Supplementary Information
In order to strongly support the results in the text, we illustrate more simulation
details in the followings:
1. Main algorithms implemented and their basic features include:
(a) Initialization: We arrange the atoms in one dimensional structure, as their initial
positions, and assign velocities from a Maxwell distribution to each atom, and rescale
them in order to obtain desired temperature and guarantee a rest center of mass.
(b) Energy and forces: Calculate the potential energy and forces between the two
atoms according to Lennard-Jones potential.
(c) Integrate the equation of motion: Knowing the forces on each atom, the next
step is to integrate the equations of motion to make the system start evolving with
time. To keep a stable temperature along the simulation, a Langevin thermostat is
added to the system.
(d) Changing atom with reservoirs: After considering the motion of atoms, some
atoms maybe go into or go out off the particle reservoirs. Then we have to compare
the number of atoms in reservoirs with that we have set. If the number is less than that
we have set, we use Monte Carlo method to generate an atom randomly in the
reservoirs. At the same time, whether this new atom can be accepted lies on the
probability in Eq.(1). On the contrary, when the number of atom is larger than that we
have set we will delete an atom according to the same method mentioned above.
(e) Collecting data: After the system reaches a stable state, measurements of the
system properties could be carried out. In this project, temperature, mass flux and
transport diffusion coefficient are measured.
2. Fixed the number of atoms in reservoirs
Average concentration (atom/nm)
To test the stability of adding or deleting atoms in reservoirs, we plot the average
concentration with time in the two reservoirs. The initial concentration in the left and
right atom reservoirs are 1.6 and 0.4 atom/nm, respectively. We get the data every 104
step (or 0.055ns), as shown in Fig. 7, the average concentration can keep to a certain
value and not change with time.
2.00
Left atom reservior
Right atom reservior
1.75
1.50
1.25
1.00
0.75
0.50
0.25
0.00
0
5
10
15
20
25
30
35
40
Time/ns
FIG 7 Average concentration in left and right atom reservoirs vs. time
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3. Stable mass flux
We calculate the mass flux by counting the number of atoms that go out off the
left reservoir and go into the right reservoir.
dn1_atom/ns
dn2_atom/ns
12.5
10.0
7.5
Flux
5.0
2.5
0.0
0
-2.5
5
10
15
20
25
30
35
40
45
Time/ns
-5.0
-7.5
-10.0
-12.5
FIG. 8 Flux in the two atom reservoirs vs. time, here transport length is 20nm, the length of the
reservoirs is100nm.
As displayed in Fig.8, dn1 is positive, which means the atom we have to add in
order to keep the left concentration of reservoir, also the number of atom that flow out
of the reservoir. dn2 is negative, which means the atom we have to delete in order to
keep the concentration of right reservoir, also the number of atom that flow into the
reservoir.
As shown in Fig.8, one can see that the flux out of the left reservoir and into the
right reservoir (numerical equality and signs are the opposite) can reach a stable value
when the time is larger than 40ns.
4. Stable concentration distribution
Average concentration (atom/nm)
The mass flux is obtained after the system reaches a stable state. Besides the
stable mass flux, one of the standards for the stable state is the steady concentration
distribution. Figure 9 illustrates the concentration distribution after running 600ns.
The two platforms are corresponding to the reservoirs, and the region (ranges from
100 to 120nm) in which the concentration perform a gradient means the transport
channel.
1.75
Distribution of the average concentration
Time=600ns
1.50
1.25
1.00
0.75
0.50
0.25
0
20
40
60
80
100
120
140
160
180
200
220
Length (nm)
FIG 9 Concentration distribution along the whole systems, the platform means the region in the
reservoirs.
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5. Impact of the Lennard-Jones parameters
𝜎∗ 12
The Lennard-Jones potential can be written as 𝑈(𝑥) = 4𝜀 ∗ [( 𝑥 )
𝜎∗ 6
− (𝑥) ] ,
where ε∗ and 𝜎 ∗ are the depth of the potential well and the finite distance at which
the inter-particle potential is zero respectively, x is the distance between the particles.
As shown in Fig. 10 that the impact of the finite distance 𝜎 ∗ on the transport
diffusion motion can be divided into two categories, small and large 𝜎 ∗ , and 1.0 is the
boundary point for those two areas. When the 𝜎 ∗ is relatively small, the transport
diffusion coefficient follows the linear law of 𝐷 ∗ ∝ 𝜎 ∗ , here the transport
coefficient changes weakly with the finite distance. However, the diffusion coefficient
deviates from the linear relationship and increases sharply at larger finite distance.
From the comparison of Figs. 4 and 10 the influential processes of the average
concentration and the parameter 𝜎 ∗ of the L-J potential on the transport diffusion are
quite similar. We can use a ratio of the finite distance 𝜎 ∗ between atoms to the
average distance r*(=1/c*) between atoms to explain their clear impacts on the
transport diffusion. As shown in Fig. 11, the green line means changing 𝜎 ∗ as r* is
fixed, and the black line means changing r* as 𝜎 ∗ is fixed. Even though D* mainly
depends on 𝜎 ∗ /r*, changing r* has more impact on D* than changing 𝜎 ∗ .
To explain why the influence of 𝜎 ∗ and c ∗ can be divided into two region, we
can use the potential of interaction between the atoms. As displayed in Fig.12, when
the distance is larger than 6 Angstom, the interaction between atoms tend to zero. In
small 𝜎 ∗ and c ∗ region, increasing 𝜎 ∗ and c ∗ only means increasing the number
of atom, which does not depend on the interaction. Therefore, the transport diffusion
coefficient follows the linear law with 𝜎 ∗ and c ∗ . In large 𝜎 ∗ and c ∗ region, the
interaction can not be omitted, and then the influences of 𝜎 ∗ and c ∗ on D* become
more complicated and nonlinear.
15
D*
12
9
L*=7.6
L*=76
L*=760
Nonlinear relationship
Linear relationship
6
3
0
0.0
0.3
0.6
0.9
1.2
*
1.5
1.8
2.1
FIG. 10 Transport diffusion coefficient with the value of 𝜎 ∗ for various transport length, where
= 38; 𝑐𝐿∗ = 0.41; 𝑙𝐵∗ = 38; 𝑐𝑅∗ = 0.10; 𝑚∗ = 4; 𝑇 ∗ = 27.7; 𝜀 ∗ = 1; 𝐿∗ =7.6, 76 and 760.
𝑙𝐵∗
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D*
6
 fixed c* changing
c* fixed  changing
4
2
0
0.00
0.15
0.30
0.45
0.60
0.75
0.90
*/r*
FIG.11 Transport diffusion coefficient depends on the ratio of 𝜎 ∗ /r*, here 𝐿∗ =7.6 and the remain
parameters are the same as for Fig.10.
FIG. 12 Lennard-Jones potential 𝑈(𝑥) vs. the distance 𝑥, the units of potential and distance are
respectively Joule and Angstrom.
Fig. 13 shows another Lennard-Jones parameter 𝜀 ∗ , which does not influence
the transport diffusion coefficient for different system sizes. Therefore, the value of
𝜀 ∗ only reflects the depth of the curve of the potential function rather than affect the
intrinsic transport ability.
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7
6
L*=7.6
L*=76
L*=760
D*
5
4
3
2
1
0
0
1
2

3
4
5
FIG. 13 Transport diffusion coefficient with 𝜀 ∗ for various transport length, where 𝑙𝐵∗ = 38;
𝑐𝐿∗ = 0.41; 𝑙𝐵∗ = 38; 𝑐𝑅∗ = 0.10; 𝑚∗ = 4; 𝑇 ∗ = 27.7; 𝜎 ∗ = 1; 𝐿∗ =7.6, 76 and 760.
6. The selection of Lennerd-Jones parameters for rare gases
In the Fig.6 of the text, the Lennard-Jones parameters we have use are given by
Tab.1. These parameters are popularly used to study the diffusion behavior of rare
gases [G. Stan, M. J. Bojan, S. Curtarolo, S. M. Gatica, and M. W. Cole, Phys. Rev. B
62, 2173 (2000)], which can reflect the thermodynamic properties of rare gases
properly.
Gas
σ(nm)
ε(K)
He
Ne
Ar
Kr
0.256
0.275
0.340
0.360
10.2
35.6
120.0
171.0
Table 1 The related parameters of L-J potential for different rare gases
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