ANALYTICAL SCIENCES JULY 2011, VOL. 27 751

Introduction

The measurement of light scattering is a powerful technique for investigations on synthetic polymers, biomaterials, drug delivery systems, and similar substances.1–4 In particular, it has led to major developments in the in situ measurement of the size of nanoparticles in liquid phases.5 In the International Organization for Standardization (ISO) standard ISO 13321:1996,6 a dynamic light scattering (DLS) method is described that involves measurements made at a single angle and a single concentration. The document recommends measuring in the low-concentration region, but that does not refer to changes in the concentrations or angles. In 2008, the ISO 13321:1996 standard developed into ISO 22412:2008,7 which is suitable for a wide range of dispersion concentrations, but not for a wide angular range of scattering. In those two standards, it is recommended that the instrument performance should be verified by using a dispersion of polystyrene latex (PSL) particles with a narrow size distribution and a certified particle size. However, the accuracy of the determined particle size of the PSL is not completely understood because of the presence of a number of unevaluated effects, such as the effects of the concentration of the suspension, the scattering angle, and the physical and chemical properties of the PSL particles.

It is worth noting the importance of extrapolations made by using various concentrations and angles. In many cases, the levels of the repeatability and accuracy are insufficiently high to permit evaluations of the effects of these factors. High-resolution DLS techniques have been developed for measuring the time

correlation function on a timescale of nanoseconds. Such nanosecond resolution is required for establishing technical standards for measuring the size of nanoparticles in liquid phases. In the current study, we measured the angular and concentration dependences of nanosized suspensions of PSL with sufficient precision to permit extrapolations of these two parameters to zero angle and zero concentration. In addition, we estimated the dependence of the apparent diffusion coefficient of PSL suspensions on the concentration and scattering angle by means of dynamic structure factors calculated from the hydrodynamic properties of PSL particles. This estimation was performed by means of a simple theory involving several parameters, including the sizes and concentrations of particles and the long-range interaction radii of the PSL suspensions. The qualitative aspects of the dependence of the apparent diffusion coefficient of PSL suspensions on the concentration of the latex and the scattering angle are clearly represented in the simulation.

Finally, the results of the recalculated particle size measured by DLS were compared with values measured by using a differential mobility analyzer (DMA) and by pulsed-field gradient nuclear magnetic resonance (PFG NMR) studies. The recalculated particle sizes measured by the three different methods are consistent with one another within their respective degrees of uncertainty.

Experimental

Measurement apparatusThe light-scattering apparatus was an ALV/compact

goniometer system (ALV-Laser Vertriebsgesellschaft m.b.H., Langen, Germany) with a YAG laser that produces a wavelength

2011 © The Japan Society for Analytical Chemistry

† To whom correspondence should be addressed.E-mail: kayori.takahashi@ni.aist.go.jp

Development of a Standard Method for Nanoparticle Sizing by Using the Angular Dependence of Dynamic Light Scattering

Kayori TAKAHASHI,† Haruhisa KATO, and Shinichi KINUGASA

Polymer Standards Section, Materials Characterization Division, National Metrology Institute of Japan (NMIJ), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Higashi, Tsukuba,  Ibaraki 305–8565, Japan

A standard method for nanoparticle sizing based on the angular dependence of dynamic light scattering was developed. The dependences of the diffusion coefficients for aqueous suspensions of polystyrene latex on the concentration and scattering angle were accurately measured by using a high-resolution dynamic light-scattering instrument. Precise measurements of the short-time correlation function at seven scattering angles and five concentrations were made for suspensions of polystyrene latex particles with diameters from 30 to 100 nm. The apparent diffusion coefficients obtained at various angles and concentrations showed properties characteristic of polystyrene latex particles with electrostatic interactions. A simulation was used to calculate a dynamic structure factor representing the long-range interactions between particles. Extrapolations to infinite dilution and to low angles gave accurate particle sizes by eliminating the effects of long-range interactions. The resulting particle sizes were consistent with those measured by using a differential mobility analyzer and those obtained by pulsed-field gradient nuclear magnetic resonance measurements.

(Received March 7, 2011; Accepted May 31, 2011; Published July 10, 2011)

752 ANALYTICAL SCIENCES JULY 2011, VOL. 27

of 532 nm. The multiple tau digital correlation was measured at a minimum sampling time of 6.25 ns by using a dual autocorrelation mode on an ALV-6010/160 correlator board (ALV-Laser Vertriebsgesellschaft m.b.H.). Static light-scattering (SLS) measurements were performed at 5° intervals by scanning the scattering angle (θ) from 30 to 150° for five suspensions of various concentrations. DLS measurements were performed at seven scattering angles between 30 and 150° (θ = 30, 70, 90, 110, 130, and 150°), and at five concentrations of the PSL. The sample cell was immersed in a bath of toluene for refractive index matching with quartz. The temperature of the toluene bath was maintained at 23.00 ± 0.01°C. The entire apparatus was set up in a clean booth, which was also maintained at 23.0 ± 0.1°C.

Sample preparationThe samples for the light-scattering measurements were PSL

suspensions supplied by JSR Corporation (Tokyo, Japan) with diameters of about 100, 70, 50, and 30 nm. The PSL suspensions were diluted with ultrapure water generated by a Milli-Q system (Nihon Millipore KK, Tokyo, Japan) using 0.1 μm filters. The ultrapure water had an electrical resistance of 18.2 MΩ cm–1 and a total organic carbon content of less than 6 ppb. The PSL suspensions were stable, and even at high dilutions did not require the presence of salts or surfactants as stabilizers.

Light-scattering measurementsThe sample cell containing the PSL suspension was set in the

toluene bath and maintained at the required temperature for at least 30 min to allow the temperature to equilibrate before the measurements were performed. SLS measurements with run times of 10 s were taken more than three times for each angle until the standard deviation of three runs of 10 s with a minimum sampling time of 6.25 ns was less than 5%. The run time for the DLS measurements was controlled between 120 and 600 s to obtain consistent statistics for the correlation function; for example, the run time required for measurements on the most concentrated suspension at a scattering angle θ = 30° was only 120 s, whereas for the lowest concentration at θ = 90°, a run time of 600 s was necessary to accumulate sufficient data.

Data analysisThe intensity of the excess scattered light, I(θ), is given by the

following equation, provided that the particle size is not much larger than the wavelength: I(θ) = KNM2P(θ)S(θ), where K is the optical constant, N the number of particles in a unit volume, and M the molecular mass of the particles. P(θ) and S(θ) are, respectively, a particle scattering function and a static structure function of the scattering angle θ. If the concentration, C (= NM), is inserted into the equation, it can be rewritten as

I(θ) = KCMP(θ)S(θ). (1)

The optical constant, K, is given by

K n n CN

= ( ) ,4 2 2 2

4

π d /dAλ (2)

where n is the refractive index of the solvent, dn/dC the refractive-index increment of the solution, NA the Avogadro constant, and λ the wavelength of incident light. In the case where the particle size is determined without determining the molecular mass, we do not need to know accurate values of dn/dC or the Rayleigh ratio. The particle size can be obtained from the radius of gyration (Rg), when Rgq << 1, by using a

Zimm-type plot,

KCI M

R q( )

( ) .θ ≈ +

1 1 13

2g (3)

In the case of larger particles, when Rgq ≈ 1, the following Berry-type plot can be used:

KCI M

R q( )

( ) ,/

/θ

≈ +

1 2

1 221 1 1

6 g (4)

where q is the scattering vector, defined as

q = (4πn/λ)sin(θ/2). (5)

In DLS measurements, the electric-field time correlation function, g(1)(τ), is calculated from the correlation function of the scattered light, g(2)(τ), by using the following Siegert relationship:

g A B g( ) ( )( ) = ( ( ) ).2 1 21τ τ+ (6)

The correlation function, g(1)(τ), was analyzed by the cumulant method, as described in ISO 13321,6 to evaluate the average decay rate Γ

g ( ) ( ) = exp .12

212

τ τ µ τ− + −

Γ � (7)

Here, μ2 is the second cumulant related to particle-size distribution. The diffusion coefficient, D, was calculated from the first cumulant, Γ, as follows:

Γ = Dq2. (8)

From the value of D, the hydrodynamic radius of the particle, Rh, was calculated by using the Stokes–Einstein relationship,

R k TDh

B= ,6πη (9)

where kB, T, and η are the Boltzmann constant, absolute temperature, and solvent viscosity, respectively. DLS measurements were performed for each size of the sample at five different concentrations and seven different angles. The measured apparent diffusion coefficients, Dapp, were extrapolated to infinite dilution (C = 0) and to zero angle (θ = 0) to obtain the true diffusion coefficients D(0).

Results and Discussion

Measurement resultsFigure 1 shows typical time correlation functions of the

scattered light at an angle of 90° for suspensions of PSL particles with average diameters of 100, 70, 50, and 30 nm. There was no obvious multimodal behavior, and a translational diffusion mode was mainly observed. In all of the measurements, the time correlations showed a single mode, and the diffusion coefficients were analyzed by the cumulant method according to Eq. (7). The results for PSL with a particle size of 70 nm (PSL 70) are shown in Fig. 2(a). Each apparent diffusion coefficient, Dapp, can be determined with sufficient precision to allow extrapolation to zero concentration and zero angle. The two lines connecting the extrapolated data to infinite dilution

ANALYTICAL SCIENCES JULY 2011, VOL. 27 753

(C → 0) and low angle (θ → 0) intersected the real value of the diffusion coefficient, D(0).8,9 For PSL 70, the values of D(0) and the hydrodynamic radius, Rh, were estimated to be 5.80 × 10–12 m2 s–1 and 4.01 × 10–8 m, respectively. In Fig. 2(b), the results of SLS measurements for PSL 70 are shown for the same suspensions that were used in the DLS measurements. The Zimm-type plot given by Eq. (3) gave a value for the radius of gyration, Rg, of 3.54 × 10–8 m, whereas the Berry-type plot given by Eq. (4) gave a value of 3.23 × 10–8 m. If the PSL particle has a spherical shape with a uniform density, the relationship Rh = √–5/3Rg holds. The value from the Zimm-type plot gives √–5/3Rg = 4.57 × 10–8 (m), whereas the Berry-type plot

gives √–5/3Rg = 4.17 × 10–8 (m). Those values are in reasonable agreement with the experimental value, Rh = 4.01 × 10–8 (m). Averaged values of √–5/3Rg are always larger than those of Rh because, as calculated by the following pair of equations,5,10,11 the moment order weighted by the polydispersity of Rg is higher than that of Rh:

Rn r r r

n r r rg

d

d=

( )

( ),∫

∫7

6 (10)

Rn r r r

n r r rh

d

d=

( )

( ),∫

∫6

5 (11)

where n(r) is the number-distribution function of particles with radius r.

In Figs. 3(a) and 3(b), typical results for DLS measurements on PSL with a diameter of 50 nm (PSL 50) are shown for two concentration regions. The values of D(0) and the hydrodynamic radius, Rh, were estimated to be 9.35 × 10–12 m2 s–1 and 2.49 × 10–8 m, respectively, in the concentrated region shown in Fig. 3(a). In the dilute region shown in Fig. 3(b), the values of D(0) and Rh were estimated to be 9.31 × 10–12 m2 s–1 and 2.50 × 10–8 m, respectively. The two extrapolations for the concentrated and dilute regions therefore gave almost identical values for the diffusion coefficient, D(0), and for the hydrodynamic radius, Rh.

Note that the dependences of the apparent diffusion coefficients of the PSL suspensions on θ were curved in the cases of both of PSL 70 and PSL 50. These curvatures were more marked at

Fig. 1　Time correlation functions of PSL 100 nm (○), PSL 70 nm (□), PSL 50 nm (△), and PSL 30 nm (◇) at scattering angles of 90° for  the concentrations of 2.4 × 10–4, 4.6 × 10–4, 6.0 × 10–4, and 7.6 × 10–4 g/mL, respectively.

Fig. 2　Dynamic light-scattering measurement for apparent diffusion coefficients (a) and static light-scattering measurement (b) of PSL 70 nm for different concentrations from 0.5 to 0.1 mg/mL. The concentrations: ●, 4.6 × 10–4; △, 3.7 × 10–4; ◆, 2.8 × 10–4; □, 1.8 × 10–4; ▲, 9.3 × 10–5 g/mL. The open circles correspond to data extrapolated to infinite dilute concentrations or to zero angles.

Fig. 3　Dynamic light-scattering measurement for apparent diffusion coefficients of PSL 50 nm for concentrated suspensions (a) and for dilute suspensions (b). The concentrations in (a): ●, 6.0 × 10–4; △, 4.7 × 10–4; ◆, 3.6 × 10–4; □, 2.4 × 10–4; ▲, 1.2 × 10–4 g/mL. The concentrations in (b): ●, 2.7 × 10–4; △, 2.2 × 10–4; ◆, 1.6 × 10–4; □, 1.1 × 10–4; ▲, 5.4 × 10–5 g/mL. The open circles correspond to the data extrapolated to infinite dilute concentrations or to zero angles.

754 ANALYTICAL SCIENCES JULY 2011, VOL. 27

lower concentrations. In a previous paper,12 we briefly described the main reason for the dependence of the apparent diffusion coefficients on the concentration and the scattering angle. It is not caused by the effect of sample polydispersity or of multiple scattering. If polydispersity had a major effect, the measured particle sizes would be larger at lower angles, but the experimental results show the opposite situation. A slight effect of the polydispersity was observed in SLS, even at small angles for more-concentrated suspensions, as shown in Fig. 2(b). The effect of multiple scattering was examined by changing the intensity of the incident laser beam, as shown in Fig. 4. If multiple scattering occurred, the decay rate should change on changing the intensity of the laser beam because of the threshold of the detector count rate. However, the most concentrated sample with the largest particle size (100 nm) did not show a shift in the decay rate as a result of multiple scattering at suitable intensities of the incident laser beam, although the count rate of the scattered light at the lowest laser intensity was insufficient to provide an adequate signal-to-noise ratio, and at the highest laser intensity, the detector was saturated beyond the limit for the count rate.

It appears that the main reason for the dependence of the measurements on the concentration and the scattering angle is the existence of electrostatic interactions between PSL particles, which affect the hydrodynamic behavior of PSL diffusing in water. In the SLS measurements shown in Fig. 2(b), no marked dependence on the scattering angle was observed at low concentrations, although a small dependence on the angle appeared at low angles in concentrated suspensions as a result of polydispersity of the particle size. In general, the dynamic structure factor of scattered light does not always agree with the static structure factor. In the next section, we attempt to estimate long-range interactions in PSL suspensions and to simulate the observed dependence of the apparent diffusion coefficients, Dapp, on the concentration and scattering angle.

Simulation of the apparent diffusion coefficientsAs described in the previous section, the diffusion coefficient

is generally given by Eq. (9); that is, the hydrodynamic radius is calculated as the same value at various angles, although the decay rate, Γ, is dependent on θ according to Eq. (8). However, in some cases where long-range interactions appear, such as in the cases of charged macromolecules,13 lattices,14,15 and colloidal silica,16,17 the diffusion coefficient and the hydrodynamic radius

show angular dependences. This long-range interaction can be expressed as the hydrodynamic function, H(q), according to the following equation:15,16,18 H(q) = (Dapp/D(0))·S(q). S(q) is a structure factor for the system, but it is often difficult to estimate this from the results of SLS. As shown in Fig. 2(b), the value of S(q) approaches unity as the suspensions become more dilute. In many cases, it is more useful to insert the dynamic structure factor, S′(q), as rewritten by using the following equation:13,14,17

D DS qapp = ( )

( ).0

′ (12)

Here, S′(q) reflects long-range interactions of particles in the suspension. We simulated the experimental results of Figs. 2 and 3, and showed that the dependences of the apparent diffusion coefficient on C and θ are not due to experimental errors or multiple-scattering effects. The simulation model is a simple hard-sphere model, but it is sufficiently useful to permit estimating the long-range interactions and to simulate the dependence of the apparent diffusion coefficient on C and θ.

Some methods for estimating S′(q) have been reported for systems of microemulsions19 and block copolymers,20–22 and by simulation.23 By applying the Percus–Yevick expression24 for the hard-sphere interaction potential, the dynamic structure factor for an interaction radius R can be obtained as

′ +S qG A A

( ) =( , )

,11 24φ φ / (13)

where A = 2qR and

G AA

A A A

AA A A

( , ) = [sin ]

[ sin ( )

φ α

β2

322 2

−

+ + −

cos

cos AA

AA A A A

A A A

−

+ − + −

+ −

2

4 3 6

6

54 2

3

]

[ cos {( )cos

( )sin

γ

++ 6}].

(14)

The parameters α, β, and γ can be written in terms of the hard-sphere volume fraction, f, as follows:

α φφ

= ( )( )

,1 21

2

4

+− (15)

β φ φφ

= ( )( )

,− +−

6 1 21

2

4

/ (16)

γ φ φφ

= ( )( )( )

./2 1 21

2

4

+− (17)

Here, f is calculated from the concentration of the suspension, C, the density of the suspension, ρs, and of the particle, ρ, the size of the particle, Rh, and the interaction radius, R, by means of the equation f = CρsR3/(ρRh

3).Figures 5(a) and 5(b) show the results of the simulation

described above for the cases of PSL 70 and PSL 50, respectively. The parameters used for the long-range interaction radii, R, are plotted in Fig. 6. The long-range interaction radii, R, are about ten-times larger than the particle sizes, and they increase with increasing dilution of the suspensions. The figure shows that long-range interactions extend over a considerable volume when no salts or surfactants are added to the suspensions, and there is little screening of the electrostatic interactions between particles. This simulation model does not assume a polydispersity in particle size, so that the curvatures are sharper than those of the

Fig. 4　Plots of the values of the measured apparent diffusion coefficient, Dapp, at various incident laser beam intensities for suspension of PSL 100 nm. The concentration is 2.4 × 10–4 g/mL and the scattering angle is 90°.

ANALYTICAL SCIENCES JULY 2011, VOL. 27 755

experimental results shown in Figs. 2(a) and 3(a). However, the effects of particle sizes and concentration on the angular dependences of DLS are similar to those shown in Fig. 5.

Although this method excludes complex formulations, it is sufficiently accurate to represent the experimental data shown in Figs. 2 and 3, and to estimate the long-range interactions by means of the Derjaguin Landau, Verwey, and Overbeek (DLVO) theory.25 This effect is always observed in the case of charged particles, such those of PSL, and extrapolation to zero angle and zero concentration is necessary for accurate sizing of particles.

Development of the nanoparticle size standardExtrapolation to infinite dilution and to lower angles by using

the high-resolution DLS technique gives accurate particle sizes that are corrected for long-range interactions. We developed a standard for nanoparticles in a liquid phase by means of this DLS technique. The traceability and validity were verified by comparison with two other techniques: DMA and PFG NMR. DMA is a powerful technique for determining particle sizes in an air phase. PFG NMR is useful for observations of chemical species in liquid phases, and can be applied to studies on the diffusion behavior of small particles. Comparisons of the results of DLS with those of DMA and PFG NMR are shown in Fig. 7. The values from DMA26,27 and PFG NMR28 were used after correcting for the distribution of particle sizes by means of Eq. (11). The extrapolation of the apparent radii to infinite dilution and to lower angles by means of DLS gave values that were consistent with those measured by DMA and PFG NMR within their respective uncertainties, which are shown by error bars in Fig. 7. Details of the evaluation of the uncertainty of DLS have been described previously.12

It is noteworthy that the application of the DLS methods stipulated in ISO 13321 or ISO 22412, which are suitable only for a narrow range of angles and concentrations, does not give very accurate particle sizes in a number of cases. For example, in Fig. 7, typical DLS results observed at a single angle and a single concentration are shown. The plots are not compatible with the values measured by other methods. To measure the size of nanoparticles accurately, it is necessary to measure the diffusion coefficients at various angles and concentrations, particularly in the case of charged particles, such as those present in PSL. However, precise DLS measurements are generally not easy to perform. The precision of our DLS measurement is sufficiently high to permit their extrapolation to finite concentrations and to lower angles, which allows the development of accurate size standards for nanoparticles.

Conclusions

Particle sizing in the nanometer size range is routinely performed by using DLS. The ISO 13321 and 22412 standards provide methodologies for photon correlation spectroscopy and

Fig. 5　Simulation results of calculated apparent diffusion coefficients expressed by Eq. (12) for PSL 70 nm (a) and PSL 50 nm (b) for different concentrations. The concentrations in (a): , 4.6 × 10–4; , 3.7 × 10–4; , 2.8 × 10–4; , 1.8 × 10–4;

, 9.3 × 10–5 g/mL. The concentrations in (b): , 6.0 × 10–4; , 4.7 × 10–4; , 3.6 × 10–4; , 2.4 × 10–4;

, 1.2 × 10–4 g/mL.

Fig. 6　Relationships between the concentrations and the long-range interaction radii R simulated for PSL 70 nm (○) and PSL 50 nm (□).

Fig. 7　Comparisons of particle sizes measured by DLS (□), DMA (○), and PFG NMR (△). The error bar shows the uncertainty of each measurement. The crosses (×) show example plots of the results of DLS measurements at one angle and one concentration. The angle and the concentration were typically selected at the widest angle (θ = 150°) and at the highest concentration.

756 ANALYTICAL SCIENCES JULY 2011, VOL. 27

DLS, respectively, but they are limited to appropriate concentrations and scattering angles. PSL particles are recommended for use as standard samples; however, their electrostatic interactions extend to long ranges and the apparent diffusion coefficients and hydrodynamic radii depend on the concentrations and scattering angles.

We performed light-scattering measurements on PSL particles with diameters from 30 to 100 nm in aqueous suspensions at various concentrations and scattering angles. The apparent diffusion coefficients obtained at various angles and concentrations showed the characteristic properties of PSL particles. A simulation based on the hard-sphere interaction potential model was used to represent the long-range interactions between PSL particles. This model was sufficiently accurate to permit simulations of the dependence of the apparent diffusion coefficient on the concentration and scattering angle, and to show that extrapolations are suitable for obtaining accurate values. The values of particle sizes obtained from the extrapolations were consistent with those obtained by the DMA and PFG NMR methods within their corresponding levels of uncertainty.

Acknowledgements

The authors gratefully acknowledge the collaboration of Dr. Kensei Ehara and Dr. Keiji Takahata (NMIJ/AIST, Tsukuba, Japan) in the current work. This work was supported by a grant from the New Energy and Industrial Technology Organization (NEDO).

References

1. B. Chu, “Laser Light Scattering”, 2nd ed., 1991, Academic Press, San Diego.

2. B. J. Berne and R. Pecora, “Dynamic  Light  Scattering”, 1976, Wiley, New York.

3. W. Brown and T. Nicolai, “Dynamic  Light  Scattering”, 1993, Clarendon Press, Oxford.

4. W. Brown, “Light Scattering: Principles and Development”, 1996, Clarendon Press, Oxford.

5. W. Schärtl, “Light  Scattering  from  Polymer  Solutions  and Nanoparticle Dispersions”, 2007, Springer, Berlin.

6. ISO 13321, “Particle  Size  Analysis—Photon  Correlation Spectroscopy”, 1996.

7. ISO 22412, “Particle  Size  Analysis—Dynamic  Light Scattering (DLS)”, 2008.

8. S. Bantle, M. Schmidt, and W. Burchard, Macromolecules, 1982, 15, 1604.

9. M. Wenzel, W. Burchard, and K. Schätzel, Polymer, 1986, 27, 195.

10. M. Schmidt, W. Burchard, and N. C. Ford, Macromolecules, 1978, 11, 452.

11. W. Burchard, Polymer, 1979, 11, 577.12. K. Takahashi, H. Kato, T. Saito, S. Matsuyama, and S.

Kinugasa, Part. Part. Syst. Charact., 2008, 25, 31.13. D. W. Schaefer and B. J. Berne, Phys. Rev. Lett., 1974, 32,

1110.14. J. C. Brown, P. N. Pusey, J. W. Goodwin, and R. H. Ottewill,

J. Phys. A: Math. Gen., 1975, 8, 664.15. J. Gapinski, A. Patkowski, A. J. Banchio, P. Holmqvist, G.

Meier, M. P. Lettinga, and G. Nägele, J. Chem. Phys., 2007, 126, 104905.

16. J.-Z. Xue, X.-L. Wu, D. J. Pine, and P. M. Chaikin, Phys. Rev. A, 1992, 45, 989.

17. B. V. R. Tata, P. S. Mohanty, J. Yamanaka, and T. Kawakami, Mol. Simul., 2004, 30, 153.

18. W. B. Russel, D. A. Saville, and W. R. Schowalter, “Colloidal  Dispersions”, 1989, Sect. 13, Cambridge University Press, Cambridge.

19. E. W. Kaler, K. E. Bennett, H. T. Davis, and L. E. Scriven, J. Chem. Phys., 1983, 79, 5673.

20. D. J. Kinning and E. L. Thomas, Maclomolecules, 1984, 17, 1712.

21. K. Mortensen and J. S. Pedersen, Maclomolecules, 1993, 26, 805.

22. M. Schwab and B. Stühn, Colloid Polym. Sci., 1997, 275, 341.

23. S. N. Thennadil and L. H. Garcia-Rubio, Part.  Part.  Syst. Charact., 2007, 24, 402.

24. J. K. Percus and G. J. Yevick, Phys. Rev., 1958, 110, 1.25. E. J. W. Verwey and J. T. G. Overbeek, “Theory  of  the 

Stability of Lyophobic Colloids”, 1948, Elsevier Publishing, New York.

26. K. Ehara, G. W. Mulholland, and R. C. Hagwood, Aerosol Sci. Technol., 2000, 32, 434.

27. K. Takahata and K. Ehara, Abstracts of Papers, the 7th International Aerosal Conference, 2006, 395.

28. H. Kato, K. Takahashi, T. Saito, M. Suzuki, and S. Kinugasa, Chem. Phys. Lett., 2008, 463, 150.