# Thermoosmosis of a near-critical binary fluid mixture:
a general formulation and universal flow direction

Youhei Fujitani<EMAIL_ADDRESS>School of Fundamental Science and
Technology, Keio University, Yokohama 223-8522, Japan  Shunsuke Yabunaka
<EMAIL_ADDRESS>Advanced Science Research Center, Japan Atomic Energy
Agency, Tokai, 319-1195, Japan

###### Abstract

We consider a binary fluid mixture, which lies in the one-phase region near
the demixing critical point, and study its transport through a capillary tube
linking two large reservoirs. We assume that short-range interactions cause
preferential adsorption of one component on the tube’s wall. The adsorption
layer can become much thicker than the molecular size, which enables us to
apply hydrodynamics based on a coarse-grained free-energy functional. For
linear transport phenomena induced by gradients of the pressure, composition,
and temperature along a cylindrical tube, we obtain the formulas of the
Onsager coefficients to extend our previous results on isothermal transport,
assuming the critical composition in the middle of each reservoir in the
reference equilibrium state. Among the linear transport phenomena, we focus on
thermoosmosis — mass flow due to a temperature gradient. We explicitly derive
a formula for the thermal force density, which is nonvanishing in the
adsorption layer and causes thermoosmosis. This formula for a near-critical
binary fluid mixture is an extension of the conventional formula for a one-
component fluid, expressed in terms of local excess enthalpy. We predict that
the direction of thermoosmotic flow of a mixture near the upper (lower)
consolute point is the same as (opposite to) that of the temperature gradient,
irrespective of which component is adsorbed on the wall. Our procedure would
also be applied to dynamics of a soft material, whose mesoscopic inhomogeneity
can be described by a coarse-grained free-energy functional.

##  I Introduction

Osmotic transport of a fluid through a channel at micrometer, or smaller
scales, has gained much attention because it is applied in lab-on-a-chip
processes breg ; lee ; shin ; shakib ; chen  and involves fundamental problems
in nonequilibrium physics piazza ; Wurger ; marbach ; ganti ; mang ; frenkel .
A gradient of temperature (concentration) along the channel induces a flow,
called thermoosmosis (diffusioosmosis), because of force density generated in
a heterogeneous layer near the wall derja ; derja2 ; anders ; thermoosmosis
does not involve the buoyancy responsible for thermal convection. Derjaguin
and his coworkers rationalize thermoosmosis and diffusioosmosis in terms of
the continuum description derja3 ; derja4 ; derja5 .

For a one-component fluid, Derjaguin and Sidorenkov (DS) derja3  derived a
formula of the thermal force density, which causes thermoosmosis. According to
DS’s formula, the direction of the flow is the same as (opposite to) that of
the temperature gradient if the excess enthalpy density is negative (positive)
near the wall. This is naively expected since the flow in this direction tends
to eliminate the temperature gradient by carrying the fluid with lower
(higher) enthalpy to the region with higher (lower) temperature. However,
there exist several difficulties when applying this formula to experimental
systems as follows. Firstly, the thickness of the heterogeneous layer is
microscopic and the continuum description cannot incorporate details inside
the layer very precisely ganti ; frenkel . Secondly, the local excess enthalpy
is not easy to access experimentally. Its evaluation based on microscopic
models has been studied fu ; ganti , whereas well-definedness of its
microscopic expression is questioned especially near the surface ganti ;
Anzini . In Ref. ganti , the authors extend DS’s formula for a multicomponent
fluid in the continuum description, while noticing the difficulties discussed
above. Therefore, it remains often hard to understand even the flow direction
in thermoosmosis piazza .

Figure 1: Schematic of a situation supposed in our formulation. A mixture is
filled in the container composed of two reservoirs and a capillary tube
connecting them. One component drawn in yellow is preferentially adsorbed by
the tube’s wall, which is impermeable and adiabatic. There may be preferential
adsorption onto the reservoir’s wall, which is not supposed in this figure.
Thick walls represent pistons. Imposing difference in pressure, composition,
and/or temperature between the reservoirs generates a mass flow through the
tube.

Thermoosmosis has not been studied in relation to critical phenomena to our
best knowledge. In the present study, for a binary fluid mixture lying in the
one-phase region close to the demixing critical point, we extend our previous
study on isothermal dynamics pipe  to cover non-isothermal dynamics of
thermoosmosis. Below, this mixture, simply referred to as a mixture, is
assumed to be filled in a container composed of two large reservoirs and a
capillary tube connecting them. The tube’s wall is impermeable and adiabatic.
Differences in temperature, composition, and pressure can be imposed between
the two reservoirs (Fig. 1). A short-range interaction is assumed between each
mixture component and tube’s wall, which in general attracts one component
more than the other. The resulting preferential adsorption (PA), becoming
remarkable owing to large osmotic susceptibility beysens1982 ; schol , has
been studied for a long time binder ; diehl86 ; diehl97 ; law . The adsorption
layer can be much thicker than the molecular size. Thus, it is expected that
we can study transport of a mixture through a tube in terms of the continuum
description, avoiding its difficulty mentioned in the last paragraph. In Ref.
pipe , applying the hydrodynamics based on a coarse-grained free-energy
functional yabufuji , the present authors studied isothermal transport in the
linear regime with respect to thermodynamic forces to calculate the involved
Onsager coefficients and conductance in diffusiooosmosis.

Order-parameter fluctuations in a mixture, being significant on length scales
smaller than the correlation length, enhance the transport coefficients to
cause universal properties kawasaki ; sighalhoh ; sengers ; folkmos ;
onukibook . However, the PA keeps the mixture composition in a tube away from
the critical one, in particular near the tube’s wall, and thus the critical
enhancement does not affect diffusioosmosis of a mixture significantly pipe .
Hydrodynamics can be applied to a flow in the tube, where the correlation
length is locally smaller than a typical length of the flow. It is also
suggested in Ref. pipe  that, in a critical regime, the mixture velocity due
to diffusioosmosis far from a flat surface should exhibit a power-law
dependence on the difference between the mixture temperature and the critical
temperature if the adsorption is sufficiently strong. This originates from the
universal order-parameter profile at equilibrium fisher-degennes ; RJ . The
same power law is numerically suggested for the diffusiophoretic mobility of a
colloidal particle in a mixture diffphore ; diffdrop .

In the present study, we extend our previous procedure of Ref. pipe  to cover
non-isothermal dynamics. Our general formulation is described in Sections II.1
and II.2. We employ the hydrodynamics under inhomogeneous temperature
formulated from a coarse-grained free-energy functional dvw ; gonn . Imposing
the no-slip condition at the tube’s wall and neglecting effects of the tube’s
edges, we discuss flow fields in the tube in the linear regime in Section
II.3. In Sections III.1 and III.2, we derive formulas for the Onsager
coefficients and a formula of the thermal force density for a cylindrical
tube, assuming the total mass density to be homogeneous inside the tube and
the mixture composition to be critical in the middle of a reservoir in the
reference equilibrium state. The former formulas include extensions of our
previous results of Ref. pipe  to non-isothermal transport, whereas the latter
can be regarded as an extension of DS’s formula to a mixture considered here.
We apply the renormalized local functional theory fisher-degennes ; rlft  to
specify the free-energy functional in Section III.3, and rewrite our formulas
in Section III.4. This theory can incorporate the effects of the critical
fluctuations when the correlation length is spatially varying as is the case
in the adsorption layer. In Section IV, we focus on thermoosmosis to show
numerical results, and predict that, irrespective of which component is
adsorbed on the wall, the flow direction is the same as (opposite to) the
direction of the temperature gradient in thermoososis of a mixture near the
upper (lower) consolute point. Further discussion and summary are given in
Section V. Some of our results are also discussed in the companion Letter
complett .

##  II Formulation

We write $\rho_{\rm a}$ ($\rho_{\rm b}$) for the mass density of a mixture
component named a (b). The sum $\rho_{\rm a}+\rho_{\rm b}$ is denoted by
$\rho$, which represents the total mass density, whereas the difference
$\rho_{\rm a}-\rho_{\rm b}$ is denoted by $\varphi$. We write $c_{n}$ for the
mass fraction $\rho_{n}/\rho$ and $\mu_{n}$ for the chemical potential
conjugate to $\rho_{n}$, where $n$ is a or b. In an equilibrium mixture with
homogeneous mass densities, $\mu_{n}$ is a function of the temperature
(denoted by $T$), pressure ($P$), and $c_{\rm a}$, and is also a function of
$T$, $\rho$, and $\varphi$. We write $\mu_{\pm}$ for $\left(\mu_{\rm
a}\pm\mu_{\rm b}\right)/2$; $\rho$ and $\varphi$ are conjugate to $\mu_{+}$
and $\mu_{-}$, respectively. Difference of a quantity in the left reservoir
subtracted from the quantity in the right is indicated by $\delta$. For
example, $\delta\mu_{\rm a}$ denotes the difference in $\mu_{\rm a}$ between
the reservoirs. If the reservoir’s wall adsorbs a component more, $\varphi$
can be inhomogeneous near the wall. For such a quantity, $\delta$ indicates
the difference between the central regions of the reservoirs. The difference
between the pressures on the pistons is given by $\delta P$.

###  II.1 Thermodynamics

We first consider entropy fluctuations of an equilibrium mixture in the
isolated container with the pistons fixed (Fig. 1). The total mass of the
component ${n}$ in the right reservoir is denoted by ${\cal M}_{{n}{\rm R}}$.
Neglecting the contribution from the mixture in the tube, we can regard the
total entropy of the mixture in the container, denoted by $S$, as a function
of ${\cal M}_{{\rm aR}}$, ${\cal M}_{{\rm bR}}$, and the internal energy of
the mixture in the right reservoir, ${\cal U}_{\rm R}$. With $t$ denoting the
time,

$\frac{dS}{dt}=\frac{d{\cal U}_{\rm
R}}{dt}\delta\left(\frac{1}{T}\right)+\frac{d{\cal M}_{{n}{\rm
R}}}{dt}\delta\left(\frac{-\mu_{n}}{T}\right)$ (1)

holds up to the second order of the magnitudes of the deviations. Repeated
indices are summed hereafter. Equation (1) is included in Eq. (XV-55) of Ref.
gromaz . The thermodynamic fluxes are given by the time derivatives on the
right-hand side (RHS) and are driven by the conjugate thermodynamic forces,
$\delta(1/T)$ and $-\delta(\mu_{n}/T)$. They are respectively the partial
derivatives of $S$ with respect to ${\cal U}_{\rm R}$ and ${\cal M}_{n{\rm
R}}$ engineer .

We assume that weak thermodynamic forces are imposed on an equilibrium
mixture. This equilibrium state is close to the critical point and is referred
to as reference state. We add a superscript (ref) to a quantity in the middle
of a reservoir in the reference state. For example, $\rho_{n}^{({\rm ref})}$
denotes $\rho_{n}$ in the central regions in the reference state. Writing $u$
for the internal energy per unit volume, we apply the Gibbs-Duhem (GD)
relation to obtain

$\left(\begin{array}[]{c}-\delta\left(P/T\right)\\\
-\delta\left(\mu_{-}/T\right)\\\
\delta\left(1/T\right)\end{array}\right)=\left(\begin{array}[]{ccc}\rho_{\rm
a}^{({\rm ref})}&\rho_{\rm b}^{({\rm ref})}&u^{({\rm ref})}\\\ 1/2&-1/2&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{c}-\delta\left(\mu_{\rm
a}/T\right)\\\ -\delta\left(\mu_{\rm b}/T\right)\\\
\delta\left(1/T\right)\end{array}\right)\ ,$ (2)

whose left-hand side (LHS) gives a new set of thermodynamic forces. We write
$\Theta$ for the coefficient matrix on the RHS above. The thermodynamic fluxes
conjugate to the respective components of the new set, denoted by ${\cal I}$,
${\cal J}$, and ${\cal K}$, are defined by the first equality of

$\left(\Theta^{-1}\right)^{\rm T}\left(\begin{array}[]{c}d{\cal M}_{\rm
aR}/(dt)\\\ d{\cal M}_{\rm bR}/(dt)\\\ d{\cal U}_{\rm
R}/(dt)\end{array}\right)=\left(\begin{array}[]{c}{\cal I}\\\ {\cal J}\\\
{\cal K}\end{array}\right)=L\left(\begin{array}[]{c}-\delta\left(P/T\right)\\\
-\delta\left(\mu_{-}/T\right)\\\ \delta\left(1/T\right)\end{array}\right)\ ,$
(3)

where the superscript T indicates the transposition. The second equality above
represents linear phenomenological equations; a matrix $L$ is here introduced
and their components are Onsager coefficients considered later.

The partial volume and partial entropy per unit mass of the component $n$, are
denoted by ${\bar{v}}_{n}$ and ${\bar{s}}_{n}$, respectively. In an
equilibrium mixture with homogeneous densities, we have

${\bar{v}}_{n}=\left.\frac{\partial\mu_{n}}{\partial P}\right)_{T,c_{\rm
a}}\quad{\rm and}\quad{\bar{s}}_{n}=-\left.\frac{\partial\mu_{n}}{\partial
T}\right)_{P,c_{\rm a}}\ ,$ (4)

where the subscript of a right parenthesis indicates the fixed variables in
the partial differentiation. Writing $s$ for the entropy per unit volume, we
have

$1={\bar{v}}_{\rm a}\rho_{\rm a}+{\bar{v}}_{\rm b}\rho_{\rm b}\quad{\rm
and}\quad s={\bar{s}}_{\rm a}\rho_{\rm a}+{{\bar{s}}_{\rm b}}\rho_{\rm b}$ (5)

at equilibrium with $\rho_{n}$ and $s$ being homogeneous. Writing
${\bar{v}}_{-}$ and ${\bar{s}}_{-}$ for $({\bar{v}}_{\rm a}-{\bar{v}}_{\rm
b})/2$ and $({\bar{s}}_{\rm a}-{\bar{s}}_{\rm b})/2$, respectively, we obtain

$\delta\mu_{-}=-{\bar{s}}_{-}^{({\rm ref})}\delta T+{\bar{v}}_{-}^{({\rm
ref})}\delta P+\left.\frac{\partial\mu_{-}}{\partial c_{\rm a}}\right)_{T,P}\
\delta c_{\rm a}\ ,$ (6)

where the partial derivative is evaluated in the middle of a reservoir in the
reference state. Thus, the thermodynamic forces in Eq. (3) are determined by
$\delta T$, $\delta P$, and $\delta c_{\rm a}$, and determine $\delta\mu_{+}$
because the GD relation gives

$\delta P=\rho_{n}^{({\rm ref})}\delta\mu_{n}+s^{({\rm ref})}\delta
T=\rho^{({\rm ref})}\delta\mu_{+}+\varphi^{({\rm ref})}\delta\mu_{-}+s^{({\rm
ref})}\delta T\ .$ (7)

###  II.2 Hydrodynamics

We consider the Helmholtz free-energy of a mixture as a functional of fields
coarse-grained up to the local correlation length of the order-parameter
fluctuations. The length is denoted by $\xi$. A part of the functional is
given by the volume integral of a function over the mixture region, $V_{\rm
tot}$. The function, denoted by $f_{\rm bulk}$, is assumed to depend on
$\rho_{\rm a}$, $\rho_{\rm b}$, their gradients, and $T$, with the dependence
on the gradients being via $|\nabla\rho_{\rm a}|^{2}$, $|\nabla\rho_{\rm
b}|^{2}$ and $(\nabla\rho_{\rm a})\cdot(\nabla\rho_{\rm b})$. The other part,
representing the wall-component interaction, is assumed to be given by the
area integral of a function over the interface, $\partial V_{\rm tot}$. This
function is denoted by $f_{\rm surf}$ and depends on $\rho_{\rm a}$,
$\rho_{\rm b}$, and $T$. Thus, the free-energy functional is given by

$F[T,\rho_{\rm a},\rho_{\rm b}]=\int_{V_{\rm tot}}d{\mbox{\boldmath$r$}}\
f_{\rm bulk}\left(T,\rho_{\rm a},\rho_{\rm b},\nabla\rho_{\rm
a},\nabla\rho_{\rm b}\right)+\int_{\partial V_{\rm tot}}dA\ f_{\rm
surf}\left(T,\rho_{\rm a},\rho_{\rm b}\right)\ ,$ (8)

where $T$ and $\rho_{n}$ depend on the position $r$. The volume element and
the area element are represented by $d{\mbox{\boldmath$r$}}$ and $dA$,
respectively.

We can formulate hydrodynamics on length scales larger than $\xi$ by using Eq.
(8). If $T$ is homogeneous, $\mu_{n}({\mbox{\boldmath$r$}})$ is given by the
functional derivative of the first term on the RHS of Eq. (8) with respect to
$\rho_{n}({\mbox{\boldmath$r$}})$ in $V_{\rm tot}$. Otherwise, it is given by

$\mu_{n}=\frac{\partial f_{\rm
bulk}}{\partial\rho_{n}}-T\nabla\cdot\left[\frac{1}{T}\frac{\partial f_{\rm
bulk}}{\partial\left(\nabla\rho_{n}\right)}\right]\ .$ (9)

The reversible part of the pressure tensor, denoted by ${\mathsf{\Pi}}$, is
given by

${\mathsf{\Pi}}=P{\mathsf{1}}+\left(\nabla\rho_{n}\right)\frac{\partial f_{\rm
bulk}}{\partial\left(\nabla\rho_{n}\right)}\ ,$ (10)

which is symmetric. Here, ${\mathsf{1}}$ is the identity tensor of order two.
The scalar pressure $P$ is given by the negative of the grand-potential
density,

$P=\mu_{n}\rho_{n}-f_{\rm bulk}=\mu_{+}\rho+\mu_{-}\varphi-f_{\rm bulk}\ .$
(11)

Equations (9) and (10) are derived for a one-component fluid in Ref. dvw  and
are applied in a straightforward way to a binary fluid mixture gonn . The
previous derivation is not applicable to the free-energy density introduced in
Section III.3 because the coefficient of its square-gradient term, $M_{-}/2$
in $f_{\rm bulk}$ of Eq. (48), depends on $T$ slightly nonlinearly. The linear
dependence is assumed in the previous derivation. In Appendix A, we show a
more general derivation, which is applicable to $f_{\rm bulk}$ of Eq. (48).
Strictly speaking, this derivation remains relevant to our later calculation
of thermoosmosis under the linear regime, because the hydrodynamic equations
including this nonlinearity must be derived before their linearization with
respect to the temperature gradient. Equations (9) and (10) yield an extended
GD formula

${\nabla\cdot{\mathsf{\Pi}}=\nabla\cdot{\mathsf{\Pi}}^{\rm
T}=\rho_{n}\nabla\mu_{n}+s\nabla T+\frac{\nabla T}{T}\cdot\frac{\partial
f_{\rm bulk}}{\partial\left(\nabla\rho_{n}\right)}\nabla\rho_{n}\ ,}$ (12)

which is consistent with principles of linear nonequilibrium thermodynamics;
Eq. (12) guarantees positive entropy production rate after combined with
irreversible terms and the Onsager’s reciprocity for osmotic fluxes through
the tube. The former is shown in Ref. dvw , whereas the latter is mentioned
below Eq. (32) in the next subsection.

The velocity field, $v$, is defined in the frame fixed to the container. The
time-derivative of $\rho$ equals $-\nabla\cdot(\rho{\mbox{\boldmath$v$}})$. In
the stationary state, we have

$0=\nabla\cdot\left(\rho{\mbox{\boldmath$v$}}\right)\ .$ (13)

Because of the mass conservation of each component, the time derivative of
$\rho_{n}$ is equal to the negative of the divergence of its flux, whose
deviation from the convective part, $\rho_{n}{\mbox{\boldmath$v$}}$, gives the
diffusion flux, denoted by ${\mbox{\boldmath$j$}}_{n}$. It is defined so that
${\mbox{\boldmath$j$}}_{\rm a}+{\mbox{\boldmath$j$}}_{\rm b}$ vanishes. In the
stationary state, we have

$0=\nabla\cdot\left(\varphi{\mbox{\boldmath$v$}}+{\mbox{\boldmath$j$}}\right)\
,$ (14)

where $j$ is defined as ${\mbox{\boldmath$j$}}_{\rm
a}-{\mbox{\boldmath$j$}}_{\rm b}$.

Assuming that $\delta T$, $\delta P$, and $\delta c_{\rm a}$ are proportional
to a dimensionless smallness parameter, $\varepsilon$, we consider the
dynamics in the tube at the order of $\varepsilon$ in Section II.3. The
superscripts (0) and (1) are used to indicate the order of $\varepsilon$. For
example, we have $\mu_{\pm}=\mu_{\pm}^{(0)}+\varepsilon\mu_{\pm}^{(1)}$ up to
the order of $\varepsilon$, $\mu_{\pm}^{(0)}=\mu_{\pm}^{({\rm ref})}$, and
$T^{(0)}=T^{({\rm ref})}$. In contrast, $\rho_{n}^{(0)}$ becomes inhomogeneous
and different from $\rho_{n}^{({\rm ref})}$ in the presence of PA. Because the
fields are assumed to be coarse-grained up to $\xi$, the mass densities at the
equilibrium specified by $T^{(\rm ref)}$ and $\mu_{n}^{({\rm ref})}$ minimize
the grand-potential functional,

${F}[T^{({\rm ref})},\rho_{\rm a},\rho_{\rm b}]-\int_{V_{\rm
tot}}d{\mbox{\boldmath$r$}}\ \left[\mu_{+}^{({\rm
ref})}\rho({\mbox{\boldmath$r$}})+\mu_{-}^{({\rm
ref})}\varphi({\mbox{\boldmath$r$}})\right]\ .$ (15)

Thus, $\rho_{n}^{(0)}$ is the solution of Eq. (9) with $T$ and $\mu_{n}$ being
replaced by $T^{({\rm ref})}$ and $\mu_{n}^{({\rm ref})}$, respectively,
together with the boundary conditions given by Eq. (79).

###  II.3 Fields in the tube

In this subsection, we consider stationary and laminar flow in the tube at the
order of $\varepsilon$. The mixture is assumed to remain in one-phase region
throughout inside the container. As in the previous study pipe , we assume
that the tube is so long and thin that effects of the tube’s edges on the flow
are negligible. We regard $\delta\mu_{n}$ and $\delta T$ as equal to the
differences in $\mu_{n}$ and $T$ between the edges, respectively; $\mu_{n}$
and $T$ are assumed to be homogeneous over the tube’s cross-section at each
edge.

Assuming the tube to extend along the $z$ axis with the same cross-section, we
take the Cartesian coordinates $(x,y,z)$ with the right reservoir lying on the
positive $z$ side. A field with the superscript (0), such as $\rho_{n}^{(0)}$,
is independent of $z$ in the tube. We apply Eq. (13) for a laminar flow to
obtain

$0=\nabla\cdot{\mbox{\boldmath$v$}}^{(1)}=\partial_{z}v_{z}^{(1)}\ ,$ (16)

without assuming the mixture to be incompressible. Here, $\partial_{z}$
denotes the partial derivative with respect to $z$. In the tube up to the
order of $\varepsilon$, the momentum conservation gives

$2\nabla\cdot\left(\eta_{\rm s}{\mathsf{E}}\right)=\nabla\cdot{\mathsf{\Pi}}\
,$ (17)

where $\eta_{\rm s}$ is the shear viscosity and ${\mathsf{E}}$ is the rate-of-
strain tensor. Owing to the critical enhancement, $\eta_{\rm s}$ depends on
the local correlation length, $\xi$. Writing $\eta_{\rm s}^{(0)}$ for
$\eta_{\rm s}$ evaluated at $\varepsilon=0$, we rewrite the LHS of Eq. (17) as
$\varepsilon$ multiplied by

$2\nabla\cdot\left(\eta_{\rm
s}^{(0)}{\mathsf{E}}^{(1)}\right)=\nabla\cdot{\left\\{\eta_{\rm
s}^{(0)}\left[\nabla{\mbox{\boldmath$v$}}^{(1)}+\left(\nabla{\mbox{\boldmath$v$}}^{(1)}\right)^{\rm
T}\right]\right\\}}\ .$ (18)

With the aid of Eq. (12), we rewrite the RHS of Eq. (17) as $\varepsilon$
multiplied by

$\nabla\cdot{\mathsf{\Pi}}^{(1)}=\rho^{(0)}_{n}\nabla\mu_{n}^{(1)}+s^{(0)}\nabla
T^{(1)}+\frac{\nabla T^{(1)}}{T^{(0)}}\cdot\frac{\partial f_{\rm
bulk}}{\partial(\nabla\rho_{n})}\left(\nabla\rho_{n}^{(0)}\right)\ ,$ (19)

where the partial derivative of $f_{\rm bulk}$ is evaluated at
$\varepsilon=0$. In the absence of PA, $\rho_{n}^{(0)}$ and $\eta_{\rm
s}^{(0)}$ are homogeneous, and thus Eq. (17) becomes the usual Stokes
equation, $\eta_{\rm s}\Delta{\mbox{\boldmath$v$}}=\nabla P$, owing to Eq.
(10).

Writing $u$ ($e$) for the internal (total) energy per unit volume, we have
$u=e-\rho\left|{\mbox{\boldmath$v$}}\right|^{2}/2$,

$s=-\left.\frac{\partial f_{\rm bulk}}{\partial
T}\right)_{\rho_{n},\nabla\rho_{n}}\ ,\ {\rm and}\quad u=f_{\rm
bulk}+Ts=-T^{2}\left.\frac{\partial}{\partial T}\frac{f_{\rm
bulk}}{T}\right)_{\rho_{n},\nabla\rho_{n}}$ (20)

in $V_{\rm tot}$. The heat flux, denoted by ${\mbox{\boldmath$j$}}_{q}$, is
defined so that the Eulerian time-derivative of $e$ equals
$-\nabla\cdot(e{\mbox{\boldmath$v$}}+{\mbox{\boldmath$v$}}\cdot{\mathsf{\Pi}}+{\mbox{\boldmath$j$}}_{q})$.
In a stationary state in the tube, we have

$0=\nabla\cdot\left(u^{(0)}{\mbox{\boldmath$v$}}^{(1)}+{\mbox{\boldmath$v$}}^{(1)}\cdot{\mathsf{\Pi}}^{(0)}+{\mbox{\boldmath$j$}}_{\rm
q}^{(1)}\right)\ ,$ (21)

where $u^{(0)}$ equals $e^{(0)}$. We can define the transport coefficients,
$\Lambda$, $\lambda$, and $\kappa$, so that

${\mbox{\boldmath$j$}}=-T\Lambda\nabla\frac{\mu_{-}}{T}+\kappa\nabla\frac{1}{T}\quad{\rm
and}\quad{\mbox{\boldmath$j$}}_{\rm
q}=-\kappa\nabla\frac{\mu_{-}}{T}+\lambda\nabla\frac{1}{T}$ (22)

hold gromaz ; engineer . The coefficients depend on $\xi$ owing to the
critical enhancement. If evaluated using $\xi$ at $\varepsilon=0$, they are
denoted by $\Lambda^{(0)}$, $\kappa^{(0)}$, and $\lambda^{(0)}$, respectively,
and are independent of $z$. In Eq. (22) at the order of $\varepsilon$, we can
use $T^{(0)}\Lambda^{(0)}$, $\kappa^{(0)}$, and $\lambda^{(0)}$ for
$T\Lambda$, $\kappa$, and $\lambda$, respectively.

With the aid of Eqs. (18) and (19), the $x$ and $y$ components of Eq. (17)
give

$0=\rho^{(0)}{\bar{\nabla}}\mu_{+}^{(1)}+\varphi^{(0)}{\bar{\nabla}}{\mu_{-}^{(1)}}+s^{(0)}{\bar{\nabla}}T^{(1)}\
,$ (23)

where ${\bar{\nabla}}$ represents the two-dimensional nabla defined on the
$(x,y)$ plane. Because $P^{(0)}$, $\rho_{n}^{(0)}$, and $u^{(0)}$ are
independent of $z$, we obtain

$\nabla\cdot{\mbox{\boldmath$j$}}^{(1)}=0\quad{\rm
and}\quad\nabla\cdot{\mbox{\boldmath$j$}}_{q}^{(1)}=0\ .$ (24)

The first entry comes from Eqs. (14) and (16), whereas the second from Eqs.
(10), (12), and (21). The components of $j$ and ${\mbox{\boldmath$j$}}_{q}$
normal to the tube’s wall vanish at the impermeable and adiabatic wall. These
conditions, the conditions at the tube’s edges mentioned in the beginning of
Section II.3, Eq. (23), and Eq. (24) are satisfied if $\mu_{n}^{(1)}$ and
$T^{(1)}$ are linear functions of $z$ and are independent of $x$ and $y$.
Then, ${\mbox{\boldmath$j$}^{(1)}}$ and ${\mbox{\boldmath$j$}}_{q}^{(1)}$ have
only $z$ components and are independent of $z$, considering Eq. (22) up to the
order of $\varepsilon$. With the aid of Eqs. (18) and (19), the $z$ component
of Eq. (17) give

${\bar{\nabla}}\cdot\left({\eta}_{\rm
s}^{(0)}{\bar{\nabla}}v_{z}^{(1)}\right)=\rho^{(0)}_{n}\partial_{z}\mu_{n}^{(1)}+s^{(0)}\partial_{z}T^{(1)}\
,$ (25)

where $v_{z}^{(1)}$ is regarded as a scalar on a cross-section. The
derivatives on the RHS above are constants determined by the thermodynamic
forces in Eq. (3). Writing $L_{\rm tube}$ for the tube length, we obtain

$\varepsilon\partial_{z}T^{(1)}=-\frac{T^{(0)2}}{L_{\rm
tube}}\delta\left(\frac{1}{T}\right)\
,\quad\varepsilon\partial_{z}\mu_{-}^{(1)}=\frac{T^{(0)}}{L_{\rm
tube}}\left[\delta\left(\frac{\mu_{-}}{T}\right)-\mu_{-}^{(0)}\delta\left(\frac{1}{T}\right)\right]\
,$ (26)

and

$\varepsilon\partial_{z}\mu_{+}^{(1)}=\frac{1}{\rho^{({\rm ref})}L_{\rm
tube}}\left(\delta P-\varphi^{({\rm ref})}\delta\mu_{-}-s^{({\rm ref})}\delta
T\right)\ ,$ (27)

with the aid of Eq. (7). Thus, we use Eqs. (11) and (20) to rewrite Eq. (25)
as

$\displaystyle\varepsilon{\bar{\nabla}}\cdot\left({{\eta}_{\rm
s}^{(0)}}{\bar{\nabla}}v_{z}^{(1)}\right)=\frac{T^{(0)}}{L_{\rm
tube}}\left\\{\frac{\rho^{(0)}}{\rho^{({\rm
ref})}}\delta\left(\frac{P}{T}\right)+\left(\varphi^{(0)}-\frac{\rho^{(0)}\varphi^{({\rm
ref})}}{\rho^{({\rm ref})}}\right)\delta\left(\frac{\mu_{-}}{T}\right)\right.$
$\displaystyle\qquad+\left.\left(\frac{\rho^{(0)}u^{({\rm ref})}}{\rho^{({\rm
ref})}}-u^{(0)}-P^{(0)}\right)\delta\left(\frac{1}{T}\right)\right\\}\ .$ (28)

This determines $v_{z}^{(1)}$, which is independent of $z$ because of Eq.
(16), together with the boundary conditions. From Eq. (22), we obtain

$\varepsilon j_{z}^{(1)}=\frac{1}{L_{\rm
tube}}\left[T^{(0)}\Lambda^{(0)}\delta\left(\frac{-\mu_{-}}{T}\right)+\kappa^{(0)}\delta\left(\frac{1}{T}\right)\right]$
(29)

and

$\varepsilon j_{qz}^{(1)}=\frac{1}{L_{\rm
tube}}\left[\kappa^{(0)}\delta\left(\frac{-\mu_{-}}{T}\right)+\lambda^{(0)}\delta\left(\frac{1}{T}\right)\right]$
(30)

in the tube. Up to the order of $\varepsilon$, $d{\cal M}_{n{\rm R}}/(dt)$ and
$d{\cal U}_{\rm R}/(dt)$ are respectively given by the area integral of
$\varepsilon(\rho_{n}^{(0)}v_{z}^{(1)}+j_{{n}z}^{(1)})$ and that of
$\varepsilon(u^{(0)}+P^{(0)})v_{z}^{(1)}+\varepsilon j_{qz}^{(1)}$ over a
tube’s cross-section, which is denoted by $S_{\rm tube}$. Thus, we use the
first equality of Eq. (3) to obtain

${\cal I}=\frac{\varepsilon}{\rho^{({\rm ref})}}\int_{S_{\rm tube}}dA\
\rho^{(0)}v_{z}^{(1)}\ ,\quad{\cal J}=\varepsilon\int_{S_{\rm tube}}dA\
\left[\left(\varphi^{(0)}-\frac{\rho^{(0)}\varphi^{({\rm ref})}}{\rho^{({\rm
ref})}}\right)v_{z}^{(1)}+j_{z}^{(1)}\right]\ ,$ (31)

and

${\cal K}=\varepsilon\int_{S_{\rm tube}}dA\
\left[\left(u^{(0)}+P^{(0)}-\frac{\rho^{(0)}u^{({\rm ref})}}{\rho^{({\rm
ref})}}\right)v_{z}^{(1)}+j_{qz}^{(1)}\right]\ .$ (32)

In Appendix B, our formulation up to here is shown to be consistent with
Onsager’s reciprocal relation, as it should be. Because of Eq. (11), $P^{(0)}$
can be inhomogeneous on a tube’s cross-section in the presence of PA. In its
absence, because $P^{(0)}$ is homogeneously equal to $P^{({\rm ref})}$, the
RHS of Eq. (28), and thus $v_{z}^{(1)}$, vanish when $\delta P$ vanishes. This
is consistent with the results in Chapter XV-5 of Ref. gromaz .

With the subscript c, we refer to the value at the critical point determined
under the pressure $P^{({\rm ref})}$. The deviation of $\varphi$ from its
value at the critical point, $\varphi-\varphi_{{\rm c}}$, plays a role of the
order parameter of phase separation and is denoted by $\psi$. At equilibrium,
correlated clusters of $\psi$ are randomly convected on length scales smaller
than $\xi$. On larger length scales, the convection is averaged out to enhance
the transport coefficient for the interdiffusion in a mixture at the critical
composition, as mentioned in Section I. The critical enhancement suppresses
the critical slowing down of the relaxation of the two-time correlation
function of $\psi$. This function follows the diffusion equation. According to
the mode-coupling theory kawasaki , the singular part of the diffusion
coefficient coincides with the self-diffusion coefficient of a rigid sphere
with the radius being equal to $\xi$. This result is slightly modified by the
dynamic renormalization-group calculation for the model H, with the weak
singularity of $\eta_{\rm s}$ taken into account halhohsig ; ohta ; sighalhoh
; sengers ; folkmos . Because multiplying the diffusion coefficient by the
osmotic susceptibility, denoted by $\chi$, gives the transport coefficient
$\Lambda$, we have

$\Lambda=\frac{{\chi}Rk_{\rm B}T_{\rm c}}{\xi\eta_{\rm sing}}\ ,$ (33)

where $R$ is a universal constant close to $1/(6\pi)$, $k_{\rm B}$ is the
Boltzmann constant, $T_{\rm c}$ is the critical temperature, and $\eta_{\rm
sing}$ represents the singular part of $\eta_{\rm s}$. The regular part of
$\Lambda$ is usually negligible in the critical regime, judging from the data
in Ref. mirz  for example. The partial enthalpy per unit mass of the component
$n$, denoted by ${\bar{H}}_{n}$, is given by

${\bar{H}}_{n}=\mu_{n}+T{\bar{s}}_{n}=-T^{2}\left.\frac{\partial}{\partial
T}\frac{\mu_{n}}{T}\right)_{P,c_{\rm a}}$ (34)

and ${\bar{H}}_{-}$ is defined as $({\bar{H}}_{\rm a}-{\bar{H}}_{\rm b})/2$.
Neglecting the regular parts in the results of Ref. mist , we use

$\kappa=\Lambda T{\bar{H}}_{-}\quad{\rm and}\quad\lambda=\Lambda
T\left({\bar{H}}_{-}\right)^{2}\ .$ (35)

Here, as shown in Appendix C, the second entry contains more approximations,
which affect only the formula for $L_{33}$ in the following, than the first
entry.

Equation (33) holds at equilibrium with the critical composition. In our
problem, to evaluate $\Lambda^{(0)}$, we simply extend this result to a
homogeneous off-critical composition and use the extended result even when the
composition is inhomogeneous. Hence, we evaluate the RHS of Eq. (33) by using
$T^{(0)}$, $\psi^{(0)}({\mbox{\boldmath$r$}})$, and the resulting local value
of $\xi$, to obtain $\Lambda^{(0)}$. This is the same procedure as used in
Refs. yabufuji ; pipe . Likewise, we obtain $\kappa^{(0)}$ and $\lambda^{(0)}$
in the dynamics by replacing $\Lambda$, $T$, and ${\bar{H}}_{-}$ with
$\Lambda^{(0)}$, $T^{(0)}$, and ${\bar{H}}_{-}^{(0)}$, respectively, in Eq.
(35).

##  III Calculation under some specifications

We specify the problem by making the following assumptions. First, we assume
$f_{\rm surf}$ to be independent of $\rho$. Thus, $\rho^{(0)}$ equals
$\rho^{({\rm ref})}$ homogeneously. Second, we assume $\psi^{({\rm ref})}=0$.
Third, we assume the tube to be a cylinder with the radius of $r_{\rm tube}$.
In the tube, a field depends only on the distance from the central axis, $r$,
on a cross-section, and we can write $\psi^{(0)}(r)$, $\eta_{\rm s}^{(0)}(r)$,
$v_{z}^{(1)}(r)$, and $j_{z}^{(1)}(r)$, for example. The LHS of Eq. (28)
becomes $\varepsilon r^{-1}\partial_{r}\left(r{\eta_{\rm
s}^{(0)}}\partial_{r}v_{z}^{(1)}\right)$; $v_{z}^{(1)}$ vanishes at $r=r_{\rm
tube}$ owing to the no-slip condition and $\partial_{r}v_{z}^{(1)}$ vanishes
at $r=0$ owing to the axissymmetry and smoothness of $v_{z}^{(1)}$. Thus, in
the tube, we obtain

$\displaystyle\varepsilon v_{z}^{(1)}(r)=\int_{r}^{r_{\rm tube}}dr_{1}\
\frac{1}{r_{1}{\eta_{\rm s}^{(0)}(r_{1})}}\int_{0}^{r_{1}}dr_{2}\ r_{2}$
$\displaystyle\quad{\frac{T^{(0)}}{L_{\rm
tube}}}\left[\delta\left(\frac{-P}{T}\right)+\psi^{(0)}(r_{2})\delta\left(\frac{-\mu_{-}}{T}\right)+\left(u^{(0)}(r_{2})-u^{({\rm
ref})}+P^{(0)}(r_{2})\right)\delta\left(\frac{1}{T}\right)\right]$ (36)

for general thermodynamic forces. Substituting Eqs. (29), (30), and (36) into
Eqs. (31) and (32) yields formulas for the components of $L$ in Eq. (3), as
shown in Section III.1.

###  III.1 Formulas for the Onsager coefficients

A dimensionless radial distance ${\hat{r}}$ is defined as $r/r_{\rm tube}$. We
define $T_{*}$ so that $\xi$ becomes $r_{\rm tube}$ for $\psi=0$ at $T=T_{*}$.
A characteristic order parameter $\psi_{*}$ is defined so that $\xi$ becomes
$r_{\rm tube}$ for $\psi=\psi_{*}$ at $T=T_{\rm c}$. A dimensionless order-
parameter at $\varepsilon=0$, ${\hat{\psi}}^{(0)}({\hat{r}})$, is defined as
$\psi^{(0)}({\hat{r}}r_{\rm tube})/\psi_{*}$. A characteristic chemical
potential, $\mu_{*}$, is defined as

$\mu_{*}=\frac{k_{\rm B}{{T_{*}}}}{3u^{*}r_{\rm tube}^{3}\psi_{*}}\ ,$ (37)

where $u^{*}$ is the scaled coupling constant at the Wilson-Fisher fixed point
and equals $2\pi^{2}/9$ at the one loop order. We define $\eta_{*}$ and
$\Lambda_{*}$ as $\eta_{\rm sing}$ and $\Lambda$ at $\psi=0$ and $T=T_{*}$,
respectively. Dimensionless transport coefficients ${\hat{\eta}}({\hat{r}})$
and ${\hat{\Lambda}}({\hat{r}})$ are defined as $\eta_{\rm s}^{(0)}/\eta_{*}$
and $T^{(0)}\Lambda^{(0)}/(T_{*}\Lambda_{*})$ evaluated at $r={\hat{r}}r_{\rm
tube}$, respectively. The flow rate of Hagen-Poiseulle flow of a fluid, with
the viscosity being $\eta_{*}$, driven by the pressure gradient
$\mu_{*}\psi_{*}/L_{\rm tube}$, is denoted by ${\cal I}_{*}$ and is given by

${\cal I}_{*}=\frac{\pi r_{\rm tube}^{4}\mu_{*}\psi_{*}}{8\eta_{*}L_{\rm
tube}}\ .$ (38)

We define a functional $\Omega[g_{1},g_{2}]$, where $g_{1}$ and $g_{2}$ are
functions, as

$\Omega[g_{1},g_{2}]=16\int_{0}^{1}dq_{1}\
q_{1}g_{1}(q_{1})\int_{q_{1}}^{1}dq_{2}\
\frac{1}{q_{2}{\hat{\eta}}(q_{2})}\int_{0}^{q_{2}}dq_{3}\ q_{3}g_{2}(q_{3})\
,$ (39)

which is found to be equal to $\Omega[g_{2},g_{1}]$ by exchanging the order of
integrals. We have

$L_{11}=\frac{{\cal I}_{*}T^{(0)}}{\mu_{*}\psi_{*}}\Omega[1,1]\ ,\quad
L_{12}=L_{21}=\frac{{\cal I}_{*}T^{(0)}}{\mu_{*}}\Omega[1,{\hat{\psi}}^{(0)}]\
,$ (40)

and

$L_{22}=\frac{{\cal
I}_{*}\psi_{*}T^{(0)}}{\mu_{*}}\Omega[{\hat{\psi}}^{(0)},{\hat{\psi}}^{(0)}]+2\pi
r_{\rm tube}^{2}\frac{\Lambda_{*}T_{*}}{L_{\rm tube}}\int_{0}^{1}d{\hat{r}}\
{\hat{r}}{\hat{\Lambda}}({\hat{r}})\ .$ (41)

These three formulas are essentially the same as obtained in Ref. pipe , where
the diffusioosmotic conductance is calculated using $L_{12}$. Introducing

${\hat{Y}}^{(0)}({\hat{r}})=\frac{1}{\mu_{*}\psi_{*}}\left[{u^{(0)}}-u^{({\rm
ref})}+P^{(0)}\right]\quad{\rm
and}\quad{{\hat{H}}_{-}^{(0)}({\hat{r}})=\frac{{\bar{H}}_{-}^{(0)}}{\mu_{*}}}\
,$ (42)

where $u^{(0)}$, $P^{(0)}$, and ${\bar{H}}_{-}^{(0)}$ are evaluated at
$r={\hat{r}}r_{\rm tube}$, we obtain new formulas,

${L}_{13}={L}_{31}={\cal I}_{*}T^{(0)}\Omega[1,{\hat{Y}}^{(0)}]\ ,$ (43)
$L_{23}=L_{32}={\cal
I}_{*}\psi_{*}T^{(0)}\Omega[{\hat{\psi}}^{(0)},{\hat{Y}}^{(0)}]+2\pi r_{\rm
tube}^{2}\frac{\mu_{*}\Lambda_{*}T_{*}}{L_{\rm tube}}\int_{0}^{1}d{\hat{r}}\
{\hat{r}}{\hat{\Lambda}}({\hat{r}}){\hat{H}}_{-}^{(0)}({\hat{r}})\ ,$ (44)

and

$L_{33}={\cal
I}_{*}\mu_{*}\psi_{*}T^{(0)}\Omega[{\hat{Y}}^{(0)},{\hat{Y}}^{(0)}]+2\pi
r_{\rm tube}^{2}\frac{\mu_{*}^{2}\Lambda_{*}T_{*}}{L_{\rm
tube}}\int_{0}^{1}d{\hat{r}}\
{\hat{r}}{\hat{\Lambda}}({\hat{r}})\left[{\hat{H}}_{-}^{(0)}({\hat{r}})\right]^{2}\
.$ (45)

The second term on the RHS of Eq. (45) is affected by the approximations
mentioned below Eq. (35).

###  III.2 Formulas for thermoosmosis

The thermal force density occurs under a temperature gradient and causes
thermoosmosis. We write $\sigma_{z}^{({\rm th})}$ for this density on a tube’s
cross-section in the linear regime. It is given by the negative of the RHS of
Eq. (28), or the $z$ component of $-\varepsilon\nabla\cdot\Pi^{(1)}$, under
$\delta T\neq 0$ and $\delta P=\delta c_{\rm a}=0$. We have

$\sigma_{z}^{({\rm th})}{(r)}=-\frac{\delta T}{T^{(0)}L_{\rm
tube}}\left(u^{(0)}(r)+{P^{(0)}(r)-u^{({\rm ref})}}-P^{({\rm
ref})}-\bar{H}_{-}^{({\rm ref})}\psi^{(0)}\right)\ .$ (46)

The factor ${\bar{H}}_{-}^{({\rm ref})}$ above comes from $\delta(\mu_{-}/T)$
via Eqs. (6) and (34). The first four terms in the parentheses above can be
interpreted as the excess enthalpy density in DS’s formula for a one-component
fluid. In the presence of PA, ${\mathsf{\Pi}}^{(0)}_{zz}=P^{(0)}$ is equal to
neither ${\mathsf{\Pi}}^{(0)}_{xx}$ nor ${\mathsf{\Pi}}^{(0)}_{yy}$. The
occurrence of ${\mathsf{\Pi}}^{(0)}_{zz}=P^{(0)}$ in Eq. (46) is consistent
with many author’s claim that the $zz$ component should be involved in the
thermal force density near the wall anders ; ganti .

Here, we compare our derivation of the thermal force density with the
corresponding part in Ref. ganti , which is mentioned in Section I. Because
the sum of the last three terms of Eq. (46) equals $-\rho_{\rm
a}{\bar{H}}_{\rm a}^{({\rm ref})}-\rho_{\rm b}{\bar{H}}_{\rm b}^{({\rm
ref})}$, thus the negative of Eq. (46), i.e., the formula for $-\sigma^{({\rm
th})}_{z}$, formally coincides with the RHS of Eq. (5) of Ref. ganti , where
the RHS is treated as the negative of the thermal force density. However, its
LHS, $\partial_{z}{\mathsf{\Pi}}_{zz}$ in our notation, is not equal to
$-\sigma^{({\rm th})}_{z}$ in general, since
$\partial_{x}{\mathsf{\Pi}}_{xz}+\partial_{y}{\mathsf{\Pi}}_{yz}$ does not
vanish in the presence of PA, up to the order of $\varepsilon$. In Ref. ganti
, this sum $\partial_{x}{\mathsf{\Pi}}_{xz}+\partial_{y}{\mathsf{\Pi}}_{yz}$
is also missing in the LHS of Eq. (2), which the authors employ as an extended
GD relation in deriving their Eq. (5). In the present study, we use our Eq.
(12), whose LHS includes
$\partial_{x}{\mathsf{\Pi}}_{xz}+\partial_{y}{\mathsf{\Pi}}_{yz}$, as an
extended GD relation to derive our formula for the thermal force density
$\sigma^{({\rm th})}_{z}$ of Eq. (46), consistently with principles of linear
nonequilibrium thermodynamics.

The superscript (th) is used to indicate a result in the linear regime for
thermoosmosis. Replacing the integrand in Eq. (36) by $\sigma_{z}^{({\rm
th})}(r_{2})$ gives $v_{z}^{({\rm th})}(r)$. Integrating this result
multiplied by $\rho^{({\rm ref})}$ over the tube’s cross-section gives the
total mass flow rate, for which we write $d{\cal M}_{\rm R}^{({\rm
th})}/(dt)$. Using the free-energy functional introduced in Section III.3, we
rewrite Eq. (46) and give an explicit expression of $d{\cal M}_{\rm R}^{({\rm
th})}/(dt)$ in Section III.4. In the absence of PA, because Eq. (46) vanishes,
$v_{z}^{({\rm th})}$ and $d{\cal M}_{\rm R}^{({\rm th})}/(dt)$ vanish. Thus,
in our formulation, thermoosmosis of a mixture occurs only in the presence of
PA.

Setting $\delta T\neq 0$ and $\delta P=\delta c_{\rm a}=0$, we use Eqs. (3),
(6), and (34) to obtain the mass flow rate of the component $n$ in
thermoosmosis as

$\frac{d}{dt}{\cal M}_{n{\rm R}}^{({\rm th})}=\frac{\delta
T}{(T^{(0)})^{2}}\left[\rho_{n}^{({\rm ref})}\left(P^{({{\rm
ref}})}L_{11}+{\bar{H}}_{-}^{({\rm
ref})}L_{12}-L_{13}\right)\pm\frac{1}{2}\left(P^{({{\rm
ref}})}L_{21}+{\bar{H}}_{-}^{({\rm ref})}L_{22}-L_{23}\right)\right]\ ,$ (47)

where the upper (lower) sign is taken for $n=$a (b) in the double sign. The
sum of Eq. (47) over $n=$a and b gives $d{\cal M}_{\rm R}^{({\rm th})}/(dt)$.
Rewriting the resultant sum with the aid of the formulas for $L_{11}$,
$L_{12}$, and $L_{13}$ shown in Section III.1, we obtain the same expression
of $d{\cal M}_{\rm R}^{({\rm th})}/(dt)$ as derived in the way mentioned in
the preceding paragraph.

###  III.3 Free-energy functional in the renormalized local functional theory

The reduced temperature $\tau$ is defined as $(T-T_{\rm c})/T_{\rm c}$, and
its characteristic magnitude $\tau_{*}$ is defined as $|T_{*}-T_{\rm
c}|/T_{\rm c}$. The scaled reduced-temperature ${\hat{\tau}}$ is defined as
$\tau/\tau_{*}$. In the one-phase region we consider, $\tau$ is positive near
a upper consolute (UC) point and is negative near a lower consolute (LC) point
kaji ; sciam ; tsori . Using the conventional notation, we write
$\alpha,\beta,\gamma,\nu,$ and $\eta$ for the critical exponents for a
mixture. We adopt $\nu=0.630$ and $\eta=0.0364$ peli ; the (hyper)scaling
relations give $2\beta+\gamma=3\nu=2-\alpha$ and $\gamma=\nu(2-\eta)$. In an
equilibrium mixture with $\psi=0$, we have
$\xi=\xi_{0}{\left|\tau\right|}^{-\nu}$ in the critical regime, where
$\xi_{0}$ is a material constant.

Neglecting coupling between $\rho$ and $\varphi$ in $f_{\rm bulk}$, we assume

$f_{\rm bulk}=-{\frac{{C}T\tau^{2}}{2}}+{u}_{\rm c}-{s}_{\rm
c}T+f_{-}(\psi)+\frac{M_{-}}{2}\left|\nabla\psi\right|^{2}+f_{+}(\rho)\ .$
(48)

Although the variable $\tau$ is dropped for conciseness, $f_{+}$ is a regular
function of $\rho$ and $\tau$ and $f_{-}$ is a function of $\psi$ and $\tau$.
The constants $u_{\rm c}$ and $s_{\rm c}$ represent the values of $u$ and $s$
at the critical point, respectively. The coefficient $M_{-}$ is described
later. In the critical regime, the singular contribution to $C$ onukibook ;
halhoma  becomes equal to $2k_{\rm B}\xi_{0}^{-3}{|\tau|^{-\alpha}}$
multiplied by a universal number, as mentioned at footnote 51 of Ref. rlft .

The reference state considered here is obtained by changing $T$ from the
critical point with $P=P^{({\rm ref})}$ and $\psi=0$ being fixed. The chemical
potentials, $\mu_{\rm a}^{({\rm ref})}$ and $\mu_{\rm b}^{({\rm ref})}$, are
tuned so that this change is realized. Thus, the $\varphi$ dependent part for
the bulk of a mixture in Eq. (15) can be obtained by coarse-graining the
$\psi^{4}$ model up to $\xi$ under no external field rlft . The bare model,
given by Eq. (89), is defined at a microscopic scale and identifies the
fluctuations with spacial resolution much smaller than $\xi$. We can regard
the coarse-grained average profile as maximizing the probability density
functional coarse-grained up to $\xi$, assuming that thermal fluctuations are
not significant after coarse-graining anymore rlft . This is consistent with
the statement given at Eq. (15).

We assume no coupling between $\rho$ and $\varphi$ in Eq. (48) because $\rho$
can be regarded as a constant approximately. The coefficient $C$ involves the
fluctuations of the internal-energy density. Some details on these points are
mentioned in Appendix D. As shown in Eq. (89), we can define $A_{0}$ so that
the bare $\psi^{4}$ model has a term $A_{0}\tau\psi^{2}/2$, which is positive
in the one-phase region. Thus, $A_{0}$ is positive near a UC point, and is
negative near a LC point onukibook . The sign is maintained in the coarse-
graining procedure. We use the coarse-grained result given by the renormalized
local functional theory (RLFT) rlft .

In the RLFT, $\omega$ is defined as $(\xi_{0}/\xi)^{1/\nu}$ and $M_{-}$ is
given by $k_{\rm B}TC_{1}\omega^{-\eta\nu}$ with $C_{1}(>0)$ being a material
constant. The self-consistent condition,
$\omega={\left|\tau\right|}+C_{2}\omega^{1-2\beta}\psi^{2}$, determines how
$\xi$ depends on $\tau$ and $\psi$, where the constant $C_{2}$ equals
$3u^{*}C_{1}\xi_{0}$. This condition gives

$\tau_{*}=\left(\frac{\xi_{0}}{r_{\rm tube}}\right)^{1/\nu}\quad{\rm
and}\quad\psi_{*}=\frac{\tau_{*}^{\beta}}{\sqrt{C_{2}}}\ .$ (49)

Defining a dimensionless function ${\hat{f}}$ as

${\hat{f}}({\hat{\psi}})={\frac{1}{2}{\hat{\omega}}^{\gamma-1}\left|{\hat{\tau}}\right|{\hat{\psi}}^{2}+\frac{1}{12}{\hat{\omega}}^{\gamma-2\beta}{\hat{\psi}}^{4}}\
,$ (50)

where ${\hat{\psi}}\equiv\psi/\psi_{*}$, ${\hat{\tau}}\equiv\tau/\tau_{*}$,
and ${\hat{\omega}}\equiv\omega/\tau_{*}$ are used, we have

${f_{-}(\psi)+\frac{M_{-}}{2}\left|\nabla\psi\right|^{2}=\mu_{-}^{({\rm
ref})}\varphi+\frac{\mu_{*}\psi_{*}T}{T_{*}}{\hat{f}}({\hat{\psi}})+\frac{\mu_{*}\psi_{*}T}{2T_{*}}{\hat{\omega}}^{-\eta\nu}\left|r_{\rm
tube}\nabla{\hat{\psi}}\right|^{2}}\ .$ (51)

As already explained, $\mu_{-}^{({\rm ref})}$ is determined so that the
reference state at $T$ is realized, being dependent on $T$. The sum of the
second and third terms on the RHS above is $k_{\rm B}T$ multiplied by the
coarse-grained result of the $\psi^{4}$ model under no external field, coming
from the RLFT. The first term on the RHS of Eq. (50) originates from
$A_{0}\tau\psi^{2}/2$. The self-consistent condition is rewritten as

${\hat{\omega}}={\left|{\hat{\tau}}\right|}+{\hat{\omega}}^{1-2\beta}{\hat{\psi}}^{2}\
,$ (52)

which means that ${\hat{\omega}}$ is a function of ${\hat{\psi}}$ and
${\hat{\tau}}$. It is even with respect to ${\hat{\psi}}$, and hence
${\hat{f}}({\hat{\psi}})$ is an even function. The function
${\hat{f}}({\hat{\psi}})$ also depends on ${\hat{\tau}}$, but the variable
${\hat{\tau}}$ is dropped for conciseness. The osmotic susceptibility $\chi$
is given by the inverse of the second partial derivative of $f_{-}$ with
respect to $\psi$, $1/f_{-}^{\prime\prime}(\psi)$; the prime indicates the
differentiation with respect the variable given explicitly. The partial
derivative $\partial{\hat{f}}/(\partial{\hat{\tau}})$, appearing in the later
calculation, equals

${\frac{\pm
1}{2}{\hat{\omega}}^{\gamma-1}{\hat{\psi}}^{2}+\frac{\partial{\hat{\omega}}}{\partial{\hat{\tau}}}}\left(\frac{\gamma-1}{2}{\hat{\omega}}^{\gamma-2}\left|{\hat{\tau}}\right|{\hat{\psi}}^{2}+\frac{\gamma-2\beta}{12}{\hat{\omega}}^{\gamma-2\beta-1}{\hat{\psi}}^{4}\right)\
,$ (53)

where Eq. (52) gives

$\frac{\partial{\hat{\omega}}}{\partial{\hat{\tau}}}=\frac{\pm
1}{1+(2\beta-1){\hat{\omega}}^{-2\beta}{\hat{\psi}}^{2}}\ .$ (54)

The same sign as $\tau$ is taken in each double sign of these equations. The
first term of Eq. (53) originates from the coarse-grained result of
$A_{0}\psi^{2}/2$. Equation (52) gives
$|{\hat{\psi}}|^{1/\beta}<{\hat{\omega}}$ and $|{\hat{\tau}}|<{\hat{\omega}}$.
For ${\hat{\psi}}\neq 0$, the sign of Eq. (53), or that of
$\partial{\hat{f}}/(\partial{\hat{\tau}})$, coincides with that of $\tau$,
considering $\beta=0.326$ and $\gamma=1.24$. If $|{\hat{\tau}}|$ is much
smaller than ${\hat{\omega}}$, Eq. (52) gives
$|{\hat{\psi}}|^{1/\beta}\approx{\hat{\omega}}$ and thus Eq. (54) is
approximately equal to $\pm 1/(2\beta)$. Then, in Eq. (53), the first term is
found to be dominant over the rest. If $|{\hat{\tau}}|$ is close to
${\hat{\omega}}$, Eq. (54) is found to be close to $\pm 1$ with the aid of Eq.
(52). Then, the first term remains dominant in Eq. (53), accounting for
approximately $80$% of the total owing to the numerator $\gamma-1=0.24$ in the
parentheses.

We assume $f_{\rm surf}$ to be a linear function of $\varphi$, or $\psi$, as
usual in studying the PA diehl97 ; rlft . The surface field $h$ is defined as
the negative of the coefficient of $\psi$. This assumption and this definition
are involved in calculating the equilibrium profile, which is used in Section
IV. The calculation procedure is mentioned below Eq. (15) and is the same as
that of Ref. rlft . Applying Eqs. (48) and (51), we find that
${\hat{\psi}}^{(0)}({\hat{r}})$ is the solution of

$0={\hat{f}}^{\prime}({\hat{\psi}})-{\frac{1}{2}\frac{\partial{\hat{\omega}}^{-\eta\nu}}{\partial{\hat{\psi}}}}\left(\partial_{\hat{r}}{\hat{\psi}}\right)^{2}-{\hat{\omega}}^{-\eta\nu}\left(\partial_{\hat{r}}^{2}+\frac{1}{{\hat{r}}}\partial_{\hat{r}}\right){\hat{\psi}}\quad{\rm
for}\ {\hat{r}}<1\ ,$ (55)

together with the boundary condition at the wall,
$({\hat{\psi}}^{0)})^{\prime}(1)={\hat{h}}{\hat{\omega}}^{\eta\nu}$. Here,
${\hat{\omega}}$ is regarded as a function of ${\hat{\psi}}$ and
${\hat{\tau}}$ via Eq. (52) and a scaled surface field ${\hat{h}}$ is defined
as ${hT_{*}/\left(T\mu_{*}r_{\rm tube}\right)}$. These equations are shown in
Appendix D of Ref. pipe ; ${\hat{\psi}}^{(0)}$ is totally determined by
$|{\hat{\tau}}|$ and ${\hat{h}}$. If we change the sign of ${\hat{h}}$, the
sign of ${\hat{\psi}}^{(0)}$ changes with the magnitude remaining the same.
Notably, $\left|h\right|$ represents the strength of the PA and vanishes in
its absence.

###  III.4 Formulas incorporating the RLFT

We apply Eq. (48) and the results of the RLFT to rewrite the RHS of Eq. (46).
Owing to Eqs. (20) and (51), we have

$u^{(0)}-u^{({\rm
ref})}=-{\frac{\left(T^{(0)}\right)^{2}\mu_{*}\psi_{*}}{T_{\rm
c}T_{*}}}\frac{\partial}{\partial\tau}\left(\hat{f}+\frac{\hat{\omega}^{-\eta\nu}}{2}\left|\partial_{\hat{r}}{{\hat{\psi}}^{(0)}}\right|^{2}\right)+\bar{H}_{-}^{({\rm
ref})}{\psi^{(0)}}\ ,$ (56)

where the partial derivative with respect to $\tau$ is done with
${\hat{\psi}}$ fixed and is evaluated at $\varepsilon=0$. The first term on
the RHS represents the difference in the internal energy density involved in
the coarse-grained result of the $\psi^{4}$ model. In deriving the second
term, we drop one term, which is proportional to the thermal expansion
coefficient. This term gives negligibly small contribution to our later
numerical results, as described in Appendix E. Owing to Eqs. (11), (48), and
(51), we have

$P^{(0)}-P^{({\rm
ref})}=-\frac{\mu_{*}\psi_{*}{T^{(0)}}}{T_{*}}{\left(\hat{f}+\frac{\hat{\omega}^{-\eta\nu}}{2}\left|\partial_{\hat{r}}\hat{\psi}\right|^{2}\right)}\
.$ (57)

We define a scaled thermal force density, ${\hat{\sigma}}_{z}^{({\rm th})}$,
so that Eq. (46) is rewritten as

$\sigma_{z}^{({\rm th})}({\hat{r}}r_{\rm tube})=\frac{\mu_{*}\psi_{*}\delta
T}{{\tau_{*}}{T_{*}}L_{\rm tube}}{\hat{\sigma}}_{z}^{({\rm th})}({\hat{r}})\
,$ (58)

and have

${\hat{\sigma}}_{z}^{({\rm
th})}={\tau_{*}}\left({\hat{f}}+\frac{1}{2{\hat{\omega}}^{\eta\nu}}\left|\partial_{\hat{r}}{\hat{\psi}}\right|^{2}\right)+{\frac{T^{(0)}}{T_{\rm
c}}}\left(\frac{\partial{\hat{f}}}{\partial{\hat{\tau}}}-\frac{\eta\nu}{2{\hat{\omega}}^{\eta\nu+1}}{\frac{\partial{\hat{\omega}}}{\partial{\hat{\tau}}}}\left|\partial_{\hat{r}}{\hat{\psi}}\right|^{2}\right)\
,$ (59)

which is evaluated at $\varepsilon=0$. The first term on the RHS of Eq. (59)
comes from Eq. (57), whereas the second term comes from the first term on the
RHS of Eq. (56). The last term in the parentheses of Eq. (46) cancels out the
last term on the RHS of Eq. (56). Thanks to this cancellation, the thermal
force density does not involve ${\bar{H}}_{-}^{({\rm ref})}$. Except for the
factors $\tau_{*}$ and $T^{(0)}/T_{\rm c}$, Eq. (59) is determined by the
scaled reduced-temperature ${\hat{\tau}}$ and the magnitude of the scaled
surface field $|{\hat{h}}|$ in the framework of the RLFT. The magnitude of the
sum in the second parentheses, in particular, is determined by
$|{\hat{\tau}}|$ and $|{\hat{h}}|$ owing to Eqs. (53) and (54).

As mentioned in Section III.2, $\varepsilon v_{z}^{(1)}$ in thermoosmosis is
given by

$v_{z}^{({\rm th})}({\hat{r}}r_{\rm tube})=\frac{8{\cal I}_{*}\delta
T}{T_{*}\tau_{*}\pi r_{\rm tube}^{2}}\int_{\hat{r}}^{1}d{\hat{r}}_{1}\
\frac{1}{{\hat{r}}_{1}{\hat{\eta}}({\hat{r}}_{1})}\int_{0}^{{\hat{r}}_{1}}d{\hat{r}}_{2}\
{\hat{r}}_{2}{\hat{\sigma}}_{z}^{({\rm th})}$ (60)

in the tube, whereas the total mass flow rate in thermoosmosis is given by

$\frac{d{\cal M}_{\rm R}^{({\rm th})}}{dt}=\frac{\rho^{({\rm ref})}{\cal
I}_{*}\delta T}{T_{*}\tau_{*}}\Omega\left[1,{\hat{\sigma}}^{({\rm
th})}_{z}\right]\ ,$ (61)

which is proportional to $\delta T$. The constant of proportionality
represents the thermoosmotic conductance. We define the dimensionless
thermoosmotic conductance, denoted by ${\hat{G}}^{({\rm th})}$, as the
quotient of the constant divided by $\rho^{({\rm ref})}{\cal
I}_{*}/(T_{*}\tau_{*})$, and have

${\hat{G}}^{({\rm th})}=\Omega[1,{\hat{\sigma}}_{z}^{({\rm th})}]\ .$ (62)

If we change the sign of ${\hat{h}}$, ${\hat{\sigma}}_{z}^{({\rm
th})}({\hat{r}})$ remains the same and is independent of which component is
adsorbed onto the tube’s wall. However, it is not the case with
${v}_{z}^{({\rm th})}$ of Eq. (60), $d{\cal M}_{\rm R}^{({\rm th})}/(dt)$ of
Eq. (61), and ${\hat{G}}^{({\rm th})}$ of Eq. (62) because ${\hat{\eta}}$ is
not always an even function of ${\hat{\psi}}$.

Some of the formulas of the Onsager coefficients are simplified using Eq. (48)
and the results of the RLFT. We have
${\hat{f}}^{\prime\prime}(0)={\left|{\hat{\tau}}\right|}^{\gamma}$ and

$f_{-}^{\prime\prime}(0)=\frac{k_{\rm
B}TC_{2}{\left|\tau\right|}^{\gamma}}{3u^{*}\xi_{0}^{3}}\ .$ (63)

Because of Eq. (48), we can replace $\chi$ in Eq. (33) with
$1/f_{-}^{\prime\prime}(\psi)$. By definition, we have

$\Lambda_{*}=\frac{3u^{*}R{T_{\rm
c}}\xi_{0}^{2}\tau_{*}^{\nu-\gamma}}{C_{2}{T_{*}}\eta_{*}}\ .$ (64)

We write $z_{\psi}$ for the dynamic critical exponent for the order-parameter
fluctuations and use $z_{\psi}=3.067$ bergmold ; bergmold2 . With the aid of
the result of the dynamic renormalization-group calculation sighalhoh ;
sengers ; folkmos , we have

${{\hat{\Lambda}}=}{\hat{\omega}}^{\nu(z_{\psi}-2)}\left[{\hat{f}}^{\prime\prime}({\hat{\psi}})\right]^{-1}$
(65)

evaluated at $\varepsilon=0$, as described in Appendix E of Ref. pipe . Using
Eq. (33) with $R=1/(6\pi)$ and Eq. (65), we can rewrite the second term on the
RHS of Eq. (41) as

$\frac{{\cal I}_{*}\psi_{*}T^{(0)}}{\mu_{*}}\frac{16\pi T_{\rm
c}}{9T^{(0)}}\int_{0}^{1}d{\hat{r}}\ {\hat{r}}{\hat{\Lambda}}({\hat{r}})\ ,$
(66)

which is essentially the same as obtained in Ref. pipe . Likewise, in the
second terms on the RHS of the new formulas (44) and (45), the coefficients
multiplied by the integrals are respectively rewritten as

${\cal I}_{*}\psi_{*}T^{(0)}\frac{16\pi T_{\rm c}}{9T^{(0)}}\quad{\rm
and}\quad{\cal I}_{*}\mu_{*}\psi_{*}T^{(0)}\frac{16\pi T_{\rm c}}{9T^{(0)}}\
.$ (67)

The integrals can be calculated if ${\hat{H}}_{-}^{(0)}$ is known. To
calculate ${\hat{Y}}^{(0)}$ contained in the first terms on the RHS’s of Eqs.
(43)–(45), we can use Eq. (56), which involves ${\bar{H}}_{-}^{({\rm ref})}$.
Thus, it is necessary to know how ${\bar{H}}_{-}$ depends on $\varphi$ to
calculate these integrals and terms. We can calculate the dependence in such a
theoretical framework as used in Refs. edison ; vanthof ; olaya .

##  IV Numerical results of thermoosmosis

In this section, we study thermoosmosis numerically with the aid of the
formulas in Section III.4 and the software Mathematica (Wolfram Research),
supposing a mixture of 2,6-lutidine and water (LW) near the LC point and a
mixture of nitroethane and 3-methylpentane (NEMP) near the UC point. In each
mixture, the former (latter) component is taken to be the component a (b). The
tube radius $r_{\rm tube}$ is set to $0.1\ \mu$m. The parameter values we use
are listed in Table 1 and are the same as used in Ref. pipe . The values of
$\xi_{0}$ are taken from the experimental data of Refs. mirz ; iwan . The
first entry of Eq. (49) gives the values of $\tau_{*}$, which appears in Eq.
(59). In Appendix C of Ref. pipe , we estimate $C_{2}$ from the data of Refs.
to ; wims . The second entry of Eq. (49) gives the value of $\psi_{*}$, and
then Eq. (37) gives that of $\mu_{*}$. In Appendix E of Ref. pipe , we obtain
the viscosity as a function of the reduced temperature and the order parameter
bhatt ; tsai  and find the value of $\eta_{*}$ from the data of Refs. gratt ;
stein ; iwan ; leis . These values give ${\cal I}_{*}L_{\rm tube}=1.04\times
10^{-2}\ \mu$m${}^{4}/$s ($4.84\times 10^{-2}\ \mu$m${}^{4}/$s) for a mixture
of LW (NEMP). The magnitude of the surface field $h$ should be smaller than
approximately $10\ $cm${}^{3}/$s2 when 2,6-lutidine adsorbs onto the solid
surface, according to the discussion in Section 6 of Ref. yabufuji . We mainly
use ${\hat{h}}=73.0$ ($66.6$) for a mixture of LW (NEMP), which amounts to
$h\approx 0.1\ $cm3/s2.

mixture |  ${T_{\rm c}}\ [{\rm K}]$ | $\ \xi_{0}\ [{\rm nm}]\ $ | $\tau_{*}\times 10^{5}\ \ $ | $C_{2}\ [{\rm cm}^{6}/{\rm g}^{2}]\ $ | $\psi_{*}\ [{\rm g}/{\rm cm}^{3}]\ $ | $\mu_{*}\ [{\rm cm}^{2}/{\rm s}^{2}]\ $ | $\eta_{*}\ [{\rm mPa}{\cdot{\rm s}}]$
---|---|---|---|---|---|---|---
LW | 307 | 0.198 | $5.12$ | $0.714$ | $0.0470$ | $137$ | $2.44$
NEMP | 300 | 0.230 | $6.49$ | $1.05$ | $0.0419$ | $150$ | $0.510$
Table 1: Parameter values: Origins of the values are described in the text.
Figure 2: Plots of the dimensionless order parameter at equilibrium,
${\hat{\psi}}^{(0)}({\hat{r}})$, against the dimensionless radial distance
${\hat{r}}$. The surface field is set to $h=0.1\ $cm${}^{3}/$s2. For a mixture
of LW (NEMP), open circles (crosses) represent ${\hat{\psi}}^{(0)}({\hat{r}})$
at $|\tau|=1.25\times 10^{-5}$, whereas closed circles (asterisks) represent
${\hat{\psi}}^{(0)}({\hat{r}})$ at $|\tau|=3.2\times 10^{-3}$. We use the
values of $\tau_{*}$ in Table 1 to find ${\hat{\tau}}=-0.24\ (0.19)$ for open
circles (crosses), and ${\hat{\tau}}=-63\ (49)$ for closed circles
(asterisks).

###  IV.1 Equilibrium profile and thermal force density

Equilibrium profiles in the tube are shown in Fig. 2. Because of
${\hat{h}}>0$, ${\hat{\psi}}^{(0)}({\hat{r}})$ increases, or the component a
is more concentrated, near the wall at ${\hat{r}}=1$. For each mixture,
${\hat{\psi}}^{(0)}({\hat{r}})$ is larger at the smaller value of $|\tau|$,
since the adsorption layer extends towards the tube’s center as the critical
temperature is approached. At $|\tau|=3.2\times 10^{-3}$, the adsorption layer
appears to localize near the tube’s wall, and ${\hat{\psi}}^{(0)}({\hat{r}})$
is larger for a mixture of NEMP than for a mixture of LW in the whole region
of $0\leq{\hat{r}}\leq 1$ although the difference is hard to see for
${\hat{r}}<0.7$ and ${\hat{r}}=1$ in the figure. This magnitude relationship
is reasonable considering that $|{\hat{\tau}}|$ is smaller for a mixture of
${\rm NEMP}$. At $|\tau|=1.25\times 10^{-5}$, the relationship holds only for
${\hat{r}}<0.7$, and $\xi$ at $\psi=0$, given by $\xi_{0}|\tau|^{-\nu}$,
exceeds the tube radius. It is approximately equal to $2r_{\rm tube}$ at
$|\tau|=1.25\times 10^{-5}$ while to $r_{\rm tube}/10$ at $|\tau|=3.2\times
10^{-3}$ for both mixtures. At ${\hat{r}}=0$ under $|\tau|=1.25\times
10^{-5}$, the composition is definitely off-critical and $\xi$ becomes
approximately equal to $r_{\rm tube}/2$ for both mixtures.

Circles in Fig. 3 represent ${\hat{\sigma}}_{z}^{({\rm th})}$ of Eq. (59) at
$|\tau|=1.25\times 10^{-5}$. Hereafter, $\tau$ ($\hat{\tau}$) represents the
(scaled) reduced temperature in the reference state. Figures 3(a) and 3(b)
show that the thermal force density becomes remarkable near the tube’s wall.
The first term of Eq. (53), which originates from $A_{0}\psi^{2}/2$ in the
term of $\psi$ squared in the bare model, contributes to the second term on
the RHS of Eq. (59) via the term $\partial{\hat{f}}/(\partial{\hat{\tau}})$.
Crosses in Fig. 3 represent this contribution, which is denoted by
${\hat{\sigma}}^{({\rm sq})}$ and is given by

${\hat{\sigma}}_{z}^{({\rm sq})}({\hat{r}})=\pm\frac{T^{(0)}}{2T_{\rm
c}}{\hat{\omega}}^{\gamma-1}{\hat{\psi}}^{2}\ .$ (68)

This is evaluated at $\varepsilon=0$ with the same sign as $\tau$ being taken.
This sign for ${\hat{\psi}}\neq 0$ comes from that of $A_{0}$, which is
negative (positive) for the LC (UC) point. The rest in the second term on the
RHS of Eq. (59) is plotted with triangles. The first term on the RHS of Eq.
(59), which originates from the scalar-pressure deviation Eq. (57), is plotted
with squares. This term gives negligibly small contributions to
${\hat{\sigma}}_{z}^{({\rm th})}$ in the whole region of ${\hat{r}}$. It
remains the case for $|\tau|$ up to $6.4\times 10^{-3}$ although data are not
shown. This can be expected because the first term contains a small positive
factor $\tau_{*}$. Near the wall in Fig. 3, we can see that
${\hat{\sigma}}_{z}^{({\rm sq})}({\hat{r}})$ is dominant in
${\hat{\sigma}}_{z}^{({\rm th})}({\hat{r}})$. The ratio
${\hat{\sigma}}_{z}^{({\rm sq})}({\hat{r}})/{\hat{\sigma}}_{z}^{({\rm
th})}({\hat{r}})$ at ${\hat{r}}=1$ is $0.88\ (0.87)$ for $|\tau|=1.25\times
10^{-5}$ in a mixture of LW (NEMP). At $|\tau|={3.2}\times 10^{-3}$, the ratio
at ${\hat{r}}=1$ remains approximately the same, $0.86\ (0.85)$, although each
of ${\hat{\sigma}}_{z}^{({\rm sq})}({\hat{r}})$ and ${\hat{\sigma}}_{z}^{({\rm
th})}({\hat{r}})$ at ${\hat{r}}=1$ is roughly halved.

Figure 3: Plots of the scaled thermal force density,
${\hat{\sigma}}_{z}^{({\rm th})}$, and its components against the
dimensionless radial distance ${\hat{r}}$ at $|\tau|=1.25\times 10^{-5}$ for a
mixture of LW (a) and a mixture of NEMP (b). Here, the reduced temperature
$\tau$ is evaluated at $T=T^{(0)}$, and is negative (positive) in the former
(latter) mixture. The surface field is set to $h=0.1\ $cm${}^{3}/$s2. Circles
represent ${\hat{\sigma}}_{z}^{({\rm th})}$ of Eq. (59), whereas squares
represent its first term. Its second term can be separated into two parts;
${\hat{\sigma}}_{z}^{({\rm sq})}$ of Eq. (68) and the rest. Crosses represent
the former, whereas triangles represent the latter.

In Fig. 4 for a mixture of NEMP, ${\hat{\sigma}}_{z}^{({\rm sq})}({\hat{r}})$
and ${\hat{\sigma}}_{z}^{({\rm th})}({\hat{r}})$ become distinctly larger near
the wall as $\tau$ is smaller and $h$ is larger. For various values of
$|\tau|$ and $h$ examined in Figs. 3 and 4, over the whole region of
${\hat{r}}$, the ratio ${\hat{\sigma}}_{z}^{({\rm
sq})}({\hat{r}})/{\hat{\sigma}}_{z}^{({\rm th})}({\hat{r}})$ remains
approximately the same as the ratio at ${\hat{r}}=1$. Thus, as far as
examined, over the whole region of ${\hat{r}}$, ${\hat{\sigma}}_{z}^{({\rm
th})}({\hat{r}})$ is negative (positive) in a mixture of LW (NEMP) and is
contributed dominantly from ${\hat{\sigma}}_{z}^{({\rm sq})}({\hat{r}})$. In
each inset of Fig. 4, ${\hat{\psi}}^{(0)}({\hat{r}})$ increase more steeply
near the wall as $h$ is larger, like ${\hat{\sigma}}_{z}^{({\rm
th})}({\hat{r}})$ in the main figure. For each value of $h$ in Figs. 4(a) and
4(b), as ${\hat{r}}$ decreases, both ${\hat{\psi}}^{(0)}({\hat{r}})$ and
${\hat{\sigma}}_{z}^{({\rm th})}({\hat{r}})$ decrease more gradually at the
smaller value of $\tau$. These behaviors can be explained by the dominance of
Eq. (68). The dominance of Eq. (68) in the term involving
$\partial{\hat{f}}/({\partial\tau})$ on the RHS of Eq. (59) is expected from
approximate estimation mentioned below Eq. (54).

Figure 4: Plots of the scaled thermal force density,
${\hat{\sigma}}_{z}^{({\rm th})}$, and its dominant term
${\hat{\sigma}}_{z}^{({\rm sq})}$ against the dimensionless radial distance
${\hat{r}}(\geq 0.8)$ for a mixture of ${\rm NEMP}$. The reduced temperature
$\tau$ evaluated at $T=T^{(0)}$ is $1.25\times 10^{-5}$ in (a) and $3.2\times
10^{-3}$ in (b). The surface field $h$ is set to $10^{-1}\ $cm${}^{3}/$s2 for
circles (${\hat{\sigma}}_{z}^{({\rm th})}$) and solid curves
(${\hat{\sigma}}_{z}^{({\rm sq})}$); these results in (a) are already shown in
Fig. 3(b). Setting $h$ to $10^{-0.5}\ (10^{-1.5})\ $cm${}^{3}/$s2, we obtain
results shown by red crosses and dash-dot curves (blue asterisks and dashed
curves); symbols represent ${\hat{\sigma}}_{z}^{({\rm th})}$ and curves
represent ${\hat{\sigma}}_{z}^{({\rm sq})}$. The change of $h$ is indicated by
arrows. (Insets) Plots of the dimensionless order parameter at equilibrium,
$\hat{\psi}^{(0)}({\hat{r}})$, against ${\hat{r}}{(\geq 0.4)}$. The parameter
values for each symbol are the same as those for the same symbol in the main
figure in each of (a) and (b). The results of the circles are already shown in
Fig. 2.

###  IV.2 Velocity field and conductance

We define ${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ as the double integral of
Eq. (60), which is plotted in Fig. 5. When $\delta T$ is positive,
${\hat{v}}_{z}^{({\rm th})}$ has the same sign as $v_{z}^{({\rm th})}$. At
$|\tau|=3.2\times 10^{-3}$, ${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ changes
only for ${\hat{r}}>0.8$ and the velocity appears to slip across a narrow
region near the wall. This is because, as shown in Fig. 4(b), the adsorption
layer and the thermal force density localize sharply in a region near the wall
${\hat{r}}>0.8$. The slip velocity is given by ${\hat{v}}_{z}^{({\rm
th})}(0)$, which is $-0.042\ (0.061)$ for a mixture of LW (NEMP) at $h=0.1\
$cm${}^{3}/$s2. Converting the value to the slip velocity with dimensions, we
find it to be $-7.09\ (38.2)\ (\mu{\rm m})^{2}/($s$\cdot$K$)$ multiplied by
$\delta T/L_{\rm tube}$, which is comparable in magnitude with typical
thermophoretic mobility far from the critical point piazza ; Jiang ; Braun ;
maeda . For each value of $h$ in Fig. 5, $|{\hat{v}}_{z}^{({\rm
th})}({\hat{r}})|$ at $|\tau|=1.25\times 10^{-5}$ is larger than
$|{\hat{v}}_{z}^{({\rm th})}({\hat{r}})|$ at $|\tau|=3.2\times 10^{-3}$ inside
the tube and increases gradually in magnitude as ${\hat{r}}$ decreases without
showing an obvious slip. In Fig. 5(b), ${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$
increases with $h$ inside the tube, as expected. The spatial resolution of our
formulation is given by $\xi$. For $h=0.1\ $cm${}^{3}/$s2 and
$|\tau|=1.25\times 10^{-5}\ (3.2\times 10^{-3})$, a mixture of LW has
$\xi/r_{\rm tube}=0.030\ (0.036)$ and a mixture of NEMP has $0.032\ (0.038)$
at ${\hat{r}}=1$. With the spatial resolution given by these values, one would
trace rapid changes of ${\hat{v}}_{z}^{({\rm th})}$ near the wall shown in
Fig. 5. In passing, the slip velocity at $|\tau|=3.2\times 10^{-3}$ can be
evaluated approximately using the Gaussian model mentioned in Appendix F.

Figure 5: The $z$ component of the dimensionless velocity in thermoosmosis,
${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$, is plotted against the dimensionless
radial distance ${\hat{r}}$. (a) Closed and open circles represent
${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ for a mixture of LW at
$\tau=-3.2\times 10^{-3}$ and $-1.25\times 10^{-5}$, respectively. The reduced
temperature $\tau$ is evaluated at $T=T^{(0)}$. Asterisks and crosses
represent ${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ for a mixture of NEMP at
$\tau=3.2\times 10^{-3}$ and $1.25\times 10^{-5}$, respectively. The surface
field is set to $h=10^{-1}\ $cm${}^{3}/$s2. (b) Asterisks and crosses
represent the same results as those in (a), respectively. The solid curve (the
dashed curve) represents ${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ for a mixture
of NEMP at $\tau=3.2\times 10^{-3}$ with $h$ set to $10^{-0.5}\ (10^{-1.5})\
$cm${}^{3}/$s2. The dash-dot curve (the dash-dot-dot curve) represents
${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ at $\tau=1.25\times 10^{-5}$ with $h$
set to $10^{-0.5}\ (10^{-1.5})\ $cm${}^{3}/$s2. The change of $h$ is indicated
by arrows.

The dimensionless thermoosmotic conductance is defined at Eq. (62), which is
rewritten as

${\hat{G}}^{({\rm th})}=16\int_{0}^{1}d{\hat{r}}\
{\hat{r}}{\hat{v}}_{z}^{({\rm th})}({\hat{r}})\ .$ (69)

When $\delta T$ is positive, ${\sigma}_{z}^{({\rm th})}{({\hat{r}})}$ has the
same sign as ${\hat{\sigma}}_{z}^{({\rm th})}{({\hat{r}})}$. For each mixture
in our numerical results, the sign of ${\hat{\sigma}}_{z}^{({\rm
th})}({\hat{r}})$ remains the same for $0\leq{\hat{r}}\leq 1$, and thus is the
same as that of ${\hat{v}}_{z}^{({\rm th})}({\hat{r}})$ and that of
${\hat{G}}^{({\rm th})}$; ${\hat{G}}^{({\rm th})}>0\ (<0)$ means that the flow
direction is the same as (opposite to) the direction of the temperature
gradient. Thus, according to our numerical results, a mixture of NEMP near the
UC point flows towards the reservoir with the higher temperature, whereas a
mixture of LW near the LC point flows in the opposite direction. This is
independent of which component is adsorbed onto the tube’s wall owing to the
independence of Eq. (59) from the sign of $h$.

Figure 6: Logarithmic plots of the absolute value of the dimensionless
thermoosmotic conductance against that of the reduced temperature $|\tau|$,
evaluated at $T=T^{(0)}$, for a mixture of LW (circle) and a mixture of NEMP
(triangle) with the surface field set to $h=0.1\ $cm${}^{3}/$s2. The
dimensionless conductance, ${\hat{G}}^{({\rm
th})}{=\Omega[1,{\hat{\sigma}}_{z}^{({\rm th})}]}$ of Eq. (62) or (69), is
negative for a mixture of LW, whereas positive for a mixture of NEMP. Red
crosses (asterisks) represent $\Omega[1,{\hat{\sigma}}_{z}^{({\rm sq})}]$,
which is dominant in ${\hat{G}}^{({\rm th})}$, for a mixture of LW (NEMP).

Logarithmic plots of $|{\hat{G}}^{({\rm th})}|$ against $|\tau|$ are shown in
Fig. 6, where the conductance increases in magnitude as $|\tau|$ decreases.
This is because larger susceptibility makes the PA stronger. For smaller
values of $|\tau|$, the increase becomes more gradual. This would represent
effect of the size of the tube, considering that the value of $\xi$ at
$\psi=0$, given by $\xi_{0}|\tau|^{-\nu}$, exceeds the tube radius
approximately for $|\tau|<5\times 10^{-5}$. Equation (68),
${\hat{\sigma}}_{z}^{({\rm sq})}$, contributes to ${\hat{G}}^{({\rm th})}$
dominantly in the range of $\tau$ examined in Fig. 6. Changing the value of
$h$ for a mixture of NEMP, we calculate the conductance, as shown in Fig. 7.
As $h$ increases, ${\hat{G}}^{({\rm th})}$ increases, as expected since
${\hat{\psi}}^{(0)}$, ${\hat{\sigma}}_{z}^{({\rm th})}$, and
${\hat{v}}_{z}^{({\rm th})}$ then increase in Figs. 4 and 5(b). In Fig. 7(b),
${\hat{G}}^{({\rm th})}$ becomes less dependent on $h$ in logarithmic scale as
$\tau$ decreases. This tendency is also observed for the dependence of
${\hat{v}}_{z}^{({\rm th})}$ on $h$ in Fig. 5(b). The contribution from
${\hat{\sigma}}_{z}^{({\rm sq})}$ to ${\hat{G}}^{\rm(th)}$ remains dominant
for the values of $\tau$ and ${\hat{h}}$ examined in Fig. 7(b).

Figure 7: (a) Logarithmic plots of the dimensionless thermoosmotic
conductance, ${\hat{G}}^{({\rm th})}$ against the reduced temperature $\tau$,
evaluated at $T=T^{(0)}$, for a mixture of NEMP. Triangles in this figure and
in Fig. 6 represent the same results with the surface field set to $h=10^{-1}\
$cm3/s2. Circles (crosses) represent ${\hat{G}}^{({\rm th})}$ for
$h$=$10^{-0.5}\ (10^{-1.5})\ $cm${}^{3}/$s2. The corresponding values of the
scaled surface field ${\hat{h}}$ are shown in the figure; the change of $h$ is
indicated by an arrow. (b) Logarithmic plots of ${\hat{G}}^{({\rm th})}$
against the scaled surface field ${\hat{h}}$ for a mixture of NEMP. Squares
and circles represent ${\hat{G}}^{({\rm
th})}=\Omega[1,{\hat{\sigma}}_{z}^{({\rm th})}]$ at $\tau=1.25\times 10^{-5}$
and $3.2\times 10^{-3}$, respectively. The values of $\tau$ are evaluated at
$T=T^{(0)}$. Red asterisks (crosses) represent
$\Omega[1,{\hat{\sigma}}_{z}^{({\rm sq})}]$, which is dominant in
${\hat{G}}^{({\rm th})}$, at $\tau=1.25\times 10^{-5}$ ($3.2\times 10^{-3}$).
The change of $\tau$ is indicated by an arrow.

###  IV.3 Prediction of universal properties

As mentioned below Eq. (68), in our numerical results, the first term on the
RHS of Eq. (59) is negligible. This would be mainly because it contains a very
small positive factor $\tau_{*}$ (Table 1). Thus, owing to $T_{\rm c}\approx
T^{(0)}$, it is strongly suggested that

${\hat{\sigma}}_{z}^{({\rm
th})}\approx\frac{\partial{\hat{f}}}{\partial{\hat{\tau}}}-\frac{\eta\nu}{2{\hat{\omega}}^{\eta\nu+1}}{\frac{\partial{\hat{\omega}}}{\partial{\hat{\tau}}}}\left|\partial_{\hat{r}}{\hat{\psi}}\right|^{2}\
,$ (70)

which is evaluated at $\varepsilon=0$, holds for any mixture. The RHS of Eq.
(70) is free from the material constants, i.e. does not suppose a special
mixture, because it is determined only by the scaled reduced-temperature
${\hat{\tau}}$ and the magnitude of the scaled surface field $|{\hat{h}}|$.
Using various values of $({\hat{\tau}},{\hat{h}})$, we numerically find that
Eq. (68) is dominant in ${\hat{\sigma}}_{z}^{({\rm th})}$. In this sense, we
have

${\hat{\sigma}}_{z}^{({\rm
th})}\approx\pm\frac{{\hat{\omega}}^{\gamma-1}}{2}{\hat{\psi}}^{2}$ (71)

evaluated at $\varepsilon=0$, where the sign is taken as that of $\tau$, i.e.,
as that of $A_{0}$ in the bare model. Notably, Eq. (71) is strongly expected
to hold in the critical regime for any mixture, which is also supported by the
approximate estimation given below Eq. (54) and by the results in Figs. 6 and
7(b). Therefore, we can predict that, for any mixture near the UC (LC) point,
the direction of thermoosmosis is the same as (opposite to) that of the
temperature gradient, irrespective of which component is adsorbed onto the
wall, if the critical composition is kept in the middle of each reservoir.

##  V Further discussion and summary

Our numerical results are based on the calculation up to the order of
$\varepsilon$, or in the linear regime with respect to $\delta T$. Obviously,
after a temperature difference $\delta T$ is imposed between the reservoirs,
the deviation of $\tau$ from the value of $\tau$ at $T=T^{(0)}$ is required to
be much smaller in magnitude than the value at $T=T^{(0)}$ throughout inside
the tube. For example, if the latter value is set to $10^{-3}$ in magnitude,
we may set $|\delta T|/T_{\rm c}$ to be smaller than its $10\ \%$, $10^{-4}$.
The resultant local changes of $T$ and ${\hat{\psi}}$ shift
${\hat{\sigma}}_{z}^{({\rm th})}$. As far as examined, the shift is roughly
smaller than $10\ \%$ in the adsorption layer. For $|\delta T|=100\ {\rm
mK}\ll|T^{(0)}-T_{\rm c}|\approx 1\ $K and $L_{\rm tube}=10\ \mu$m, we find
from the results in Fig. 5(a) that the slip velocity is approximately $0.1\
\mu$m$/$s, which would be measured experimentally. In passing, in the
experiments on the Brownian motion of colloidal particles in a mixture,
$|\tau|$ is set to be smaller than $10^{-4}$ homogeneously omari ; beys2019 .

The RLFT succeeds in describing several phenomena of a mixture rlft . However,
in the theory, crossover to the regular part of the free energy wegner ; CAS ;
CATS ; edison ; vanthof ; olaya  is not considered, the results up to the one-
loop order approximation are used, and validity of the definition of the local
correlation length in the inhomogeneous composition is not fully discussed.
The regular parts of the transport coefficients folkmos2  are considered only
for the viscosity $\eta_{\rm s}$, whose singularity is very weak, in the
present study. These points are to be improved in future for quantitatively
better numerical results for transport properties, not only of thermoosmosis
but also of various phenomena, such as cross effects, described by the Onsager
coefficients. Still, the qualitative property on the flow direction in
thermoosmosis of a mixture, predicted by the present study, should be robust
to changes of details in the formulation, considering that it originates from
the sign of the coefficient, $A_{0}$, in the bare model.

Equation (71) leads to a possible power-law dependence with respect to
$|\tau|$ for the slip velocity far from a flat wall in thermoosmosis, as shown
in Appendix F. Thermophoresis would occur for a particle in a mixture in the
presence of PA onto the particle surface; the direction of the particle motion
is expected to be the same as (opposite to) that of the imposed temperature
gradient if the mixture is near the LC (UC) point and has the the critical
composition far from the particle. These points clearly require further
investigation in future.

Our present study is summarized as follows. We consider transport of a binary
fluid mixture, lying in the one-phase region near the demixing critical point,
through a capillary tube. One component is assumed to be adsorbed onto the
tube’s adiabatic wall and the adsorption layer can be much thicker than the
molecular size. We formulate the hydrodynamics from a coarse-grained free-
energy functional using an extended Gibbs-Duhem relation, Eq. (12),
consistently with principles of linear nonequilibrium thermodynamics. This
relation is originally derived in Ref. dvw , and is derived on a more general
condition in Appendix A. Assuming the critical composition in the middle of
each reservoir in the reference equilibrium state, we derive the Onsager
coefficients in Section III.1. Among various phenomena described by the
coefficients, we focus on thermoosmosis of a mixture. The formula for the
thermal force density, Eq. (46), is explicitly derived as an extension of
Derjaguin and Sidorenkov’s formula for one-component fluids and is rewritten
as Eq. (59) in terms of the renormalized local functional theory fisher-
degennes ; rlft . We predict that the direction of thermoosmotic flow of a
mixture near the upper (lower) consolute point is the same as (opposite to)
that of the temperature gradient, irrespective of which component is adsorbed
onto the tube’s wall. The magnitude of the thermoosmotic conductance
increases, with the increase being more gradual owing to the size effect, as
the critical point is approached. The thermal force density is given in a
scaled form by Eq. (70), which depends only on the scaled reduced-temperature
and the scaled surface field, and is dominantly contributed from Eq. (71).

Such mesoscopic inhomogeneity as is generated in a mixture by the surface
field can occur in many soft matter systems — polymer solutions,
polyelectrolytes, and liquid crystals onukibook ; degennes . In particlular,
their dynamics driven by a temperature gradient would be studied by applying
our procedure to a suitable set of hydrodynamic equations based on a coarse-
grained free-energy functional. Also, for thermoosmosis of a solution far from
the critical point, our results may help as a guide regarding properties
independent of the microscopic details. Hence, our present study would lay
solid foundations on non-isothermal hydrodynamics in the presence of
mesoscopic inhomogeneity and predict universal properties on thermoosmosis of
a near-critical binary fluid mixture.

##  Appendix A Non-dissipative part of the stress tensor

For conciseness, we here write $(T,\rho_{n},\nabla\rho_{n})$ for the variables
of $f_{\rm bulk}$ in Eq. (8). They are also variables of $s$ and $u$ because
of Eq. (20). The entropy density is also a function of $u$, $\rho_{n}$, and
$\nabla\rho_{n}$, and we define ${\tilde{s}}$ so that

$s(T,\rho_{n},\nabla\rho_{n})={\tilde{s}}(u(T,\rho_{n},\nabla\rho_{n}),\rho_{n},\nabla\rho_{n})$
(72)

holds. Explicit expressions of ${\tilde{s}}$, although given in special cases
dvw ; gonn , are not required in a general argument given below. We have

$\frac{\partial{\tilde{s}}}{\partial u}=\frac{1}{T}\quad{\rm
and}\quad\frac{\partial{\tilde{s}}}{\partial\zeta}=-\frac{1}{T}\frac{\partial{f_{\rm
bulk}}}{\partial\zeta}$ (73)

for ${\zeta}=\rho_{n}$ or $\nabla\rho_{n}$. As mentioned above Eq. (8), we
assume the dependence of $f_{\rm bulk}$ on $\nabla\rho_{n}$ to be through a
quadratic form, whose coefficients may depend on $T$ and $\rho_{n}$. Below, as
in Appendix A of Ref. pipe , we consider a quasistatic deformation of a
mixture to derive Eqs. (9) and (10). We write $V_{t}$ for a small region co-
moving with the deformation. Here, $t$ is not time but a parameter of the
deformation. In general, an infinitesimal change in the entropy are
contributed independently from the mechanical work, from the change in the
composition, and from the change in the internal energy. Thus, regarding $T$,
${\mathsf{\Pi}}$, and $\mu_{n}$ as homogeneous over a small region $V_{t}$, we
have

$T\frac{d}{dt}\int_{V_{t}}d{\mbox{\boldmath$r$}}\
{\tilde{s}}={\mathsf{\Pi}}:\int_{\partial V_{t}}dA\
{\mbox{\boldmath$n$}}_{\partial V_{t}}{\mbox{\boldmath$v$}}-\mu_{n}\
\frac{d}{dt}\int_{V_{t}}d{\mbox{\boldmath$r$}}\
\rho_{n}({\mbox{\boldmath$r$}},t)+\frac{d}{dt}\int_{V_{t}}d{\mbox{\boldmath$r$}}\
u\ .$ (74)

Here, the symbol $:$ is defined so that
${\mathsf{A}}:{\mathsf{B}}={\mathsf{A}}_{ij}{\mathsf{B}}_{ji}$ holds for two
tensors ${\mathsf{A}}$ and ${\mathsf{B}}$, and
${\mbox{\boldmath$n$}}_{\partial V_{t}}$ is the outward facing unit normal
vector of the surface of $V_{t}$.

Each locus of a mixture is assumed to have each bath of particles and heat. We
here write ${\mbox{\boldmath$j$}}_{n}$ and ${\mbox{\boldmath$j$}}_{u}$ for
their respective fluxes to the bath, and write $v$ for a displacement vector
per unit value of $t$. The meanings of ${\mbox{\boldmath$j$}}_{n}$ and $v$ are
different from the ones in the text, respectively; ${\mbox{\boldmath$j$}}_{\rm
a}+{\mbox{\boldmath$j$}}_{\rm b}$ does not always vanish here. Although $t$ is
not the time, we can treat $t$ as the time formally to define the Eulerian
time-derivative $\partial/(\partial t)$ and Lagrangian time-derivative
$D/(Dt)$. We have

$\frac{D\rho_{n}}{Dt}=-\rho_{n}\nabla\cdot{\mbox{\boldmath$v$}}-\nabla\cdot{\mbox{\boldmath$j$}}_{n}\quad{\rm
and}\quad\frac{Du}{Dt}=-u\nabla\cdot{\mbox{\boldmath$v$}}-\nabla\cdot{\mbox{\boldmath$j$}}_{\rm
u}\ .$ (75)

The whole region occupied by the mixture, $V_{\rm tot}$, is deformable here,
unlike in the text. The LHS of Eq. (74) is rewritten as the integral of
$T[D{\tilde{s}}/(Dt)+{\tilde{s}}\nabla\cdot{\mbox{\boldmath$v$}}]$ over
$V_{t}$. Rewriting the last two terms on the RHS similarly and applying the
divergence theorem for the first term, we obtain an equation for the
integrands owing to arbitrariness of $V_{t}$. With the aid of this equation,
the change in the entropy in $V_{\rm tot}$ per unit value of $t$ is found to
be

$\int_{V_{\rm tot}}d{\mbox{\boldmath$r$}}\
\left[\frac{D{\tilde{s}}}{Dt}+{\tilde{s}}\nabla\cdot{\mbox{\boldmath$v$}}\right]=\int_{V_{\rm
tot}}d{\mbox{\boldmath$r$}}\
\left[\frac{{\mathsf{\Pi}}}{T}:\nabla{\mbox{\boldmath$v$}}+\frac{\mu_{n}}{T}\nabla\cdot{\mbox{\boldmath$j$}}_{n}-\frac{1}{T}\nabla\cdot{\mbox{\boldmath$j$}}_{\rm
u}\right]\ ,$ (76)

where $T$, ${\mathsf{\Pi}}$, and $\mu_{n}$ can be inhomogeneous. The factor
$\nabla\cdot{\mbox{\boldmath$v$}}$ in Eqs. (75) and (76) comes from the change
rate of the Jacobian between the Eulerian and Lagrangian coordinates. We have

$\frac{D{\tilde{s}}}{Dt}=\frac{\partial{\tilde{s}}}{\partial
u}\frac{Du}{Dt}+\frac{\partial{\tilde{s}}}{\partial\rho_{n}}\frac{D\rho_{n}}{Dt}+\frac{\partial{\tilde{s}}}{\partial\left(\nabla\rho_{n}\right)}\cdot\nabla\left(\frac{D\rho_{n}}{Dt}\right)-\frac{\partial{\tilde{s}}}{\partial\left(\nabla\rho_{n}\right)}\cdot\left(\nabla{\mbox{\boldmath$v$}}\right)\cdot\left(\nabla\rho_{n}\right)\
,$ (77)

which can be rewritten using Eqs. (73) and (75). Substituting the result into
the LHS of Eq. (76) and applying integration by parts, we find the LHS to be
the sum of

$-\int_{\partial V_{\rm tot}}d{\mbox{\boldmath$r$}}\
\frac{1}{T}\frac{D\rho_{n}}{Dt}\frac{\partial f_{\rm
bulk}}{\partial(\nabla\rho_{n})}\cdot{\mbox{\boldmath$n$}}_{\partial V_{\rm
tot}}$ (78)

and the RHS of Eq. (76) with $\mu_{n}$ and ${\mathsf{\Pi}}$ being replaced by
the RHS’s of Eqs. (9) and (10), respectively. This means that $\mu_{n}$ and
${\mathsf{\Pi}}$ are given by these equations, respectively. Because
${\mathsf{\Pi}}$ is symmetric, we can derive Eq. (12), or equivalently,
$\nabla\cdot({\mathsf{\Pi}}/T)=\rho_{n}\nabla({\mu_{n}}/{T})-u\nabla(1/{T})$,
which is of the same form as Eq. (2.44) of Ref. dvw . Equation (9) can be used
in calculating not only $\varphi^{(0)}$ but also $\varphi^{(1)}$; the latter
need not be obtained in the present study.

We next consider thermodynamics of the mixture in a thin interfacial region
regarded as $\partial V_{\rm tot}$. The free energy per unit area of this
region is denoted by $f_{\rm surf}$ in Eq. (8), and is here denoted by
$f^{({\rm s})}$. The superscript (s) in general indicates a thermodynamic
quantity in $\partial V_{\rm tot}$; a density with the superscript represents
a quantity per unit area. As in Eq. (20), we can introduce internal energy
${u}^{({\rm s})}$ and entropy ${s}^{({\rm s})}$ from $f^{({\rm s})}$. These
three quantities are functions of $T^{({\rm s})}$ and $\rho_{n}^{({\rm s})}$.
As Eq. (74) yields Eq. (76), an equation for a small co-moving area on
$\partial V_{\rm tot}$ yields an equation representing the change of the
entropy on $\partial V_{\rm tot}$. Two points are to be noted in this
derivation. First, the mechanical contribution consists of a term involving
the two-dimensional pressure tensor ${\mathsf{\Pi}}^{({\rm s})}$ and a term
involving the force normal to the small area. Because $f^{({\rm s})}$ includes
no gradients of mass densities, the pressure tensor is written as the two-
dimensional scalar pressure multiplied by the identity tensor on $\partial
V_{\rm tot}$. The scalar pressure denoted by $P^{({\rm s})}$. We define
$P_{\rm n}^{({\rm s})}$ so that the normal force is $P_{\rm n}^{({\rm
s})}{{\mbox{\boldmath$n$}}_{\partial V_{\rm tot}}}$ per unit area. Second, the
factor coming from the change rate of the Jacobian is not
$\nabla\cdot{\mbox{\boldmath$v$}}$, appearing Eqs. (75) and (76), but
$\nabla_{\parallel}\cdot{\mbox{\boldmath$v$}}_{\parallel}-2H_{\rm
m}{\mbox{\boldmath$v$}}\cdot{\mbox{\boldmath$n$}}_{\partial V_{\rm tot}}$,
where ${\mbox{\boldmath$v$}}_{\parallel}$ is the projection of $v$ on the
plane tangential to $\partial V_{\rm tot}$,
$\nabla_{\parallel}\cdot{\mbox{\boldmath$v$}}_{\parallel}$ indicates the
divergence defined on $\partial V_{\rm tot}$, and $H_{\rm m}$ denotes the mean
curvature of $\partial V_{\rm tot}$ physica . The curvature is defined so that
it is positive when the center of curvature lies on the side directed by
${\mbox{\boldmath$n$}}_{\partial V_{\rm tot}}$.

The temperature at a local area on $\partial V_{\rm tot}$, $T^{({\rm s})}$,
should be equal to $T$ at its adjacent local region of $V_{\rm tot}$.
Similarly, $\rho_{\alpha}^{({\rm s})}$ is determined by $\rho_{\alpha}$ at the
adjacent region so that the former equals the latter multiplied by the
interfacial region’s width. No other factor is involved in determining
$\rho_{\alpha}^{({\rm s})}$, which means $\mu_{\alpha}^{({\rm s})}=0$. Through
these relationships, $f_{\rm surf}(T,\rho_{n})$ equals $f^{({\rm s})}(T^{({\rm
s})},\rho_{n}^{({\rm s})})$. Taking Eq. (78) into account, we find

$0=\frac{\partial f_{\rm surf}}{\partial\rho_{n}}+\frac{\partial f_{\rm
bulk}}{\partial\left(\nabla\rho_{n}\right)}\cdot{\mbox{\boldmath$n$}}_{\partial
V}\quad{\rm at}\quad\partial V$ (79)

from the equation representing the change of the entropy on $\partial V_{\rm
tot}$. We also find $P^{({\rm s})}=-f_{\rm surf}$, which gives the Laplace
pressure ${P_{\rm n}^{({\rm s})}}=-2f_{\rm surf}H_{m}$ effvis . Notably,
$f_{\rm surf}$ equals the grand-potential density of $\partial V_{\rm tot}$
owing to $\mu^{({\rm s})}_{n}=0$. We need not consider these interfacial
forces and the force exerted on the mixture by the tube’s wall in calculating
the velocity field in the tube because the latter force is determined so that
the no-slip condition is realized.

##  Appendix B Reciprocal relations

We consider two sets of flow fields, each being driven by the thermodynamic
forces $\left(-\delta(P/T)_{k},-\delta(\mu_{-}/T)_{k},\delta(1/T)_{k}\right)$,
with $k$ being i or ii. The resultant thermodynamic fluxes and fields in the
tube are also indicated by the subscript k. Different ways of applying the
divergence theorem to the volume integral of ${\eta_{\rm
s}^{(0)}}E^{(1)}_{{\rm i}}:E^{(1)}_{\rm ii}$ over the tube interior, denoted
by $V_{\rm tube}$, give

$\int_{V_{\rm tube}}d{\mbox{\boldmath$r$}}\ {\mbox{\boldmath$v$}}_{\rm
ii}^{(1)}\cdot\left[\nabla\cdot\left({\eta_{\rm s}^{(0)}}E_{\rm
i}^{(1)}\right)\right]=\int_{V_{\rm tube}}d{\mbox{\boldmath$r$}}\
{\mbox{\boldmath$v$}}_{\rm i}^{(1)}\cdot\left[\nabla\cdot\left({\eta_{\rm
s}^{(0)}}E_{\rm ii}^{(1)}\right)\right]$ (80)

with the aid of Eq. (16) and the no-slip condition at the tube’s wall. Here,
we neglect effects of tube’s edges on the laminar flow. Substituting Eq. (28)
into Eq. (80), we find that

$\displaystyle\int_{S_{tube}}dA\ v_{{\rm
ii},z}^{(1)}\left[\frac{\rho^{(0)}}{\rho^{({\rm
ref})}}\delta\left(\frac{P}{T}\right)_{{\rm
i}}+\left(\varphi^{(0)}-\frac{\varphi^{({\rm ref})}}{\rho^{({\rm
ref})}}\right)\delta\left(\frac{\mu_{-}}{T}\right)_{\rm i}\right.$
$\displaystyle\qquad\qquad\left.-\left(P^{(0)}+e^{(0)}-\frac{\rho^{(0)}e^{({\rm
ref})}}{\rho^{({\rm ref})}}\right)\delta\left(\frac{1}{T}\right)_{{\rm
i}}\right]$ (81)

equals the above equation with the subscripts i and ii exchanged. Putting
$\delta(P/T)_{\rm i}$, $\delta(\mu_{-}/T)_{\rm i}$, $\delta(P/T)_{\rm ii}$,
and $\delta(1/T)_{\rm ii}$ equal to zero, we use Eqs. (29), (30), (31), and
(32) to find $L_{23}=L_{32}$. Likewise, we can obtain $L_{13}=L_{31}$ by
putting $\delta(P/T)_{\rm i}$, $\delta(\mu_{-}/T)_{\rm i}$,
$\delta(\mu_{-}/T)_{\rm ii}$, and $\delta(1/T)_{\rm ii}$ equal to zero. The
other reciprocal relations can be derived similarly, as shown in Appendix B of
Ref. pipe .

##  Appendix C Dissipative fluxes

In an equilibrium mixture, we consider a region where the mass densities are
homogeneous. There, $f_{\rm bulk}$ is a function of $T$, $\rho$, and
$\varphi$, and we have

$\left.\frac{\partial\mu_{-}}{\partial\varphi}\right)_{T,P}=\left.\frac{\partial\mu_{-}}{\partial\varphi}\right)_{T,\rho}+\left.\frac{\partial\mu_{-}}{\partial\rho}\right)_{T,\varphi}\left.\frac{\partial\rho}{\partial\varphi}\right)_{T,P}\
.$ (82)

The first partial derivative of the second term on the RHS above equals
$\partial^{2}f_{\rm bulk}/(\partial\rho\partial\varphi)$, which vanishes
because Eq. (48) is assumed. The second derivative does not diverge, as
mentioned in Appendix D. Thus, whether $T$ and $P$ are fixed or $T$ and $\rho$
fixed, $\partial\mu_{-}/(\partial\varphi)$ are the same and are regarded as
equal to the inverse of $\chi$, which appears in Eq. (33). We have

$\frac{1}{\chi}=\left.\frac{\partial\mu_{-}}{\partial c_{\rm
a}}\right)_{T,P}\left.\frac{\partial c_{\rm
a}}{\partial\varphi}\right)_{T,P}=\frac{1}{2\rho^{2}{\bar{v}}_{+}}\left.\frac{\partial\mu}{\partial
c_{\rm a}}\right)_{T,P}\ ,$ (83)

where ${\bar{v}}_{+}$ denotes $({\bar{v}}_{\rm a}+{\bar{v}}_{\rm b})/2$. The
second equality above comes from Eq. (34) of Ref. pipe . Because a mixture we
consider has $\rho{\bar{v}}_{+}\approx 1$ pipe , Eq. (33) is consistent with
the result in Refs. mist ; luet .

With ${\bar{\delta}}$ indicating the deviation from the average, the
thermodynamic forces are ${\bar{\delta}}(1/T)$, $-{\bar{\delta}}(\mu_{+}/T)$,
and $-{\bar{\delta}}(\mu_{-}/T)$ in Eq. (22). This equation is rewritten as

${\mbox{\boldmath$j$}}=-4{\tilde{\alpha}}\nabla\mu_{-}+2{\tilde{\beta}}\nabla
T\quad{\rm and}\quad{\mbox{\boldmath$j$}}_{\rm
q}-\mu_{-}{\mbox{\boldmath$j$}}=2T{\tilde{\beta}}\nabla\mu_{-}-{\tilde{\gamma}}\nabla
T\ ,$ (84)

whereby ${\tilde{\alpha}}$, ${\tilde{\beta}}$, and ${\tilde{\gamma}}$ are
defined. We write ${\check{s}}(\equiv s/\rho)$ for entropy per unit mass. As
can be seen from Ref. landau , the irreversible fluxes of $c_{\rm a}$ and
${\check{s}}$ are respectively given by the quotient of the first entry in Eq.
(84) divided by $2\rho$ and that of the second divided by $\rho T$, whereas
the conjugate thermodynamic forces are respectively given by
$-2\rho({\bar{\delta}}\mu_{-})/T$ and $-\rho({\bar{\delta}}T)/T$. After the
division, the second term on the RHS of the second entry becomes equal to the
product of $-\rho(\nabla T)/T$ multiplied by ${\tilde{\gamma}}/\rho^{2}$,
which is one of the Onsager coefficients. Similarly, we can obtain the other
Onsager coefficients. Comparing Eq. (22) with (84), we obtain

$\Lambda=4{\tilde{\alpha}}\
,\quad\kappa=2T\left(2\mu_{-}{\tilde{\alpha}}-T{\tilde{\beta}}\right)\
,\quad{\rm
and}\quad\lambda=4\mu_{-}T\left(\mu_{-}{\tilde{\alpha}}-T{\tilde{\beta}}\right)+T^{2}{\tilde{\gamma}}\
.$ (85)

In Refs. mist ; jetp ; luet , the singular parts of the LHS’s above, indicated
by the subscript sing, are shown to satisfy

$\frac{{\tilde{\beta}}_{\rm sing}}{2{\tilde{\alpha}}_{\rm
sing}}=-\left.\frac{\partial c_{\rm a}}{\partial
T}\right)_{P,\mu_{-}}\left.\frac{\partial\mu_{-}}{\partial c_{\rm
a}}\right)_{T,P}=-{\bar{s}}_{-}\ ,$ (86)

whose second equality comes from Eq. (4), and

$\frac{{\tilde{\gamma}}_{\rm sing}}{4{\tilde{\alpha}}_{\rm sing}}=\frac{\rho
T}{\chi}\left.\frac{\partial{\check{s}}}{\partial T}\right)_{P,\mu_{-}}\approx
T{\bar{s}}^{2}_{-}\ .$ (87)

The approximate equality of Eq. (87) is explained in the next paragraph. The
background parts of ${\tilde{\alpha}}$ and ${\tilde{\beta}}$ are negligible in
the critical regime mirz ; iwan ; gigl ; folkmos2 . Because of the singular
properties, the Ludwig-Soret effect has universal properties in a near-
critical binary fluid mixture mist ; gigl ; luet ; ryzh ; kohl . For
${\tilde{\gamma}}$, neglecting the background part and adopting the
approximate equality, we obtain Eq. (35), which leads to neglect of the
thermal conductivity not exhibiting the critical enhancement luet .

The partial derivative in Eq. (87) equals

$\left.\frac{\partial{\check{s}}}{\partial T}\right)_{P,c_{\rm
a}}+\left.\frac{\partial{\check{s}}}{\partial c_{\rm
a}}\right)_{T,P}\left.\frac{\partial c_{\rm a}}{\partial
T}\right)_{P,\mu_{-}}=\frac{c_{P}}{\rho T}+2{\bar{s}}_{-}\left.\frac{\partial
c_{\rm a}}{\partial T}\right)_{P,\mu_{-}}\ ,$ (88)

where $c_{P}$ denotes the isobaric specific heat under constant $c_{\rm a}$.
The equality between the second terms on both sides above comes from a Maxwell
relation and Eq. (4). The second partial derivative in Eq. (86) can be
rewritten using Eq. (34) of Ref. pipe . As a result, the last partial
derivative in Eq. (88) equals ${\bar{s}}_{-}\chi/(2\rho^{2}{\bar{v}}_{+})$. As
mentioned in Appendix D, $c_{P}$ diverges more weakly than $\chi$ in the
critical regime, where the approximate equality in Eq. (87) is valid. Without
the above-mentioned approximations for ${\tilde{\gamma}}$, the area integral
of $T^{2}$ multiplied by the sum of the thermal conductivity and $\Lambda
c_{P}/\chi$, evaluated at $\varepsilon=0$, over $S_{\rm tube}$ should be added
to the RHS of Eq. (45).

An alternative explanation is as follows. In the mode-coupling theory, the
singular part of an Onsager coefficient is calculated in terms of the time-
integral of the two-time correlation function of the reversible fluxes, as
mentioned in Section 6.5 of Ref. onukibook . The Onsager coefficients
mentioned above Eq. (85) are found to be ${\tilde{\gamma}}/\rho^{2}$,
$-{\tilde{\beta}}T/\rho^{2}$, and ${\tilde{\alpha}}T/\rho^{2}$, and their
singular parts are linked with the autocorrelation of
$({\bar{\delta}}{\check{s}}){\mbox{\boldmath$v$}}$, the crosscorrelation of
$({\bar{\delta}}{\check{s}}){\mbox{\boldmath$v$}}$ and $({\bar{\delta}}c_{\rm
a}){\mbox{\boldmath$v$}}$, and the autocorrelation of $({\bar{\delta}}c_{\rm
a}){\mbox{\boldmath$v$}}$, respectively. This means that, in the critical
regime, the ratio of ${\tilde{\gamma}}_{\rm sing}$ to $-{\tilde{\beta}}_{\rm
sing}T$ and that of $-{\tilde{\beta}}_{\rm sing}T$ to ${\tilde{\alpha}}_{\rm
sing}T$ are given by $\partial{\check{s}}/(\partial c_{\rm a})$ with $T$ and
$P$ fixed, which is consistent with Eqs. (86) and (87).

##  Appendix D Internal-energy fluctuations and approximate incompressibility

Here, we discuss some backgrounds of Eq. (48). In the bare model, without the
contribution from $\partial V_{\rm tot}$ taken into account, we define the
effective Hamiltonian ${\cal H}$ so that the equilibrium probability density
functional (EPDF) of $\rho$ and $\psi$ is proportional to $e^{-{\cal H}}$. As
mentioned in the text, ${\cal H}$ includes the $\psi^{4}$ model,

$\int_{V_{\rm tot}}d{\mbox{\boldmath$r$}}\
\left[\frac{1}{2}A_{0}\tau\psi^{2}+\frac{\lambda_{0}}{4!}\psi^{4}+\frac{a_{0}^{2}}{2}\left|\nabla\psi\right|^{2}\right]\
,$ (89)

where $\lambda_{0}(>0)$ and $a_{0}$ are constants, and the integrand above
becomes a part of Eq. (48) divided by $k_{\rm B}T$ after coarse-grained. The
value of $T_{\rm c}$ in the definition of $\tau$ depends on the stage of the
coarse-graining. Writing $m$ for $(u-u_{\rm c})/(k_{\rm B}T_{\rm c})$, we can
also consider the EPDF of $\rho$, $\psi$, and $m$. We define ${\cal H}_{\rm
s}$ so that this EPDF is proportional to $e^{-{\cal H}_{\rm s}}$ in the bare
model. Integrating out $m$ from this EPDF should yield the EPDF of $\rho$ and
$\psi$. In other words, the latter’s Legendre transform is the former, and
vice versa. Thus, owing to a term $A_{0}\tau\psi^{2}/2$ in ${\cal H}$, ${\cal
H}_{\rm s}$ has a term proportional to $m\psi^{2}$ and ${\cal H}$ has a term
proportional to $\tau^{2}$ halhoma ; onukibook . We define $C_{0}$ so that
this term equals $-C_{0}\tau^{2}/(2k_{\rm B})$, and the variance of $m$ is
proportional to $C_{0}$. Coarse-graining the $\psi^{4}$ model and imposing the
self-consistent condition for off-critical compositions set up the RLFT rlft .
We can also set it up by coarse-graining ${\cal H}_{\rm s}$, imposing a self-
consistent condition, and integrating out $m$. In this procedure, $C_{0}$
becomes $C$ of Eq. (48) because fluctuations of $m$ are affected by those of
$\psi$ via their coupling term halhoma ; onukibook . Instead of using Eq. (48)
as it is, we can calculate ${\hat{\sigma}}^{({\rm th})}$ by evaluating the
dependence of $C$ on $\xi$ locally. Although data are not shown, the resultant
changes from the results of Fig. 3 are negligibly small.

The isochoric specific heat $c_{V}$, given by $-T\partial^{2}f_{\rm
bulk}/(\partial T^{2})$ with $\rho$ and $\varphi$ being fixed, remains finite
at the critical point, although it appears to diverge in the regime accessible
to usual experiments anisim72 ; onukibook . Linked with the fluctuations of
$m$, the isobaric specific heat $c_{P}$ becomes proportional to $C\
(\propto|\tau|^{-\alpha}$ with $\alpha=0.11)$ in the critical regime with
$c_{\rm a}$ being the value at the critical point anisim95 ; onukibook . The
same power-law dependence is shared by the isothermal compressibility
$\kappa_{T}$ and the thermal expansion coefficient $\alpha_{P}$, which are
given by

${\kappa_{T}}=\frac{1}{\rho}\left.\frac{\partial\rho}{\partial
P}\right)_{T,c_{\rm a}}\quad{\rm
and}\quad{\alpha_{P}}=-\frac{1}{\rho}\left.\frac{\partial\rho}{\partial
T}\right)_{{P},c_{\rm a}}={\kappa_{T}}\left.\frac{\partial P}{\partial
T}\right)_{\rho,c_{\rm a}}\ .$ (90)

These quantities are related with each other via
$\kappa_{T}\left(c_{P}-c_{V}\right)=\alpha_{P}^{2}T$. The singularity of
$\kappa_{T}$ is generated by coupling between $\rho-\rho_{\rm c}$ and
$\psi^{2}$ in the $\rho$-dependent part in the EDPF. We neglect this coupling
in Eq. (48), considering that the singularity is not accessible to usual
experiments clerke ; onukibook . Observed values of $\kappa_{T}$ and
$\alpha_{P}$ are typically $10^{-9}\ $Pa-1 and $10^{-3}\ $K-1, respectively,
near the critical point clerke ; pous .

In this paragraph, we show that the last partial derivative of Eq. (82) is
finite. In the region mentioned at the beginning of Appendix C, we simply
write $f$ for $f_{\rm bulk}$ and refer to its derivatives by adding
subscripts. For example, $f_{\rho\varphi}$ represents $\partial^{2}f_{\rm
bulk}/(\partial\rho\partial\varphi)$, which vanishes because Eq. (48) is
assumed. We have

$\delta P=\rho\left(f_{\rho\rho}\delta\rho+f_{\rho T}\delta
T\right)+\varphi\left({f_{\varphi\varphi}\delta\varphi}+f_{\varphi T}\delta
T\right)+s\delta T$ (91)

owing to Eq. (7). Here, unlike in the text, $\delta$ indicates an
infinitesimal change. Using Eq. (91) and $\varphi=\rho(2c_{\rm a}-1)$, we find
$\delta\varphi$ ($\delta\rho$) equal to $\varphi$ ($\rho$) multiplied by

$\delta P\frac{1}{\rho^{2}f_{\rho\rho}+\varphi^{2}f_{\varphi\varphi}}$ (92)

when $T$ and $c_{\rm a}$ are fixed. The fraction is found to equal
$\kappa_{T}$ because of the first entry of Eq. (90) and Eq. (91). Considering
that the sum in the first (second) parentheses of Eq. (91) equals
$\delta\mu_{+}$ ($\delta\mu_{-})$, we use Eq. (92) to find that the first
entries of Eqs. (4) and (5) give ${\bar{v}}_{-}=\varphi
f_{\varphi\varphi}\kappa_{T}$ and ${\bar{v}}_{+}=\rho f_{\rho\rho}\kappa_{T}$.
Thus, we use Eq. (91) to find that the last partial derivative in Eq. (82)
equals $-{\bar{v}}_{-}/{\bar{v}}_{+}$, which result holds if $f_{\rho\varphi}$
does not vanish. The limit of this fraction obtained as the critical point is
approached can be written in terms of $\rho_{\rm c}$ and $\varphi_{\rm c}$ and
is finite, which supports the description below Eq. (82).

Finally, we consider validity of $\rho^{({\rm ref})}\approx\rho_{\rm c}$;
$\rho^{({\rm ref})}$ appears in Eq. (61) and is involved in returning the
dimension to ${\hat{G}}^{({\rm th})}$. For definiteness, we here write
$\tau^{({\rm ref})}$ for the value of $\tau$ in the reference state. At the
state we reach by changing $\tau$ from zero to $\tau^{({\rm ref})}$ with $P$
and $c_{\rm a}$ being fixed, how $\rho$ changes from $\rho_{\rm c}$ can be
approximately calculated from the regular part of $\alpha_{P}$ clerke . The
difference between the value of $\rho$ at this state and $\rho^{({\rm ref})}$
can be calculated using the last partial derivative in Eq. (82). Thus, we can
calculate the difference $\rho^{({\rm ref})}-\rho_{\rm c}$, and find that the
difference divided by $\rho_{\rm c}$ is smaller than $10^{-3}$ for
${\tau^{({\rm ref})}}=1/300$ in magnitude. Thus, we can use $\rho^{({\rm
ref})}\approx\rho_{\rm c}$.

##  Appendix E Approximation in the derivation of Eq. (56)

By using Eq. (91) to calculate the second entry of Eq. (4), we obtain
${\bar{s}}_{-}=\varphi f_{\varphi\varphi}\alpha_{P}-f_{\varphi T}$. The term
$\mu_{-}^{({\rm ref})}\varphi$ is included in $f_{-}$, as shown in Eq. (51).
Its contribution to the second entry of Eq. (20) is

$-T^{2}\left.\frac{\partial}{\partial
T}\right)_{\rho,\varphi}\frac{\mu_{-}^{({\rm
ref})}\varphi}{{T}}=\mu_{-}^{({\rm
ref})}{\varphi-T\varphi}\left.\frac{\partial}{\partial
T}\right)_{\rho,\varphi}{\mu_{-}^{({\rm ref})}}\ .$ (93)

The value of the last partial derivative above equals that of $f_{\varphi T}$
in the reference state owing to $f_{\rho\varphi}=0$. Thus, because of Eq.
(34), Eq. (93) consists of ${\bar{H}}_{-}^{({\rm ref})}\varphi$ and the other
term. This term gives an extra term

${\frac{T_{*}\tau_{*}}{\mu_{*}}}\left(\varphi f_{\varphi\varphi}\alpha_{\rm
P}\right)^{({\rm ref})}{\hat{\psi}}^{(0)}$ (94)

to Eq. (59), where the superscript (ref) implies that the product in the
parentheses should be evaluated in the reference state. We can use Eq. (63) to
evaluate $f_{\varphi\varphi}$ and find Eq. (94) to be smaller than
$30|\tau|^{\gamma}|{\hat{\psi}}^{(0)}|$ in magnitude. This magnitude is found
to be much smaller than the corresponding magnitude given by the circles in
Fig. 3, with the aid of values of ${\hat{\psi}}^{(0)}$ in Fig. 2. Thus, Eq.
(94) is negligible in deriving Eq. (56).

##  Appendix F Thermoosmotic flow far from a flat surface

For a mixture occupying a semi-infinite space bounded by a flat wall surface,
we consider imposing a temperature gradient along the $z$ axis, which is
parallel to the wall surface. The equilibrium profile $\psi^{(0)}$ is regarded
as a function of the distance from the surface, denoted by $X$, and is assumed
to approach zero as $X\to\infty$. The velocity field can be calculated
similarly to Eq. (60). Assuming that $\eta_{\rm s}$ to be $\eta_{*}$
homogeneously and using Eq. (71), we find that the $z$ component of the
mixture velocity far from the surface, or the slip velocity, is approximately
given by

$\frac{k_{\rm
B}C_{1}{|\tau|}}{2\eta_{*}\xi_{0}^{2}\tau}\int_{0}^{X}dX_{1}\int_{X_{1}}^{\infty}dX\
\omega^{\gamma-1}\left[\psi^{(0)}(X)\right]^{2}$ (95)

multiplied by the $z$ component of the temperature gradient.

When the second term is much smaller than the first term on the RHS of Eq.
(52), ${\hat{\omega}}\approx{|{\hat{\tau}}|}$ holds and the second term on the
RHS of Eq. (50) is negligible. We further approximate $M_{-}$ to be $k_{\rm
B}T_{\rm c}C_{1}$ to obtain the free-energy density in the Gaussian model,
where $\xi$ becomes homogeneous and $\psi^{(0)}(X)$ equals $h\xi
e^{-X/\xi}/M_{-}$ undul ; EPJ . Substituting this into Eq. (95) with
$\omega=|\tau|$ and $h=0.1\ $cm${}^{3}/$s2, we find that the slip velocity in
terms of ${\hat{v}}_{z}^{({\rm th})}$ for a mixture of LW (NEMP) is $-0.082\
(0.12)$ at $|\tau|=3.2\times 10^{-3}$, and $-0.017\ (0.025)$ at
$|\tau|=6.4\times 10^{-3}$. Here, for the critical exponents, we use the
values mentioned in Section III.3, not the values in the Gaussian model. These
values of the slip velocity are comparable with the corresponding values
calculated in the same procedure as used for Fig. 5, which are $-0.042\
(0.061)$ at $|\tau|=3.2\times 10^{-3}$ and $-0.014\ (0.021)$ at
$|\tau|=6.4\times 10^{-3}$.

The equilibrium profile $\psi^{(0)}(X)$ becomes universal in the adsorption
layer as the critical point is approached beyond the regime of the Gaussian
model fisher-degennes ; RJ ; diehl97 . As is done for the diffusioosmosis in
Appendix D of Ref. pipe  and in Section VC of Ref. diffdrop , we use the
universal profile $\psi^{(0)}(X)\propto X^{-\beta/\nu}$ in Eq. (95). With the
aid of Eq. (52), we find that the slip velocity in thermoosmosis becomes
proportional to $|\tau|^{\nu-1}$ as $\tau$ approaches zero.

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