle tracking (SPT)10–15 through mean squared displacement as a function of time. The slope of asymptotic regime corresponds to the diffusion coefficient based on the Stokes-Einstein relation. Space distribution of diffusion coefficient can be evaluated if one divides the images consisting the microscopy movie into regions of interest (ROIs) and attribute each particle locations. However, such kinds of analyses require the prerequisite condition that the particle tracking analysis is possible.16–18 It has long been impossible to evaluate the space distribution of diffusion coefficients for particle dispersion with high concentration and/or small particle sizes where particles cannot be tracked individually.

In contrast to the fact that there exists PIV when PTV is not feasible for the evaluation of the velocity field, the methodology corresponding to “particle image diffusometry” (PID) has long been missing. The diffusion coefficient of particle dispersion can be evaluated without tracking the particles if one uses the dynamic light scattering (DLS).19 However, DLS basically yields only one scalar value of diffusion coefficient to represent the whole sample. In addition, DLS requires special optical setup that is often expensive as a whole system. The requirement of special hardware setup and sufficiently high time resolution hinders the wide-spread application of parallelization technique to spatially resolve the diffusion coefficient of samples of interest.

In order to overcome the gap between the SPT and DLS, the differential dynamic microscopy (DDM)20 has been developed. The DDM makes use of the optical microscopy movie data, but does not track particles. Instead, DDM makes use of the same principle as the DLS at the fundamental level. The difference is that the DDM makes use of the space resolution to analyze the decay of time correlation of light intensity in the time-series images due to Brownian motion of particles suspended in fluid. Consequently, the DDM uses space resolution of the microscopy and camera to yield a single scalar just like DLS. As far as the DDM is simply employed to yield a single diffusion coefficient to represent the whole system of interest, the advantage of the DDM to DLS is the lower cost and the ease of hardware setup.

If we divide the image data into many ROIs analogous to PIV, DDM appears promising for the PID. However, there is a bottleneck to this extension of the DDM. The DDM is based on the fitting of scaling law for the wave number q dependence of characteristic time. In practice, this fitting holds for a limited range of q. One needs to decide the range of fitting for a set of movie data to obtain a diffusion coefficient. As far as we know, there has been no report on the objective quantitative protocol to determine this range where the scaling law is applicable.

In this study, we extend the DDM to enable the evaluation of diffusion coefficient field by developing a simple new algorithm to automatically decide the range of fitting for the scaling law of q-dependence of characteristic time scale τ for the correlation of light intensity in the sequential images. We demonstrate our novel technique by applying it to the system of heterogeneous field of diffusion coefficient prepared by an interface of different sample dispersions.

First, we briefly explain the principle of the DDM20 from the data processing point of view. The raw data to be analyzed is obtained from a camera mounted on an optical microscopy with Koehler illumination as the time-sequential images. The image intensity |FD(q, Δt)| of the Fourier transform as a function of the wave number q and time difference Δt, where the minimum and maximum is limited by the frame interval of the camera and the number of sequential frames, respectively, is expressed as20 

|FD(q,Δt)|2=A(q)1expΔtτ(q)+B(q)
(1)

within a certain range of experimental conditions, where A(q) and B(q) depends on the system properties. The first fitting procedure to Eq. (1) for the whole range of available q yields the following relation19,20

τ(q)=1Dq2,
(2)

where D is the diffusion coefficient of the suspended Brownian particles of interest.

The scaling law of Eq. (2) holds for a certain range of q, and the applicable range is conventionally determined by visual inspection of the plotted data points in a figure. Defining T ≡ log10τ and Q ≡ log10q, Eq. (2) gives T = −2Q − log10D. Thus, the suitable range [Q1, Q2] to apply this scaling has to be determined. Although visual inspection is the conventional protocol, application to many ROIs calls for an automated decision of this range of fitting. It is beneficial not only for this specific purpose in this article but also for the objective evaluation.

Conceptually, we consider employing the fundamental principle of the least square fitting. Whereas a conventional usage of the least square fitting principle is to determine the numerical value of coefficients in the prescribed model equation for an already specified set of measurement data, we define a problem in a different manner. Namely, the set of data to apply the fitting of Eq. (2) is not specified at the beginning of the analysis but the range [Q1, Q2] itself is determined by the least square principle with respect to the error that depends on [Q1, Q2].

The following quantity f(Q1, Q2) represents the error of fitting that depends on the fitting range parameters Q1 and Q2:

f(Q1,Q2)σ(Q1,Q2)2n(Q1,Q2),
(3)
σ21nQ1Q2TQ(2)2,
(4)
nQ2Q1ΔQ.
(5)

The above definition of f(Q1, Q2) is based on the notion of the standard error σ/n where n is the number of samples and σ is the error. Eq. (4) is defined to consider the validity of the fitting regarding the slope ∂T/∂Q as -2. As far as the slope is the same, the mean value σ2 of its squared difference remains the same.

Combining Eqs. (3), (5), and (4), the following equation is derived as the criterion function:

f(Q1,Q2)=1n2Q1Q2TQ+22.
(6)

Basically, too wide range of [Q1, Q2] gives large values of f(Q1, Q2), and the suitable fitting range [Q1, Q2] is given by Q1 and Q2 that yields the minimum f(Q1, Q2). However, as there are scattering of the experimental data, we assign a practical constraint that the range of fitting must consist of at least three points of data in order to take into account the possibility of accidentally obtained minimum value of f(Q1, Q2). In addition, we reduce the computational cost of this evaluation procedure by limiting the candidate range of Q within the values smaller than that gives minimum T.

We examine the validity of this protocol of the fitting range decision based on Eq. (6) by applying it to the evaluation of the heterogeneous field of the diffusion coefficients by and interface of two kinds of high-concentration suspension of the colloidal particles in fluid. More specifically, one of the sample particle dispersion consists of polystyrene particle with a diameter of 500 nm (Micromer, Mcromod. GmbH) at the concentration of 0.1 wt% in water. Another sample also contains 0.5 wt% of cellulose nanofibers (CNFs)21,22 (FMa-10002, BiNFi-s, Sugino Machine Ltd.) in addition to the particles and water. The use of CNFs was intended for the increase of effective viscosity. However, it is nontrivial to determine this effect quantitatively in advance because of the filament structure with many hydrogen bonding sites. Therefore, rheological behaviors of CNFs have been actively discussed until today.23–26 We prepared the sample dispersions in the domain with a thickness of 100 μm using a silicone sheet as a spacer and sealed with cover glasses on top and bottom sides as shown in Fig. 1. Then, we obtained the microscopy movie data consisting of 2000 frames of 512×512 px images at a frame rate of 203.8 fps with an exposure time of 9 μs. We used an inverted microscope (IX73, Olympus) with an objective lens of ×20 under phase contrast setting, and a sCMOS camera (Zyla, Andor), leading to 0.325 μm/px of image resolution for a image size of 512 px corresponding to 166 μm.

FIG. 1.

Schematic illustration of the sample configuration. Two kinds of sample dispersion is contained in a shallow cylindrical domain. The subdomain “A” consists of the aqueous dispersion of 500-nm polystyrene particle at 0.1 wt%, and the subdomain “C” also contains 0.5 wt% of cellulose nanofibers (CNFs). These two samples faces with each other at the subdomain “B”. The observation height is ca. 15 μm from the bottom wall.

FIG. 1.

Schematic illustration of the sample configuration. Two kinds of sample dispersion is contained in a shallow cylindrical domain. The subdomain “A” consists of the aqueous dispersion of 500-nm polystyrene particle at 0.1 wt%, and the subdomain “C” also contains 0.5 wt% of cellulose nanofibers (CNFs). These two samples faces with each other at the subdomain “B”. The observation height is ca. 15 μm from the bottom wall.

Close modal

Fig. 2 shows the results of this test for subdomains of A and C in Fig. 1. Each of the points plotted in this figure indicates the experimentally evaluated relation of Eq. (2) based on the assumption of Eq. (1) regardless of the validity range. The validity is evaluated afterwards by the criterion of Eq. (6), and the result of fitting line is shown as the solid line in the figure. The diffusion coefficient from the scaling shown in Fig. 2(a) for subdomain A, i.e., particle dispersion in water, is 7.7 × 10−13 m2/s, which is in reasonable agreement with the assumption of the Stokes-Einstein relation. The diffusion coefficient in the subdomain C was 3.7 × 10−13 m2/s based on Fig. 2(c), which is significantly smaller than that in A because of the addition of CNFs. It should be noted that this level of difference is often nontrivial to determine in practice, but Figs. 2(a) and 2(c) show clear scaling behaviors. These scaling behaviors have been easily determined by our protocol based on Eq. (6). Furthermore, our protocol robustly works even for extremely small size of the image as shown in Figs. 2(b) and 2(d).

FIG. 2.

Performance of the automated fitting range decision based on Eq. (6) with different samples and size of images. The sample of interest for (a) and (b) corresponds to “A” in Fig. 1, and that for (c) and (d) corresponds to “C” in Fig. 1. The image size for (a) and (c) is 512×512 px, and that for (b) and (d) is 32×32 px. The points in these four figures indicate the results of fitting to Eq. (1), and the solid lines indicate the finally determined fitting to Eq. (2).

FIG. 2.

Performance of the automated fitting range decision based on Eq. (6) with different samples and size of images. The sample of interest for (a) and (b) corresponds to “A” in Fig. 1, and that for (c) and (d) corresponds to “C” in Fig. 1. The image size for (a) and (c) is 512×512 px, and that for (b) and (d) is 32×32 px. The points in these four figures indicate the results of fitting to Eq. (1), and the solid lines indicate the finally determined fitting to Eq. (2).

Close modal

Now that we have confirmed the validity of the automated fitting-range decision based on Eq. (6), we demonstrate its usefulness by the characterization of diffusion field, i.e., the space distribution of the diffusion coefficient by the analysis of microscopy movie data obtained including the subdomain B in Fig. 1. Figs. 3(a), 3(c), and 3(e) show the microscopy images for subdomains A, B, and C consisting the movie data to be analyzed by our method. The individual particles cannot be visually determined from the captured image because of the high concentration in addition to the small particle size. This characteristic hinders the SPT analysis to obtain the trajectory data. On the other hand, the DDM can extract the diffusion coefficient of the system without tracking the particles, at the cost of using space resolution. The ROI binning can map the diffusion coefficients in principle, but the visual inspection of the range of scaling law for many ROIs are unrealistic. The number of ROIs amounts to 3,721 when we define the 32×32 px of a ROI size with an increment of 8 px for each of the x and y directions in the image of 512×512 px.

FIG. 3.

Optical microscopy images where (a), (c), and (e) correspond to the subdomains A, B, and C in Fig. 1, respectively, and the diffusion field of these samples were evaluated as (b), (d), and (f), respectively, from the microscopy movie data including these images. Each 512×512 px image of 2000 frames were divided into 3,721 ROIs of 32×32 px with grid increment of 8 px.

FIG. 3.

Optical microscopy images where (a), (c), and (e) correspond to the subdomains A, B, and C in Fig. 1, respectively, and the diffusion field of these samples were evaluated as (b), (d), and (f), respectively, from the microscopy movie data including these images. Each 512×512 px image of 2000 frames were divided into 3,721 ROIs of 32×32 px with grid increment of 8 px.

Close modal

Our proposed approach can automatically determine the diffusion coefficient and reveals the mapping of diffusion coefficient as shown in Figs. 3(b), 3(d), and 3(f). First, it can be seen in Figs. 3(b) and 3(f) that roughly constant diffusion coefficient field is confirmed at the subdomains A and C at significantly different values with each other. In contrast, Fig. 3(d) reveals the heterogeneous field of diffusion coefficient with high resolution. Comparison between the interface and the bulk subdomains clarifies that this is the significant heterogeneity revealed from our novel approach. Diffusion coefficient is empirically well-known to be a physical quantity that is difficult to evaluate precisely. This is to some extent objectively understood from the fact that the variance is the higher order characteristic compared to the mean value of the displacement. Considering this fact, our proposed approach of the precise evaluation of the spatially non-uniform field of the diffusion coefficient is expected to be highly appreciated in the near future.

We have realized the experimental evaluation of non-uniform diffusion coefficient field for high-concentration particle dispersion by the development of a simple mathematical criterion to be applied to the DDM. This new technique is especially promising for understanding new physics in the phase transition such as glass formation, gelation and crystal nucleation of colloidal particles and molecules in fluid. Since fluorescent labeling is not required in this technique, its usefulness is likely to be highly appreciated in the industrial applications ranging from materials and chemistry to foods, where non-invasive observation is desired. In addition, it is also useful in the evaluation of local viscosity of non-colloidal material or fluid from the diffusion coefficient of spherical tracer particles with a known diameter and the Stokes-Einstein relation.

This work was partly supported by the Japan Society for the Promotion of Science (JSPS) through a Grant-in-Aid for Young Scientists (A), No. 26709008, and a Grant-in-Aid for Scientific Research on Innovative Areas, “Nano-Material Optical-Manipulation”, No. 17H05463.

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