Making Sense of Brownian Motion: Colloid Characterization by

Invited Historical Review pubs.acs.org/Langmuir

Making Sense of Brownian Motion: Colloid Characterization by Dynamic Light Scattering Puthusserickal A. Hassan,* Suman Rana, and Gunjan Verma Chemistry Division, Bhabha Atomic Research Centre, Mumbai-400085, India ABSTRACT: Dynamic light scattering (DLS) has evolved as a fast, convenient tool for particle size analysis of noninteracting spherical colloids. In this historical review, we discuss the basic principle, data analysis, and important precautions to be taken while analyzing colloids using DLS. The effect of particle interaction, polydispersity, anisotropy, light absorption, and so forth, on measured diffusion coefficient is discussed. New developments in this area such as diffusing wave spectroscopy, particle tracking analysis, microrheological studies using DLS, and so forth, are discussed in a manner that can be understood by a beginner.

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INTRODUCTION The colloidal domain, which typically falls in the range of nanometers to a few microns, deals with an intermediate length scale between true solutions and bulk materials. Paints, pigments, pharmaceuticals, food and beverages, cosmetics, ceramics, personal care products, and so forth, are some of the industrial sectors that deal with colloidal matter at one stage or another of their product development. Average particle size, size polydisperisty, shape and surface characteristics of particulate materials can play a vital role in dictating the efficacy of the formulation or products. Thus, particle size characterization of colloids plays an important role in product development. Several methods such as microscopic imaging, electrical sensing (Coulter) counters, hydrodynamic or field flow fractionation, disc centrifuge particle sizing, size exclusion chromatography, scattering techniques, and so forth, are employed for submicron particle size analysis. Among these, microscopic imaging provides direct estimate of size, shape and texture in high resolution. However, it is important to count a large number of particles from the images, in order to get good statistical average. Further, when electron microscopy is employed for imaging, it can only analyze dry specimens after solvent removal, unless the cryofixation technique is used. This may affect the native structure of colloids, as they are prone to structural changes, aggregation, and growth during solvent evaporation. This is especially true for self-assembled materials, certain polymers, and biological materials. Thus, in situ colloid characterization techniques such as dynamic light scattering (DLS) becomes an important tool for screening of several formulations. Some of the advantages of DLS for particle sizing are its ability to provide ensemble averaged estimate of particle size in suspensions, wide accessible range of particle sizes in the submicron range and fast data acquisition. This review presents © 2014 American Chemical Society

a brief introduction to the technique of DLS with special emphasis on data analysis and what to look for in interpreting the DLS data.

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THEORY OF DYNAMIC LIGHT SCATTERING DLS makes use of two common characteristics of colloids, i.e., the Tyndall effect (scattering) and Brownian motion. Among other factors, the intensity of scattered light from a colloidal suspension depends on the scattering angle (θ) and the observation time (t). This leads to the development of two types of scattering experiments, namely, the static light scattering (SLS) in which the time averaged scattering intensity is measured at various scattering angles and the DLS in which the time dependence of the intensity is measured. To obtain particle size information using SLS, the dimension of the colloids should be in the same range as that of the wavelength (λ) of light or at least greater than λ/20. However, DLS can measure particles that are much smaller than the wavelength of light. The time dependence of scattering intensity arises from the fact that, at ambient conditions, colloidal particles are no longer stationary in the suspension medium, rather they move in a random walk fashion by the Brownian motion process. When a coherent monochromatic radiation is impinged on a collection of particles, each particle acts as a secondary source due to scattering of the radiation. Since particles are moving randomly in space, the distance traveled by the scattered waves from the particle to the detector varies with time. Due to the interference of scattered waves from particles, the net intensity fluctuates randomly in time as the relative positions of the particles change. This produces random “speckles” in space, Received: May 9, 2014 Revised: July 16, 2014 Published: July 22, 2014 3

dx.doi.org/10.1021/la501789z | Langmuir 2015, 31, 3−12

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Invited Historical Review

which appear as randomly fluctuating dark or bright spots in a detector. In order to observe such speckles, it is necessary that the spatial resolution of the photodetector is high enough to measure these fluctuations in time. The area (or size) of a single speckle known as the “coherence area” (Acoh) depends on λ and the distance between the scattering object and detector (r), as given by the relation1 Acoh =

λ 2r 2 πx 2

(1)

where x is the radius of the scattering volume. The resolution of the measurement depends on the spatial coherence factor, which is decided by the ratio of the aperture area of the photodetector to the coherence area (Acoh). Thus, it is necessary to use appropriate pin holes (aperture) in front of the detector in order to observe the random fluctuations in the signal in DLS. In other words, a high value of the coherence factor can be achieved by decreasing the detector aperture size. In DLS, the characteristic time of fluctuations in the scattered intensity is measured, and it depends on the diffusion coefficient of the particles undergoing Brownian motion. Small particles diffuse in the medium relatively rapidly resulting in a rapidly fluctuating intensity signal as compared to the large particles, which diffuse more slowly. Quantitative information regarding the time scale of these fluctuations in the scattered intensity is obtained by a signal processing technique known as autocorrelation. Because of this reason, this technique is known also under the name photon correlation spectroscopy (PCS). To understand the autocorrelation of scattered intensity, consider a time varying signal, I(t) monitored at different intervals of time. If we represent the scattered intensity at an arbitrary time as I(t) and those after a delay time τ as I(t + τ), the normalized autocorrelation function of the scattered intensity, g2(τ) can be written as g 2 (τ ) =

I (t )I (t + τ ) I (t ) 2

Figure 1. Schematic representation of fluctuations in the intensity of scattered light and the corresponding autocorrelation functions from suspensions of different size particles. 2

g 1(τ ) = A ·e−Dq τ + B

(4)

where A is the amplitude of the correlation function, which depends on the spatial coherence factor discussed above, B is the baseline, D is the translational diffusion coefficient of the particles, and q is the magnitude of the scattering vector given as (4πn/λ) sin(θ/2). Here, n is the refractive index of the medium. For spherical particles, the hydrodynamic radius Rh can be obtained from the translational diffusion coefficient (D) using the Stokes−Einstein relationship

D = kT /(6πηR h)

(5)

where k is the Boltzmann’s constant, η is the solvent viscosity, and T is the absolute temperature. If the particle is nonspherical, then Rh is often taken as the apparent hydrodynamic radius or equivalent sphere radius. This is how the particle size information is arrived in a typical DLS experiment. Figure 2 shows a representative electric field

(2)

It can be seen that when the delay time τ is zero, the signal is perfectly correlated and the numerator in eq 2 (unnormalized correlation function) yields a value of ⟨I2⟩, and when τ is infinity, it is perfectly uncorrelated yielding a value of ⟨I⟩2. From statistics, one can see that ⟨I2⟩ is greater than or equal to ⟨I⟩2. The decay of this function follows a characteristic time scale depending on the diffusion coefficient of the scatterers. For photo counts obeying Gaussian statistics, g2(τ) is related to the first-order correlation function of the electric field g1(τ) by the relationship g 1(τ ) = [g 2(τ ) − 1]0.5

(3) Figure 2. Electric field autocorrelation function of scattered light from polystyrene particles of size 54 nm and the corresponding fit to the data using single exponential decay.

Figure 1 shows a schematic representation of the fluctuations in intensity for two different sized particles and the corresponding correlation functions. It may be noted that for a rapidly fluctuating signal, the correlation function decays faster than the one obtained from slow fluctuations.

correlation function obtained from a colloidal suspension and its corresponding fit using a single exponential decay (solid line). The obtained hydrodynamic radius of the colloids (assuming as spheres) is 54 nm. The nearly single exponential decay of the correlation function suggests a narrow distribution of particle sizes in the suspension. For an unknown suspension, the correlation function is usually calculated over wide range of time starting from microseconds to seconds, as the characteristic relaxation time can fall anywhere in this range. Since the

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ANALYSIS OF AUTOCORRELATION FUNCTION The essence of the technique lies in analyzing the correlation function to obtain the particle size information. For a suspension of monodisperse, spherical particles undergoing Brownian diffusion, the autocorrelation function decays exponentially with the delay time τ and is given as 4


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time is varied over a wide range, the x-axis is displayed in log scale for better visualization of the curve. Note that the maximum amplitude of the correlation function observed in the present instrumental setup is 0.83, which is lower than the theoretical limit of 1. It may be noted the amplitude of the correlation function depends on the instrument geometry (in particular the aperture size) and should be kept in such a way to attain maximum amplitude for better resolution of measurements. The effect of aperture size on amplitude of the correlation function is exemplified in Figure 3, by measuring

There are different ways one can take to extract the particle size (diffusion coefficient) information from the measured data in polydisperse samples. One approach is to assume a known analytical distribution and adjust the variables so as to get a best fit to the experimental data. However, the main problem with this approach is the assumption of a given form for the distribution which often leads to ambiguous results. The second approach is to calculate the mean and variance of the distribution, without considering the higher moments. Under this approach, the method of cumulants2 has been widely used to analyze narrow polydispersity in DLS experiments and is the only method that is currently recommended in the international standard ISO 13321.3 Finally, one can also obtain the full distribution by performing an inversion of the Laplace integral equation. Some of the commonly used methods, under this category, are non-negatively constrained least-squares (NNLS),4 exponential sampling,5 and CONTIN.6,7 A brief description about different techniques employed commonly in DLS data analysis is given below: (a). Method of Cumulants. This is the most common method used in DLS analysis and is applicable only for narrow monomodal distribution. The term exp(−Γτ) in eq 6 can be expanded about a mean value Γ̅ such that exp( −Γτ ) = exp(−Γ̅ τ )· exp(− [Γ − Γ̅ ]τ ) = exp( −Γ̅ τ ) ·[1 − (Γ − Γ̅ )τ + (Γ − Γ̅ )2 τ 2/2 ! + ....]

Figure 3. Correlation function from 10% w/w aqueous Pluronic P123 solution at different aperture sizes of the instrumental set up. Note that the amplitude of the correlation function decreases with an increase in aperture size.

(7)

The relative magnitude of contributions of the terms in the bracket falls of rapidly with increasing order. When G(Γ) is relatively narrow, terms with order 3 and above can be neglected, and substituting it in eq 6, we get ⎡ g 1(τ ) = e(−Γ̅ τ)⎢ ⎣

the autocorrelation function in a micellar solution formed by Pluronic P123 triblock copolymer. With an increase in the aperture size, the amplitude of the correlation function decreases systematically, highlighting the importance of the coherence area discussed above.

∫0

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POLYDISPERSITY IN DIFFUSION COEFFICIENT In many practical systems, there exists a distribution of particle sizes and scattering from such polydisperse population of particles leads to deviation from the expected single exponential decay of the correlation function. For a polydisperse distribution of diffusion coefficient, the RHS in eq 4 can be written as a sum of exponentials, weighted by their amplitudes. For a continuous distribution, each D contributes its own exponential, and the observed correlation function can be represented as an integral weighted by the distribution function, which can be expressed as 1

g (τ ) =

∫0

∞

G(Γ) dΓ − τ

∞

G(Γ) dΓ +

τ2 2

∫0

∞

∫0

∞

G(Γ)Γ dΓ + Γ̅ τ

⎤ (Γ − Γ̅ )2 G(Γ) dΓ⎥ ⎦ (8)

By definition, the mean (Γ̅ ) and variance (μ2) of the distribution G(Γ) are Γ̅ =

∫0

μ2 =

∞

∫0

G(Γ)Γ dΓ ∞

(9)

G(Γ)(Γ − Γ̅ )2 dΓ

(10)

Thus, eq 8 can be written as ⎡ μ τ2 ⎤ g 1(τ ) = e(−Γ̅ τ)⎢1 + 2 ⎥ ⎢⎣ 2 ⎥⎦

∞

G(Γ) exp( −Γτ ) dΓ

∫0

(6)

where Γ = Dq is the decay constant for a given size. The distribution G(Γ) represents the relative intensity of light being scattered with decay constant Γ and will depend on the volume fraction and size of scatterers. Note that, eq 5 is the Laplace transform of G(Γ) with respect to Γ. Thus, in principle, G(Γ) can be obtained by performing an inverse Laplace transform on the measured correlation function. However, the solution to this Laplace transform inversion is nontrivial. The occurrence of measurement noise, baseline drifts, dust interference and the exponential function under the integral all put together make this equation ill-conditioned.

(11)

When μ2τ ≪1, eq 11 can be simplified as

2

2

g 1(τ ) = e−Γ̅ τ + μ2 τ

2

/2

(12)

Taking the natural logarithm, adding the background B in eq 12 gives ln[g 1(τ )] = ln B − Γ̅ τ +

μ2 τ 2 2

(13)

This equation is the one that is used traditionally in cumulants analysis. By fitting ln(g1(τ)) to a quadratic in τ, one 5


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can get the mean, Γ̅ and the variance, μ2. The ratio of variance to the square of the mean is a measure of the polydispersity of diffusion coefficient and this is very often represented as a polydispersity index (PI). Figure 4 shows a plot of ln[g1(τ)]

selected. The coefficients bi of Γi are obtained from the best fit to the data within the constraint of non-negativity of size distribution, which inherently corresponds to the physical meaning of the distribution. The coefficients bi are obtained by minimization of the expression N

χ2 =

M

∑ [g(1)(τj) − ∑ bi exp(−Γiτj)]2 j=1

i=1

(14)

where M is the number of decay constants in the distribution with the constraint that coefficients bi of Γi are positive or zero values, and N is the number of data points considered in the fit. (c). Exponential Sampling. Exponential sampling is another method used for reconstructing the whole decay distribution function G(Γ) in an iterative way by calculating a set of bi of Γi values. This method is based on the eigen functions ψω(Γ) and eigen values λω of the Laplace transform integral equation

∫0

∞

ψω(Γ) exp( −Γτ ) dΓ = λωψω(τ )

(15)

and leads to an algorithm for finding an appropriate grid of exponentially spaced Γ values according to Γn + 1 = Γ1 exp(nπ /ωmax )

Figure 4. Variation of autocorrelation function with delay for a polydisperse population of particles and its corresponding fit using the method of cumulants. A nonlinear variation of the plot is an indication of polydispersity in size of the particles.

(16)

where ωmax is the highest possible value of ω, determined by the experimental noise that does not need negative values for bi to minimize expression 14. This value is determined by trial and error by increasing ω until further changes yield nonphysical (negative) values for the coefficients of the distribution. (d). CONTIN. CONTIN is the most widely applied Laplace integral transform algorithm developed by Provencher that uses a regularized NNLS technique.6,7 During the fitting process, instead of minimizing the residual as shown in eq 14, the algorithm minimizes the regularized residual, and an appropriate weighting function is included. So the minimization equation takes the form

versus τ for a polydisperse population of polystyrene particles and its corresponding fit using a quadratic equation. The linear variation (blue line) is the one expected for monodisperse particles while the nonlinear variation (red line) with a deviation at long time indicates polydispersity in the sample. The fitted values of mean relaxation time and PI are 616 μs and 0.34, respectively. Application of the cumulant analysis (eq 12) to polydispersed system requires that the criterion μ2τ2 ≪ 1 is satisfied. As PI increases, at large τ values, μ2τ2 becomes comparable to 1 and leads to deviation from the expected decay of the correlation function. This aspect has been discussed in detail elsewhere8 and a modified cumulants approach is used to minimize such errors. Simulation of the correlation function for a polydisperse system and its analysis using traditional cumulants show that there is significant deviation of the fitted curve from the correlation function at long correlation time. The correlation time above which such deviation occurs depends on the polydispersity of the system. This necessitates the truncation of data at long correlation time or use of an appropriate weighting function so as to eliminate the data at large τ. Such elimination of data at large τ can lead to an underestimate of the mean and variance of the distribution thereby limiting its use to small polydispersity, say PI < 0.1. A modified nonlinear equation can be employed to circumvent this problem that does not require any truncation of the data. By this approach, it could be possible to extend the maximum limit of PI that can be evaluated from DLS data without any truncation, provided the distribution is still unimodal in nature. (b). NNLS. This method is used for broad monomodal distribution or multimodal distributions. In this method, a set of a discrete number of decay constants Γi that represent the continuous distribution G(Γ), spaced in a particular pattern (either linear or logarithmic) over a chosen ranges of Γ, is

N

χ2 =

∑ (1/σi2)[g(1)(τj) − ∫ G(Γ)e−Γτ dΓ]2

+ α 2 || LG(Γ ||2

j=1

(17)

where α is the regularization parameter and L is the operator of the regularizer, which is usually taken as the second derivative. It uses a nonlinear statistical technique to smooth the solution, and the regularization parameter is chosen on the basis of F-test and confidence levels. In addition to the non-negativity constraint and regularization, it uses an additional principle known as the parsimony principle that prefers the distribution function, which reveals the least amount of detail or oscillation. The usefulness of this technique in inverting the correlation function to particle size distribution can be understood from Figure 5, where the correlation function for two monodisperse particles (open symbols) and a bimodal mixture of the two particles (closed symbols) are shown. The individual particles have a diameter of ∼50 nm and ∼220 nm, respectively. The data from individual particles can be fitted well using a single exponential decay (fit not shown), while the mixture showed deviation from single exponential decay. However, the data can be fitted well by including a polydispersity using cumulants approach (red line). This approach assumes a monomodal (single peak) distribution and gives an average radius of 121 nm with polydispersisty index 0.13. To verify this result, the 6


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(3)

(4) Figure 5. DLS data from polystyrene particles of average sizes 50 nm, 200 nm, and their mixture. The red line is a fit to the data using the cumulants method.

data were analyzed by CONTIN, and it reveals the presence of two peaks in the distribution (Figure 6), indicating that

(5)

Figure 6. Size distribution of particles in a mixture containing polystyrene spheres of sizes 50 and 200 nm, as revealed by CONTIN analysis of the data presented in Figure 5.

(6)

cumulants analysis is not valid in this case. Thus, for samples with large PI, say >0.1, it is important to analyze by other multimodal methods before concluding the average size obtained from the cumulants method.

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WHAT TO DO AND WHAT NOT TO DO IN DLS There are several precautions to be taken during sample preparation and data analysis for DLS measurements. These are discussed in detail in the ISO standard.3 Some of the important points are (1) The scattering intensity from the sample (represented as the count rate) shall be at least 10 times that of the solvent. This will ensure a good signal-to-noise ratio. The scattering intensity depends on particle size as well as particle concentration, among other factors. Small particles scatter weakly, as compared to large particles and hence the particle concentration should be optimized based on the nature of the particles (2) For weakly scattering suspensions, especially those with particle size less than 50 nm, it is desirable that the dispersion medium be filtered through 0.2 micron filters

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to avoid interference from dust particles or other contaminants. In the analysis of DLS data using the Stokes−Einstein relationship, it is assumed that there is no effect of interparticle interaction on diffusion of the particles. Moreover, the scattering intensity decreases for strongly charge stabilized dispersions. In such cases it is desirable to perform the experiments at different concentrations of electrolytes, say NaCl, to reduce the effect of particle interactions. Too high electrolyte concentration can lead to attraction between particles and leads to coagulation. Thus, an optimum electrolyte concentration may be identified from trial and error. It is desirable that the amplitude of the correlation function be not less than 80% of the ideal value as determined using a calibration standard. Poor scattering intensity, increase in aperture size of the detector, nonergodicity of the samples, and so forth, can lead to decrease in amplitude. For each sample, at least six separate measurements be performed and check for the reproducibility of measurements by overlay of the correlation function from different measurements. The counting time for each run can be adjusted based on the scattering intensity such that a smooth correlation function is obtained. Too small measurement duration will lead to poor averaging, and too large time duration can sometimes introduce spurious results due to interference from dust particles, if the sample is not clean. Thus, multiple measurements with duration of 60 seconds is preferred. For good results, the standard deviation in the average count rate be less than 5%. From several successive runs, outliers can be rejected using standard statistical methods. Among several repeated runs, the runs with low count rates will be preferred, as any interference from dust particles can lead to increase in the count rate. For narrow polydisperse samples, say PI less than 0.1, the average intensity weighted particle diameter and PI are reported as obtained from cumulant analysis. The standard deviation in the average diameter is less than 5%. When the PI is greater than 0.1, the data may be analyzed further by any of the multimodal analysis methods and check for multiple peaks in the distribution. If the distribution is unimodal in nature, the values obtained from cumulants are still valid. However, for multimodal distribution, the intensity weighted size distribution obtained from NNLS, CONTIN, and so forth, can be used as a guide to infer the nature of polydispersity in the sample and the approximate range of particle sizes in each peak. For precise estimate of the particle size in such dispersions, the particles corresponding to different peaks be separated using appropriate methods (such as ultracentrifugation, size exclusion chromatography, etc.) and analyzed again independently.

INTENSITY WEIGHTING: NUMBER-AVERAGE VERSUS Z-AVERAGE The hydrodynamic size distribution obtained from the analysis of DLS data are primarily intensity weighted distributions and hence care should be taken while comparing the size distributions obtained from other methods such as number or volume distributions. The intensity of light scattered by 7


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particles in a given class is proportional to the number of particles (N) in that class, square of the particle mass (M) and the intraparticle interference (form factor) term, P(q,R) which is a function of particle size and scattering vector. Taking these factors into account, the average diffusion coefficient (D̅ ) obtained by DLS can be written as D̅ =

2 ∑i NM i i P(q , R )Di 2 ∑i NM i i P(q , R )

(18)

This suggests that the average diffusion coefficient of a polydisperse population, obtained by DLS, is biased toward large sizes, as the diffusion coefficient is weighted by M2. Since the particle scattering power also depends on the refractive index of the particles, the distribution by weight or number can only be obtained if the particle refractive index is known. In the limiting case of particles that are small compared with the wavelength of light, the form factor can be approximated to unity, and the measured diffusion coefficient becomes the zaverage diffusion coefficient. Since the diffusion coefficient is inversely proportional to the diameter (d) and M is proportional to d3, the average diameter (d̅) obtained by DLS can be written as

d̅ =

Figure 7. Scattering vector dependence of the relaxation rate observed from different size micellar solutions. A linear variation of Γ versus q2 is an indication of pure translational diffusion. The slope decreases with a decrease in diffusion coefficient.

dependence of relaxation rate in highly polydisperse systems. For this reason, the interference from dust or other large particles which are present only in small volume fractions can be minimized by performing experiments at large scattering angles. However, this procedure may mask the presence of small fractions of large particles actually present in a bimodal distribution.

6 ∑i Nd i i 5 ∑i Nd i i

(19)

By definition, the number-average (dn), area average (da) and weight-average (dw) diameter of polydisperse spherical particles follow the order dn ≤ da ≤ d w ≤ d ̅

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ANISOTROPIC PARTICLES Anisotropic particles can undergo a rotational diffusion motion in addition to translational diffusion and may affect the decay of the intensity correlation function. One of the consequences of coupling between translation and rotation motion is the nonlinear variation of Γ versus q2 plots. For rigid rod-like particles, the resultant correlation function can be expressed as9

(20)

The equality sign occurs for monodisperse samples. This suggests that, for a polydispersed system, the average size obtained by DLS will be significantly larger than the number or volume average diameter usually determined by TEM. The higher the polydispersity, the larger will be the deviation from number-average or volume average.

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2

g1(τ ) = S0(qL)e−(Dq τ) + S1(qL)e−(Dq

ANGLE DEPENDENCE OF RELAXATION RATE For particles undergoing translational diffusion, the mean relaxation rate (Γ) follows a q2 dependence with respect to the scattering vector, q. Figure 7 shows the variation of average relaxation rate as a function of q2, for a suspension of different sized micelles. A linear variation of the plot is an indication of pure translation diffusion of the micelles and from the slope of the plot the average diffusion coefficient is obtained as 164 × 10−8 cm2/s and as 26.4 × 10−8 cm2/s, respectively. However, for a broad distribution of particles, deviation from the q2 dependence can occur since the angular dependence of scattered intensity for different size particles are different. For particles much smaller than the wavelength of light, say 30 nm or less, the scattering follows Rayleigh law and hence the scattering is isotropic. However, when particle dimension becomes comparable or larger than the wavelength, the forward scattering is more than backscattering. This leads to an increase in the relative contribution of scattering intensity from large particles at small angles. This means that, for polydispersed particles, the measured intensity weighted average diffusion coefficient can yield different results at different scattering angle. This will result in comparatively large average size, when measurements are performed at low scattering angle. This factor should be kept in mind while comparing the angular

2

+ 6Dr )τ

+ ...

(21)

where Dr is the rotational diffusion coefficient, L is the length of the rod and the functions Si(qL) are weighting factors that depend on the magnitude of qL. The weighting factors Si(qL) that represent the contribution to the total scattered intensity from rotation and translation are derived by Pecora.9 As q → 0, S0(qL) approaches 1 and S1(qL) vanishes. Thus, in the limit of q → 0, the extrapolated value of Γ̅ /q2 gives the translational diffusion coefficient. Incorporation of translational-rotational coupling at high q values leads to a nonlinear variation of Γ̅ veruss q2 for rod-like particles. Using depolarized dynamic light scattering (DDLS) in which the incident light is vertically polarized with respect to the scattering plane and horizontally polarized scattered light is monitored, the contributions from translational and rotational diffusion coefficients of anisotropic particles can be separated out. For such geometry, the correlation function can be represented as1 g(1)(τ )VH = A e−(Dq

2

+ 6DR )τ

(22)

This shows that, in DDLS, a plot of Γ̅ versus q is linear with a finite value of intercept. From the slope and intercept of this plot, both rotational and translational diffusion coefficients can be obtained and hence estimate the length (L) and diameter (d) of the rod-like particles. 2

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that are moving in and out of the scattering volume. These effects become important when the average number of particles N in the scattering volume does not satisfy the condition N2 ≫ N. In practice, it is found that for N larger than about 1000, this condition is fulfilled. For a given volume fraction, the number density of the particle decreases with increasing particle size and thus the effect of number fluctuations are to be expected for higher particle sizes. Number fluctuation lead to an additional time decaying term in the measured correlation function which is much slower than the decay due to Brownian motion of the particles, thus the measured size will be overestimated when neglecting this effect. Sedimentation of large particles in the suspension also put constraint in particle sizing using DLS. Micron sized particles can undergo appreciable sedimentation motion during the experiment if their difference in density is large enough. Due to this reason, DLS is not advisible for particles whose sizes are above a few microns.

The average translational diffusion coefficient obtained from polarized DLS can be used to obtain the length of rod-like or ellipsoidal particles, if the diameter of the particles is known. For example, the translational diffusion coefficient can be related to the axial ratio, p = L/d by the Broersma relationship10 D=

kBT [ln p + ς] 3πηL

(23)

where ζ is a function of the axial ratio, p, which are derived independently by several approaches. For small axial ratios (in the range of 2 to 30), the parameter ζ can be obtained as11

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ζ = 0.312 + 0.5656/p − 0.05/p2

(24)

EFFECT OF INTERPARTICLE INTERACTIONS IN DLS The discussions made so far with respect to the analysis of DLS data rests on the assumption that the particles are dilute enough to neglect the effect of any interparticle interactions. When interparticle interactions are present, the diffusion of one particle is affected by the presence of neighboring particles. The measured apparent diffusion coefficient, Dapp can be related to the infinite dilution diffusion coefficient, Do as Dapp = Do[1 + kDc]

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DLS OF ABSORBING PARTICLES: THERMAL LENSING EFFECT Absorption of light by the particle suspensions can complicate DLS measurements and analysis. One way to overcome the problem of absorption is to choose the wavelength of incident radiation such that the absorption of light is as small as possible. However, certain colloids such as pigments, conjugated polymer particles, some of the metal or semiconductor nanoparticles etc. absorb strongly in the visible range. To minimize absorption, a low particle concentration is favorable, while good scattering intensity favors high concentration. These two opposing effects put restrictions in accessible concentration range. The major sources of errors in absorbing particles arise from the thermal lensing effect and convective flow in the scattering volume.13 The absorption of light from an intense source like laser results in the local heating of the solution. This increase in temperature leads to decrease in the density and refractive index of the fluid in the scattering volume. The change in refractive index creates a “lens” in the solution and refraction of the illuminating beam leads to changes in the beam divergence. This transforms the cylindrical incident beam to a conical geometry. This effect is referred to as thermal lensing or coning effect. Another effect of local heating of the solution is the convective flow of liquid in the scattering volume due to changes in density. Both these effects can influence the nature of the correlation function in dynamic light scattering, and a new analytical treatment has been proposed to circumvent this problem.13 The occurrence of convective motion in suspensions of polydisperse particles can be identified from an additional oscillatory term superposed on the exponential decay of the correlation function. The thermal lensing or coning effect leads to a spread in the angle between the incident and scattering beam and hence a spread in the scattering vector. Incorporation of this effect shows that the plot of Γ̅ versus q2 gives a positive intercept and a modified analytical expression is proposed for Γ̅ . It follows that

(25)

where kD is often called the diffusion virial coefficient and c is the concentration. For a particulate system, Do can be evaluated by measuring Dapp at different concentrations, c, and extrapolating the data to zero concentration. However, for self-assembled structures, such extrapolation becomes difficult unless the aggregation number remains practically constant in the concentration range of investigation. The magnitude and sign of kD depends on the nature of particle interaction and hydrodynamic factors. In the absence of any hydrodynamic effects, repulsive interactions such as hard sphere, electrostatic or steric factors lead to an increase in the diffusion coefficient while for attractive interaction (van der Waals, depletion, etc.) diffusion coefficient decreases with concentration. For spherical particles, at low volume fractions, the effect of particle interaction can be related to the inter particle structure factor S(q) by the relation Dapp = Do/S(q), which can be approximated as Do/S(0) for DLS. This accounts for the thermodynamic contribution to the diffusion virial coefficient. In addition to this, the diffusion coefficient can also be affected by the hydrodynamic interactions. The hydrodynamic interaction is significant for nonspherical particles even at low particle volume fractions and this correction needs to be applied before data analysis. Combining the thermodynamic and hydrodynamic interactions, the diffusion virial coefficient can be expressed as12 kD = (2MB2 − k f − v ̅ ) (26) where M is the aggregate molecular weight, B2 is the osmotic second virial coefficient, kf is the hydrodynamic friction virial coefficient, and v ̅ is the specific volume of the aggregate.

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NUMBER FLUCTUATIONS AND SEDIMENTATION Since DLS measures a collective phenomenon from an ensemble of particles, it is important to ensure that the number density of particles in the scattering volume is large enough to remain constant during the experiments. If the particle number density is small, it may complicate the intensity correlation function due to number fluctuations that are caused by particles

Γ̅ = C − Cαx

(27)

where C = D/2((4πn)/λ)2 and α = cos2(Δ/2); x = cos (θ); D is the diffusion coefficient, Δ is the coning angle. Thus, a plot of Γ̅ against cos(θ) yields the value of D and the coning angle Δ. 9


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dependent viscosity of the medium. In DLS, the first-order autocorrelation function, g(1)(τ) is related to the MSD, Δr2(τ), by the relation:

EXTENSION OF DLS: DIFFUSING WAVE SPECTROSCOPY, PARTICLE TRACKING AND MICRORHEOLOGY To extract particle size information using traditional DLS, it is necessary that the particle concentrations be low enough that only single scattering events are involved. At high particle concentrations, multiple scattering can occur, and efforts have been made to analyze correlation function from multiply scattered light. In the extreme case of strong multiple scattering, light waves propagate through the medium by several scattering events, and this can be viewed as a diffusion of photons through the medium. Analysis of correlation function from such medium led to the development of a new technique called diffusing wave spectroscopy (DWS) which is essentially a DLS of strong multiply scattered light.14 Experimentally, DWS uses the same data acquisition and processing as that of DLS, and the photon detection is carried out either as transmission or backscattering geometry. The light path in the medium should be such that only completely randomized diffused light arrives at the detector. By knowing the transport mean free path of the photon in the medium (which depends on the particle concentration) and the sample thickness, the diffusion coefficient of the particles can be estimated. One advantage of DWS over traditional DLS is that it can probe the dynamics (motion) of the particles at very small length scales as compared to DLS because the phase difference introduced at every single scattering event gets added up in the multiply scattered light. A new variant in traditional particle size analysis from Brownian motion is the tracking of particles using digital microscopy.15 Here, the light scattered from individual particles is captured by a high-resolution digital camera attached to a microscope, and each particle is considered as a point source of scattered light. A series of images are captured at specific time intervals decided by the frame rate of video imaging. From the location of the particles in each image, the Brownian trajectories of the particles can be reconstructed. The diffusion coefficients of the particles are obtained from the trajectories of the particles and a number-weighted size distribution can be calculated. Since this is a direct counting technique, based on the number of scattering objects in the image, there is no need to apply any model to convert from intensity weighting to number weighted results. However, in order to image the scattering from individual particles, the scattering intensity should be sufficiently higher than solvent scattering. This depends on the refractive index difference between particle and solvent, and thus high refractive index of the particle is preferred as the particle size decreases. Moreover, it employs much lower concentrations compared to DLS, and hence a high dilution of the suspension is needed, which limits its use in studying structures that are susceptible to changes upon dilution. Another application of DLS or particle tracking is its ability to probe the rheological properties of complex fluids at a microscopic length scale, often called microrheology.16 In this approach, one measures the local displacement of embedded colloidal probe particles and converting this displacement into rheological properties of the medium, from the known particle dimension. In microrheological measurements using DLS, the mean squared displacement (MSD) of probe particles of known size in the medium is measured as a function of time and converts this time-dependent MSD into frequency

⎛ 1 ⎞ g(1)(τ ) = exp⎜ − q2Δr 2(τ )⎟ ⎝ 6 ⎠

(28)

This allows us to calculate the MSD of the particles from the correlation function and the influence of the fluid on the thermal motion of the scatterers will be reflected in the MSD. Figure 8 shows the variation of MSD of polystyrene particles of

Figure 8. Variation of mean square displacement (MSD) of particles dispersed in a Newtonian fluid and a viscoelastic fluid. The nonlinear variation of MSD with time is evident for particles in a viscoleastic fluid.

100 nm diameter dispersed in two different micellar fluids; one spherical micelles with a Newtonian behavior, and second elongated micelles with viscoelastic behavior, at the same volume fractions of micelles. In Newtonian fluid, the MSD is linear with time while a nonlinear variation is observed as the fluid became viscoelastic. Assuming the complex fluid to be an isotropic, incompressible continuum around a sphere, the complex shear modulus G(s) of the fluid can be obtained from the MSD using a generalized Stokes−Einstein relationship17 G (s ) =

kBT πas⟨Δr 2(s)⟩

(29)

where Δr (s) is the Laplace transform of the MSD, kBT is the thermal energy, a is the radius of the probe particle, and s is the Laplace frequency. The above equation assumes that the Stokes−Einstein relation valid for Newtonian fluids can be generalized to viscoelastic fluids, and the inertial effects on probe motion can be neglected. The obtained G(s) can be fitted to a polynomial in s, which can then be used to estimate the complex modulus G*(ω) in the frequency (ω) domain, by the method of analytic continuation17 (substituting s = iω in the fitted form). The real and imaginary parts of G*(ω) are the storage and loss moduli of the fluid, respectively. The storage modulus G′ represents the elastic component of the stress, while the loss modulus G″ represents the viscous component. The acceptability of the generalized Stokes−Einstein relationship rests on the assumption that the inertial effects on the probe particles can be neglected, and the treatment of the solution as a continuous, incompressible viscoelastic medium is valid. This has been addressed in detail by Levine and Lubensky.18 To neglect the inertial effects, the viscoelastic 2

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Notes

penetration depth should be larger than the particle radius, a. For the motion of particle having density ρ in a medium of modulus G, the above condition is satisfied for frequencies (ω) much lower than (9G/2a2ρ)0.5. For the assumption of continuum viscoelasticity to be valid, the probe particle radius should be larger than the characteristic dimension of the microstructure in the medium (for example, the mesh size (ξ) of entangled polymers). Several systems have been investigated for its microrheological properties using DLS and DWS.19−26 The advantage of this technique is that rheological experiments can be performed on relatively small sample volumes as compared to that required for macrorheology. This can be a major advantage in characterizing many costly biological fluids. Moreover, this technique can be used to probe complex moduli over an extended frequency range that is inaccessible by conventional rheometery.

The authors declare no competing financial interest. Biographies

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SUMMARY DLS has emerged as a nondestructive technique for particle size characterization of colloidal suspensions, mainly in the submicron range. DLS is an ensemble averaged technique, and it measures the time dependence of scattering intensity to compute the autocorrelation function. The decay of the correlation function is fitted with appropriate models to extract diffusion coefficient of particles undergoing Brownian motion. Obtaining particle size distribution of spherical particles with narrow polydispersity from the measured correlation function is rather straightforward. However, it is important to consider various factors that affect the measured diffusion coefficient while reporting the size distribution. Interparticle interaction, particle anisotropy, and so forth, affect the measured diffusion coefficient and hence should be used as an apparent diffusion coefficient. Appropriate corrections to the diffusion coefficient can be applied to account for interparticle interactions. Anisotropic particle dimensions can be calculated using analytical expressions that relate the dimensions of the particles to the average diffusion coefficient. Measurement of the relaxation rate of the correlation function at different angles can identify true translational diffusion of the particles. Particle polydispersity, anisotropy, absorption of light by the particles, and so forth, affect the angular dependence of the relaxation rate. DLS is a low-resolution technique and hence cannot resolve distribution peaks that are closely spaced. Different algorithms are available for analysis of suspensions comprising broad and multimodal distributions. This provides intensityweighted particle size distribution and can be used as a guide to assess the presence of agglomerates or widely different particle sizes. To obtain true number distribution of the particles, it is desirable to separate the various fractions and analyze independently so as to reduce the number of variables during model fitting of the autocorrelation function. Direct counting of number distribution of particles became feasible through particle tracking light scattering, within a limited range of particle sizes, and it depends on the scattering power of the materials. Concentrated suspensions that undergo multiple scattering of incident light can be analyzed by diffusive wave spectroscopy. DLS can also be applied to extract information about local rheological behavior of complex fluids.

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Dr. P. A. Hassan earned his M.Sc. Degree in Chemistry from Mahatma Gandhi University, Kottayam, Kerala, India, in 1991, securing first rank. In 1993, he joined Bhabha Atomic Research Centre (BARC), Mumbai, India, after one year orientation training at BARC Training School. Currently, he is Head of the Thermal & Interfacial Chemistry Section, Chemistry Division, BARC. He was a visiting researcher at LUDFC, University of Louis Pasteur, Strasbourg, France, in 1995. He received his Ph.D. degree in Chemistry from the University of Mumbai in 1998 based on his work on hydrotrope-induced structural transitions in surfactant assemblies, under the guidance of Dr. C. Manohar. He pursued his postdoctoral research at the Department of Chemical Engineering, University of Delaware, USA in 2000−2002, under Prof. Eric Kaler. He was instrumental in setting up the light scattering and other colloid characterization facilities for interfacial chemistry research at BARC. His research contributions include vesicle-to-micelle transition in surfactant mixtures, design of viscoelastic fluids through self-assembly, nanoparticle synthesis in micelles, microrheology using light scattering and interfacial engineering for immunoassays and drug delivery. His current research interests are structural transitions in organized assemblies, polyelectrolyte− surfactant interactions, and biotechnological applications of nanomaterials.

Dr. Gunjan Verma obtained her M.Sc. degree in Chemistry from the Indian Institute of Technology, Roorkee, and obtained her Ph.D. from the Indian Institute of Technology Delhi. She joined Bhabha Atomic Research Centre (BARC), Mumbai, in the year 2005 as a scientific officer. Since then, she has been working on the microstructure and dynamics of association colloids such as micelles, their application in materials synthesis, and rheological studies of complex fluids.

AUTHOR INFORMATION

Corresponding Author

*Tel: + 91- 22-25595099; Fax: + 91- 22-25505151; E-mail: [email protected]. 11


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(15) Anderson, W.; Kozak, D.; Coleman, V.A.; Jämting, T.K.; Trau, M. A Comparative Study of Submicron Particle Sizing Platforms: Accuracy, Precision and Resolution Analysis of Polydisperse Particle Size Distributions. J. Colloid Interface Sci. 2013, 405, 322−330. (16) MacKintosh, F. C.; Schmidt, C. F. Microrheology. Curr. Opin. Colloid Interface Sci. 1999, 4, 300−307. (17) Mason, T. G.; Weitz, D. A. Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids. Phys. Rev. Lett. 1995, 74, 1250−1253. (18) Levine, A. J.; Lubensky, T. C. One- and Two-Particle Microrheology. Phys. Rev. Lett. 2000, 85, 1774−1777. (19) Hassan, P. A.; Bhattacharya, K.; Kulshreshtha, S. K.; Raghavan, S. R. Microrheology of Wormlike Micellar Fluids from the Diffusion of Colloidal Probes. J. Phys. Chem. B 2005, 109, 8744−8748. (20) Hassan, P. A.; Manohar, C. Diffusion of Probe Particles in Surfactant Gels: Dynamic Light Scattering Study. J. Phys. Chem. B 1998, 102, 7120−7125. (21) Phillies, G. D. J.; Lacroix, M. Probe Diffusion in Hydroxypropylcellulose−Water: Radius and Line-Shape Effects in the Solutionlike Regime. J. Phys. Chem. B 1997, 101, 39−47. (22) Streletzky, K. A.; Phillies, G. D. J. Coupling Analysis of Probe Diffusion in High Molecular Weight Hydroxypropylcellulose. J. Phys. Chem. B 1999, 103, 1811−1820. (23) Narita, T.; Knaebel, A.; Munch, J. P.; Candau, S. J.; Zrinyi, M. Diffusing-Wave Spectroscopy Study of the Motion of Magnetic Particles in Chemically Cross-Linked Gels under External Magnetic Fields. Macromolecules 2003, 36, 2985−2989. (24) Sood, A. K.; Bandyopadhyay, R.; Basappa, G. Linear and Nonlinear Rheology of Wormlike Micelles. Pramana 1999, 53, 223− 235. (25) Cardinaux, F.; Cipelletti, L.; Scheffold, F.; Schurtenberger, P. Microrheology of Giant-Micelle Solutions. Europhys. Lett. 2002, 57, 738−744. (26) vanZanten, J. H.; Rufener, K. P. Brownian Motion in a Single Relaxation Time Maxwell Fluid. Phys. Rev. E 2000, 62, 5389−5396.

Suman Rana received her B.Sc. and M.Sc. degrees in Chemistry from Maharshi Dayanand University (MDU) and her M.Phil. degree from Chaudhary Devi Lal University (CDLU), Haryana. Currently, she is pursuing her Ph.D. in Chemistry under the supervision of Dr. P. A. Hassan at Bhabha Atomic Research Centre (BARC), Mumbai, India. Her research interests are in the self-assembly of amphiphiles and interfacial modification of magnetic nanoparticles for drug delivery.

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REFERENCES

(1) Pecora, R. In Dynamic Light Scattering: Application of Photon Correlation Spectroscopy; Plenum Press: New York, 1985. (2) Koppel, D. E. Analysis of Macromolecular Polydispersity in Intensity Correlation Spectroscopy: The Method of Cumulants. J. Chem. Phys. 1972, 57, 4814−4820. (3) Particle Size Analysis - Photon Correlation Spectroscopy; International Standard ISO 13321:1996(E); International Standards Organization: Geneva, Switzerland, 1996. (4) Morrison, I. D.; Grabowski, E. F.; Herb, C. A. Improved Techniques for Particle Size Determination by Quasi-Elastic Light Scattering. Langmuir 1985, 1, 496−501. (5) Bertero, M.; Pike, E. R. Exponential-Sampling Method for Laplace and Other Dilationally Invariant Transforms: II. Examples in Photon Correlation Spectroscopy and Fraunhofer Diffraction. Inverse Probl. 1991, 7, 21−41. (6) Provencher, S. W. A Constrained Regularization Method for Inverting Data Represented by Linear Algebraic or Integral Equations. Comput. Phys. Commun. 1982, 27, 213−227. (7) Provencher, S. W. CONTIN: A General Purpose Constrained Regularization Program for Inverting Noisy Linear Algebraic and Integral Equations. Comput. Phys. Commun. 1982, 27, 229−242. (8) Hassan, P. A.; Kulshreshtha, S. K. Modification to the Cumulant Analysis of Polydispersity in Quasielastic Light Scattering Data. J. Colloid Interface Sci. 2006, 300, 744−748. (9) Pecora, R. Spectral Distribution of Light Scattered by Monodisperse Rigid Rods. J. Chem. Phys. 1968, 48, 4128−4130. (10) Lehner, D.; Lindner, H.; Glatter, O. Determination of the Translational and Rotational Diffusion Coefficients of Rodlike Particles Using Depolarized Dynamic Light Scattering. Langmuir 2000, 16, 1689−1695. (11) Tirado, M. M.; Martinez, C. L.; de La Torre, J.G. Comparison of Theories for the Translational and Rotational Diffusion Coefficients of Rod-like Macromolecules. Application to short DNA fragments. J. Chem. Phys. 1984, 81, 2047−2052. (12) Russo, P. S.; Karasz, F. E.; Langley, K. H. Dynamic Light Scattering Study of Semidilute Solutions of a Stiff-Chain Polymer. J. Chem. Phys. 1984, 80, 5312−5325. (13) Sehgal, A.; Seery, T. A. P. Anomalous Dynamic Light Scattering from Solutions of Light Absorbing Polymers. Macromolecules 1999, 32, 7807−7814. (14) Pine, D. J.; Weitz, D. A.; Chaikin, P. M.; Herbolzheimer, E. Diffusing Wave Spectroscopy. Phys. Rev. Lett. 1988, 60, 1134−1137. 12


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