Suppression of viscosity enhancement around a Brownian particle in a near-critical binary fluid mixture

Abstract

We consider the Brownian motion of a rigid spherical particle in a binary fluid mixture, which lies in the homogeneous phase near the demixing critical point, assuming that neither component is more attracted by the particle surface. In a recent study, it was experimentally shown that the self-diffusion coefficient first decreases and then reaches a plateau as the critical point is approached. The decrease should reflect the critical enhancement of the viscosity, while the plateau was interpreted as representing the suppression of the enhancement due to the shear around the particle. To examine this interpretation, we take into account the local shear rate to calculate the dependence of the drag coefficient on the particle speed, and then utilize a Langevin equation to calculate the self-diffusion coefficient.

1. Introduction

We consider the Brownian motion of a colloidal rigid particle in a binary fluid mixture lying in the homogeneous phase near the demixing critical point. In some combinations of the mixture and particle material, one of the components is preferentially attracted by the particle surface and the preferred component is remarkably adsorbed near the particle surface because of the near-criticality (Beysens & Leibler 1982; Beysens & Estève 1985). The particle motion deforms the adsorption layer, which affects the force exerted on the particle (Lee 1976; Omari, Grabowski & Mukhopadhyay 2009; Okamoto, Fujitani & Komura 2013; Fujitani 2018; Tani & Fujitani 2018; Yabunaka & Fujitani 2020). In other combinations exhibiting negligible preferential adsorption, the particle motion remains still influenced by the near-criticality because of the critical enhancement of the viscosity (Ohta 1975; Ohta & Kawasaki 1976). This enhancement can also be influenced by the particle motion, as is pointed out in a recent experimental work (Beysens 2019). We briefly mention its background in some paragraphs below.

Let us first assume there are no particles in an equilibrium near-critical binary fluid mixture. The composition can be represented by the difference between (or the ratio of) the mass densities of the two components. The order parameter, which we can take to be proportional to the deviation of the local composition from the critical one, fluctuates about the equilibrium value on length scales smaller than the correlation length, $\xi$. Correlated clusters, where the order parameter keeps the same sign on average, range over these scales, and are convected to enhance the interdiffusion of the components on larger length scales (Kawasaki 1970; Onuki 2002). Thus, $\xi$ affects how the two-time correlation function of the order-parameter fluctuation decays. Writing $\varGamma _k$ for the relaxation coefficient of its spatial Fourier transform, with $k$ denoting the magnitude of the wavenumber vector, we have

(1.1)\begin{equation} \varGamma_k=\varOmega(k\xi) \times k^{z}, \end{equation}

for small $k$ with $k\xi$ being finite. Here, $z$ denotes the dynamic critical exponent for the order-parameter fluctuation and $\varOmega$ represents a scaling function, which approaches a constant multiplied by $(k\xi )^{2-z}$ as $k\xi$ becomes much smaller than unity (Siggia, Hohenberg & Halperin 1976; Hohenberg & Halperin 1977). This leads to $\varGamma _k\propto k^2$ for sufficiently small $k$, which is expected for the hydrodynamic mode of a conserved quantity. We write $k_{B}$ for the Boltzmann constant and $T$ for the temperature of the mixture. The mode-coupling theory for a three-dimensional mixture gives

(1.2)\begin{equation} \varOmega(k\xi)=\frac{k_{B}T}{6{\rm \pi}\tilde{\eta}} \frac{K(k\xi)}{(k\xi)^3}, \end{equation}

where $K$ denotes the Kawasaki function with $K(x)\approx 3{\rm \pi} x^3/8$ for $x\gg 1$ and $K(x)\approx x^2$ for $x\ll 1$, and $\tilde {\eta }$ represents the shear viscosity (Kawasaki 1970; Onuki 2002). In this theory, the weak critical singularity of the viscosity is neglected, and the dynamic critical exponent is found to be three. This theoretical result turns out to be in good agreement with the experimental results (Swinney & Henry 1973). In the refined calculation of the dynamic renormalization group, the critical enhancement of $\tilde {\eta }$ is considered, and the value of $z$ is found to be slightly larger than three (Folk & Moser 2006).

The mixture is assumed to be at equilibrium in the preceding paragraph. The critical enhancement of the transport coefficients, i.e. the Onsager coefficient for the interdiffusion and the shear viscosity, can be suppressed when a shear is imposed on the mixture. Influences of a simple shear flow are studied theoretically (Onuki & Kawasaki 1979; Onuki, Yamazaki & Kawasaki 1981) and experimentally (Beysens, Gbadamassi & Boyer 1979; Beysens & Gbadamassi 1980). In an example of this flow, the $x$ component of the velocity is $y$ multiplied by the constant shear rate $s\,(>\!0)$, with $(x,y)$ denoting two of the three-dimensional Cartesian coordinates. A correlated cluster of the order-parameter fluctuation would be deformed by the shear when its lifetime is longer than a typical time scale of the shear, $1/{s}$. The lifetime for a cluster with the size of $1/k$ is evaluated to be $1/\varGamma _k$, while the cluster size ranges up to $\xi$. Hence, if the shear is strong enough to satisfy

(1.3)\begin{equation} {s}>\varGamma_{1/\xi}, \end{equation}

the enhancement is suppressed. This condition of strong shear is also derived in terms of the renormalization group, as mentioned in appendix A.

A simple shear flow is a kind of linear shear flow, where the velocity $\boldsymbol {V}$ at a position is proportional to the positional vector, and can be regarded as a linear combination of a stagnation-point flow and a purely rotational flow (Rallison 1984). These linear shear flows are two-dimensional. A pure-extension flow, being a three-dimensional linear shear flow, and a stagnation-point flow are referred to as elongational flows in Onuki & Kawasaki (1980a,c), where the suppression is studied for some linear shear flows. In a linear shear flow, the time derivative of a directed line segment $\boldsymbol {X}$ linking two fluid particles is equal to $(\boldsymbol {\nabla } \boldsymbol {V})^\textrm {T}\boldsymbol {\cdot } \boldsymbol {X}$, where the matrix $\boldsymbol {\nabla } \boldsymbol {V}$ represents the homogeneous velocity gradient and superscript $\textrm {T}$ indicates the transposition. Thus, the exponential of the product of $(\boldsymbol {\nabla } \boldsymbol {V})^\textrm {T}$ and the time $t$ determines how $\boldsymbol {X}$ is stretched and shrunk with time. In the elongational flow, the shear rate $s$ in (1.3) is given by the largest stretching rate, i.e. the largest eigenvalue of $\boldsymbol {\nabla } \boldsymbol {V}$; some details are mentioned in the penultimate paragraph of appendix A.

The mean square displacement of a Brownian particle becomes proportional to the time interval as it becomes sufficiently long. The self-diffusion coefficient of the particle is defined as the constant of proportionality divided by twice the spatial dimension. In the study mentioned at the end of the first paragraph (Beysens 2019), it is shown that the self-diffusion coefficient of a Brownian particle in a near-critical binary fluid mixture first decreases and then reaches a plateau as $T$ approaches the critical temperature $T_{c}$ along the critical isochore in the homogeneous phase. The first decrease should reflect the critical enhancement of $\tilde {\eta }$, while the plateau can be regarded as representing the suppression of the enhancement due to the shear caused by the particle motion. Using (1.2) and replacing $s$ in (1.3) with the average particle speed divided by the particle radius, Beysens (2019) estimates the temperature range exhibiting the suppression; the estimated range appears consistent with the observed one. In the present study, we calculate the self-diffusion coefficient for direct comparison with the experimental results.

In the first three subsections of § 2, we calculate the hydrodynamic force exerted on a rigid spherical particle moving translationally in a fluid mixture quiescent far from the particle. Assuming a typical length scale of the flow to be much larger than $\xi$, we need not consider dynamics of the order-parameter fluctuation, which is significant only on length scales smaller than $\xi$ (Furukawa et al. 2013; Okamoto et al. 2013). The mixture is assumed to be incompressible, as in the previous studies mentioned above (Folk & Moser 1998; Onuki 2002). This assumption usually works well in a near-critical mixture prepared experimentally (Anisimov et al. 1995; Onuki 2002; Pérez-Sanchez et al. 2010). When the viscosity is homogeneous, the magnitude of the force is proportional to the particle speed. The constant of proportionality (the drag coefficient) is given by Stokes’ law (Stokes 1851) and is linked with the self-diffusion coefficient of its Brownian motion through the Sutherland–Einstein relation (Einstein 1905; Sutherland 1905), although the Brownian motion is not always translational. This relation can be derived from the Langevin equation for the particle velocity (Bian, Kim & Karniadakis 2016), and is further founded on the fluctuating hydrodynamics (Bedeaux & Mazur 1974), even near the critical point (Mazur & van der Zwan 1978). In our problem, the suppression of the viscosity enhancement is locally determined by the inhomogeneous shear around the particle, and the drag coefficient can depend on the particle speed in its range to be considered in the Brownian motion. Neither Stokes’ law nor the Sutherland–Einstein relation is applicable when the suppression occurs. Assuming the suppression to remain weak even if it is brought about by the local strong shear, we calculate the drag coefficient. In § 2.4, we use a one-dimensional Langevin equation to link the drag coefficient with the self-diffusion coefficient. Our results are shown and discussed in § 3, and summarized in § 4.

2. Formulation and calculation

We write $d$ for the spatial dimension; our calculations in the text are limited to the case of $d=3$. The values of the static critical exponents are shown in Pelisetto & Vicari (2002); we use $\nu \approx 0.630$ and $\eta \approx 0.0364$. The exponent $\eta$ represents the deviation from the straightforward dimensional analysis of the static, or equal time, correlation function of the order-parameter fluctuation at the critical point. When the shear is not so strong as to suppress the critical enhancement, the correlation length $\xi$ is homogeneously given by $\xi =\xi _0\tau ^{-\nu }$ on the critical isochore, where $\tau$ is defined as $|T-T_{c}|/T_{c}$ and $\xi _0$ is a non-universal constant. Then, the singular part of the shear viscosity is proportional to $\tau ^{\nu (d-z)}$ in a flow whose typical length is much larger than $\xi$, as described at (A 2). This exponent is measured to be around $-0.042$ (Berg & Moldover 1989, 1990), which leads to $z=3.067$. Because $|\nu (d-z)|$ is small, the viscosity exhibits a very weak critical singularity. Thus, for the viscosity, the dependence of the regular part on $\tau$ is also significant unless the mixture is very close to the critical point, unlike for the Onsager coefficient of the interdiffusion. As in Beysens (2019), we use

(2.1)\begin{equation} \tilde{\eta}_{B}(T) \tau^{\nu (d-z)} \end{equation}

as the viscosity free from the shear effects. In this form of multiplicative anomaly, the regular part $\tilde {\eta }_{B}$ is defined as

(2.2)\begin{equation} \tilde{\eta}_{B}(T)=\tilde{\eta}_0 \exp{\left(\frac{E_{a}}{k_{B}T}\right)}, \end{equation}

where $\tilde {\eta }_0$ is a non-universal constant and $E_{a}$ denotes the activation energy (Sengers 1985; Mehrotra, Monnery & Svrcek 1996). Molecules would be required to overcome some energy barrier to shift their locations in a dense liquid. Equation (2.1) supposes $\tau <1$ and the singular part represents the enhancement.

We define $\tau _s$ so that a given shear rate ${s}$ affects the critical enhancement for $\tau <\tau _s$, and define $s^*$ so that

(2.3)\begin{equation} \tau_s= \left(\frac{s}{s^*}\right)^{1/(\nu z)} \end{equation}

holds. Because of (1.1) and (1.3), $s^*$ is independent of the imposed shear. We will later apply our results to a mixture of isobutyric acid and water. For this mixture under no shear, measured values of $\varGamma _k/k^2$ for small $k$ in the neighbourhood of the critical point are shown in figure 10 of Chu, Schoenes & Kao (1968). These values and $\xi _0=0.3625\ \textrm {nm}$ (Beysens, Bourgou & Calmettes 1985) give $s^*=3.7\times 10^8\ \textrm {s}^{-1}$ with the aid of (A 4) and (A 6).

From § 2.1 to § 2.3, we calculate the drag coefficient of a spherical rigid particle with radius $r_0$, by assuming it to move translationally with the velocity $U\boldsymbol {e}_z$ in a binary fluid mixture in the absence of the preferential adsorption (figure 1). Here, $\boldsymbol {e}_z$ denotes a unit vector. The mixture is on the critical isochore in the homogeneous phase with $T$ being close to $T_{c}$, and is quiescent far from the particle. Assuming $\xi$ to be much smaller than a typical length scale that the flow changes, we regard the local velocity field as a linear shear flow having the same velocity gradient to determine the local viscosity.

Figure 1. A drawing of a situation for our calculation of the drag coefficient from § 2.1 to § 2.3. A particle with the radius $r_0$ moves translationally with the velocity $U\boldsymbol {e}_z$ in a mixture fluid quiescent far from the particle. A part of a cross-section containing the $z$ axis is shown; the dashed curve represents half of the cross-section of the particle surface. The velocity field, represented by arrows outside the particle, is calculated with a homogeneous viscosity, although the viscosity becomes inhomogeneous when the suppression of the critical enhancement occurs somewhere. A magnified view of the smaller rectangular region is given in the larger one, where clusters are schematically drawn in black and white with some being deformed by the local shear. The correlation length $\xi$ is assumed to be sufficiently small as compared with a typical length of the flow.

2.1. Assumption on the viscosity

We assume that the suppression of the critical enhancement of $\tilde {\eta }$ is perfect if it occurs, and thus the viscosity is assumed to be given by

(2.4)\begin{equation} \tilde{\eta}=\begin{cases} \tilde{\eta}_{B}(T) \tau^{\nu (d-z)} & \textrm{for}\ \tau>\tau_s,\\ \tilde{\eta}_{B}(T) \tau_s^{\nu (d-z)} & \textrm{for}\ \tau< \tau_s, \end{cases}\end{equation}

which supposes $\tau <1$ and $\tau _s<1$ because (2.1) underlies (2.4). This assumption is discussed in § 3. We write $\tilde {\eta }^{(0)}$ for (2.1), and define $\tilde {\eta }^{(1)}$ so that $\tilde {\eta }=\tilde {\eta }^{(0)}+\tilde {\eta }^{(1)}$ holds. Introducing $\check {\tau }_s\equiv \tau _s/\tau$, we have

(2.5)\begin{equation} \tilde{\eta}^{(1)}\equiv \tilde{\eta}^{(0)}\left( \check{\tau}_s^{\nu (d-z)}- 1\right)\varTheta(\check{\tau}_s-1), \end{equation}

where $\varTheta$ denotes the step function; $\varTheta (x)$ vanishes for $x<0$ and equals unity for $x>0$. The shear rate is inhomogeneous, as shown later. Thus, the suppression makes the viscosity inhomogeneous. Subtracting the homogeneous part $\tilde {\eta }^{(0)}$ from the whole viscosity $\tilde {\eta }$ gives $\tilde {\eta }^{(1)}$, which is non-positive because of $d=3<z$.

The velocity and pressure fields, $\boldsymbol {v}$ and $p$, satisfy the incompressibility condition and Stokes’ equation, i.e.

(2.6a,b)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}=0\quad\textrm{and}\quad -\boldsymbol{\nabla} p+ 2\boldsymbol{\nabla}\boldsymbol{\cdot} \left(\tilde{\eta} \boldsymbol{\boldsymbol{\mathsf{E}}}\right)=0, \end{equation}

where $\boldsymbol{\boldsymbol{\mathsf{E}}}$ is the rate-of-strain tensor. Here, a low Reynolds number is assumed, as discussed in § 2 of Yabunaka & Fujitani (2020). The no-slip boundary condition is imposed at the particle surface, while $\boldsymbol {v}$ tends to zero and $p$ approaches a constant, denoted by $p_{\infty }$, far from the particle.

2.2. Approximation for a weak suppression

We consider a particular time and take the spherical coordinates $(r,\theta ,\phi )$ so that the origin is at the particle's centre and that the polar axis ($z$ axis) is along $\boldsymbol {e}_z$; the coordinate $z$ should not be confused with the dynamic critical exponent. The unit vectors in the directions of $r$ and $\theta$ are denoted by $\boldsymbol {e}_r$ and $\boldsymbol {e}_{\theta }$, respectively. The no-slip condition gives $\boldsymbol {v}=U\boldsymbol {e}_z$ at $\rho =1$, where $\rho$ denotes a dimensionless radial distance, $r/r_0$. We can assume $v_{\phi }=0$. The drag force is along the $z$ axis; its $z$ component, denoted by $\mathcal {F}_z$, is given by the surface integral of $(2\tilde {\eta } \boldsymbol{\boldsymbol{\mathsf{E}}} \boldsymbol {\cdot } \boldsymbol {e}_r - p\boldsymbol {e}_r)\boldsymbol {\cdot }\boldsymbol {e}_z$ over the particle surface. The drag coefficient is given by $-\mathcal {F}_z/U$, and can depend on $U$ in our problem. Thus, we write $\gamma (U)$ for the drag coefficient.

We, respectively, write $\boldsymbol {v}^{(0)}$ and $p^{(0)}$ for the velocity and pressure fields obtained when the viscosity is forced to be $\tilde {\eta }^{(0)}$ homogeneously. Equation (2.6a,b) yields

(2.7a,b)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}^{(0)}=0 \quad\textrm{and}\quad -\boldsymbol{\nabla} p^{(0)}+ \tilde{\eta}^{(0)} \Delta \boldsymbol{v}^{(0)}=0. \end{equation}

The boundary conditions are $\boldsymbol {v}^{(0)}=U\boldsymbol {e}_z$ at $\rho =1$, and $\boldsymbol {v}^{(0)}\to \boldsymbol {0}$ and $p^{(0)}\to p_{\infty }$ as $\rho \to \infty$. The solution is well known (Stokes 1851) and is given by

(2.8a–c)\begin{align} p^{(0)}=p_{\infty}+\frac{3\tilde{\eta}^{(0)}}{2r_0 \rho^2}U\cos{\theta},\quad v_r^{(0)}=\frac{3\rho^2-1}{2\rho^3}U\cos{\theta},\quad v_{\theta}^{(0)}=-\frac{3\rho^2+1}{4\rho^3}U\sin{\theta} \end{align}

and $v_{\phi }^{(0)}=0$. The arrows outside the particle in figure 1 represent $\boldsymbol {v}^{(0)}$ for $U>0$. The superscript ${(0)}$ is also added to a quantity calculated from $\boldsymbol {v}^{(0)}$ and $p^{(0)}$. The drag coefficient calculated from (2.8a–c) is given by

(2.9)\begin{equation} \gamma^{(0)}= -\frac{\mathcal{F}_z^{(0)}}{U} = 6{\rm \pi}\tilde{\eta}^{(0)}r_0, \end{equation}

which is independent of $U$ and represents Stokes’ law. We define $\boldsymbol {v}^{(1)}$ and $p^{(1)}$ as $\boldsymbol {v}-\boldsymbol {v}^{(0)}$ and $p-p^{(0)}$, respectively. They satisfy $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {v}^{(1)}=0$ and

(2.10)\begin{equation} -\boldsymbol{\nabla} p^{(1)}+ \tilde{\eta}^{(0)} \Delta \boldsymbol{v}^{(1)}=-2\boldsymbol{\nabla}\boldsymbol{\cdot} \left(\tilde{\eta}^{(1)}\boldsymbol{\boldsymbol{\mathsf{E}}}\right). \end{equation}

The boundary conditions are

(2.11)\begin{equation} \left. \begin{gathered} \boldsymbol{v}^{(1)}=\boldsymbol{0} \quad \textrm{at}\ \rho=1 \quad\textrm{and}\\ \boldsymbol{v}^{(1)}\to \boldsymbol{0}\quad \textrm{and}\quad p^{(1)}\to 0\quad \textrm{as}\ \rho\to\infty. \end{gathered} \right\} \end{equation}

In the flow field of $\boldsymbol {v}$ and $p$, we define $\kappa$ as the maximum value of a dimensionless ratio $|\tilde {\eta }^{(1)}/\tilde {\eta }^{(0)}|$. Equation (2.5) is proportional to $\kappa$. From (2.10) and (2.11), $\boldsymbol {v}^{(1)}$ is also proportional to $\kappa$. The particle speed supposed here lies in the range involved in the Brownian motion. We assume that $\tau$ is not so small as to cause strong suppression, and assume $\kappa$ to be so small that the calculation up to the order of $\kappa$ makes sense. At this order, (2.10) becomes

(2.12)\begin{equation} -\boldsymbol{\nabla} p^{(1)}+ \tilde{\eta}^{(0)} \Delta \boldsymbol{v}^{(1)}=-\tilde{\eta}^{(1)}\Delta\boldsymbol{v}^{(0)} -\left(2\boldsymbol{\nabla}\tilde{\eta}^{(1)}\right)\boldsymbol{\cdot} \boldsymbol{\boldsymbol{\mathsf{E}}}^{(0)}. \end{equation}

Here, $s$ contained in $\tilde {\eta }^{(1)}$ is replaced by $s^{(0)}$, which is the shear rate calculated from $\boldsymbol {v}^{(0)}$. Likewise, we can evaluate $\kappa$ by using $s^{(0)}$ instead of $s$ in (2.5). We define $\mathcal {F}_z^{(1)}$ so that $\mathcal {F}_z=\mathcal {F}_z^{(0)}+\mathcal {F}_z^{(1)}$ holds. At the order of $\kappa$, $\mathcal {F}_z^{(1)}$ equals the surface integral of

(2.13)\begin{equation} 2\tilde{\eta}^{(1)} \boldsymbol{\boldsymbol{\mathsf{E}}}^{(0)}_{rz}+2\tilde{\eta}^{(0)} \boldsymbol{\boldsymbol{\mathsf{E}}}^{(1)}_{rz}-p^{(1)}\boldsymbol{e}_r\boldsymbol{\cdot}\boldsymbol{e}_z \end{equation}

over the particle surface, where $\boldsymbol{\boldsymbol{\mathsf{E}}}^{(1)}$ is the rate-of-strain tensor for $\boldsymbol {v}^{(1)}$ and $p^{(1)}$.

On the $z$ axis, the components of $\boldsymbol {\nabla } \boldsymbol {v}^{(0)}$ with respect to the Cartesian coordinates $(x,y,z)$ is expressed by $\partial _zv^{(0)}_z$ multiplied by a traceless diagonal matrix, whose diagonal elements are $-1/2, -1/2$ and $1$ from the top. Here, $\partial _z$ indicates the partial derivative with respect to $z$. Thus, noting the description at the end of the preface of § 2, we can regard $\boldsymbol {v}^{(0)}$ on the $z$ axis as a pure-extension flow locally. In particular, at a point with $\theta ={\rm \pi}$ for $U>0$, the largest stretching rate occurs in the $z$ direction, i.e. the radial direction, and is given by $\partial _r v^{(0)}_r$. As $\theta$ approaches ${\rm \pi} /2$, periodic motion becomes predominant over elongational motion in $\boldsymbol {v}^{(0)}$, as is suggested by figure 1 and is explicitly shown in appendix B. A rotational flow is found to be weak in suppressing the critical enhancement (Onuki & Kawasaki 1980c). Thus, considering the discussion on the elongational flow in the fourth paragraph of § 1, we assume that $s^{(0)}$ is given by the largest real-part of the eigenvalues of $\boldsymbol {\nabla }\boldsymbol {v}^{(0)}$. Calculating them directly from (2.8a–c), we find $s^{(0)}$ to be given by

(2.14)\begin{equation} s^{(0)}(\rho,\theta)=\left| \frac{3{c} U}{4r_0} \left(\frac{1}{\rho^{2}}-\frac{1}{\rho^{4}}\right)\cos{\theta}\right|, \end{equation}

where a positive factor ${c}$, depending on $\theta$, ranges from $1/2$ to $2$. Equation (2.14) with ${c}=2$ equals $\vert \partial _r v^{(0)}_r\vert$. We proceed below with calculations by regarding ${c}$ as a constant, in spite of the actual dependence of $c$ on $\theta$. It will be shown later that our result of the self-diffusion coefficient is rather insensitive to the value of ${c}$.

2.3. Expansions with respect to the spherical harmonics

The flow we consider here is symmetric with respect to the polar axis, and thus the angular part of $\boldsymbol {v}^{(1)}$ can be expanded in terms of the vector spherical harmonics,

(2.15a,b)\begin{equation} \boldsymbol{P}_{J0} = Y_{J0} \boldsymbol{e}_r\quad\textrm{and}\quad \boldsymbol{B}_{J0}=\frac{r\boldsymbol{\nabla} Y_{J0}}{\sqrt{J(J+1)}} \end{equation}

for $J=1,2,\ldots$ (Morse & Feshbach 1953; Barrera, Estévez & Giraldo 1985; Fujitani 2007). Here, $Y_{J0}(\theta )$ is one of the spherical harmonics, $\sqrt {(2J+1)/(4{\rm \pi} )}P_J(\cos {\theta })$, with $P_J$ denoting the Legendre polynomial, e.g. $P_1(x)=x$. The mode $J=0$ need not be considered because of the incompressibility. We define functions $\varPi _J$, $R_J$ and $T_J$ so that

(2.16a,b)\begin{equation} p^{(1)}= \sum_{J} \varPi_J(\rho) Y_{J0}(\theta)\quad \textrm{and}\quad \boldsymbol{v}^{(1)}=\sum_J \left[R_J(\rho) \boldsymbol{P}_{J0}(\theta,\phi) +T_{J}(\rho) \boldsymbol{B}_{J0}(\theta,\phi) \right] \end{equation}

hold. We expand the negative of the right-hand side of (2.12) as

(2.17)\begin{equation} \sum_{J} \left[ F_{J}(\rho) \boldsymbol{P}_{J0}(\theta,\phi) +H_{J}(\rho)\boldsymbol{B}_{J0}(\theta,\phi) \right], \end{equation}

whereby $F_J$ and $H_J$ are defined. They are obtained with the aid of the orthogonality of the vector spherical harmonics. The incompressibility condition gives

(2.18)\begin{equation} T_{J}(\rho)=\frac{1}{\rho\sqrt{J(J+1)}}\partial_{\rho} \rho^2R_{J}(\rho). \end{equation}

We use (2.18) to delete $T_J$ from the $r$ and $\theta$ components of (2.12), which are combined to give

(2.19)\begin{equation} \left(\rho\partial_{\rho}+J\right)\left(\rho\partial_{\rho}+J+2\right) \left(\rho\partial_{\rho}-J+1\right) \left(\rho\partial_{\rho}-J-1\right)R_{J}(\rho)=-X_J(\rho). \end{equation}

Here, $X_J$ is defined as

(2.20)\begin{equation} X_J(\rho) =\frac{r_0^2\rho^2}{\tilde{\eta}^{(0)}}\left[ \sqrt{J(J+1)}\partial_{\rho} \rho H_{J}(\rho)-J(J+1)F_{J}(\rho)\right]. \end{equation}

Similar calculations can be found in deriving (2.17) of Fujitani (2007) and in deriving (3.20) of Yabunaka & Fujitani (2020). Equations (2.11) and (2.18) give

(2.21)\begin{equation} R_J(\rho)=\partial_{\rho} R_J(\rho)=0\quad \textrm{at}\ \rho=1 \quad \textrm{and}\quad R_J(\rho)\to 0\quad \textrm{as}\ \rho\to\infty . \end{equation}

Applying the method of variation of parameters, we can rewrite (2.19) as

(2.22)\begin{equation} R_{J}(\rho)=\int_1^{\infty} \textrm{d}\sigma\ \varGamma_J(\rho,\sigma)X_J(\sigma), \end{equation}

where the kernel $\varGamma_{J}$ is given in appendix C.

The first term of (2.13) does not contribute to $\mathcal {F}_z^{(1)}$ because (2.14), and thus (2.5), vanish at the particle surface. With the aid of (2.18), we use the $\theta$ component of (2.12) to delete $\varPi _J$ and $T_J$ from the last two terms of (2.13). These terms are thus rewritten as the sum of terms involving $R_J$ and $H_J$ over $J$. Only the terms involving $R_1$ are left after the surface integration of (2.13), as shown by (C 4) and described below (C 5). Substituting (2.22) into the result of the surface integration, we use

(2.23)\begin{equation} \left(\frac{1}{2}\partial_{\rho}^3+2\partial_{\rho}^2\right)\varGamma_1\left(\rho,\sigma\right) =\frac{3\sigma^2-1}{4\sigma^3}\quad \textrm{at}\ \rho=1 . \end{equation}

The right-hand side above is related to the fraction appearing in (2.8b) because of the Lorentz reciprocal theorem (Lorentz 1896), as shown in appendix B of Fujitani (2018) and mentioned at (A2) of Yabunaka & Fujitani (2020). We thus arrive at

(2.24)\begin{equation} \mathcal{F}_z^{(1)}= \mathcal{F}_z^{(0)}\int_1^{\infty}\,\textrm{d}\rho\ \frac{3\rho^2-1}{\rho^3}\check{X}(\rho) , \end{equation}

where $\check {X}$ is defined as

(2.25)\begin{equation} \check{X}(\rho)\equiv \sqrt{\frac{3}{\rm \pi}}\frac{X_1(\rho)}{36U}. \end{equation}

As shown by (C 7), $\check {X}$ is given in terms of the integral with respect to $\theta$ because (2.20) involves $F_1$ and $H_1$, which are calculated from the right-hand side of (2.12) with the aid of the orthogonality of the vector spherical harmonics. Thus, (2.24) contains a double integral with respect to $\rho$ and $\theta$. We have analytical results for the integrals with respect to $\rho$, as described at the end of appendix C, and thus what to calculate numerically is the integration with respect to $\theta$.

We find $\check {X}$ to depend on $U$ only through $\tau _{s^{(0)}}$. With $U^*$ denoting $s^*r_0$, the ratio $\gamma (U)/\gamma ^{(0)} = 1+(\mathcal {F}_z^{(1)}/\mathcal {F}_z^{(0)})$ is a function of a dimensionless speed $|U|/U^*$ because of (2.3) and (2.14), and is denoted by $\check {\gamma }(U/U^*)$. Using the value of the critical exponents stated in the preface of § 2, we numerically calculate the integral of (2.24) to obtain $\check {\gamma }(u)$. In figure 2, $\check {\gamma }(u)$ decreases as $u$ increases, which represents that the critical enhancement of $\tilde {\eta }$ is suppressed more as the faster particle causes stronger shear. At the smaller value of $\tau$, the suppression is shown to be stronger, which can be explained by the existence of larger clusters deformable for smaller $u$.

Figure 2. We plot $\check {\gamma }(u)$ for $\tau =1.008\times 10^{-3}\ (\circ )$ and $1.26\times 10^{-4}\ (+)$. As mentioned in the text, $\check {\gamma }$ represents the drag coefficient non-dimensionalized by (2.9), while $u$ represents the particle speed non-dimensionalized by $U^{\ast }\equiv s^{\ast } r_0$.

2.4. Description of the Brownian motion

When the viscosity is homogeneous in the absence of the suppression, as mentioned in § 1, a simple description of the Brownian motion is given by the Langevin equation for the particle velocity, where the force exerted on the particle is separated into the thermal noise and the instantaneous friction force (Bian et al. 2016). The former represents the force varying much more rapidly than the latter and vanishes after being averaged over a macroscopic time interval (Sekimoto 2010), while the friction coefficient in the latter equals the drag coefficient given by Stokes’ law. This is founded in terms of the fluctuating hydrodynamics (Bedeaux & Mazur 1974; Mazur & van der Zwan 1978) and is numerically verified (Keblinski & Thomin 2006). The components of the thermal noise in the three orthogonal directions are statistically independent, and thus the self-diffusion coefficient can be calculated in one dimension. To calculate the self-diffusion coefficient in our problem, we still use the drag coefficient $\gamma (U)$ as the friction coefficient in the Langevin equation, considering that the viscosity can be only weakly inhomogeneous depending on the particle speed. This amounts to assuming $\gamma (U)$ to be most probable friction coefficient when the particle velocity is $U$ in the Brownian motion at the time resolution of the Langevin equation.

The effective mass, denoted by $m$, is the sum of the particle mass and half the mass of the displaced fluid (Lamb 1932; Bian et al. 2016; Fujitani 2018). Here, unlike in the preceding subsections, $U$ is a stochastic variable depending on the time $t$. The Langevin equation is

(2.26)\begin{equation} m\frac{\textrm{d}U}{\textrm{d}t} =-\gamma(U) U+ b(U)\circ \frac{\textrm{d}W}{\textrm{d}t}, \end{equation}

where $W$ represents the Wiener process and the symbol $\circ$ indicates that (2.26) should be interpreted in the Stratonovich sense (Risken 2002; Sekimoto 2010). The positive function $b(U)$ is fixed so that (2.26) is consistent with the Boltzmann distribution, as shown in appendix D. The self-diffusion coefficient of the particle $D$ is given by (Bian et al. 2016)

(2.27)\begin{equation} D=\int_0^{\infty} \textrm{d}t\ \langle U(t)U(0) \rangle, \end{equation}

where $\langle \cdots \rangle$ means the equilibrium average. Defining $M$ as $mU^{*2}/(k_{B}T)$, we utilize the Laplace transformation to obtain

(2.28)\begin{equation} D=\frac{k_{B}T}{\gamma^{(0)}}\sqrt{\frac{1}{2{\rm \pi} M}}\int_{0}^{\infty}\,\textrm{d}u\ \left[\int_{u}^{\infty}\,\textrm{d}u_1\ \check{\gamma}(u_1)u_1 \textrm{e}^{-Mu_1^2}\right]^{-1}{\rm e}^{-3Mu^2/2}, \end{equation}

as shown in appendix D. This equation can be also derived from (2.26) by using not the Laplace transformation but some of the equations in § S.9 of Risken (2002). Converting the integration variables $u$ and $u_1$ to $u\sqrt {M}$ and $u_1\sqrt {M}$, respectively, we find that $D$ depends on $M$ only through the variable of $\check {\gamma }$.

Equation (2.28) involves $\check {\gamma }(u)$ for infinitely large $u$ because (2.26) formally supposes any particle speed, including the particle speed larger than assumed in the hydrodynamic formulation. This is also the case with the Langevin equation supposing a constant drag coefficient, where Stokes’ law is assumed even for particle speeds so large as to break the validity of the Stokes approximation. In either case, we can avoid this inconvenience in computing the self-diffusion coefficient, to which such large speed never contributes. An effective cutoff speed is implicitly imposed on the Langevin equation. This point is discussed in the next section.

3. Results and discussion

Latex beads with radius $80\ \textrm {nm}$ are put in a mixture of isobutyric acid and water in the experiment of Beysens (2019), where we have $m=3.32\times 10^{-18}\ \textrm {kg}$ and $U^*=29.6\ \textrm {m}\ \textrm {s}^{-1}$. The mixture can be regarded as incompressible near the demixing critical point (Clerke & Sengers 1983; Onuki 2002). The thermal average of $U$ is $3.53\times 10^{-2}\ \textrm {m}\ \textrm {s}^{-1}$, which is denoted by $\bar {U}$. The improper integrals in (2.28) can be replaced by definite integrals involving only particle speeds smaller than approximately $4\bar {U}$, as described in the latter half of appendix D. Because the viscosity of the mixture is around $2.5\times 10^{-3}\ \textrm {kg}\ \textrm {m}^{-1}\ \textrm {s}^{-1}$ (Allegra, Stein & Allen 1971), the Reynolds number is $1.4\times 10^{-2}$ for $U=\bar {U}$, and remains sufficiently small as compared with unity even if multiplied by four. This is consistent with our hydrodynamic formulation. The variable $u$ in figure 2 represents $U/U^*$, which equals $1.19\times 10^{-3}$ for $U=\bar {U}$. Thus, the range of the horizontal axis in figure 2 approximately coincides with the integration interval of the definite integrals used in our numerical calculation.

The data of the self-diffusion coefficient in Beysens (2019), ranging over $6.31\times 10^{-5}\le \tau \le 6.81\times 10^{-2}$, are replotted with open circles in figure 3. The viscosity of the near-critical mixture of isobutyric acid and water, containing no particles, is measured in Allegra et al. (1971). From the data in their table 2, with the ones for four values of $\tau$ from the smallest being excluded according to Oxtoby (1975), we calculate the self-diffusion coefficient by applying Stokes’ law and the Sutherland–Einstein relation, i.e. by dividing $k_{B}T/(6{\rm \pi} r_0)$ by the viscosity, and plot the results with crosses in figure 3. The crosses, ranging over $1.14\times 10^{-4}\le \tau \le 2.78\times 10^{-2}$, agree with the open circles for $\tau >7\times 10^{-3}$. These open circles and the crosses should be explained by using (2.1), i.e. $\tilde {\eta }^{(0)}$, which is free from the suppression due to the shear.

Figure 3. Plot of the self-diffusion coefficient against $\tau$. Open circles represent the experimental data of Beysens (2019), where $T_{c}$ is estimated to be 301.1 K. Crosses come from Allegra et al. (1971), with $T_{c}$ being estimated to be 299.4 K. The dashed curve represents $k_{B}T/(6{\rm \pi} \tilde {\eta }^{(0)}r_0)$, which we calculate by using the parameter values estimated from the open circles for $\tau >7\times 10^{-3}$ and the crosses. Solid circles represent our results of (2.28) for $c=1$. The red short bar above (below) each of the solid circles for $\tau \le 1.61\times 10^{-2}$ represents (2.28) for $c=4\ (0.25)$.

Conversely, we can calculate the viscosity from the open circles for $\tau >7\times 10^{-3}$ by applying Stokes’ law and the Sutherland–Einstein relation. In a graph (not shown here) where these results and the data of Allegra et al. (1971) with the exclusion above are linearly plotted against $\tau$, we perform the curve-fit to $\tilde {\eta }^{(0)}$ with the aid of ‘NonlinearModelFit’ of Mathematica (Wolfram Research) by using $T_{c}=300.1\ \textrm {K}$ (Toumi & Bouanz 2008) and the values of the critical exponents stated in the preface of § 2. Estimated values are $\tilde {\eta }_0=3.38\times 10^{-6}\ \textrm {kg}\ \textrm {m}\ \textrm {s}^{-1}$ and $E_{a}/(k_{B}T_{c})=6.35$ with the standard deviations being $5.11\times 10^{-7}\ \textrm {kg}\ \textrm {m}\ \textrm {s}^{-1}$ and $1.53\times 10^{-1}$, respectively. We use the estimated values to calculate $\tilde {\gamma }^{(0)}=6{\rm \pi} \tilde {\eta }^{(0)}r_0$ and plot $k_{B}T/\gamma ^{(0)}$ with the dashed curve in figure 3; the curve agrees well with the data applied to the curve-fit. Employing ${\tilde \gamma }^{(0)}$ thus obtained, we calculate the prefactor of (2.28), whose integrals are numerically calculated after being replaced by the definite integrals mentioned above. Our results of $D$ for $c=1$ are plotted with solid circles in figure 3. It appears that the open circles saturate to reach a plateau as $\tau$ decreases below $3.2\times 10^{-3}$, although they are distributed rather widely in the direction of the vertical axis. Our results pass through the middle of the distribution. This strongly suggests that the saturation should represent the suppression of the critical enhancement of $\tilde {\eta }$ due to the local shear caused by the particle motion, as claimed by Beysens (2019).

Figure 4 shows that our calculation results of $D$ increase as $c$ increases, which can be expected because the shear is then evaluated to be larger. In this figure, the ratios of the change in $D$ lie within $5\,\%$ when $c$ changes from unity to $0.1$ or to $10$; dependences of the ratio on $c$ are almost the same for the two values of $\tau$. For comparison with the data of Beysens (2019), we plot (2.28) for $c=0.25$ and $4$ with red short bars in figure 3. It is clear for $\tau >10^{-3}$ that the two bars at the same $\tau$ are closer to each other as $\tau$ is larger; (2.28) depends on $c$ only when the suppression occurs. The range of $c$ is considered as $1/2\le c\le 2$ in § 2.2. For $c=1/2\ (2)$, each of the results indicated by solid circles for $\tau \le 1.61\times 10^{-2}$ is shifted to the middle between the solid circle and the lower (upper) short bar. These slight shifts show that our results for any value of $c$ in the interval of $1/2\le c\le 2$ explain the experimental data well. It is also suggested that, if we take into account the dependence of $c$ on $\theta$ in this interval, the calculation results should remain in good agreement with the experimental data.

Figure 4. Ratios of $D$ for some $c$ values to $D$ for $c=1$ are calculated with the aid of (2.28) and plotted. The values of $\tau$ are $1.26\times 10^{-4}\ (+)$ and $1.008\times 10^{-3}\ (\circ )$.

Let us examine where the suppression occurs around a particle moving in the way supposed in figure 1. In the approximation mentioned below (2.12), we use $s^{(0)}$ instead of $s$ in (2.5) to calculate $|\tilde {\eta }^{(1)}/\tilde {\eta }^{(0)}|$ for $c=1$ and $U=\bar {U}$, and show the results in figure 5, where the suppression occurs in coloured regions. The maximum of (2.14) is taken at $(\rho , \theta )= (\sqrt {2}, 0)$ or $(\sqrt {2},{\rm \pi} )$. The value of $\vert \tilde {\eta }^{(1)}/\tilde {\eta }^{(0)}\vert$ equals $\kappa$ at these points, and becomes smaller at a point more distant from these points, as shown in figure 5. The maximum of ${\tau }_{s^{(0)}}$ is $0.0129\times c^{0.518}$ for $U=\bar {U}$, and is $0.0264\times c^{0.518}$ for $U=4\bar {U}$, which is approximately the effective cutoff in our numerical integration. The latter yields $\kappa \approx 0.13$ at $\tau =10^{-3}$ and $\approx 0.21$ at $\tau =10^{-4}$ for $c=1$ and $U=4\bar {U}$. Thus, $\kappa$ is adequately small as compared with unity in the range of $\tau$ examined in figure 3, which would make our formulation globally meaningful.

Figure 5. Colour gradation represents the largest of zero and $1-\check {\tau }_{s^{(0)}}^{\nu (d-z)}$ for $c=1$ and $U=\bar {U}$ at each of $\tau =1.26\times 10^{-2}$ $(a)$ and $8.064\times 10^{-3}$ $(b)$. The particle is assumed to move translationally with the velocity $\bar {U}\boldsymbol {e}_z$. The region shown in each figure is the same as in figure 1; the dotted curve represents the half of the cross-section of the particle surface. The curves outside the particle are the stream lines of $\boldsymbol {v}^{(0)}$, which are represented by arrows in figure 1.

From the maximum of ${\tau }_{s^{(0)}}$ for $U=\bar {U}$ mentioned above, $\kappa$ is found to become non-zero when $\tau$ is smaller than $1.29\times 10^{-2}$ for $c=1$ and $U=\bar {U}$. This value of $\tau$ approximately agrees with the onset temperature of the suppression estimated in Beysens (2019), where the shear rate, with its inhomogeneity and dependence on $U$ being neglected, is evaluated to be a typical shear rate, $\bar {U}/r_0$. The value of $\tau$ is slightly smaller than $1.29\times 10^{-2}$ in figure 5(a), where a very weak suppression occurs in narrow regions around $(\rho , \theta )= (\sqrt {2}, 0)$ and $(\sqrt {2},{\rm \pi} )$ for $c=1$ and $U=\bar {U}$, as expected. However, at this temperature, the suppression cannot be read in figure 3. In figure 5(b) with smaller $\tau$, $|\tilde {\eta }^{(1)}/\tilde {\eta }^{(0)}|$ becomes larger in wider regions, which means that the suppression occurs more strongly and extensively, although the suppression can be read only slightly from the solid circle at this temperature and cannot be read from the open circle closest to this temperature in figure 3. It is for $\tau <3.2\times 10^{-3}$ in figure 3 that the suppression can be read explicitly from the experimental data ($\circ$); the suppression for $c=1$ and $U=\bar {U}$ should occur more strongly and extensively in this range of $\tau$ than in figure 5(b). Thus, we overestimate the onset temperature of the suppression in the data of the self-diffusion coefficient if we evaluate the shear rate to be $\bar {U}/r_0$.

For $U=4\bar {U}$, as mentioned above, the maximum of $\tau _{s^{(0)}}$ is $0.0264\times c^{0.518}$. The maximum is smaller than unity in the range of $c$ examined in figure 4, as supposed in our formulation. This also shows that our results are free from the details of a formal rule for large particle speed in appendix D. In the absence of the suppression, we have $\xi =\xi _0\tau ^{-\nu }=28\ \textrm {nm}$ and $120\ \textrm {nm}$ for $\tau =10^{-3}$ and $10^{-4}$, respectively. The correlation length should be reduced by the strong shear, which suppresses the order-parameter fluctuation with small wavenumber. The correlation length under the shear effect is dependent on the direction, and proportional to $\tau ^{-0.5}$ at the largest in the stagnation-point flow (Onuki & Kawasaki 1980c). This exponent is the same as in the mean-field approximation. The curvature of the stream line in figure 3 suggests that a typical length of the flow is several times larger than the particle diameter. Thus, a typical length of the flow is sufficiently large as compared with $\xi$ in the range of $\tau$ examined in figure 3, as supposed in our formulation.

It is assumed in (2.4) that the suppression is perfect if it occurs. In terms of the renormalization-group calculation, the singular part of the viscosity changes in the coarse-graining procedure, which makes sense until the resolution reaches $\xi$, and the way of the change is altered when the resolution exceeds a threshold determined by the shear rate. Equation (1.3) can be derived from the condition of whether the threshold comes before $\xi$ or not, as mentioned in the latter half of appendix A. In the procedure after the alteration, the singular part of the viscosity continues to be changed and becomes anisotropic. Thus, the assumption in (2.4) does not hold exactly. The ratio of the correction in the later stage, i.e. in the stage after the alteration, to the one in the earlier stage is evaluated by averaging the former correction over the directions in appendix E. If $d=3$ is substituted into these results valid up to the linear order with respect to $4-d$, the evaluated ratio is smaller than $4\ (7)$ per cent for a pure-extension flow (a simple shear flow). These small values would support the appropriateness of (2.4), which can explain the data for the three-dimensional mixture of Beysens (2019) in figure 3.

In deriving (1.3), the lifetime of a correlated cluster and the range of the cluster size are evaluated. However, some deviations are possible in these evaluations, and may be required to compensate for the approximation in (2.4) mentioned above. For example, let us consider replacing $\xi$ with $1.5 \xi$ in (1.3). This replacement is equivalent with putting $c$ equal to $1.5^{z}\approx 3.5$. A change of (1.3) to this extent cannot be denied from the data of Beysens (2019) in figure 3, considering that the red short bars above the solid circles still lie in the middle of the distribution of the data.

We simply link the drag coefficient with the self-diffusion coefficient by means of the one-dimensional Langevin equation for the particle velocity. Similar nonlinear Langevin equations are used for different problems in Klimontovich (1994) and Lindner (2007). Validity of the Langevin equation with the Stratonovich interpretation in our problem, where the viscosity can be inhomogeneous and dependent on the particle speed, remains to be founded on the fluctuating hydrodynamics, unlike in the cases studied by Bedeaux & Mazur (1974) and Mazur & van der Zwan (1978). It still appears that the Langevin equation can describe the data of Beysens (2019) well in figure 3, where a rather large distribution of the data for small $\tau$ in the direction of the vertical axis may come from the properties of the viscosity in our problem.

4. Summary and concluding remarks

Correlated clusters of the order-parameter fluctuation are generated in a near-critical binary fluid mixture lying in the homogeneous phase near the demixing critical point. The upper size of clusters, the correlation length, becomes larger as the critical point is approached. Then, as is well known, the convection of large and long-lived clusters enhances the transport coefficients in the coarse-grained dynamics (Kawasaki 1970). It is also well known that a sufficiently strong shear, if imposed, can deform long-lived clusters to suppress the critical enhancement (Onuki & Kawasaki 1979). In a recent experiment (Beysens 2019), shear around a Brownian particle in a near-critical mixture on the critical isochore was suggested to cause this suppression to influence the motion. Deviation of the self-diffusion coefficient from the Stokes–Sutherland–Einstein formula was observed in the temperature range where the suppression is judged to occur from a typical shear rate around a particle moving with a typical speed.

How the deviation depends on the temperature is calculated in the present study. We first calculate the drag coefficient of a particle moving translationally in a mixture which is quiescent far from the particle. The suppression is simply assumed to occur perfectly when the cluster with the size of the correlation length becomes so long-lived as to be deformed by the shear. The shear rate is inhomogeneous and depends on the particle speed. Hence, the suppression makes the viscosity inhomogeneous and dependent on the particle speed. The calculation supposes a low Reynolds-number and a sufficiently weak influence of the shear on the viscosity, which are realized in the experiment. We next employ the drag coefficient thus calculated, dependent on the particle speed, as the frictional coefficient in a one-dimensional Langevin equation of the Stratonovich type to calculate the self-diffusion coefficient. The calculation results agree well with the experimental data, which is rather robust to changes in the threshold for the occurrence of the suppression.