A molecular theory of large-solute diffusion
The limit of a large solute in the molecular theory of diffusion developed by Yamaguchi et al. [Yamaguchi T. et al., J. Chem. Phys., 2005, 123, 034504] is studied. By the limit, the Stokes approximation to the hydrodynamic equations is derived in the outside region of a diffusing solute. The limit...
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irk-123456789-1189482017-06-02T03:03:20Z A molecular theory of large-solute diffusion Yoshimori, A. The limit of a large solute in the molecular theory of diffusion developed by Yamaguchi et al. [Yamaguchi T. et al., J. Chem. Phys., 2005, 123, 034504] is studied. By the limit, the Stokes approximation to the hydrodynamic equations is derived in the outside region of a diffusing solute. The limit of a large solute also leads to equations in the inside region of the solute. The analytical solution of the inside equation allows one to derive the boundary condition, which is needed on the surface of the solute when the hydrodynamic equations are calculated. The boundary condition includes stick and slip boundary conditions employed by the Stokes law, in the special case. Besides stick and slip conditions, other conditions can be expressed. The boundary condition depends on properties of a solvent. Вивчається границя великої розчинюваної частинки в рамках молекулярної теорiї, розвинутої Ямагучi. В цiй границi наближення Стокса до гiдродинамiчних рiвнянь отримано за межами дифузiї розчинюваної частинки. Границя великої частинки також приводить до рiвнянь всерединi областi розчинюваної частинки. Аналiтичний розв’язок дозволяє отримати граничнi умови на поверхнi частинки, необхiднi при розрахунку гiдродинамiчних рiвнянь. Граничнi умови в спецiальному випадку включають граничнi умови змiщення, що використовуються в законi Стокса. Крiм цих, можуть бути отриманi також iншi умови. Граничнi умови залежать вiд властивостей розчинника. 2007 Article A molecular theory of large-solute diffusion / A. Yoshimori // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 563-571. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 66.10.-x, 05.20.Jj, 05.60.Cd, 87.15.Vv, 47.10.-g DOI:10.5488/CMP.10.4.563 http://dspace.nbuv.gov.ua/handle/123456789/118948 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The limit of a large solute in the molecular theory of diffusion developed by Yamaguchi et al. [Yamaguchi T.
et al., J. Chem. Phys., 2005, 123, 034504] is studied. By the limit, the Stokes approximation to the hydrodynamic
equations is derived in the outside region of a diffusing solute. The limit of a large solute also leads to
equations in the inside region of the solute. The analytical solution of the inside equation allows one to derive
the boundary condition, which is needed on the surface of the solute when the hydrodynamic equations
are calculated. The boundary condition includes stick and slip boundary conditions employed by the Stokes
law, in the special case. Besides stick and slip conditions, other conditions can be expressed. The boundary
condition depends on properties of a solvent. |
| format |
Article |
| author |
Yoshimori, A. |
| spellingShingle |
Yoshimori, A. A molecular theory of large-solute diffusion Condensed Matter Physics |
| author_facet |
Yoshimori, A. |
| author_sort |
Yoshimori, A. |
| title |
A molecular theory of large-solute diffusion |
| title_short |
A molecular theory of large-solute diffusion |
| title_full |
A molecular theory of large-solute diffusion |
| title_fullStr |
A molecular theory of large-solute diffusion |
| title_full_unstemmed |
A molecular theory of large-solute diffusion |
| title_sort |
molecular theory of large-solute diffusion |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118948 |
| citation_txt |
A molecular theory of large-solute diffusion / A. Yoshimori // Condensed Matter Physics. — 2007. — Т. 10, № 4(52). — С. 563-571. — Бібліогр.: 22 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT yoshimoria amoleculartheoryoflargesolutediffusion AT yoshimoria moleculartheoryoflargesolutediffusion |
| first_indexed |
2025-07-08T14:57:18Z |
| last_indexed |
2025-07-08T14:57:18Z |
| _version_ |
1837091142032687104 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 4(52), pp. 563–571
A molecular theory of large-solute diffusion
A.Yoshimori∗
Department of Physics, Kyushu University, Fukuoka, 812–8581, Japan
Received September 4, 2007, in final form October 11, 2007
The limit of a large solute in the molecular theory of diffusion developed by Yamaguchi et al. [Yamaguchi T.
et al., J. Chem. Phys., 2005, 123, 034504] is studied. By the limit, the Stokes approximation to the hydrody-
namic equations is derived in the outside region of a diffusing solute. The limit of a large solute also leads to
equations in the inside region of the solute. The analytical solution of the inside equation allows one to de-
rive the boundary condition, which is needed on the surface of the solute when the hydrodynamic equations
are calculated. The boundary condition includes stick and slip boundary conditions employed by the Stokes
law, in the special case. Besides stick and slip conditions, other conditions can be expressed. The boundary
condition depends on properties of a solvent.
Key words: microscopic theory, solvent-solute interaction, hydrodynamic limit, exclusive effect, momentum
conservation, generalized Langevin equation
PACS: 66.10.-x, 05.20.Jj, 05.60.Cd, 87.15.Vv, 47.10.-g
1. Introduction
When a solute particle diffuses in a solution, a spatial distribution of solvent particles around
the solute can largely affect the diffusion. Solvent particles distribute around the solute even when
the solute does not move in the equilibrium state. A high density of solvent particles around the
solute causes a decrease in a value of the diffusion coefficient. The distribution is directly affected
by interaction between solute and solvent particles. Thus, by observing diffusion of the solute in
the solvent, one can study solute-solvent interaction if the relationship between the interaction and
diffusion is established.
Such effects of a solvent distribution have been observed by recent experiments on diffusion
of proteins [1–3]. In the experiments, diffusion of Cytochrome c in water has been observed using
the transient grating technique [1,2]. Diffusion of Cytochrome c was three times slower after the
formation of hydrogen bonds [1]. A distribution of solvent particles changed when the hydrogen
bonds formed. One can expect this change to have caused a decrease in the value of the diffusion
coefficient.
When a solute particle is large, a hydrodynamic effect is also important. Motions of a solute
cause motions of solvent particles by collisions. Solvent particles moved by the solute collide with
a solute, again. The collision affects diffusion of the solute. The effect turns out to be long ranged
because the momentum of solvent and solute particles is conserved.
The hydrodynamic effect is described by the Stokes law, which can be derived by applying
the Stokes approximation to the hydrodynamic equations. The Stokes law can be derived from
the Navier-Stokes equation including viscosity [4]. The approximation is given by the neglect of
nonlinear advection terms.
The diffusion coefficient calculated using the Stokes law depends only on the size of a solute
particle [4]. The expression does not include the other parameters of a solute. This shows that the
hydrodynamic effect depends little on the particularities of interaction between solvent and solute
particles.
The hydrodynamic effect has been shown to be dominant when a solute is ten times as large
as a solvent particle, by recent simulations [5]. By the simulations, the diffusion coefficients were
∗E-mail: yosi3scp@mbox.nc.kyushu-u.ac.jp
c© A.Yoshimori 563
A.Yoshimori
calculated in the systems of hard spheres with various size ratios between the solute and solvent.
When the ratio was larger than about 10, the Stokes law was valid. A dominant hydrodynamic
effect is shown by the validity of the Stokes law, because the other effects are not considered by
the Stokes law.
The simulation results contrast with the experimental results. The size ratio of 10 employed
by the simulation is similar to the ratio of Cytochrome c to water. The simulations with the ratio
showed the validity of the Stokes law, which is not affected by a distribution of solvent particles.
Thus, the experimental results contrast with the validity of the Stokes law because the hydrogen
boding is important in the experiments.
Comparison of the simulations with the experiments on Cytochrome c clarifies the importance of
attractive interaction between solute and solvent particles. Attractive interaction, such as hydrogen
boding, was not considered by the simulations. The systems employed by the simulations consisted
of hard spheres. In the experiments, however, attractive interaction plays an important role. Thus,
one can expect that attractive interaction brings in experimental results in contrast to the results
obtained by simulations.
To study the effects of attractive interaction theoretically, one needs a molecular theory giving
the correct limit of a large solute. Whenever one studies diffusion of a large solute theoretically, the
theories should have an exact asymptotic expression in the limit of a large solute. If the exact limit
is not given, the diffusion coefficient calculated by the theory deviates from the exact value when
a solute is large. The exact limit of a large solute is given by the diffusion coefficient proportional
to 1/R where R is the radius of the solute. Apart from the correct limit, in order to study the
effects of attractive interaction, one needs to explicitly deal with the interaction between solute
and solvent particles.
A molecular theory of this kind turns out to be the mode coupling theory [6,7]. The mode
coupling theory has been widely employed when calculating the dynamical properties of liquids
such as a diffusion coefficient [8–11]. The mode coupling theory gives the correct limit of a large
solute by the transverse coupling term [7]. Using the theory, one can consider a solvent distribution
through the radial distribution function and the direct correlation function between the solute and
solvent.
The mode coupling theory, however, cannot adequately deal with the effects of a solute on
motions of solvent particles. In the mode coupling theory, motions of solvent particles are included
by the dynamic structure factor and transverse current autocorrelation function in a bulk solvent
[7]. They do not have the effects of a solute. One cannot neglect the effects of a solute on motions
of solvent particles, when the interaction between solute and solvent particles is strong.
To deal with the effects of a solute on motions of solvent particles, Yamaguchi et al. have
developed a new molecular theory [12]. The theory was formulated using the generalized Langevin
equation [13] in an inhomogeneous system. They considered an inhomogeneous system, regarding
the interaction between solute and solvent particles as an external field.
The theory developed by Yamaguchi et al. [12] can be obtained using the time dependent
density functional method (TDDFM) [14–18]. One can derive the theory by Yamaguchi et al. [12]
from the TDDFM, adding the memory effects and momentum conservation. In the same way as the
theory by Yamaguchi et al. [12], the TDDFM adequately deals with the attractive interaction. The
equation of the TDDFM, however, does not include the terms of the memory effect. In addition,
when the TDDFM is calculated, momentum does not conserve.
Using the TDDFM, the diffusion coefficients have been calculated [19–21]. The calculations did
not agree with molecular dynamics simulations [19]. The disagreement was caused by the neglect
of the momentum conservation. The basic equation of the TDDFM does not have the terms of
memory effects as well. The diffusion coefficient, however, is not affected by the memory terms
because one can calculate the diffusion coefficient only in case of obtaining the density field in the
steady state.
Though the theory by Yamaguchi et al. [12] adequately deals with the solute effect on solvent
motions, the correct limit of a large solute has never been proved so far. In the limit of a large
solute, the diffusion coefficient should be proportional to 1/R. By their theory, Yamaguchi et al.
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A molecular theory of diffusion
have numerically studied the diffusion of a large solute on their own [22]. These numerical studies
showed the dependence of 1/R when the ratio between the solute and solvent radii was 8. The
infinite limit, however, has never been exactly studied.
On the other hand, when considering diffusion of a large solute, one has a problem of the
boundary condition. The boundary condition on the surface of the solute should be imposed to
calculate the diffusion coefficient using the hydrodynamic equations. One usually assumes the stick
or slip boundary condition. Whether the stick or slip boundary condition is assumed, one can obtain
the dependence of 1/R, though the proportional coefficient is different.
None has clearly understood the relationship between the boundary condition and molecular
theories. Though the boundary condition should be assumed in the hydrodynamic equations, they
are not needed by molecular theories. This is because the molecular theories explicitly deal with
interaction between the solute and solvent particles. On the other hand, molecular theories should
agree with the hydrodynamic equations in the limit of a large solute. Thus, the boundary conditions
can be derived from molecular theories by the limit of a large solute.
The theory developed by Yamaguchi et al. [12] turns out to be adequate to the study of the
boundary conditions because the hydrodynamic equations can be derived from their theory. Using
the molecular theory by Yamaguchi et al. [12], one can derive the hydrodynamic equations with the
boundary condition in the system of solute and solvent particles. Since the boundary condition is
expressed by molecular parameters, one can understand the boundary condition at the molecular
level.
The derivation of the hydrodynamic equations from the theory by Yamaguchi et al. [12] is easy
because the basic equations are the same types as the hydrodynamics. They are the continuum
equation for the density field and the equation of motion for the current density field. The equations
of motion agree with the momentum conservation.
When the hydrodynamic equation is derived, the molecular theory by Yamaguchi et al. [12]
contrasts with the mode coupling theory. Using the mode coupling theory, one can directly calcu-
late the diffusion coefficient without equations of motion. The hydrodynamic equations cannot be
derived from the mode coupling theory.
The purpose of the present study is to obtain the exact large-solute limit in the theory developed
by Yamaguchi et al. [12]. In addition, the boundary condition is derived at the molecular level.
The system consists of a spherical solute particle dissolved in spherical solvent particles. The limit
of a large solute in the theory is analytically obtained in the system. The large-solute limit gives
the boundary condition on the surface of the solute without any other assumptions.
2. A generalized Langevin equation theory in an inhomogeneous system
In the present study, a solute is fixed in a flowing solvent. The solvent particles have the velocity
u at an infinite distance from the solute. The present system is equivalent to a system of a solute
moving with the velocity of −u. Solute and solvent particles are simple liquids with spherical
interaction. One can obtain the diffusion coefficient, calculating the density field for the flowing
solvent particles in a steady state.
The steady-state density field of the flowing solvent particles is given using the theory developed
by Yamaguchi et al. [12] by
∇ · J(r) = 0, (1)
ρeq(r)∇
P (r)
ρeq(r)
+
∫
dr′M(r − r
′)
J(r′)
ρeq(r′)
= 0, (2)
where
P (r) = kBTδρ(r) − kBTρeq(r)
∫
dr′c(r − r
′)δρ(r′). (3)
Here, J(r) is the current density field at the steady state. It satisfies the boundary condition:
J(r) → ρu at r → ∞, where ρ is the density of the solvent and r is the distance from the solute.
565
A.Yoshimori
In equations (2) and (3), ρeq(r) is the inhomogeneous equilibrium density field (u = 0). It
depends explicitly on r because the solute affects the density field of the solvent. In equation (3),
δρ(r) = ρ(r) − ρeq(r) where ρ(r) is the density field at the nonequilibrium steady state.
In addition, M(r) is the tensor obtained by a random force, and c(r) is the direct correlation
function of solvent particles. One can obtain M(r), integrating the time correlation function of
a random force from t = 0 to ∞. The tensor M(r) and the direct correlation function c(r) are
approximately given in the homogeneous system without the solute. Then, M(r) is explicitly
expressed by [12]
∫
M(r)eikrdr = k ⊗ kK̃L(k) + (k2
U − k ⊗ k)K̃T (k), (4)
where k = |k| and U is the unit tensor. Here, K̃L(k) and K̃T (k) are the longitudinal and transverse
parts of the time integral for the correlation function of a random force [12]. They are related to
the dynamical structure factor and transverse current correlation function.
Equations (1) and (2) are the generalized Langevin equations [13] in an inhomogeneous system.
The present inhomogeneity is caused by the solute. The density of solvent particles depends on the
distance from the solute.
Equations (1) and (2) describe the inhomogeneous system by the equilibrium density field
ρeq(r). Because of ρeq(r), equation (2) does not have the translational invariant. This contrasts
with usual homogeneous generalized Langevin equations extensively employed in the study of the
dynamics of liquids [8]. The inclusion of ρeq(r) in equation (2) allows one to adequately deal with
strong attractive interaction using the theory by Yamaguchi et al. [12].
To derive equations (1) and (2), Yamaguchi et al. [12] had two approximations to the exact
framework of the generalized Langevin equations [13]. The first approximation shows that a den-
sity fluctuation in the inhomogeneous system is given at the level of the hyper-netted chain theory
[8]. From this approximation, one can calculate the direct correlation function c(r) by a homo-
geneous system without the solute. The second approximation is given by the random force in a
homogeneous system. Thus, M(r) is given by equation (4).
The theory developed by Yamaguchi et al. [12] is applied to the present solute-solvent system
with assumptions on the inside of the solute. The assumption is given by ρeq(r) = 0 for r 6 R,
where R is the radius of the solute. In addition,
f(r) ≡ 1
ρeq(r)
dρeq(r)
dr
→ ∞ (5)
is also assumed at r 6 R. These are assumed because interaction between the solvent and solute
particles diverges in all regions of r 6 R.
Besides the assumptions about the inside, equations (1) and (2) are assumed to be valid in the
whole space of the system. The validity on the inside of the solute, in particular, is assumed. In
addition, M(r − r
′) is assumed to be a smooth function. Then, the value of M(r − r
′) is not zero
even if a value of |r − r
′| is large.
From the assumptions, one can show that the velocity field v(r) defined by J(r)/ρeq(r) does
not diverge on the inside of the solute r < R. If v(r) diverges for r < R, the second term on the
left hand side of equation (2) diverges in all regions. This is because M(r − r
′) has a small value
even for a large value of |r − r
′|. Then, equation (2) shows that the first term also diverges in
all regions. Thus, v(r) does not diverge for r < R so that the first term in equation (2) does not
diverge for r > R.
By the definition of the velocity field, one finds that the components except the radial direction
can have a non-zero value on the inside of the solute. On the inside of the solute, the current
density field J(r) and density field ρeq(r) vanish. Since the velocity field v(r), however, is defined
by the ratio of J(r) to ρeq(r), it can have a value that is not zero.
On the other hand, the velocity field of the radial direction should vanish on the inside of the
solute so that the velocity field does not diverge on the inside. If the velocity field of the radial
direction is denoted by vr(r), using f(r) in equation (5), one can rewrite equation (1) by
f(r)vr(r) = −∇ · v(r). (6)
566
A molecular theory of diffusion
Since v(r) does not diverge on the inside of the solute, equation (6) leads to
vr(r) = 0 (7)
for r < R.
In addition, the pressure P (r) in equation (2) should vanish on the inside of the solute. Equa-
tion (2) is written in the form
−r̂f(r)P (r) + ∇P (r) +
∫
dr′M(r − r
′)v(r′) = 0, (8)
where r̂ is the unit vector of the radial direction. The first term diverges with f(r), though the
third term
∫
dr′M(r − r
′)v(r′) does not diverge. Thus, to satisfy equation (8), one has
−f(r)P (r) +
∂P (r)
∂r
= 0. (9)
Solving the equation, one can obtain P (r) ∝ ρeq(r) and P (r) = 0 for r < R.
3. Limits of a large solute in the molecular theory
The present section shows the limit of a large solute in the molecular theory introduced in the
previous section. In the previous section, equations (1) and (2) were presented to study the steady
state using the theory developed by Yamaguchi et al. [12]. In the present section, the limit of a
large solute in equations (1) and (2) is given by R→ ∞ with r̄ ≡ r/R fixed when v(r) = v̄(r̄) and
P (r) = P̄ (r̄)/R are assumed.
The term of integration in equation (2) is first calculated in the limit of a large solute. The
term in equation (2) is written in the form
∫
dr′M(r − r
′)
J(r′)
ρeq(r′)
=
∫
dr′M(r′)v̄
(
r̄ − r
′
R
)
. (10)
In the limit of R→ ∞, from equation (4), one has
lim
R→∞
R2
∫
dr′M(r′)v̄
(
r̄ − r
′
R
)
= −η∇̄2
v̄(r̄) − γ∇̄
{
∇̄ · v̄(r̄)
}
, (11)
where ∇̄ denotes the gradient of r̄ and
η = K̃T (0), (12)
γ = K̃L(0) − K̃T (0) = ζ +
η
3
. (13)
Here, η and ζ are the shear and bulk viscosity. The substitution of equation (11) into equation (2)
yields
ρeq(r)∇̄
P̄ (r̄)
ρeq(r)
− η∇̄2
v̄(r̄) − γ∇̄{∇̄ · v̄(r̄)} = 0. (14)
Using the limit in the integration term, one can obtain the continuity of the velocity field on
the surface of the solute. Equation (14) is the second differential equation of v(r). Thus, if the
equations are valid in all regions, v(r) and its first derivative should continue even on the surface
of the solute. The first derivative of the radial component vr(r), however, can be discontinuous
because P (r) in equation (14) can be also discontinuous.
One can explicitly express the continuity of the velocity field by equations. To express the
equations, the polar coordinate is introduced where the origin is the center of the solute with the
567
A.Yoshimori
z-axis parallel to the direction of u. Using the coordinate, one has
lim
r→+R
vr(r) = lim
r→−R
vr(r) = 0, (15)
lim
r→+R
vθ(r) = lim
r→−R
vθ(r), (16)
lim
r→+R
∂vθ(r)
∂r
= lim
r→−R
∂vθ(r)
∂r
, (17)
lim
r→+R
vψ(r) = lim
r→−R
vψ(r), (18)
lim
r→+R
∂vψ(r)
∂r
= lim
r→−R
∂vψ(r)
∂r
, (19)
where vr(r), vθ(r) and vψ(r) are the r, θ, and ψ-components of the velocity field. The limits of
r → +R and r → −R denote the approach from the outside and inside of the solute to the surface.
Equation (15) was obtained from equation (7).
The next task for obtaining the limit of a large solute in equations (1) and (2) is to calculate
the limit in the equilibrium density ρeq(r). It is written in the form ρeq(r) = ρ̄({r − R}/σ) =
ρ̄({r̄ − 1}R/σ) where σ is the radius of the solvent particle. Since ρeq(r) = ρ0 for r → ∞ and
ρeq(r) = 0 for r < R, ρeq(r) is the step function given by
lim
R→∞
ρeq(r) = ρ0H(r̄ − 1) ≡
{
ρ0 r̄ > 1,
0 r̄ < 1,
(20)
where ρ0 is the homogeneous density and r̄ = |r̄|.
In the large-solute limit, equations (1) and (2) agree with the hydrodynamic equations derived
using the Stokes approximation on the outside of the solute. For r > R or r̄ > 1, from equation
(20), one has ρeq(r) = ρ0. Thus, equations (1) and (14) are rewritten by
∇ · v(r) = 0, (21)
∇P (r) − η∇2
v(r) = 0. (22)
Here, r̄ = r/R, v̄(r̄) = v(r), and P̄ (r̄) = P (r)R were employed. These are exactly the same
equations as the equations by the Stokes approximation.
On the other hand, one can also obtain the equations on the inside of the solute. Since P (r) = 0
for r < R, one can obtain
r̂Q(r) − η∇2
v(r) − γ∇{∇ · v(r)} = 0. (23)
Here, r̂ is the unit vector of the radius direction, and Q(r) ≡ f(r)P (r) can have a finite value
because f(r) → ∞ for r < R though P (r) = 0. Besides equation (23), one has equation (7) on the
inside of the solute. By these equations, one can obtain the velocity field v(r) on the inside of the
solute. The inside of the solute is connected with the outside using equations (15)–(19).
The equations on the inside of the solute can be solved by consideration of the symmetry. From
the symmetry, one finds that only u and r are vectors given by the present system. Since v(r) is a
vector, v(r) should be expressed by a linear combination of u and r. In addition, v(r) should be
a linear function of u. To satisfy these conditions, one has
v(r) = g(r)(r · u)r + h(r)u, (24)
where g(r) and h(r) are functions of only r.
One can explicitly obtain an analytical solution of the equations on the inside of the solute.
Since equation (7) gives (r · v(r)) = 0, one has h(r) = −r2g(r). To calculate g(r), the tangental
component of equation (23) is calculated by
−η(eθ · ∇2
v(r)) − γ(eθ · ∇{∇ · v(r)}) = 0, (25)
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A molecular theory of diffusion
where eθ is the unit vector of the θ-direction. The substitution of equation (24) into equation (25)
yields g(r) = crα−2, where c is a constant independent of r,
α =
−1 +
√
9 + 8γ̄
2
, (26)
and γ̄ = γ/η.
The solution on the inside of the solute leads to the boundary condition for the hydrodynamic
equations with the Stokes approximation. From equation (24) with g(r) = crα−2 and h(r) = −crα,
one has
∂vθ
∂r
=
α
r
vθ, (27)
for r < R. From equations (16) and (17), one can obtain
∂vθ
∂r
=
α
R
vθ, (28)
on r = R.
The obtained boundary condition can be different from the stick or slip condition, though it can
agree with one of them. The stick boundary condition is obtained by equation (28) when α→ ∞.
In addition, the slip condition is obtained when α = 1. The present boundary condition, however,
does not agree with that assumed in the Stokes law when 1 < α <∞.
The present boundary condition is independent of solute properties. The boundary condition
in equation (28) depends only on the value of α, which is given only by η and γ, as shown by
equation (26). The boundary condition, however, should depend on the state of the solute surface
from the hydrodynamic viewpoint. This point will be discussed in section 4.
4. Discussion
The present study has shown the limit of a large solute in the molecular theory developed by
Yamaguchi et al. [12], on the inside and outside of the solute. On the outside of the solute, in the
large-solute limit, the theory has agreed with the hydrodynamic equation derived by the Stokes
approximation. One finds that their theory has a correct limit of a large solute in the diffusion
coefficient, by the agreement. Thus, their theory can be employed to study diffusion of a large
particle. On the other hand, on the inside of the solute, one can obtain an analytical solution,
which leads to the boundary condition on the surface of the solute.
On the outside of the solute, the theory developed by Yamaguchi et al. [12] has agreed with the
hydrodynamic equation derived by the Stokes approximation. Thus, the calculation by the theory
gives the diffusion coefficient proportional to 1/R in the limit of a large solute. The asymptotic
form is independent of the boundary condition on the surface of the solute. In spite of the stick,
slip, and other boundary conditions, the same asymptotic form is obtained.
The agreement with the Stokes approximation allows one to employ the theory by Yamaguchi
et al. [12] to study diffusion of a large solute. To study diffusion of a large solute, one needs a
theory that is exact as R→ ∞. The theory by Yamaguchi et al. [12] is satisfied with the condition.
On the other hand, on the inside of the solute, an analytical solution has been obtained using
equations (7) and (9) showing that vr(r) and P (r) vanish on the inside. The limit of a large solute
does not affect the validity of the equations (7) and (9). Thus, the solution can be obtained even
when equation (20) is not valid. The limit of M(r) in equation (2), however, is required.
The analytical solution on the inside has led to some kinds of boundary conditions. They include
not only the stick and slip boundary conditions but also other ones. The boundary conditions
depend on the value of α in equation (26). The value is given by the property of the solvent. The
boundary condition is not stick or slip if the value of α is not the unity or infinity.
When the diffusion coefficient is proportional to 1/R in the limit of a large solute, these bound-
ary conditions affect the proportional coefficient. Denoting the proportional coefficient by a, one
569
A.Yoshimori
can express the asymptotic form of the diffusion coefficient by akBT/(πηR). The slip and stick
boundary conditions have a = 1/4 and 1/6.
The boundary condition different from the slip or stick condition has also been obtained in
the previous studies. The proportional coefficient a = 1/5 was obtained using the mode coupling
theory of hard spheres [6]. That is, the boundary condition was different from the slip or stick
condition because the value of a was not a = 1/4 or 1/6. In addition, Yamaguchi et al. have
numerically calculated the diffusion coefficient using the theory of their own [22]. The calculated
diffusion coefficient deviated from that by the slip and stick boundary conditions.
The present boundary condition is independent of the solute property because the limiting
theory of a large solute does not include any parameters of the solute except R. The property of
the solute is presented by ρeq(r) in the theory developed by Yamaguchi et al. [12]. When a solute
is large, ρeq(r) is approximately the step function given by equation (20). It does not include the
properties of the solute except R, which represents the position of the boundary. This is because
the solute size does not affect the range of interaction between solute and solvent particles. Thus,
the ratio of the range to R approaches zero as R→ ∞.
The present result of the boundary condition contrasts with the hydrodynamic viewpoint. From
the hydrodynamic viewpoint, the boundary condition depends on the state of the solute surface.
The state of the solute surface is often expressed by the strength of friction on the surface of
the solute.
From the hydrodynamic viewpoint, the stick or slip boundary condition is determined by friction
on the surface of the solute. When friction is strong, the boundary condition approaches the stick
condition. The slip condition is caused by weak friction. Friction of the solute surface depends on
the interaction between solute and solvent particles at the molecular level.
One can study the effects of the solute on the boundary condition by the perturbative expansion
from the limit of a large solute. If R→ ∞, the boundary conditions are independent of the solute
property. Thus, the boundary conditions can depend on the properties of the solute when R has a
finite value. One can study the effects of the finite value of R if the basic equations are expanded
in terms of 1/R. The boundary condition depends on the properties of the solute from the terms
of the first order.
In addition, one can also study the diffusion of proteins by the perturbative expansion in 1/R.
Since the present theory is formulated at R → ∞, one cannot consider the effects of the solvent
distribution. The perturbative expansion allows one to consider the effects. One cannot calculate
the diffusion coefficient of a protein unless the effects are considered, because the experiments
showed the importance of the effects.
Numerical studies of the theory obtained by the perturbative expansion are a future work. The
numerical studies are interesting because the calculated results can be compared with experiments
on diffusion of proteins. In addition, the numerical calculations are easier than the original theory
developed by Yamaguchi et al. [12]. The perturbative expansion theory requires studies on the
limit of R → ∞. Thus, the present study is a first step in the development of the perturbative
expansion theory.
Acknowledgements
I wish to thank Professor T. Yamaguchi for helpful comments. This work was supported by
the Grant-in-Aid for Scientific Research on Priority Area and for Scientific Research (c) from the
Japanese Ministry of Education, Science, Sports, and Culture.
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Молекулярна теорiя дифузiї великих частинок в розчинi
А.Йошiморi
Фiзичний факультет, Кюшу унiверситет, Фукуока, Японiя 812–8581
Отримано 4 вересня 2007 р., в остаточному виглядi – 11 жовтня 2007 р.
Вивчається границя великої розчинюваної частинки в рамках молекулярної теорiї, розвинутої Яма-
гучi. В цiй границi наближення Стокса до гiдродинамiчних рiвнянь отримано за межами дифузiї
розчинюваної частинки. Границя великої частинки також приводить до рiвнянь всерединi областi
розчинюваної частинки. Аналiтичний розв’язок дозволяє отримати граничнi умови на поверхнi ча-
стинки, необхiднi при розрахунку гiдродинамiчних рiвнянь. Граничнi умови в спецiальному випадку
включають граничнi умови змiщення, що використовуються в законi Стокса. Крiм цих, можуть бути
отриманi також iншi умови. Граничнi умови залежать вiд властивостей розчинника.
Ключовi слова: мiкроскопiчна теорiя, взаємодiя розчинник-розчинювана речовина,
гiдродинамiчна границя, ексклюзивний ефект, момент збереження, узагальнене рiвняння
Ланжевена
PACS: 66.10.-x, 05.20.Jj, 05.60.Cd, 87.15.Vv, 47.10.-g
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