1. Introduction

Particle size analysis is widely used in the fields of pharmaceutics, ceramics, and biotechnology, as well as in the semiconductor industry. Static light scattering, more commonly referred to as the laser diffraction method [1–4] and based on the angular variation of scattering intensity, is successfully applied for particles greater than sub-micrometer-size. For much smaller particles than the wavelength of light, the angular variation of scattering intensity becomes unable to provide information on particle size; thus, dynamic light scattering (DLS) [5–7], which is also referred to as photon correlation spectroscopy, is generally used. DLS is based on the measurement of the fluctuation in scattering intensity due to the Brownian motion of the particles. It determines the diffusion coefficient, D, which is then converted to the diameter. However, there is an important drawback to DLS: the sensitivity of detection changes drastically depending on the particle size. This is because the efficiency of light scattering is proportional to the diameter to the sixth power (per particle) [2], or proportional to the diameter to the third power (per unit volume of particles).

Here we report an alternative method that is expected to have less sensitivity dependence on particle size. The new method determines the diffusion coefficient D as in DLS; however, it includes an activation procedure by dielectrophoresis (DEP). It first generates a periodic density modulation of the particles, which are then released to diffuse until they reach a steady state, as shown in the following description. Although many reports on DEP [8–11] have already discussed such subjects as trapping nano- or microparticles or applying such trapping to the separation of substances, to date there has been no report discussing particle-size analysis. The idea of density modulation was previously reported [12], however, it was for different applications from our purpose.

2. Description of the method

Figure 1 shows the general schematics of the system. The heart of the system is a sample cuvette having a pair (L and R) of gold comb electrodes, patterned by lithography on one of the inner quartz windows.

 

Fig. 1. Schematics of the overall system.

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Fig. 2. Schematics of arrangement of electrodes.

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The teeth of the comb-electrodes belonging to the L or R side are specially arranged as R-RL-L-R-R-L-L-R-R, (not as R-L-R-L-R-L), as in Fig. 2, where the pitch of the teeth (a) is 20 µm and the open space between adjacent teeth is 10 µm. When a radio frequency (r-f) voltage is introduced between the R and L sides, it causes an intense electric field between the teeth of R and L. Hence, the particles in the cuvette migrate by DEP force toward the space between R and L, because the DEP exerts forces on the particles with induced dipole moment toward the more intense electric field [8–11]. Thus, the generated periodic modulation density in the particles functions as a grating whose pitch (Λ) is twice as large as that of the teeth of the electrodes; i.e., a=20µm but Λ=40µm.

Therefore, once again, as in Fig.1, as the collimated beam illuminates the induced grating from the normal direction, the light diffracted by the induced grating appears at all orders, while that from the electrode appears only at the angles of even orders. This is because the electrode grating has twice as many grooves as the induced one and the angle of the first order diffraction by the electrode agrees with that of the second order by the induced grating. We use the light of the first order to minimize the undesirable effect from the electrodes. The effectiveness of this technique is further explained in Fig. 4 of the subsequent section.

When the r-f voltage is turned off, the diffraction intensity I starts to decrease according to

as derived in the following section, where I ₀ is the initial intensity at the instance when DEP is turned off, q is defined using Λ (the pitch of the density grating) as

and D is the diffusion coefficient given by the Einstein-Stokes relation

where k_B is the Boltzmann’s constant, T is the temperature in Kelvin, η is the viscosity of the dispersion medium, and d is the diameter of the particles. From the exponential factor of the decay curve, D is obtained using Eq. (1), and then is converted to d by Eq. (3) based on the known η and T.

3. Derivation of the basic decay equation

Let us assume a thin transmission grating in the x-y plane with the period Λ formed by the DEP. The x-y-z coordinates are defined as follows: x, the distance normal to the grooves; y, the distance along the grooves; and z, the distance normal to the grating surface. The complex transmission amplitude can be defined as T_amp(x)·e^iφ(x), where T_amp(x) and φ(x) are the transmittance and the phase due to the periodic density of the particles, respectively. Then, defining absorption µ(x) by T_amp(x)=e-^µ(x), we have e-^µ(x)·e^iφ(x) as the complex transmission amplitude of the grating.

 

Fig. 3. Density grating illuminated by a collimated beam and viewed at an angle θ. The density grating (shown in red color) turns in a short time to be a simple sinusoid according to Eq. (6) of m=1.

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We also assume that the particle density u(x) is an integrated value over the z direction because the diffusion toward the z direction does not contribute to the change in the complex amplitude. The particle density u(x) is a function of the period Λ, and may thus be expressed by the Fourier cosine series as

where the origin of x is properly shifted to make u(x) symmetric, and m is a positive integer. Next, we introduce a term for the relaxation due to the diffusion of particles. The density expression including time u(x,t) should satisfy the one-dimensional diffusion equation

Using the variable separation method, it is shown that the relaxation term should be exp(-m²q²Dt). Therefore, Eq. (4) becomes

In Eq. (6) the Fourier component with m≥2 decreases very rapidly; we therefore conclude that the dominant term is only from the m=1 component.

The diffracted light intensity from the density grating is further modulated by the periodic narrow groove apertures formed by electrodes, as in Fig. 3. Note here that µ(x,t) and φ(x,t) are both proportional to u(x,t). Let us assume that the groove position coincides with the maximum [cos(qx)=1] or the minimum [cos(qx)=-1] of the periodic function. Then the electric field E at an angle θ and at infinite distance can be calculated as the sum of the contribution of N grooves at the maxima and the other N grooves at the minima, the latter being shifted by Λ/2 from the maxima. Hence the electric field E is written as

In Eq. (7a) n is the refractive index of the dispersion medium and λ ₀ is the wavelength in a vacuum. The equation (7) can be explained as follows. The last factor, [sin(Nδ/2)/sin(δ/2)], is a familiar textbook formula [13] for a field by a linear array of N grooves. The term exp(-µ+iφ) represents the contribution from N grooves at the density maxima. The other term, exp(µ-iφ)·exp(iδ/2), is the contribution from the other N grooves at the minima, which are displaced from the maxima by Λ/2, corresponding to the phase factor exp(iδ/2). A graphical representation of (7) with a typical set of parameters is shown in Fig. 4. Increases in intensity are found at δ/2=π (first diffraction order) and δ/2=3π (third diffraction order), in accordance with the increases in φ, as described above for Fig. 1.

 

Fig. 4. Dependence of |E|² on δ. Three values of φ are calculated for the case of N=10, µ=0.

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Next, let us consider the case of the first diffraction order in Eq. (7). For this, we adopt the diffraction angle in Eq. (7a):

From Eq. (7a), δ=2π, so eiδ ^/2=-1 and the last term of Eq. (7) becomes N. Therefore, the intensity I for the first diffraction order is simplified to

After further calculation of Eq. (8), including Taylor’s expansion of exponential functions to the fourth order, we have (omitting a constant)

Then, introducing µ=µ₀exp(-q²Dt), φ=φ₀exp(-q²Dt), where µ ₀ and φ ₀ are the initial values for the absorption and phase, respectively, we finally obtain

The value of I derived in Eq. (10) shows the dominant decay of exp(-2q²Dt). The value F in Eq. (11) is the factor of deviation from dominant decay, and has a dependence on time (time being multiplied by q²D) as shown in Fig. 5. It is approximately unity if µ₀ and φ₀ are small, and approaches unity exactly when q²Dt≥2.

 

Fig. 5. Graph of the deviation factor.

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4. Experiment and results

In order to validate the theory, decay curve measurements were conducted for silica particles (Cataroid-S Series from Catalysts & Chemicals Industries Co., Ltd. (CCIC), Kawasaki Japan) and for polystyrene particles (NIST standards from Duke Corporation, Palo Alto, CA).

All the sample particles had been dispersed in water by the manufacturer; they were then further diluted by water and introduced into the cuvette. The final concentration of silica samples in the cuvette was 1% (weight percent) except for the particles of 45nm and 80nm in diameter, in the latter case the concentration being 0.5%. The cuvette was 4 mm high, 8 mm wide, and 1 mm thick. The wavelength of the diode laser was 658 nm and the diffraction intensity was monitored by a silicone photodiode. The r-f voltage for DEP was 500 kHz and 20 V peak-to-peak, except for the case of the smallest diameter of 5 nm. In this smallest-diameter case, we increased the voltage to 30 V peak-to-peak. Figure 6 presents a typical raw intensity trace. The r-f voltage is turned on for 0.1 sec, and during this period the signal increases; i.e., the particles migrate to form a density grating. The time axis is shifted so that the timing of the end of the r-f voltage reaches “0” in the figure.

 

Fig. 6. Typical raw trace of the diffraction intensity for silica particles 8 nm in diameter. The ordinate is the response of the photodiode (in arbitrary units).

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Thereafter the signal decreases exponentially, and finally returns to the baseline level. Similar raw traces are converted to the normalized decay signals, as shown in Fig. 7, where the ordinate is the change in the signal from the original level after normalization to unity at t=0. The results for seven similar silica particles of different diameters ranging from 5 nm to 80 nm are plotted. In the experiments four traces have been recorded for each sample to estimate the reproducibility of measurements.

By fitting Eq. (1) to the decay curves in Fig. 7 in the range between 80 % and 10 % of their intensity, the corresponding D values are obtained. Then the diameters d of the particles are calculated using Eq. (3) assuming T=298 K and η=0.89x10^-3 Pa·s.

A comparison of the measured diameters of silica particles and their nominal values is summarized in Fig. 8. Here the four plots at each nominal diameter represent the diameter values, which correspond to the four measurement traces to indicate the reproducibility of the measurements. For the nominal diameters we used the values determined by the manufacturer based on methods depending on particle size: neutralizing titration using NaOH for particles of 28 nm or less and the BET (Brunauer-Emmett-Teller) method for 45, 80 nm in diameter.

Measurement for particles of polystyrene (NIST standard) with three different mean diameters of 33±1.4, 50±2.0 and 73±2.6 nm was conducted in a similar way but with a different apparatus. In this case, the sample concentration was 0.1% (weight percent), the wavelength of the diode laser was 785 nm, the r-f voltage of DEP was 1 MHz, 15 V peak-to-peak, and the duration was 0.2 sec. The cuvette measured 5 mm high, 7 mm wide, and 0.2 mm thick. The results are also presented in Fig. 8. In this case we have three traces for the same sample, thus the three plots for each nominal diameter indicate the measurement reproducibility, although some of them are overlapped.

 

Fig. 7. Normalized changes in intensity for silica particles. The nominal diameter of the particle given by the manufacturer is shown for each curve.

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Fig. 8. Summary of the measurement by comparison of diameters; nominal vs. measured. To indicate the repeatability of measurements four plots (for silica) or three plots (for polystyrene) are presented for each nominal diameter. Some of them are overlapped in cases when the ordinate values are very close to each other.

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These experimental results show that all measured diameters of silica and polystyrene particles are in good agreement with their nominal values.

5. Discussion

The proposed method has both advantages and disadvantages in relation to DLS, DEP, and the density grating.

DLS measures the fluctuation of scattered light intensity due to the Brownian motion of the particles and determines the diffusion coefficient D. Here DLS is based on the so-called intensity auto-correlation function [5], which includes the factor

Where τ is the time difference and K is a value called the modulus of scattering vector, which is given by

where n is the refractive index of the dispersion medium, λ ₀ is the wavelength, and θ_s is the angle of observation used in scattering experiments. The decay factor (12) appearing in DLS is formally similar to exp(-2q²Dt) in the decay function Eq. (1) of the proposed method.

Here the first comment on the difference between the proposed method and DLS is that Eq. (1) is a direct decay function of time while Eq. (12) is an auto-correlation function of τ calculated from the fluctuating time signal. Secondly it is important to note the difference between K and q. The value K in Eq. (13) includes a refractive index, n, while q in Eq. (2) does not. Note here that the value of n in Eq. (7b) is related to sinθ but not to q. Therefore the decay speed parameter 2q²D is dependent only on Λ and D. Accordingly, when we obtain a decay speed 2q²D from an experiment, D is calculated independently of the refractive index, based on the value of Λ, which is defined as twice the known pitch of the electrodes.

Another difference exists in the magnitude of the values of 2q²D and 2K²D. For example, under typical conditions, d=10 nm, T=298 K, η=0.89x10^-3 Pa·s, Λ=40 µm, λ=0.6 µm (wavelength of DLS apparatus), and n=1.33; using Eq. (2), (3) and (13), these values are calculated as

The value 2K ² D for DLS is ten thousand times larger than 2q²D; therefore, DLS requires a fast data collection system with micro-second resolution, while an inexpensive detection system with millisecond resolution is sufficient for the new method.

The previous reports on DEP [8–11] have described two types of DEP action: i.e., positive and negative DEP. In the former case, particles with larger induced dipole moment (i.e., larger than that of the dispersion medium) move toward a more intense electric field. Therefore, while in the case of positive DEP the particles gather between the electrodes as described in Fig. 2, in negative DEP, the electric field pushes particles away from the electrodes. This means that in negative DEP u(x) has a particle density that is opposite that in Fig. 3; i.e., the polarity of µ(x) and φ(x) are inverted. However, because Eq. (9) and Eq. (10) are unchanged even if µ or φ is replaced by (-µ) or (-φ), the signals of raw traces always increase as in Fig. 5 regardless of the type of DEP. The drawback of using DEP for particle analysis is the influence of the electric conductivity of the medium of dispersion. If the medium has significant conductivity, the strength of the electric field decreases and the DEP force that moves the particles also decreases. The resulting low modulation of the grating leads to low sensitivity. Development of a countermeasure against this conductivity effect would thus be an important subject for the future improvement of the system.

The idea of density grating was previously reported [12] as the transient grating method, and was successfully applied to the analysis of the diffusion coefficient of materials in chemical reactions. In this case, periodic excitation was achieved by photochemical reactions in the region of the optical fringe generated by the interference of crossed laser beams. Use of photochemical reactions, however, limits the type of material to be analyzed. Our method uses simple comb electrodes instead of photochemical reactions; thus it has fewer limitations in regard to the type of particles to be analyzed.

The accuracy of dominant decay (1) can be examined using Eq. (6) and Eq. (11). Equation (6) shows that the higher Fourier components decay much faster; for example, the second component decreases four times faster than the main component. The use of the main component is sufficient as long as its latter portion is used. On the other hand, equation (11) indicates that the error in the dominant decay caused by a large initial signal also vanishes after some time. Therefore, it is concluded that the initial variations in particle density generated by the DEP procedure do not lead to significant errors as long as the latter portion of the dominant decay is used. At this stage we would like to mention the discussion on the problem for polydisperse samples. In the test measurements shown in Fig. 7, all the solutions are monodisperse and the decay curves are composed of single exponential functions, therefore the above discussion is valid. We are also interested in the case for polydisperse samples, which shows the sum of exponential curves with different decays. In DLS researchers already discussed how to guess the particle size distribution from the autocorrelation function consisting of many exponential curves [14–17].

However, in our case, there could be a problem that the higher order terms in the Fourier series of Eq. (6) for larger particles might interfere with a fundamental term for smaller particles. We would like to study DEP and find an improved condition, which induces only the fundamental term of the Eq. (6) to solve the problem.

Another possible drawback of the new method would be adhesion of particles to the electrode or the wall. For obtaining the test data of Fig. 7, we performed four (or three) successive measurements for a same sample without rinsing, then introduced a procedure of wiping followed by cleaning with ethanol between the measurements of different samples. If particles remain on the electrode periodically, the baseline would change during successive measurements due to the change in the grating performance, on the other hand, spatially uniform adhesion would not cause a drift. As a simple drift test, twenty measurement cycles were repeated without rinsing, however in this case, the drift of the baseline was not found. We have not examined further details on the maintenance of the cuvette. In any case improvement of the cuvette for long time use should be an important subject of development.

The reason for the reduction in the sensitivity dependence on particle size may be explained in the following way. If we denote the number of particles in unit volume by ρ and the volume of a particle by V, the particle volume fraction can be expressed by ρV. Then the delay in the phase φ (=2πn/λ₀) is proportional to ρV for particles sufficiently small compared to the wavelength [2]. The diffracted light intensity from a grating is, therefore, proportional to (ρV)² from Eq. (9) for a non-absorptive solution. In the case of scattering measurements (DLS), on the other hand, the scattering intensity for a particle is known to be proportional to V ² for small particles. Hence, a solution including ρ particles should exhibit a signal intensity proportional to ρV ² because the phases of light scattered by many particles are independent of each other [2]. Here let us estimate the error of measurement for a typical case where sample particles 10 nm in diameter are contaminated by a small amount of larger particles 100 nm in diameter. Assume that the particle volume fraction (ρV) for the sample particles is 10^-3 (i.e., 0.1%) and that for the larger particles (ρ’V’) is 10^-6. If V and V’ represent the volumes of the sample particle and contaminant respectively, then ρ’V’/ρV=10^-3 and V’/V=10³. In this condition a comparison of errors can be estimated. The signal S_G (for the grating method) and the signal S_S (for the scattering method) can be written using constants C ₁ and C ₂ as

The second term in Eqs. (16) and (17) represents the estimated error due to contamination of larger particles. The error for the grating methods is only 0.2 %, while that in the scattering method is as large as the sample signal. This example demonstrates the advantage of the new method of induced grating that emphasizes the capability of suppression of sensitivity dependence on particle size.

6. Conclusion

We proposed a new method for particle size analysis and discussed its potential advantage and limitation. Special comb electrodes were designed to generate periodic modulation of particle density. For this purpose an innovative electrode system, which ensured an induced grating pitch twice as large as that of the electrode grating, was developed and was proved to separate successfully the required diffracted light from undesirable electrode signals.

We showed the importance of the parameter q (=2π/Λ), where Λ is the pitch of the density grating. This parameter corresponds to the modulus of the scattering vector K in DLS; however, it is as small as one hundredth of K, and thus may lead to a small decay constant 2q²D permitting easy signal detection. In addition, q does not include the refractive index and depends only on Λ. Another expected advantage demonstrated in the present study is the reduction in the sensitivity dependence on particle size. The system is very simple and features to obtain an exponential decay signal directly, not as an autocorrelation function of DLS. In the test measurement of polystyrene and silica, good agreement is shown with the nominal diameters. The subject of future improvement would be the problems related with polydispersity and the counter measures to avoid the influence of the electric conductivity of samples as well as to avoid adhesion of particles to the electrodes and the wall of the cuvette.