Supplementary Material (I)
Derivation of the theoretical model for complex-formation in an aqueous bulk.
In the derivation of the kinetic model, we considered the following stepwise processes: I. diffusion of DHA-Et
from organic bulk to the interface across the organic laminar film, II. diffusion of DHA-Et across the aqueous
laminar film to the aqueous bulk and III. complex-formation reaction between DHA-Et and silver ion in the
aqueous bulk.
Diffusion of DHA-Et through organic film to interface
With a linear driving force approximation, the mass transfer rate of DHA-Et through the organic film, Worg,
can be expressed as the product of an interfacial area, A, and a mass transfer rate based on unit interface, RA,org.
Worg  A  RA,org  A
Dorg
 org
[DHA-Et ]
b
 [DHA -Et ]i

(I-1)
where Dorg and δorg mean the diffusion coefficient of DHA-Et in the organic film and the thickness of the organic
film, respectively. Subscripts “b” and “i” denote the bulk phase and near the interface, respectively. Overbar
denotes the organic phase.
Diffusion of DHA-Et through aqueous film to aqueous bulk
The mass transfer rate of DHA-Et through the aqueous film, Waq, can be expressed with a linear driving force
approximation as follows.
Waq  A  R A,aq  A
Daq
 aq
[DHA -Et ]i  [DHA -Et ] b 
(I-2)
where RA,aq is a mass transfer rate based on unit interface. Daq and δaq mean the diffusion coefficient of DHA-Et in
the aqueous film and the thickness of the aqueous film, respectively. Here, it can be considered that the distribution
of DHA-Et between organic and aqueous phases near the interface is in local equilibrium. The distribution of
DHA-Et between organic and aqueous phases is written as follows.
DHA-Et DHA-Et

Kd 
[DHA -Et ] b,eq
[DHA -Et ] b,eq
(I-3)
; Kd

[DHA -Et ]i
(I-4)
[DHA -Et ]i
where Kd is the distribution coefficient. Substituting Eq. I-4 into Eq. I-2, the following equation is obtained.
Waq  A
Daq
 aq
K
d [ DHA -Et ] i
 [DHA -Et ] b

(I-5)
Complex-formation reaction between DHA-Et and silver ion in the aqueous bulk
Considering stepwise complex-formation reaction, the complex-formation rate of DHA-Et with silver ion in
the aqueous bulk, Wcom, can be written as follows.

Wcom  A k1 [DHA-Et ] b [Ag  ] b m  k 2 [com] b

(I-6)
where com denotes the extracted complex. m is the apparent reaction order. k1 is the apparent forward reaction rate
constant and k2 is the apparent reverse reaction rate constant. The derivation of Eq. I-6 is written in Supplementary
Material (II). Regarding the silver ion concentration, the concentration of silver ion actually decreased until the
reaction reaches equilibrium. However, it can be assumed constant because the amount of silver ion is significantly
larger than DHA-Et and the complex under the investigated experimental condition. That is, [Ag+]b = [Ag+]b,eq =
[Ag+]b,0, where subscripts “eq” and “0” denote the equilibrium and initial state, respectively.
Now, we consider the extraction equilibrium state. At equilibrium state, Wcom = 0. Hence from Eq. I-6,
k2 
k1 [DHA -Et ] b,eq [Ag  ] b,0 m
(I-7)
[com] b,eq
The [com]b,eq in Eq. I-7 can be expressed with [DHA -Et ]b,eq and [Ag+]b,0 with the extraction equilibrium
constant of DHA-Et with silver ion. Here, we consider the extraction equilibrium relationship as the following;
DHA-Et  nAg 
K ex 



(I-8)
com ; K ex
[com] b,eq
(I-9)
[DHA -Et ] b,eq [Ag  ] b,0 n
where Kex is the extraction equilibrium constant. n is the number of Ag+ coordinated with a DHA-Et, which is
corresponding to the number of unsaturated double bonds of DHA-Et. Substituting Eqs. I-7 and I-9 into Eq. I-6,
Wcom is rewritten as follows:

[com] b
Wcom  Ak1 [Ag  ] b,0 m  [DHA -Et ] b 
n

K ex [Ag  ] b,0





(I-10)
Here, we consider the mass-balance of DHA-Et. The mass-balance of DHA-Et can be written as follows.
[DHA-Et ]b,0Vorg  [DHA-Et ]bVorg   DHA-Et A  [DHA-Et ]b Vaq  [com]b Vaq
(I-11)
where Vorg and Vaq are the volume of organic and aqueous phases, respectively. ΓDHA-Et is adsorption amount of
DHA-Et on the interface. Because the amount of DHA-Et distributed in an aqueous phase and that adsorbed on the
interface are much smaller than the amount of DHA-Et in an organic phase (Kd < 1×10-7)8, [DHA-Et]bVaq and
ΓDHA-EtA can be ignored. Therefore, Eq. I-11 can be rewritten as follows.

[com] b  [DHA -Et ] b,0  [DHA -Et ] b
VV
org
(I-12)
aq
By substituting Eq. I-12 into Eq. I-10, Wcom can be expressed as follows:

[DHA -Et ] b,0  [DHA -Et ] b Vorg
Wcom  Ak1 [Ag  ] b,0 m  [DHA -Et ] b 
n

Vaq
K ex [Ag  ] b,0





(I-13)
Rate equation for overall process
By assuming a quasi-steady-state for the overall process of the DHA-Et extraction, it is considered that Eqs.
I-1, I-5 and I-13 are equal. In addition, they are also considered equal to the reaction rate of DHA-Et, -rDHA-Et.
 rDHA-Et  A
Dorg
 org
[DHA-Et ]

 Ak1 [Ag ] b,0
b

 [DHA -Et ] i  A
Daq
 aq

K
d [DHA -Et ] i
 [DHA -Et ] b
[DHA -Et ] b,0  [DHA -Et ] b Vorg 
[DHA -Et ] b 

Vaq 
K ex [Ag  ] b,0 n


m
Equation I-14 can subsequently be rewritten to Eq. I-15:

(I-14)
 rDHA- Et 
[DHA -Et ] b  [DHA -Et ] i
 org
[DHA -Et ] i 

[DHA -Et ] b
Kd
 aq
ADorg
ADaq K d
(I-15)

[DHA -Et ] b [DHA -Et ] b,0  [DHA -Et ] b Vorg

Kd
Vaq
K ex K d [Ag  ] b,0 n
1

Ak1 [Ag ] b,0 m K d
By adding the denominator of the second, third and forth terms together and also adding the numerator of the
second, third and forth terms together, Eq. I-15 can be rewritten as follows:
 rDHA- Et  Vorg
d[DHA -Et ] b
dt
[DHA -Et ] b 

 org

ADorg

[DHA -Et ] b,0  [DHA -Et ] b Vorg
K ex K d [Ag  ] b,0 n
 aq
ADaq K d

Vaq
(I-16)
1
Ak1 [Ag  ] b,0 m K d
(1   )[DHA -Et ] b   [DHA -Et ] b,0

where

 org
ADorg

 aq
ADaq K d

1

Ak1 [Ag ] b,0 m K d
(I-17)

Vorg
1
K ex K d [Ag  ] b,0 Vaq
n
Integrating Eq. I-16 under the initial condition; [DHA -Et ] b = [DHA-Et ]b,0 at t = 0, we derive the following
equation.
[DHA -Et ]b 
  exp (1   )t (  Vorg )
1 
[DHA-Et ]b,0
(I-18)
The relationship between [DHA -Et ] b and t can be simulated with Eq. I-18. The parameters used in the
calculation are as follows. [Ag+]b,0, A, Vorg and Vaq are the experimental condition. Kex was calculated from the
following equation.
log K ex 
S 
2.303R

H 
1
2.303R T

(I-19)
where R and T are the gas constant and temperature, respectively. ΔS° and ΔH° are entropy and enthalpy of the
extraction of DHA-Et with silver ion. They were determined in the previous paper13. The n was also determined in
the previous paper13. Dorg and Daq are the established from the literature20. We took 2.510-9 m2/s for them, because
the diffusion coefficient of several solutes in several solvents are in the range between 110-9 m2/s and 410-9 m2/s
as reported by Wilke and Chang20. In addition, it was assumed that δorg and δaq are the same. The calculation of the
boundary layer thickness is very difficult, because the phases are in turbulent flow in a geometry which is not
usually well defined. However, both δorg and δaq have to be larger than 110-10 m, because it is hard to consider that
the liquid film is thinner than molecular size. So we calculated the uptake curves with more than 110-10 m of δ.
The kinetic parameters, m and k1, were determined from the experimental results for the extraction dynamics
obtained by using stirred transfer cell. The m was determined as 2.5 from the plot based on the effect of silver ion
concentration on the initial extraction rate shown in Figure 7 in the main text. The k1 was determined from the plot
based on the theoretical equation derived below.
Determination of the apparent rate constant for complex-formation reaction between DHA-Et and silver ion
Before the modeling of the uptake curve with Eq. I-18, we determined the apparent rate constant for
complex-formation reaction between DHA-Et and silver ion, k1, according to the following theory. It should be
noted that the theory described below is available for the case that chemical reaction, i.e. the complex-formation,
controls the extraction. Such condition can be realized by using a stirred transfer cell for the extraction. Therefore,
k1 was determined from the experimental results for the extraction dynamics obtained by using the stirred transfer
cell.
As mentioned above, the complex-formation rate of DHA-Et with silver ion in an aqueous bulk, Wcom, can be
written as Eq. I-10. In such case, Wcom is the overall mass transfer rate. Therefore, the mass transfer rate based on
unit volume, RV, can be written as follows:
RV  

d[DHA -Et ] b
[com] b
A

k1 [Ag  ] b,0 m  [DHA -Et ] b 
n

dt
Vaq
K ex [Ag  ] b,0





(I-20)
In addition, it can be considered that the distribution of DHA-Et between organic and aqueous phases is in
equilibrium.
[DHA -Et ] b  K d [DHA -Et ] b
(I-21)
Substituting Eqs. I-9, I-12 and I-21 into Eq. I-20 and after simplification, RV can be rewritten as follows.
RV  
d[DHA -Et ] b
dt
m

A k1 K d [Ag ] b,0 [DHA -Et ] b,0

[DHA -Et ] b  [DHA -Et ] b,eq
Vaq [DHA -Et ] b,0  [DHA -Et ] b,eq


(I-22)
Integrating Eq. I-22 under the initial condition; [DHA -Et ] b = [DHA-Et ]b,0 at t = 0, the following equation
is led.
 ln
[DHA -Et ] b  [DHA -Et ] b,eq
[DHA -Et ] b,0  [DHA -Et ] b,eq
According to Eq. I-23, a plot of  ln
m

A k1 K d [Ag ] b,0 [DHA -Et ] b,0

t
Vaq [DHA -Et ] b,0  [DHA -Et ] b,eq
[DHA -Et ] b  [DHA -Et ] b,eq
[DHA -Et ] b,0  [DHA -Et ] b,eq
(I-23)
versus t gives a proportional relationship
when the complex-formation reaction controls. Therefore, the slope of the plot gives k1. Figure I shows the plot
based on Eq. I-23 for different temperatures. As shown in Figure I, the plots based on Eq. I-23 show straight
relationships. With the determined k1, the Arrhenius plot was made and the activation energy and the frequency
factor were determined. They were used to predict the uptake curves by using Eq. I-18.
2
278 K
[Ag+] = 2.7103 mol/m3
[DHA-Et]org = 10 mol/m3
283 K
1.5
298 K
308 K
1
313 K
0.5
0
0
5000
10000
t [s]
Figure I. Plots based on Eq. I-23 for different temperatures.
Table I. Experimental Conditions and Parameters Used for Calculation of Uptake Curves of DHA-Et
Extraction Shown in Figure II.
condition 1
condition 2
condition 3
condition 4
unit
[DHA-Et]org,0
9.98
10.2
9.25
30.7
mol/m3
[Ag+]aq,0
2700
2700
1600
2700
mol/m3
T
298
313
298
298
K
ΔH°
-122.0
-122.0
-122.0
-122.0
kJ mol-1
ΔS°
-795.5
-795.5
-795.5
-795.5
J K-1 mol-1
-8
-8
-8
-8
Kd
1.3310
1.3310
1.3310
1.3310
n
2.5
2.5
2.5
2.5
activation energy
-54.3
-54.3
-54.3
-54.3
kJ/mol
frequency factor
2.31
2.31
2.31
2.31
m8.5/(mol2 s)
Daq
m2/s
2.510-9
2.510-9
2.510-9
2.510-9
Dorg
m2/s
2.510-9
2.510-9
2.510-9
2.510-9
-10
-10
-10
-10
δaq
m
110
110
110
110
δorg
m
110-10
110-10
110-10
110-10
condition 1
condition 2
calced for 1
calced for 2
12
9
6
3
50
15
[DHA-Et] org mol/m3
[DHA-Et] org mol/m3
15
condition 3
calced for 3
condition 4
calced for 4
12
9
30
6
20
3
10
0
0
0
0
5000
10000
t [s]
15000
20000
40
0
5000
10000
15000
20000
t [s]
Figure II. Experimental and predicted uptake curves of DHA-Et extraction measured by using the
stirred transfer cell. Plots are the experimental data and lines are the calculated results.
Figure II shows the extraction uptakes resulted from the experiments using the stirred transfer cell. The
experimental condition is same to those shown in Table 3 in the main text. In Figure II, the calculated results with
Eq. I-18 were also shown. The parameters used for the calculation are shown in Table I. As shown in Figure II,
although the δ, which is the solely fitting parameter in the calculation, is enough thin (110-10 m), the calculated
uptake curves shows slower extraction rate than the experimentally obtained ones. From Eq. I-4, it is clear that the
concentration of DHA-Et in the aqueous phase near the interface is smaller than Kd [DHA-Et ]i,0 (≈
[DHA-Et ]i,0 10-7). Therefore, the concentration gradient of DHA-Et in an aqueous laminar film near the interface
becomes quite small. That is why the calculated uptake curves show quite slow extraction rate.
From the above investigation, we concluded that DHA-Et did not react with silver ion in an aqueous bulk.
Supplementary Material (II)
Derivation of Eq. 2 (in the main text) and Eq. 6 (in Supplementary Material (I))
Regardless where DHA-Et reacts with silver ion, the extracted complex must generated via the following
stepwise reactions.
DHA-Et  Ag 
ki




ki '
com1
; r1  k i [DHA-Et][Ag  ]  k i ' [com1]
com1  Ag 
kii




kii '
com2
; r2  k ii [com1][Ag ]  k ii ' [com2]
com2  Ag 
kiii



kiii '
com3
; r3  k iii [com2][Ag ]  k iii ' [com3]
(II-3)
com3  Ag 
kiv



kiv '
com4
; r4  k iv [com3][Ag ]  k iv ' [com4]
(II-4)
com4  Ag 
kv




kv '
com5
; r5  k v [com4][Ag ]  k v ' [com5]
com5  Ag 
k
vi


k vi '
com6
; r6  k vi [com5][Ag ]  k vi ' [com6]
(II-1)
(II-2)
(II-5)
(II-6)
where com1 – com5 denote the 1:1 – 1:5 intermediate complexes, respectively. The com6 denote the extracted
complex, which is same to com in the main text and Supplementary Material (I). From Eqs. (II-1) – (II-6),
formation rate of com1 – com6 are written as follows.
rcom1  r1  r2  k i [DHA -Et][Ag  ]  k i ' [com1]  (k ii [com1][Ag  ]  k ii ' [com2])
0
rcom2  r2  r3  k ii [com1][Ag  ]  k ii ' [com2]  (k iii [com2][Ag  ]  k iii ' [com3])
0
rcom3  r3  r4  k iii [com2][Ag  ]  k iii ' [com3]  (k iv [com3][Ag  ]  k iv ' [com4])
0
rcom4  r4  r5  k iv [com3][Ag  ]  k iv ' [com4]  (k v [com4][Ag  ]  k v ' [com5])
0
rcom5  r5  r6  k v [com4][Ag  ]  k v ' [com5]  (k vi [com5][Ag  ]  k vi ' [com6])
0
 k vi [com5][Ag ]  k vi ' [com6]
rcom6  r6
(II-7)
(II-8)
(II-9)
(II-10)
(II-11)
(II-12)
In the above equations, steady state was assumed for com1 – com5. From Eq. II-11, [com5] can be written as
follows:
[com5] 
k v [com4][Ag ]  k vi ' [com6]
(II-13)
k v 'k vi [Ag  ]
Substituting Eq. II-13 into Eq. II-10, [com4] can be written as follows:
k iv [com3][Ag ] 
[com4] 

k iv ' k v [Ag ] 
k v ' k vi ' [com6]
k v ' k vi [Ag  ]
k v k v ' [Ag  ]
k v ' k vi [Ag  ]
where
A1 
kv '
k v ' k vi [Ag  ]
(II-15)

k iv [com3][Ag ]  A1 k vi ' [com6]
k iv ' k v [Ag  ]  A1 k v [Ag  ]
(II-14)
In the same way, [com3] – [com1] can be written as the followings:
[com3] 
[com2] 
[com1] 
k iii [com2][Ag ]  A1 A2 k vi ' [com6]
(II-16)
k iii 'k iv [Ag  ]  A2 k iv [Ag  ]
k ii [com1][Ag ]  A1 A2 A3 k vi ' [com6]
(II-17)
k ii 'k iii [Ag  ]  A3 k iii [Ag  ]
k i [DHA -Et][Ag  ]  A1 A2 A3 A4 k vi ' [com6]
(II-18)
k i ' k ii [Ag  ]  A4 k ii [Ag  ]
where
A2 
A3 
A4 
k iv '
(II-19)

k iv ' k v [Ag ]  A1 k v [Ag  ]
k iii '
(II-20)

k iii ' k iv [Ag ]  A2 k iv [Ag  ]
k ii '
(II-21)

k ii ' k iii [Ag ]  A3 k iii [Ag  ]
Rearranging Eq. II-18, [com1] can be rewritten as the following simple form.
[com1]   com1 [DHA -Et]   com1 [com6]
(II-22)
where
 com1 
k i [Ag  ]


k i ' k ii [Ag ]  A4 k ii [Ag ]
,
 com1 
A1 A2 A3 A4 k vi '
k i ' k ii [Ag  ]  A4 k ii [Ag  ]
(II-23)
In the same way, [com2] – [com5] can be rewritten as the followings:
[com2]   com2 [DHA -Et]   com2 [com6]
(II-24)
[com3]   com3 [DHA -Et]   com3 [com6]
(II-25)
[com4]   com4 [DHA -Et]   com4 [com6]
(II-26)
[com5]   com5 [DHA -Et]   com5 [com6]
(II-27)
where
 com2 
 com3 
 com4 
 com5 
 com1 k ii [Ag  ]


,  com2 
 com2  com1
A1 A2 A3 k vi '

 com1
k ii 'k iii [Ag  ]  A3 k iii [Ag  ]
(II-28)

,  com3 
 com3  com2
A1 A2 k vi '

 com2
k iii 'k iv [Ag  ]  A2 k iv [Ag  ]
(II-29)
k ii 'k iii [Ag ]  A3 k iii [Ag ]
 com2 k iii [Ag  ]

k iii 'k iv [Ag ]  A2 k iv [Ag ]
 com3 k iv [Ag  ]


k iv 'k v [Ag ]  A1 k v [Ag ]
 com4 k v [Ag  ]
,  com5 

k v 'k vi [Ag ]
,  com4 
 com4  com3
A1 k vi '

 com3
k iv 'k v [Ag  ]  A1 k v [Ag  ]
 com5  com4
k vi '

 com4
k v 'k vi [Ag  ]
(II-30)
(II-31)
Next, we consider the overall reaction rate. The overall reaction of the stepwise reactions of II-1 – II-6 is
written as following stoichiometric equation.
DHA-Et  6Ag 



com6
(II-32)
; r
The relationship among rDHA-Et, rAg+, rcom6 and the overall reaction rate of Eq. II-32, r, is as follows.
r  rDHA-Et  
r
Ag
6
 rcom6
(II-33)
Therefore, substituting Eqs. II-12 and II-27 into Eq. II-33 leads to the following equation.
r  rcom6
  com6 [DHA -Et]   com6 [com6]
(II-34)
where
 com6   com5 k vi [Ag  ] ,  com6  k vi  com5 [Ag  ]  k vi '
(II-35)
The αcom6 consists of the rate constants of each elementary reaction and silver ion concentration; i.e. αcom6 is
the function of silver ion concentration. If we express αcom6 as an exponential function of [Ag+];
 com6  k1 [Ag  ] m ,  com6  k 2
(II-36)
the overall reaction rate is rewritten as follows.
r  k1 [DHA -Et ][Ag  ] m  k 2 [com6]
(Eq. 2 in the main text and Eq. I-6 in Supplementary Material (I))