Effective diffusion coefficient including the Marangoni effect Hiroyuki Kitahata∗1 and Natsuhiko Yoshinaga2 1
Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan 2 WPI-AIMR, Tohoku University, Miyagi 980-8577, Japan
arXiv:1604.01108v1 [nlin.PS] 5 Apr 2016
Surface-active molecules supplied from a particle fixed at the water surface create a spatial gradient of the molecule concentration, resulting in Marangoni convection. Convective flow transports the molecules far from the particle, enhancing diffusion. We analytically derive the effective diffusion coefficient associated with the Marangoni convection rolls. The resulting estimated effective diffusion coefficient is consistent with our numerical results and the apparent diffusion coefficient measured in experiments. PACS numbers: 05.45.-a, 47.55.dk, 82.40.Ck
I.
Self-propelled active materials have attracted increasing attention as a method for understanding biological systems from the viewpoint of physics [1–3]. The studies in this field include not only real biological systems in vivo and in vitro but also synthesized physico-chemical systems. Physico-chemical systems are advantageous because the parameters can be controlled and specifically designed in order to clarify the desired mechanism in a quantitative manner. The camphor-water system is one of the most well-studied physico-chemical systems for self-propulsion [4–12]. Camphor is a volatile organic material, which reduces the surface tension of water. When a camphor particle is placed onto the water surface, camphor molecules spread at the water surface, inducing a decrease in the surface tension around the camphor particle. The surface tension gradient at the water surface can drive the motion of the camphor particle itself. A mathematical model for camphor motion, which is composed of a partial differential equation for the surface concentration of camphor particles, has been proposed in Ref. [13]. The model is simple, mathematically tractable, and reproduces the self-propulsion of camphor particles. Recently, considerable experimental efforts have been made to determine the parameters in the model [13], such as the sublimation rate, friction constant, and supply rate of camphor molecules from a camphor particle [14]. The apparent diffusion coefficient for the camphor molecules at the surface was also estimated to be approximately 10−3 m2 · s−1 [14]. Considering the fluctuation-dissipation theorem, however, the diffusion coefficient of the molecules should be on the order of 10−9 m2 · s−1 , which is six orders of magnitude smaller than the observed value. To understand this discrepancy, we suggest that the Marangoni effect, which drives flow at the surface under the existence of the surface tension gradient [15, 16], plays an important role. In other words, the apparent diffusion coefficient observed in experiments
∗ Corresponding
y
INTRODUCTION
author. E-mail:
[email protected].
camphor molecules c(x)
camphor particle x
velocity field v(x,y) water
FIG. 1: Schematic illustration of the considered system. The x-axis corresponds to the water surface, and the y-axis is set in the vertical direction.
corresponds to the effective diffusion coefficient enhanced by the Marangoni effect. In fact, Marangoni convection was experimentally observed in the camphor-water system [17, 18]. In the present article, we theoretically investigate the dynamics of the concentration profiles of camphor molecules and the flow profile of the aqueous phase. We show that the effect of the flow in the aqueous phase may be described as the effective diffusion coefficient, which depends on the wave number [19]. The enhancement of the diffusion coefficient is understood as the effect of convective flow associated with the scale of the Marangoni convection. Numerical calculations based on the NavierStokes equation are compared with the analytical results.
II.
MODEL
We consider a two-dimensional camphor-water system, where the x-axis corresponds to the water surface, and the y-axis is in the vertical direction (Fig. 1). We set c(x) as the surface concentration of camphor molecules and v(x, y) = vx (x, y)ex +vy (x, y)ey as the flow profile in the aqueous phase, where ex and ey are the unit vectors
2
(1)
where D is the diffusion coefficient, a is the sublimation rate, and f0 is the supply rate of camphor molecules from the particle. The flow field v obeys the Navier-Stokes equation ∂ ρ + v · ∇ v = −∇p + η∇2 v, (2) ∂t where p is the pressure, and ρ and η are the density and viscosity of the fluid, respectively. In addition, we assume incompressibility: ∇ · v = 0.
(3)
The surface tension is a decreasing function of the camphor surface concentration. For simplicity, we assume a linear relation between the surface tension γ and the surface concentration of camphor c as γ = γ0 − Γc,
which is derived from the stress balance at the surface [20–22]. In this theoretical analysis, we apply the Stokes approximation; that is, we neglect the inertia term ρv · ∇v in the Navier-Stokes equation [23]. The justification of this approximation will be discussed later. Here, we consider the steady state; ∂c/∂t = 0Cand ∂v/∂t = 0. The flow profile can be written as a functional of the surface tension profile γ(x). When γ(x) can be expanded in Fourier space, Z ∞ (γc (k) cos kx + γs (k) sin kx) dk, (6) γ(x) = γc0 + 0
the flow velocity at the surface V (x) is described as Z ∞ 1 V (x) = (−γc (k) sin kx + γs (k) cos kx) dk. (7) 2η 0 The derivation of Eq. (7) is shown in Appendix A. Without the effect of the Marangoni flow, the concentration field is obtained as the solution of (8)
Γ0 = 3 Γ0 = 4 Γ0 = 5 Γ0 = 6
2
1
0
0
1
2
3 k/ a/D
4
6
5
(b) 105 Γ0 = 0 Γ0 = 1 Γ0 = 10
104 103
Γ0 = 102 Γ0 = 103 Γ0 = 104 Γ0 = 105
102 10 1
10-1
(4)
where Γ (> 0) is a proportionality constant, and γ0 is the surface tension of pure water. The surface tension gradient induces flow through the boundary condition as ∂γ ∂vx , (5) = η ∂y y=0 ∂x
∂2c D 2 − ac + f0 δ(x) = 0. ∂x
Γ0 = 0 Γ0 = 1 Γ0 = 2
Deff(k) / D
∂c ∂ ∂2c + (V c) = D 2 − ac + f0 δ(x), ∂t ∂x ∂x
(a) 3
Deff(k) / D
in the x and y directions, respectively. The x-component of the flow velocity at the surface is set as V (x), that is, V (x) = vx (x, 0). The time evolution equation for c is written as
0
1
2
3 k/ a/D
4
6
5
FIG. 2: Plot of Deff (k)/D against k for various Γ0 . (a) Linear plot for small Γ0 . (b) Semi-logarithmic plot for large Γ0 .
That is, r f0 a c(x) = √ exp − |x| . D 2 aD
(9)
It is noted that this steady-state concentration profile can be described as a Fourier series: Z f0 ∞ 1 c(x) = cos kxdk. (10) πa 0 1 + Dk 2 /a When the camphor particle stops at the origin, both the concentration and flow profile should be symmetric, reflecting the symmetry of the system. Therefore, we assume that the concentration profile c(x) is expanded in the Fourier space as Z 1 f0 ∞ cos kxdk. (11) c(x) = πa 0 1 + Deff (k)k 2 /a III.
EFFECTIVE DIFFUSION COEFFICIENT
In the presence of advection, Eq. (1) cannot be solved analytically. Nevertheless, we may use an ansatz where the concentration profile c(x) is expanded as in Eq. (11),
3
Γf0 , πηDa
0 0
(12)
the equation for determining the effective diffusion coefficient becomes ! r Deff (k) Γ0 Deff (k)k 2 = G − 1, (13) 2 a D
y
Γ0 =
0.5 c
using the effective diffusion coefficient Deff (k) that depends on wave number k [19]. We have used the fact that both the concentration and flow profiles are symmetric when the camphor particle stops at the origin. By substituting Eq. (11) into Eq. (1) with ∂c/∂t = 0, we may obtain Deff (k) self-consistently. By defining the following nondimensionalized parameter
-500
where (1 + ξ 2 ) ξ arctan ξ + ln(1 + ξ 2 ) G(ξ) = . ξ 2 (4 + ξ 2 )
(14)
The plot of the nondimensionalized effective diffusion coefficient Deff (k)/D aspa function of the nondimensionalized wave number k/ a/D is shown in Fig. 2. For small wave numbers, limξ→0 G(ξ) = 1/2, and the effective diffusion coefficient becomes Γ0 . (15) Deff (0) = D 1 + 4 Another property of the effective diffusion coefficient is a shift of the peak as shown in Fig. 2. In fact, the wave number kmax at which Deff has a maximum value goes to +0 when Γ0 goes to infinity. This behavior occurs be2 cause kmax Deff (kmax )/a = ξmax = const., where ξmax is the positive root of dG/dξ = 0. Here, ξmax is numerically estimated as ≃ 1.50856. The effective diffusion coefficient Deff (k) at k = kmax can be calculated as Deff (kmax ) = D(1 + αΓ0 ),
(16)
where the constant α = G(ξmax )/2 is evaluated as α ≃ 0.306554. For the small wave length limit, i.e., k → ∞, Deff /D − 1 → 0 because G(ξ) ∼ 1/ξ as ξ → +∞. Expanding Deff around this limit as Deff = D(1 + f (k)), we obtain r Γ0 a 1 p = f (k). (17) 2 D k 1 + f (k)
Because f (k) is infinitesimally small when k is sufficiently large, we obtain r Γ0 a 1 Deff (k) ∼ D + . (18) 2 Dk This means that the gradient of the concentration field around the camphor particle is not significantly affected by the convective flow. We now consider the diffusion-like phenomenon at the large spatial scale. For this purpose, Deff (k) for large k is not suitable. Instead, it is natural to take the effective diffusion coefficient as Deff ∼ D(1 + αΓ0 ).
-1000 -500
0 x
500
FIG. 3: Stationary concentration and flow profiles for Γ0 = 100, obtained by numerical calculation.
IV.
NUMERICAL CALCULATION
To confirm the validity of the present approach, we performed numerical calculations. The reaction-diffusion equation Eq. (1) is calculated using the Euler method with Neumann boundary conditions at x = ±Lx . The hydrodynamics, given in Eqs. (2) and (3) are solved by the stream function-vorticity method supplemented with the boundary conditions in Eq. (5) at y = 0 and non-slip boundary conditions for v at x = ±Lx and y = −Ly . In the theoretical analysis, we consider an infinite half plane, but we adopt a sufficiently large plane, −Lx ≤ x ≤ Lx and −Ly ≤ y ≤ 0, in the numerical calculation. We set Lx = 500 and Ly = 1000 so that the system size does not affect the results. We set the time step as ∆t = 10−4 , and the space grid as ∆x = 1. The numerical calculations were performed until the concentration and flow profiles approximately reach stationary solutions. The representative steady-state concentration and flow profiles are shown in Fig. 3. We calculated results for various nondimensionalized parameters of Γ0 by changing the parameter Γ. The other parameters were set as D = 1, a = 0.1, f0 = 1, ρ = 1, and η = 1000. The delta function for the supply of the camphor molecules in Eq. (1) is approximated by the supply at three discretized points. The steady-state concentration fields of c for various Γ0 are shown in Fig. 4(a) on a linear scale and in Fig. 4(b) on a logarithmic scale.
4
(a) 1.5
(a) 3 Γ0 = 0 Γ0 = 1 Γ0 = 2
c
1
Deff(k) / D
Γ0 = 0 Γ0 = 10 Γ0 = 102 Γ0 = 103
0.5
0 -500 (b) 10
0
1
2
3 k/ a/D
(b) 105 Γ0 = 0 Γ0 = 10 Γ0 = 102 Γ0 = 103
c
10-1 -2
10
10-3
4
Γ0 = 0 Γ0 = 1 Γ0 = 10
104 Deff(k) / D
1
10-4 -500
1
0
500
0 x
2
Γ0 = 3 Γ0 = 4 Γ0 = 5 Γ0 = 6
103
5
6
Γ0 = 102 Γ0 = 103 Γ0 = 104 Γ0 = 105
102 10 1
0 x
500
FIG. 4: Stationary concentration profile of c for each Γ0 : (a) linear plot, and (b) semi-logarithmic plot.
The normalized effective diffusion coefficient p Deff /D as a function of the normalized wave number k/ a/D is obtained from the results of the numerical calculation using the Fourier cosine transformation. The results are shown in Fig. 5, where the numerical results are shown by solid curves and the analytical results by dashed curves. There were some discrepancies between the numerical and analytical results. The reasons for these discrepancies are considered to be a result of the following aspects: (i) discretization in the numerical calculation, (ii) system size of the numerical calculation, and (iii) higher-order wave number dependence on the effective diffusion coefficient. The effect of discretization should appear strongly at large wave numbers, and the discrepancies at higher wave numbers seem to be due to the effect of discretization. As for the second aspect, the numerical simulations are performed in a finite domain. Although we use the largest possible system size, the convective flow reaches the boundary for very large Γ0 . Discrepancies at small wave numbers may be due to the effect of the boundary. As for the third aspect, our analysis is valid around dDeff /dk ≪ 1. As Γ0 is increased, such dependence becomes stronger and causes a worse estimate away from the peak value. When Γ0 is equal to zero, Marangoni convection does
10-1
0
1
2
3 k/ a/D
4
5
6
p FIG. 5: Plots of Deff (k)/D as a function of k/ a/D for each Γ0 . Stationary concentration profile of c for each Γ0 was obtained, from which Deff /D was calculated: (a) linear plot for small Γ0 and (b) semi-logarithmic plot for large Γ0 .
not occur. Therefore, the stationary concentrationp profile is proportional to exp (− |x − x0 | /λ), where λ = D/a, as shown in Fig. 4(b). As Γ0 increases, the concentration profile deviates from the exponential form. This difference may lead to the difference between the results between analytical and numerical calculations as shown in Fig. 5. Moreover, this difference is consistent with the discussion that the effects of the higher-order terms should increase with an increase in the nondimensionalized parameter Γ0 .
V. COMPARISON WITH THE EXPERIMENTAL RESULTS
Here, we discuss the validity of our estimation based on the experimental results [14]. The rate of sublimation to air and dissolution to the aqueous phase a, the supply rate of camphor molecules from the camphor grain f0 , and the proportionality constant between the surface tension and the camphor concentration, Γ, are estimated to be a ∼ 10−2 s−1 , f0 ∼ 10−11 mol · s−1 , and Γ ∼ 10−3 Pa · s, respectively. The viscosity of water η,
5 and the diffusion coefficient of the camphor molecule D, are known as η ∼ 10−3 Pa · s and D ∼ 10−9 m2 · s−1 , respectively, at room temperature. From these values, the nondimensionalized number Γ0 is estimated as 103 [Pa · s] · 10−11 [mol · s−1 ] ∼ 106 . 3.14 · 10−3 [Pa · s] · 10−9 [m2 · s−1 ] · 10−2 [s] (19) With this value, the effective diffusion coefficient Deff is calculated as Γ0 ≃
Deff ∼ D(1 + αΓ0 ) ∼ 10
−3
2
[m · s
−1
].
(20)
In the previous paper, the apparent diffusion coefficient was estimated as 4×10−3 m2 ·s−1 [14], which is consistent with our theoretical estimation. VI.
In this section, we consider the relationship between Γ0 and other nondimensional numbers [24]. The Reynolds number Re = ρRU/η is πρΓf0 , η2 a
from the characteristic length R, r R ∼ x0 =
D . a
(21)
(22)
The characteristic velocity U is U∼
Γf0 Γf0 √ ∼ . ηax0 πη Da
(23)
The characteristic velocity is obtained from the mechanical balance at the surface as in Eq. (5): η
U γ Γc ∼ ∼ . R R R
(24)
The Reynolds number can also be described using the nondimensional constant Γ0 defined in Eq. (12) as Re ∼
ρΓf0 πΓ0 = , η2 a Sc
(25)
where Sc is Schmidt number, i.e., Sc = η/(Dρ). The nondimensional number Γ0 is nothing but the Peclet number, because Pe = ReSc, Γ0 =
Pe . π
VII.
COMPARISON WITH PERTURBATIVE EXPANSION
To discuss the meaning of our derivation of Deff , we compare it with the perturbation method. The diffusion coefficient under the perturbation method D′ is D′ Γ0 =1+ G(x0 k) + O(ǫ2 ). D 2
NONDIMENSIONAL NUMBERS
Re ∼
calculation was also set to be 1000 by setting ρ = D = 1 and η = 1000. A small Schmidt number means the nonlinear term, ρ(v · ∇)v, in the Navier-Stokes equation in Eq. (2), is insignificant compared with the nonlinear term, v · ∇c, in the evolution equation for the concentration in Eq. (1). Therefore, the Stokes approximation for the hydrodynamics is justified.
(26)
The Schmidt number of water is calculated to be
(28)
The detailed derivation is shown in Appendix C. This result looks similar to Eq. (16) in that both have additional terms proportional to Γ0 whose coefficient is proportional to the wave number k. However, in Eq. (16), the wave number kmax for the maximum Deff depends on Γ0 while D′ (k) in Eq. (28) has a maximum value at a constant k = ξmax /x0 even when Γ0 changes. Comparing Eqs. (13) and (28), our method allows x0 in Eq. (28) to be dependent on the wave number k. The physical meaning of this is the rescaling of the wave number. To obtain an effective diffusion coefficient for enhanced diffusion due to Marangoni convection, it is natural to take the value with ∂Deff /∂k = 0, which is equivalent to the peak value of Deff . Therefore, we used the effective diffusion coefficient given in Eq. (16). To confirm the validity of the present method, we plotted the numerical results with rescaled p axes. The wave number k should be rescaled as k Deff /a. Considering that Deff is approximately √ described by Deff = D(1 + αΓ0 ), we rescale k as k 1 + αΓ0 . In Fig. 6, (Deff /D − 1)/Γ0 obtained p from the numerical calculation is plotted against k D/a in (a) and against p k (1 + αΓ0 )D/a in (b). The plots in (a) do not collapse, while those in (b) tend to collapse to a universal curve, even though there are some discrepancies at larger wave numbers. In Fig. 6(c), (Deff /D − 1)/Γ0 obtained from the analytical calculation is plotted against p k (1 + αΓ0 )D/a. The curves approximately collapse to a universal curve. These results show that our method is applicable for diffusion phenomena at the scale of the size of the convective roll.
VIII.
SUMMARY
−3
Sc ≃
10 [Pa · s] ∼ 103 , 10−9 [m2 /s] · 103 [kg/m3 ]
(27)
where we use D ∼ 10−9 m2 /s, ρ ∼ 103 kg/m3 , and η = 10−3 Pa · s. The Schmidt number in the numerical
When a surface-active chemical compound is supplied from a particle placed at the water surface, Marangoni convection is induced, which accelerates the transport of the surface active chemical compounds. This process
6
(Deff / D − 1) / Γ0
(a) 0.4 Γ0 = 1 Γ0 = 2 Γ0 = 5
0.3 0.2 0.1 0
0
1
(b) 0.4
(Deff / D − 1) / Γ0
In the present calculation, we consider only a stationary camphor particle at a certain position. Therefore, the effective diffusion coefficient obtained in this paper may be different from that for a spontaneously moving camphor particle. Nevertheless, we expect that the effective diffusion coefficient is valid when the system is near the drift bifurcation point and the particle is moving at a low velocity. The situation in which the particle is moving at a finite constant velocity is left for future work.
Γ0 = 10 Γ0 = 102 Γ0 = 103 Γ0 = 104
2
3 k D/a
4
Γ0 = 1 Γ0 = 2 Γ0 = 5
0.3
5
6
Acknowledgments
6
The authors thank Nobuhiko J. Suematsu (Meiji University, Japan), Alexander S. Mikhailov (Fritz-Haber Institute), and Yutaka Sumino (Tokyo University of Science, Japan) for their helpful discussion. This work was supported in part by Grant-in-aid for Scientific Research (C) to H. K. (No. 15K05199), Grant-in-aid for Young Scientists (B) to N. Y. (No. 26800219), Grantin-aid for Scientific Research on Innovative Areas “Fluctuation & Structure” to H. K. (No. 25103008) and to N. Y. (No. 26103503), and the Core-to-Core Program “Nonequilibrium dynamics of soft matter and information” to H. K. from MEXT, Japan.
Γ0 = 10 Γ0 = 102 Γ0 = 103 Γ0 = 104
0.2 0.1 0
0
1
2 3 4 k (1 + α Γ0 ) D / a
5
(Deff / D − 1) / Γ0
(c) 0.4 Γ0 = 1 Γ0 = 2 Γ0 = 5
0.3
Γ0 = 10 Γ0 = 102 Γ0 = 103 Γ0 = 104
Appendix A: Derivaton of Eq. (7)
We show the derivation of the steady state solution of Eqs. (2) and (3) with the boundary condition in Eq. (5) and v = 0 at y = −H. The surface tension γ is related to the concentration c by Eq. (4). By defining the stream function ψ as
0.2 0.1 0
∂ψ , ∂y ∂ψ vy = − , ∂x
vx =
0
1
2 3 4 k (1 + α Γ0 ) D / a
5
6
FIG. p 6: (a) Numerical results of (Deff /D − 1)/Γ0 against k D/a for various Γ0 . The peak positions strongly depend on p Γ0 . (b) Numerical results of (Deff /D − 1)/Γ0 against k (1 + αΓ0 )D/a for various Γ0 . The peak positions are almost independent of Γ0 except for the data with Γ0 = 104 . The curves go up at high wave numbers for small Γ0 due to the effect of the spatial mesh. The number of data points for large Γ0 is small due to the rescaling p effect. (c) Analytical results of (Deff /D − 1)/Γ0 against k (1 + αΓ0 )D/a for various Γ0 . The curves almost collapse to a universal curve.
can be described using the effective diffusion coefficient, which is derived analytically in this paper. The estimated effective diffusion coefficient is compatible with previously reported experimental results [14].
(A1) (A2)
the incompressibility given in Eq. (3) is always satisfied. We can obtain the equation for ψ as ∇2 ∇2 ψ = 0.
(A3)
We assume that the surface tension profile can be expanded in Fourier space as γ(x) = γc0 +
Z
∞
(γc (k) cos kx + γs (k) sin kx) dk. (A4)
0
Because the equation is linear, the general solution satisfying periodicity in the x-direction is obtained as
7
ψ = Ac (k)eky + Bc (k)e−ky + Cc (k)yeky + Dc (k)ye−ky cos kx + As (k)eky + Bs (k)e−ky + Cs (k)yeky + Ds (k)ye−ky sin kx,
(A5)
where Ac (k)CBc (k)CCc (k)CDc (k)CAs (k)CBs (k)CCs (k)Candconditions, we obtain Ds (k) are integration constants. Using the boundary
ψ=
2kH 2 sinh ky + 2kHy sinh ky + y cosh ky − y cosh(2kH + ky) (−γc (k) sin kx + γs (k) cos kx) , 2η(2kH − sinh 2kH) p = p0 +
2k 2 H sinh ky − k cosh(2kH + ky) + k cosh ky (γc (k) cos kx + γs (k) sin kx) , 2kH − sinh 2kH
where p0 is a constant. In this work, we consider the aqueous phase with an infinite depth, and take the limit of H → ∞. Then, the stream function ψ and the pressure p converge to 1 ky ye (−γc (k) sin kx + γs (k) cos kx) , 2η
(A8)
p = p0 + keky (γc (k) cos kx + γs (k) sin kx) .
(A9)
ψ=
By adding all modes, we obtain the stream function ψ, the pressure p, and the velocity field (vx , vy ) as follows: 1 ψ= 2η
∞
Z
yeky (−γc (k) sin kx + γs (k) cos kx) dk,
0
p = p0 +
(A10) Z
vx =
1 2η
vy =
keky (γc (k) cos kx + γs (k) sin kx) dk, (A11)
∞
(1+ky)eky (−γc (k) sin kx + γs (k) cos kx) dk,
0
1 2η
(A12) Z
(A7)
Appendix B: Derivation of Eq. (13)
The nonlinear term ∂(V c)/∂x is calculated by considering the coupling between two modes in Fourier space Z ∞ Γf0 2 ∂ sin kx ∂ (V c) = 2 2 dk ∂x 2π ηa ∂x 1 + Deff (k)k 2 /a 0 Z ∞ cos kx dk × 1 + Deff (k)k 2 /a 0 ! r Z ∞ Γf0 2 Deff (k)k 2 = k 2 cos kxdk, (B1) F 2π 2 ηa2 0 a where F (ξ) =
ξ arctan ξ + ln(1 + ξ 2 ) . ξ 2 (4 + ξ 2 )
(B2)
∞
0
Z
(A6)
Therefore, by comparing the coefficient of cos kx in Eq. (1), we obtain ! r Dk 2 /a + 1 Deff (k)k 2 Γ0 Dk 2 /a = − F +1 2 a Deff (k)k 2 /a + 1 (B3) leading to Eq. (13).
∞
kyeky (γc (k) cos kx + γs (k) sin kx) dk.
0
(A13) Finally, we determine the following flow velocity in the x-direction at the water surface: V (x) = vx (x, 0) Z ∞ 1 = (−γc (k) sin kx + γs (k) cos kx) dk. 2η 0 (A14)
Appendix C: Diffusion coefficient obtained with the perturbation method
We calculate the diffusion coefficient using the perturbation method. To do this, Γ in Eq. (4) is treated as an infinitesimally small parameter ǫ. The concentration field c(x) is expanded with regard to ǫ as c(x) = c0 (x) + ǫc1 (x) + O(ǫ2 ).
(C1)
8 In Fourier space, c˜(k), the Fourier transform of c(x), is also expanded with regard to ǫ as Z ∞ c(x) = (˜ c0 (k) + ǫ˜ c1 (k)) cos kxdk + O(ǫ2 ). (C2)
The perturbed diffusion coefficient D′ is also expanded with regard to ǫ as
D′ = D0′ + ǫD1′ + O(ǫ2 ) = D + ǫD1′ + O(ǫ2 ).
0
Eq. (7) is also written as Z ∞ ǫ c˜0 (k) sin kxdk + O(ǫ2 ). V (x) = 2η 0
(C3)
At the order of ǫ0 , − Dk 2 c˜0 (k) − a˜ c0 (k) +
f0 , π
Using the relation between c˜n (k) and Dn′ ,
c˜0 (k) + ǫ˜ c1 (k) + O(ǫ2 ) = (C4)
(C8)
1 f0 + O(ǫ2 ), π (D0′ + ǫD1′ )k 2 + a (C9)
we obtain
and the solution is obtained as f0 1 1 f0 f0 . = = π(a + Dk 2 ) πa 1 + k 2 D/a πa 1 + x0 2 k 2 (C5) At the order of ǫ1 , Z ∞ Z ∞ ∂ ǫ c˜0 (k) sin kxdk c˜0 (k) cos kxdk ∂x 2η 0 0
c˜1 (k) = −
c˜0 (k) =
= −Dk 2 ǫ˜ c1 (k) − aǫ˜ c1 (k),
(C6)
This calculation is performed in the same manner as that in Eq. (B1), and we obtain c˜1 (k) = −
x0 k arctan x0 k + ln(1 + x0 2 k 2 ) f0 2 . 2π 2 ηDa2 (1 + x0 2 k 2 ) (4 + x0 2 k 2 ) (C7)
[1] A. S. Mikhailov and V. Calenbuhr, From Cells to Societies (Springer, Berlin, 2002). [2] S. Ramaswamy, The Mechnics and Statistics of Active Matter, Annu. Rev. Cond. Mat. Phys. 1, 323 (2010). [3] T. Vicsek and A. Zafeiris, Collective Motion, Phys. Rep. 517, 71 (2012). [4] C. Tomlinson, On the Motions of Camphor on the Surface of Water, Proc. R. Soc. London 11, 575 (1860). [5] S. Nakata, Y. Iguchi, S. Ose, M. Kuboyama, T. Ishii, and K. Yoshikawa, Self-Rotation of a Camphor Scraping on Water: New Insight into the Old Problem, Langmuir 13, 4454 (1997). [6] Y. Hayashima, M. Nagayama, and S. Nakata, A Camphor Grain Oscillates While Breaking Symmetry, J. Phys. Chem. B 105, 5353 (2001). [7] H. Kitahata and K. Yoshikawa, Chemo-Mechanical Energy Transduction through Interfacial Instability, Physica D 205, 283 (2005). [8] S. Soh , K. J. M. Bishop, and B. A. Grzybowski, Dynamic Self-Assembly in Ensembles of Camphor Boats, J. Phys. Chem. B 112, 10848 (2008). [9] S. Soh , M. Branicki, and B. A. Grzybowski, Swarming in Shallow Waters, J. Phys. Chem. Lett. 2, 770 (2011). [10] E. Heisler, N. J. Suematsu, A. Awazu, and H. Nishimori,
f0 x0 2 k 2 D1′ + O(ǫ2 ). 2 πa (1 + x0 2 k 2 ) D
(C10)
From Eqs. (C7) and (C10), we derive
D1′ f0 (1 + x0 2 k 2 ) x0 k arctan x0 k + ln(1 + x0 2 k 2 ) = D 2πηDa x0 2 k 2 (4 + x0 2 k 2 ) f0 G(x0 k). (C11) = 2πηDa
Thus Eq. (28) is obtained.
[11]
[12]
[13]
[14]
[15] [16]
Collective Motion and Phase Transitions of Symmetric Camphor Boats, J. Phys. Soc. Jpn. 81, 074605 (2012). E. Heisler, N. J. Suematsu, A. Awazu, and H. Nishimori, Swarming of Self-Propelled Camphor Boats, Phys. Rev. E 85, 055201 (2012). S. Nakata, M. Nagayama, H. Kitahata, N. J. Suematsu, and T. Hasegawa, Physicochemical Design and Analysis of Self-Propelled Objects That are Characteristically Sensitive to Environments, Phys. Chem. Chem. Phys. 17, 10326 (2015). M. Nagayama, S. Nakata, Y. Doi, and Y. Hayashima, A Theoretical and Experimental Study on the Unidirectional Motion of a Camphor Disk, Physica D 194, 151 (2004). N. J. Suematsu, T. Sasaki, S. Nakata, and H. Kitahata, Quantitative Estimation of the Paramters for Self-Motion Driven by Difference in Surface Tension, Langmuir 30, 8101 (2014). L. E. Scriven and C. V. Sternling, The Marangoni Effects, Nature 187, 186 (1960). H. Linde, P. Schwartz and H. Wilke, Dissipative Structures and Nonlinear Kinetics of the MarangoniInstability, in Dynamics and Instability of Fluid Interfaces, ed. T. S. Sørensen, (Springer-Verlag, Berlin, 1979).
9 [17] H. Kitahata, S. Hiromatsu, Y. Doi, S. Nakata, and M. R. Islam, Self-Motion of a Camphor Disk Coupled with Convection, Phys. Chem. Chem. Phys. 4, 2409 (2004). [18] Y. S. Ikura, R. Tenno, H. Kitahata, N. J. Suematsu, and S. Nakata, Suppression and Regeneration of CamphorDriven Marangoni Flow with the Addition of Sodium Dodecyl Sulfate, J. Phys. Chem. B 116, 992 (2012). [19] D. Forster, D. R. Nelson, and M. J. Stephen, LargeDistance and Long-Time Properties of a Randomly Stirred Fluid, Phys. Rev. A 16, 732 (1977). [20] A. A. Nepomniashchy, M. G. Velarde, P. Colinet, Interfacial Phenomena and Convection (Chapman & Hall/CRC,
Boca Raton, 2002). [21] L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, Oxford, 1959). [22] N. O. Young, J. S. Goldstein, and M. J. Block, The Motion of Bubbles in a Vertical Temperature Gradient, J. Fluid Mech. 6, 350 (1959). [23] J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media (Prentice-Hall, Englewood Cliffs, 1965). [24] L. G. Leal, Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes (Cambridge University Press, 2007).
