Particle size measurement of spherical Brownian particles

One of the commonly-seen application of DLS is the measurement of particle sizes in colloidal dispersion. The main idea is to observe the diffusive nature of par-ticles undergoing Brownian motions. The thermal motions of spheres in dispersion determine the diffusion coefficient D in Stoke-Einstein Equation,

D = k_BT

6πηa (2.24)

where k_B is the usual Boltzmann Constant; T is the absolute temperature; η is the dynamic viscosity; a is the radius of the spheres, or the hydrodynamic radius (RH), which is the effective radius calculated with Eq. (2.24) for the hard sphere with the same diffusion coefficient. The fact that we can usually estimate the particles as spheres makes Eq. (2.24) powerful for particle size analysis. The model system for dilute spherical particles hence is often used in DLS experiments.

We start our discussion from the polarization of the scattered light from spheres in a dilute dispersion that no collision is concerned. The polarizability of a uniform particle is determined by its geometric configuration. For a sphere, the three-by-three tensor of second-rank representing the polarizability (α_sphere)_αβ is diagonal, and moreover, (αsphere)xx = (αsphere)yy = (αsphere)zz = α. That is

αsphere = α1 or (αsphere)αβ = α∆αβ (2.25)

This is expected for the symmetric property of a sphere in geometry. The dipole moment from a sphere µ_sphere is then

µ_sphere = α_sphere−→

E_i = α−→

E_i (2.26)

The dipole moment in Eq. (2.26) is proportional to the incident field −→

E_i with the factor alpha. Hence, the scattered field from the sphere is also directly proportional to −→

E_i. For the study of scattered field at the detector in Fig. 2.1, The selection of

−

→n_i and −n→_f contributes to the scattered field as shown in Eq. (2.9). Substitute Eq.

(2.26) to Eq. (2.9), we find

Es(−→q , t) ∝ (−→ni · −n→f)X

exp[i−→q · −→rj(t)] (2.27)

The scattered field strength Es is proportional to −→ni · −n→f in Eq. (2.27) indicates no polarization of −→

E_s comes from the scattering.

The intensity autocorrelation function for spherical particles in a light scatter-ing experiment is the product of two components, one is the correlation of particles going in and out the scattering volume V_s; the other is the intensity correlation due to the fluctuation of dielectric property. The former one corresponds to the time scale in how long a particle can go across Vs,

τ_v = (V_s)^2/3

D (2.28)

The later one corresponds to the wave number component in Es, which is |−→q | = q.

The characteristic length q⁻¹ represents the scale seen by the scattering. The time scale for intensity fluctuations from the diffusive thermal motions is hence

τq = q⁻²

D (2.29)

The comparison of the two time scale is τ_v

τ_q = (qV_s^1/3)² (2.30)

In common cases, q ∼ 0.1um and Vs^1/3 ∼ 100um, making τ_v/τ_q ∼ 10⁶. It is reason-able to discard the correlation of particle entering and exiting V_s if we choose the time scale to observe the diffusive motions of the particles.

Now we can focus on the Brownian motion part of the intensity correlation function. We start from the results obtained in Section 2.1.3. The generalized inten-sity autocorrelation function in Eq. (2.23) shows that the term hδ_if(−→q , t)δ_if(−→q , t+

τ )i_tfully determines the τ dependency of the correlation function. δ_if(−→q , t) comes from the integration inside V_s of δ_if(−→r , t), which represents the dielectric constant fluctuation in sub-region centered at −→r . Each of the sub-regions are to be small to contain only one spherical particle. This constrain arises for the assumption that each sub-region is large-enough to isolate the movements of the scatterer inside. The other assumption that the sub-regions are small compared with the wavelength of incident light indicates that all molecules inside the j^th sub-region at position −→rj

and time t interact with −→

E_i(−→r_j, t). So all the molecules composing the spherical particle in sub-region at position −→r_j and time t also interact with −→

E_i(−→r_j, t). The scattered field from the spherical particle can be described with Eq. (2.27). On the other hand, if the sub-region contains no particle at position −→r_j and time t,

δif(−→r , t) = 0. Only the sub-regions with particles inside contributes to the scat-tered field, and each sub-region with a particle inside represents the position and the incident field interacts with the particle inside. If all spherical particles are uniform in material and size, we find the relationship between particle positions and the term of dielectric constant fluctuation autocorrelation in Eq. (2.6) as

hδ_if(−→q , t)δ_if(−→q , t + τ )i

where the position −→r_j is confined to the positions of sub-regions with a sphere. In the right-most part of Eq. (2.31), the polarization components are discarded as the polarization of −→

E_i and −→

E_f is in the same direction. The analyzer at the near end to the detector only take effect as a neutral density filter. The quantity

F_D^(j) ≡ hexp[i−→q · ∆−→rj(τ )]i (2.32) is named the “self-intermediate scattering function”, where ∆−→r_j(τ )] ≡ −→

R_j(t + τ ) −

−→

R_j(t) . There is no t dependency left in ∆−→r_j(τ ) because no particular t should be unique statistically. For all identical spherical particles in our model, F_D^(j) should be the same since the number j is arbitrarily marked. The summation in Eq. (2.31) becomes a multiple by the number of sphere in V_s. The term with τ dependency in hδ_if(−→q , t)δ_if(−→q , t + τ )i_t is now ∆−→r_j(τ ).

If we take the self-intermediate scattering function as the special Fourier trans-form of a function GD(−→

The statistical meaning for Eq. (2.34) is the probability distribution of a particle traveled the displacement −→

R time τ . In our model, the spherical particles are un-dergoing Brownian motions that causes the self-diffusion behavior. We can apply the diffusion equation

∂ G_D(−→

R , τ ) = D∇²G_D(−→

R , τ ) (2.35)

to Eq. (2.33), and obtain the partial differential equation describing F_D^(j) as

Substituting Eq. (2.37) with Eq. (2.32) back to Eq. (2.31), we now can predict hδ_if(−→q , t)δ_if(−→q , t + τ )i ∝ exp(−q²Dτ ) (2.38) That is with Eq. (2.24), the generalized intensity autocorrelation function for spher-ical particles is

G¯⁽²⁾(τ ) = 1 + ˜B exp(−2q²Dτ ) (2.39) We can use this result in Eq. (2.39) with Eq. (2.24) to analyze our DLS data for particle size or RH .