3. Results

The single-particle self-diffusion coefficient ${D_s}(n)$ can be obtained from the mean square particle displacement $\langle \Delta \boldsymbol{\boldsymbol{\mathsf{S}}_i^2(\tau )\rangle = 4{D_s}(n)\tau}$, where $\Delta {\boldsymbol{\boldsymbol{\mathsf{S}}_i}(\tau ) = {\boldsymbol{\boldsymbol{\mathsf{S}}_i}(t + \tau ) - {\boldsymbol{\boldsymbol{\mathsf{S}}_i}(t)}}}$, as shown in figure 1(c). Here, $\tau $ is the lag time, and n is the area fraction of the particles. Figure 2(a) shows the normalized curves of ${D_s}(n)/{D_0}$ as functions of the particle concentration n for the three samples, where ${D_0}$ is the diffusion coefficient for a single particle in the bulk water, ${D_0} = {k_B}T/6\pi {\eta ^{(b)}}a$. The value for each data point in figure 2(a) was obtained by averaging more than ${10^6}$ particles. The solid lines in figure 2(a) illustrate the results of fitting the data to ${D_s}(n)/{D_0} = \alpha (1 - \beta n - \gamma n^2)$ (Chen & Tong Reference Chen and Tong2008). When $n \to 0$, ${D_s}(0)/{D_0} = \alpha $, where ${D_s}(0)$ is the single-particle diffusion coefficient in the monolayer in the dilute limit. The fitted values of the parameters $\alpha $, $\beta $ and $\gamma $ for the three samples are given in table 1. As seen from this table, a larger particle size corresponds to a larger value of $\alpha $ and a smaller value of $\beta $. Here, the value of the parameter $\alpha $ reflects the strength of the viscosity experienced by a single particle in the local environment in the dilute solution limit. A large $\alpha $ implies a low viscosity (Zhang et al. Reference Zhang, Chen, Li, Zhang and Chen2013a). The value of the parameter $\beta $ represents the strength of the effective hydrodynamic interactions between two particles, excluding the effects of the local environment. A large $\beta $ reflects strong hydrodynamic interactions. The value of the parameter $\gamma $ represents the strength of the many-body effect among the particles in the monolayer.

Figure 2. (a) Normalized self-diffusion coefficients of a single particle. The normalized self-diffusion coefficient ${D_s}(n)/{D_0}$ is plotted as a function of the area fraction n for samples Si1, Si2 and Si3. The solid lines represent the second-order polynomial fits to the formula ${D_s}(n)/{D_0} = \alpha (1 - \beta n - \gamma {n^2})$ for each sample. (b) The dependence of the Saffman length ${\lambda _s}$ of monolayers on the area fraction n for each sample.

Table 1. Parameters of the three samples.

The separation z is calculated according to (3.1) (Wang et al. Reference Wang, Prabhakar and Sevick2009, Reference Wang, Prabhakar, Gao and Sevick2011b):

(3.1)\begin{equation}\frac{{{D_s}(0)}}{{{D_0}}} = 1 + \frac{3}{{16}}\left( {\frac{{2{\eta^{(b)}} - 3{\eta^{(a)}}}}{{{\eta^{(b)}} + {\eta^{(a)}}}}} \right)\left( {\frac{a}{z}} \right),\end{equation}

where ${\eta ^{(b)}}$ is the viscosity of the bulk water, ${\eta ^{(a)}}$ is the viscosity of the air and z is the separation between the particle centre and the liquid interface. The calculated values of z are shown in table 1. It should be noted that the thermal motion of the particles prevents them from staying at a fixed location, and rather causes them to sway around their equilibrium positions, while their vertical positions follow a Boltzmann distribution. The measured value of z presented here is an average one, with a fluctuation width $\Delta z$ in the order of ${k_B}T/\Delta mg$, where $\Delta m$ is the difference of the mass between the particle and a water sphere with the same size. This is true for a 1 μm radius silica particle, $\Delta z\sim 0.1\;{\rm \mu} \textrm{m}$, for example. All our measurements are the results of HIs averaged in the water film with a thickness of ${\sim} \Delta z$. As shown in table 1, a large $\alpha $ corresponds to a small separation $z/a$, which suggests that the local viscosity is higher when the particle monolayer in the water is farther from the water–air interface.

The water–air interface is perfectly slipping, incapable of supporting a shear stress. Such a slip boundary corresponds to a decreased particle friction (increased mobility) and decreased energy dissipation. Intuitively, the closer a Brownian particle is to the water–air interface, the less surrounding water can be driven by the particle's moving, because the water film between the particle and the interface is thinner. The particle will thus feel less viscosity when approaching the interface, unlike that in the bulk of the water. From (3.1), we can know that for a particle that is far away from the water–air interface, the particle's friction is increased with respect to its bulk value, and the diffusion constant is decreased, so the value of the measured diffuse constant tends to its bulk values.

According to the work of Sickert, Rondelez & Stone (Reference Sickert, Rondelez and Stone2007) and Fischer, Dhar & Heinig (Reference Fischer, Dhar and Heinig2006), the single-particle diffusion coefficient can be written as

(3.2)\begin{equation}{D_s}(n) \cong \frac{{{k_B}T}}{{{\kappa ^{(0)}}{\eta ^{(b)}}a + {\kappa ^{(1)}}{\eta ^{(s1)}}(n)}},\end{equation}

where ${k_B}$ is the Boltzmann constant and ${\eta ^{(s1)}}(n)$ is the n-dependent surface viscosity of the particle monolayer. Equation (3.2) indicates that the friction of a particle experienced in a monolayer (or a film) comes from two terms: ${\kappa ^{(0)}}{\eta ^{(b)}}a$ and ${\kappa ^{(1)}}{\eta ^{(s1)}}$. The first term represents the friction due to the viscosity experienced by the particle in its local environment, where ${\eta ^{(b)}}$ is the viscosity of the bulk of fluid and ${\kappa ^{(0)}}$ is related to the position of the monolayer. The second term represents the friction coming from the monolayer (or film) itself, which is a function of the area fraction n. The dimensionless coefficients ${\kappa ^{(0)}}$ and ${\kappa ^{(1)}}$, which are the functions of z, can be calculated by following equations (Fischer et al. Reference Fischer, Dhar and Heinig2006),

(3.3)\begin{gather}{\kappa ^{(0)}} \approx 6\pi \sqrt {\tanh \left[ {\frac{{32(z/a + 1)}}{{9{\pi^2}}}} \right]} ,\end{gather}

(3.4)\begin{gather}{\kappa ^{(1)}} \approx{-} 4 \times \ln \left( {\frac{2}{\pi }\arctan \left( {\frac{2}{3}} \right)} \right)\frac{{{{(3a)}^{3/2}}}}{{{{(z + 2a)}^{3/2}}}}\quad (z \ge a).\end{gather}

The calculated values of ${\kappa ^{(0)}}$ and ${\kappa ^{(1)}}$ for the three samples are listed in table 1. More details about the method to obtain the values of ${\kappa ^{(0)}}$ and ${\kappa ^{(1)}}$ are presented in the supplementary material available at https://doi.org/10.1017/jfm.2020.693. Given the results of ${D_s}(n)$, the surface viscosity ${\eta ^{(s1)}}$ of the monolayers can be estimated by (3.2). The Saffman length ${\lambda _s}$ can be obtained by ${\lambda _s} = {\eta ^{(s1)}}/{\eta ^{(b)}}$ (Saffman & Delbrück Reference Saffman and Delbrück1975; Saffman Reference Saffman1976). The curves of ${\lambda _s}$ vs. n are shown in figure 2(b), which indicates that ${\lambda _s}$ increases with n monotonically.

The correlated diffusion reflects the response function of the HIs between two particles (Crocker et al. Reference Crocker, Valentine, Weeks, Gisler, Kaplan, Yodh and Weitz2000; Gardel, Valentine & Weitz Reference Gardel, Valentine and Weitz2005; Prasad et al. Reference Prasad, Koehler and Weeks2006). In terms of the particle displacement, the correlated diffusion coefficient is defined as

(3.5)\begin{equation}{D_{||, \bot }}(r) = \frac{{{{\langle \Delta s_{||, \bot }^i(t,\tau )\Delta s_{||, \bot }^j(t,\tau )\delta (r - {R^{i,j}}(t))\rangle }_{i \ne j}}}}{{2\tau }}.\end{equation}

Here, $\Delta s_{||, \bot }^i$ is the displacement of the $i\textrm{th}$ particle in the longitudinal $(||)$ or transverse $( \bot )$ direction during a time interval $\tau $, as shown in figure 1(c). The longitudinal displacement $\Delta {s_{||}}$ and transverse displacement $\Delta {s_ \bot }$ are the components of $\Delta {\boldsymbol{s}_i}(\tau )$ that are parallel and perpendicular, respectively, to the line connecting the centres of two particles i and j. The average ${\langle \rangle _{i \ne j}}$ is taken over all pairs consisting of the $i\textrm{th}$ and $j\textrm{th}$ particles with a separation distance r for $i \ne j$. The correlated diffusion coefficients ${D_{||, \bot }}(r)$ for sample Si1 are plotted in figure 3(a,b), where the results are normalized with respect to the single-particle diffusion ${D_s}(n)$ to eliminate the effects of the local viscosity. In figure 3(a,b), from bottom to top, the area fraction n varies from 0.03 to 0.59, and the black squares and green crosses correspond to the smallest and largest n values, respectively. The curves of ${D_{||, \bot }}(r)$ for samples Si2 and Si3 are plotted in figures 3(c,d) and 3(e,f), and exhibit behaviours similar to those of ${D_{||\bot }}(r)$ for sample Si1.

Figure 3. Measured correlated diffusion coefficients ${D_{||}}/{D_s}(n)$ (a) and ${D_ \bot }/{D_s}(n)$ (b) as a function of the distance $r/(2a)$ for sample Si1; (c and d) for sample Si2; (e and f) for sample Si3. The different symbols represent measurements at different area fractions n. The highest n is labelled in each figure. The dashed lines corresponding to ${\sim} 1/r$ (a) and ${\sim} 1/{r^2}$ (b) are plotted for reference.

Curves of ${D_{||\bot }}(r)$ in figure 3(a–f) collapse to a single master curve ${\tilde{D}_{||, \bot }}({R_{||, \bot }})$ for each direction when an effective diffusion coefficient $D_s^m$ and adjustable parameters ${\chi _{||, \bot }}$ are used as the scaling factors in vertical and horizontal axis respectively, i.e. ${\tilde{D}_{||}} \equiv {D_{||}}/D_s^m$ and ${R_{||, \bot }} \equiv r/{\chi _{||, \bot }}$ (see figure 4). The effective diffusion coefficient $D_s^m$ is obtained by the single-particle diffusion coefficient ${D_s}(n)$ in the following method. As shown in (3.2), the retardation experienced by a Brownian particle involves two factors: the local viscosity experienced by the particle in the dilute limit ${\kappa ^{(0)}}{\eta ^{(b)}}a$ and the many-body effect of the particles ${\kappa ^{(1)}}{\eta ^{(s)}}$ in the particle monolayer. Based on (3.2), two effective diffusion coefficients can be defined by ${D_s}(0) \equiv {k_B}T/{\kappa ^{(0)}}{\eta ^{(b)}}a$ and $D_s^m(n )\equiv {k_B}T/{\kappa ^{(1)}}{\eta ^{(s1)}}$. Hence, (3.2) can be rewritten as

(3.6)\begin{equation}\frac{1}{{{D_s}(n)}} = \frac{1}{{{D_s}(0)}} + \frac{1}{{D_s^m(n)}}.\end{equation}

Here, ${D_s}(0)$ is the diffusion coefficient of a single particle in the monolayer in the dilute limit $(n \to 0)$, which can be obtained from figure 2(a), and $D_s^m(n)$ is a function of n because the viscosity ${\eta ^{(s1)}}$ stems from the HIs between particles in the monolayer. Since the measured correlated diffusion coefficients ${D_{||,\; \bot }}(r)$ describe the HIs between particles, $D_s^m(n)$ is a more suitable scaling factor than ${D_s}(n)$ for this scenario.

Figure 4. Scaled correlated diffusion coefficients ${\tilde{D}_{||}}({R_{||}})$ (a) and ${\tilde{D}_ \bot }({R_ \bot })$ (b) for sample Si1; (c and d) for sample Si2; (e and f) for sample Si3. The symbols used are same as in figure 3.

The scaling factors ${\chi _{||, \bot }}$, which are the functions of n, can be regarded as characteristic lengths of the particle monolayer. Based on the concept of the Saffman length, the values of ${\chi _{||, \bot }}(n)$ are determined by the ratio of the surface viscosity of the monolayer and the bulk viscosity of water (Saffman & Delbrück Reference Saffman and Delbrück1975). When ${\chi _{||}}(n)$ obtained in figure 4(a,c,e) are regarded as

(3.7)\begin{equation}{\chi _{||}} = \eta _{||}^{(s2)}/{\eta ^{(b)}},\end{equation}

the surface viscosity $\eta _{||}^{(s2)}$ of the monolayer can be estimated from the value of ${\chi _{||}}$. The comparison between such $\eta _{||}^{(s2)}$ and ${\eta ^{(s1)}}$ is plotted in figure 5(a), which reads that $\eta _{||}^{(s2)}$ and ${\eta ^{(s1)}}$ agree with each other. The values of $\eta _{||}^{(s2)}$ and ${\eta ^{(s1)}}$, which are obtained in one-particle and two-particle measurements respectively, should agree with each other for the same homogeneous monolayer (Prasad et al. Reference Prasad, Koehler and Weeks2006; Zhang et al. Reference Zhang, Li, Bohinc, Tong and Chen2013b). Usually, the agreement between $\eta _{||}^{(s2)}$ and ${\eta ^{(s1)}}$ will be satisfied when n is small. The monolayer can turn into a heterogeneous state with an increase in $n.$ Then, there will be a gradual deviation between $\eta _{||}^{(s2)}$ and ${\eta ^{(s1)}}$ with an increase in n, (Prasad et al. Reference Prasad, Koehler and Weeks2006) as shown in figure 5(a). The results in figure 5(a) suggest that ${\chi _{||}}$ is the Saffman length indeed, i.e. ${\chi _{||}} = {\lambda _s}$.

Figure 5. (a) Plots of $\eta _{||}^{(s2)}$ vs. ${\eta ^{(s1)}}$ for three samples. (b) Plots of $\eta _ \bot ^{(s2)}$ vs. ${\eta ^{(s1)}}$ calculated from ${\chi _ \bot } = {\lambda _s}$ for three samples. The cyan curves represent fits to $\eta _ \bot ^{(s2)}\sim {({\eta ^{(s1)}})^{2/3}}$. (c) Plots of $\eta _ \bot ^{(s2)}$ vs. ${\eta ^{(s1)}}$ calculated from (3.7). (d) Plots of $\eta _ \bot ^{(s2)}$ vs. $\eta _{||}^{(s2)}$ from (3.7) for three samples. The navy blue line in (a,c,d) is shown for reference, and the slope of the line is 1.0.

However, we found that ${\chi _ \bot }(n)$ obtained in figure 4(b,d,f) cannot be regarded as the Saffman length ${\lambda _s}$. If ${\chi _ \bot }(n) = {\lambda _s}$ was regarded, the values of $\eta _ \bot ^{(s2)}$ estimated from ${\chi _ \bot }(n) = \eta _ \bot ^{(s2)}/{\eta ^{(b)}}$ would disagree with the values of ${\eta ^{(s1)}}$. The comparison between such $\eta _ \bot ^{(s2)}$ and ${\eta ^{(s1)}}$ is plotted in figure 5(b), indicating that the viscosity $\eta _ \bot ^{(s2)}$ obtained from ${\chi _ \bot } = {\lambda _s}$ follows the power-law relationship $\eta _ \bot ^{(s2)} \sim {({\eta ^{(s1)}})^{2/3}}$. Considering this power-law relationship, the dependence of the scaling length ${\chi _ \bot }$ on the Saffman length ${\lambda _s} = \eta _ \bot ^{(s2)}/{\eta ^{(b)}}$ should be expressed as

(3.8)\begin{equation}{\chi _ \bot } = \frac{1}{2}a{\left( {\frac{{{\lambda_s}}}{a}} \right)^{2/3}},\end{equation}

for $\eta _ \bot ^{(s2)} = {\eta ^{(s1)}}$ to be satisfied. The viscosity $\eta _ \bot ^{(s2)}$ that is obtained using (3.8) is plotted against ${\eta ^{(s1)}}$ in figure 5(c). Once again, the values of ${\eta ^{(s1)}}$ and $\eta _ \bot ^{(s2)}$ agree with each other when n is small. The comparison between the viscosity $\eta _{||}^{(s2)}$ and $\eta _ \bot ^{(s2)}$ is plotted in figure 5(d), which shows $\eta _{||}^{(s2)} = \eta _ \bot ^{(s2)}$ all the time.

It should be noted that there exist sets of values of ${\chi _{||, \bot }}$ and their multiples such that all of these values can make ${\tilde{D}_{||, \bot }}$ collapse to a single curve. However, there is only one special pair of ${\chi _{||, \bot }}$ values for which the calculated viscosity ${\eta ^{(s2)}} \equiv \eta _{||, \bot }^{(s2)}$ agrees with ${\eta ^{(s1)}}$. This constraint allows for one to determine a unique pair of ${\chi _{||, \bot }}$ values with which to determine the positions of the master curves of ${\tilde{D}_{||, \bot }}$.

The dependence of ${\eta ^{(s2)}}$ on n is plotted in figure 6, showing that it follows the Krieger–Dougherty equation (Krieger & Dougherty Reference Krieger and Dougherty1959) as follows:

(3.9)\begin{equation}{\eta ^{(s2)}} = {\eta ^{(s1)}}(0)\left[ {{{\left( {1 - \frac{n}{{{n_m}}}} \right)}^{ - [\eta ]{n_m}}} - 1} \right],\end{equation}

where ${\eta ^{(s1)}}(0)$ is a characteristic scale for the surface viscosity, which is affected by the confining boundary. In the above equation, ${n_m} \cong 0.84$ is the maximum random packing fraction in two dimensions (Berryman Reference Berryman1983; O'Hern et al. Reference O'Hern, Langer, Liu and Nagel2002), and the intrinsic viscosity $[\eta ]$ is the only fitting parameter. The fitted values of the intrinsic viscosity $[\eta ]$ are shown in table 1.

Figure 6. The surface viscosities ${\eta ^{(s2)}}$ as functions of the particle area fraction n. The cyan curves represent fits to the Krieger–Dougherty equation.

In the vicinity of a fluid–fluid interface, the mobility of particles and the hydrodynamic interactions between particles will be different from those in the bulk, which show a complex dependence on the separation z (Jones, Felderhof & Deutch Reference Jones, Felderhof and Deutch1975; Bickel Reference Bickel2007; Wang et al. Reference Wang, Prabhakar and Sevick2009, Reference Wang, Prabhakar, Gao and Sevick2011b). In our experiments, ${\tilde{D}_{||}}$ and ${\tilde{D}_ \bot }$ indeed depend on the separation z. The larger z is, the weaker the influence of the water–air interface. The master curves of ${\tilde{D}_{||}}({R_{||}})$ and ${\tilde{D}_ \bot }({R_ \bot })$ with different z each degenerate to a single curve when ${\tilde{D}_{||}}({R_{||}})$ and ${\tilde{D}_ \bot }({R_ \bot })$ are multiplied by a factor of ${(z/a)^{2/3}}$ (as shown in figure 7). This degeneracy of ${\tilde{D}_{||, \bot }}$ by a factor of ${(z/a)^{2/3}}$ at $z \gt 0$ indicates that, with the exception of the boundary effect, no dynamic mechanisms are introduced into the system by the water–air interface.

Figure 7. (a) Universal master curve of ${\tilde{D}_{||}} {(z/a)^{2/3}}$ as a function of ${R_{||}}$ for three samples. (b) Universal master curve of ${\tilde{D}_ \bot } {(z/a)^{2/3}}$ as a function of ${R_ \bot }$ for three samples.

It can be seen from figure 7(b) that the value for Si3 is slightly lower than the other two. This deviation between them looks more obvious in the half-log plot (see supplementary figure S1). We think that such deviation comes from fluctuations in the vertical position of the particles, which will weaken HIs between the particles at given projection distance r. The smaller the mass of the particles, the greater the fluctuations experienced in the vertical position. The correlated diffusion, ${D_{||\bot }}(r)$, of Sample Si3 was smaller than that of the other two samples. Another feature of such deviation is that a significant deviation appears in a short distance, as the strength of such an influence is supposed to be proportional to $\Delta z/r$.