Here we show the principal derivation of the intensity autocorrelation function of the scattered light from an ensemble of particles undergoing Brownian diffusional motion as well as an external-induced flow motion with the velocity \(\varvec{v}\). The particle concentration at point \(\varvec{r}\) and time \(t\) is defined by \(c(\varvec{r},t)\). The continuity equation which describes how the particles flow and diffuse (having a diffusion coefficient \(\text{D}\)) in the system can be written as:

$$\frac{\partial c}{\partial t}+\nabla \bullet \left(\varvec{v}c\right)=D{\nabla }^{2}c$$

1

According to Berne and Pecora [19], whose notation we adopted in this study, it is reasonable to assume that the *probability distribution function*, \({G}_{s}(\varvec{r},t)\), satisfies the same equation. Therefore, we have:

$$\frac{\partial {G}_{s}}{\partial t}+\nabla \bullet \left({\varvec{v}G}_{s}\right)=D{\nabla }^{2}{G}_{s}$$

2

We consider the *characteristic function of distribution*, \({F}_{s}(\varvec{q},t)\) as the Fourier transform of \({G}_{s}\), based on the following definitions:

$${F}_{s}\left(\varvec{q},t\right)=\int {G}_{s}\left(\varvec{r},t\right)\text{exp}\left(i\varvec{q}\bullet \varvec{r}\right){d}^{3}r,$$

3

$${G}_{s}\left(\varvec{r},t\right)={\left(2\pi \right)}^{-3}\int {F}_{s}\left(\varvec{q},t\right)\text{exp}\left(-i\varvec{q}\bullet \varvec{r}\right){d}^{3}q$$

4

where \(q=\frac{4\pi n\text{ sin}\left(\frac{\theta }{2}\right)}{{\lambda }_{l}}\) is the scattering wave vector with a wavelength of \({\lambda }_{l}\) and scattering angle of \(\theta\) and the medium refractive index of *n*. In case the system is subjected to US waves having a wave vector of \(\left|\varvec{k}\right|=2\pi /{\lambda }_{u}\) (\({\lambda }_{u}\) is the US wavelength) and angular frequency of \(\omega\), the velocity of the fluid at point \(\varvec{r}\) and time \(t\), having a mean value \({\varvec{v}}_{0}\) can be written as:

$$\varvec{v}\left(\varvec{r},t\right)={\varvec{v}}_{0}\text{exp}\left(i\varvec{k}\bullet \varvec{r}-i\omega t\right).$$

5

By taking the spatial Fourier transform from Eq. (2), the first term on the left-hand side and the term on the right-hand side yield to \(\frac{\partial {F}_{s}(\varvec{q},t)}{\partial t}\) and \(-D{q}^{2}{F}_{s}\left(\varvec{q},t\right)\), respectively. The second term on the left-hand side can be written as (Ft: Fourier transform):

$$Ft\left[\varvec{v}\bullet \nabla G+G\nabla \bullet \varvec{v}\right]=\int \varvec{v}\bullet \frac{\partial {G}_{s}}{\partial r}\text{exp}\left(i\varvec{q}\bullet \varvec{r}\right){d}^{3}r+\int {G}_{s}\frac{\partial \varvec{v}}{\partial r}\text{exp}\left(i\varvec{q}\bullet \varvec{r}\right){d}^{3}r.$$

6

Using the Eq. (5), the Eq. (6) yields to:

$${\varvec{v}}_{0}\text{exp}\left(-i\omega t\right)\int \frac{\partial {G}_{s}}{\partial r}{\text{e}}^{i\left(\varvec{q}+\varvec{k}\right)\bullet \varvec{r}}{d}^{3}r+i\varvec{k}{\bullet \varvec{v}}_{0}\text{exp}\left(-i\omega t\right)\int {G}_{s}{\text{e}}^{i\left(\varvec{q}+\varvec{k}\right)\bullet \varvec{r}}{d}^{3}r,$$

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where \(\int \frac{\partial {G}_{s}}{\partial r}{\text{e}}^{i\left(\varvec{q}+\varvec{k}\right)\bullet \varvec{r}}{d}^{3}r=-iq\int {G}_{s}{\text{e}}^{i\left(\varvec{q}+\varvec{k}\right)\bullet \varvec{r}}{d}^{3}r\). The reason is that taking a derivative from Eq. (4), leads to:

$$\frac{\partial }{\partial r}{G}_{s}\left(\varvec{r},t\right)={-iq\left(2\pi \right)}^{-3}\int {F}_{s}\left(\varvec{q},t\right)\text{exp}\left(-i\varvec{q}\bullet \varvec{r}\right){d}^{3}q=-iq{G}_{s}.$$

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Therefore, Eq. (7) can be written as:

$$-i\varvec{q}\bullet {\varvec{v}}_{0}\text{exp}\left(-i\omega t\right)\int {G}_{s}{\text{e}}^{i\left(\varvec{q}+\varvec{k}\right)\bullet \varvec{r}}{d}^{3}r+i\varvec{k}\bullet {\varvec{v}}_{0}\text{exp}\left(-i\omega t\right)\int {G}_{s}{\text{e}}^{i\left(\varvec{q}+\varvec{k}\right)\bullet \varvec{r}}{d}^{3}r=-i\left(\varvec{q}-\varvec{k}\right)\bullet {\varvec{v}}_{0}\text{exp}\left(-i\omega t\right){F}_{s}\left(\varvec{q}+\varvec{k},t\right).$$

9

Finally, the spatial Fourier transform of Eq. (2) leads to:

$$\frac{\partial {F}_{s}(\varvec{q},t)}{\partial t}-i\left(\varvec{q}-\varvec{k}\right)\bullet {\varvec{v}}_{0}\text{exp}\left(-i\omega t\right){F}_{s}\left(\varvec{q}+\varvec{k},t\right)=-D{q}^{2}{F}_{s}\left(\varvec{q},t\right).$$

10

In the case of a uniform flow with the velocity of \({\varvec{v}}_{0}\) instead of ultrasonic vibration (i.e., \(k=\omega =0\)), Eq. (10) reduces to the same equation derived by Berne and Pecora [19] as:

$$\frac{\partial {F}_{s}(\varvec{q},t)}{\partial t}-i\varvec{q}\bullet {\varvec{v}}_{0}{F}_{s}\left(\varvec{q},t\right)=-D{q}^{2}{F}_{s}\left(\varvec{q},t\right).$$

11

With the initial condition of \({F}_{s}\left(\varvec{q},0\right)=1\), we have \({F}_{s}\left(\varvec{q},t\right)=\text{e}\text{x}\text{p}(-D{q}^{2}t)\text{e}\text{x}\text{p}(i\varvec{q}\bullet {\varvec{v}}_{0}t)\).

At sufficiently low US frequencies (i.e., less than 1 GHz) in water, as the medium, \(q\gg k\). Therefore, the Eq. (10) can be simplified as:

$$\frac{\partial {F}_{s}(\varvec{q},t)}{\partial t}-i\varvec{q}\bullet {\varvec{v}}_{0}\text{exp}\left(-i\omega t\right){F}_{s}\left(\varvec{q},t\right)=-D{q}^{2}{F}_{s}\left(\varvec{q},t\right).$$

12

By solving the Eq. (12) with the initial condition of \({F}_{s}\left(\varvec{q},0\right)=1\),\({F}_{s}\) can be obtained as:

$${F}_{s}\left(\varvec{q},t\right)=\text{exp}\left\{-D{q}^{2}t\right\}\text{e}\text{x}\text{p}\{-\frac{\varvec{q}\bullet {\varvec{v}}_{0}}{\omega }\text{e}\text{x}\text{p}(-i\omega t\left)\right\}.$$

13

Assume that \({\varvec{v}}_{0}={\varvec{r}}_{0}\omega\), where \({r}_{0}\) is the amplitude of vibration, Eq. (13) can be rewritten as:

$${F}_{s}\left(\varvec{q},t\right)=\text{exp}\left\{-D{q}^{2}t\right\}\text{e}\text{x}\text{p}\{-\varvec{q}\bullet {\varvec{r}}_{0}\text{exp}\left(-i\omega t\right)\}.$$

14

From Berne and Pecora [19], the homodyne correlation function can be obtained as:

$${F}_{2}\left(\varvec{q},t\right)={⟨N⟩}^{2}[1+\left|{F}_{s}{\left(\varvec{q},t\right)}^{2}\right|]+⟨\delta N\left(0\right)\delta N\left(t\right)⟩= {⟨N⟩}^{2}[1+Re(\text{exp}\left\{-D{q}^{2}t\right\}\text{exp}\left\{-2\varvec{q}\bullet {\varvec{r}}_{0}\text{exp}\left(-i\omega t\right)\right\}]+⟨\delta N\left(0\right)\delta N\left(t\right)⟩$$

15

Therefore, in the homodyne experiment, the autocorrelation function for the time lag, \(\tau\), can be written as:

$${g}_{2}\left(\tau \right)= A\left[1+B\text{exp}\{-2D{q}^{2}\tau \}\text{e}\text{x}\text{p}\{-2q{r}_{0}\text{cos}\left(\omega \tau \right)\}\right].$$

16

The B is called intercept, related to light-collection efficiency, and can be omitted by normalizing the autocorrelation function [20]. Therefore, the normalized autocorrelation function (NACF) can be written as:

$$NACF=\frac{{g}_{2}\left(\tau \right)-A}{B}=\text{exp}\left\{-2D{q}^{2}\tau \right\}\text{exp}\left\{-2q{r}_{0}\text{cos}\left(\omega \tau \right)\right\}.$$

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