Kr Adsorption in Porous Carbons: Temperature
Dependent Experimental and Computational
Studies
Ziyi Wang,,
∗
Cullen M. Quine,
∗
Claire N. Saunders,
∗
Camille M.
Bernal-Choban,
∗
Channing C. Ahn,
∗
and Brent T. Fultz
∗
California Institute of Technology, Pasadena, California 91125, United States
E-mail: ziyiwang@alumni.caltech.edu; cquine@alumni.caltech.edu; csaunders@caltech.edu;
cmbchoban@gmail.com; cca@caltech.edu; btf@caltech.edu
Henry’s law analysis
Fig. S1 shows graphs used for the Henry’s Law analysis, appropriate for low coverage, or low
n
. Henry’s constants are
K
= lim
n
→
0
n/P ,
(S1)
or intercepts at the vertical axes. For adsorption at high temperatures, the linear fits were
acquired from the first 15 data points.
Phonon dispersions
In Fig. S2, the green lines show the phonon dispersion relationship of graphite calculated by
the sTDEP method. The sTDEP results are in good agreement with experimental results
reported by Mohr.
?
1
0
0.1
0.2
0.3
n
·N
A
-1
(mmol)
0
4
8
12
n·N
A
-1
/ P (mmol·g
-1
·Pa
-1
)
0
0.1
0.2
0.3
n
·N
A
-1
(mmol)
0
2
4
6
n·N
A
-1
/ P (mmol·g
-1
·Pa
-1
)
×10
-5
×10
-5
(a)
(b)
CNS-201
MSC-30
253
K
259
K
268
K
276
K
285
K
294
K
303
K
312
K
322
K
331
K
250
K
255
K
258
K
260
K
264
K
268
K
272
K
276
K
280
K
284
K
289
K
293
K
298
K
302
K
307
K
311
K
316
K
321
K
325
K
330
K
Figure S1:
n/P
results of CNS-201 and MSC-30 at different temperatures.
0
40
80
120
160
200
Γ
M
K
Γ
Frequency (meV)
Frequency (cm
-1
)
200
600
1000
1400
Figure S2: Phonon dispersions of graphite calculated by the sTDEP method (green), com-
pared to data from Mohr 2007
?
(solid curves with circles) and inelastic X-ray scattering
data (red and blue points).
Monte Carlo calculation
Fig. S3 shows the generation of MC configurations. The sTDEP method generated multiple
equal-probability snapshots, and they were used to build slit-pores with a krypton atom
inserted. The vdW potential curves of each configuration were calculated, and then the
probabilities of these configurations were calculated with Boltzmann factors.
2
3×3×2
graphite
0
K
a)
c)
3×3×2
graphite
500
K
TDEP
Layer
Distance
b)
slit
pore
500
K
Figure S3: The generation process of thermally displaced configurations. (a) The graphite
supercell without thermal displacement. (b) One simulated thermally displaced configuration
of the cell at 500 K. (c) One system used for the MC calculation.
Partition functions
Fig. S4 shows the relationship between the layer distance
H
and the partition function
q
z
.
Three results merit further discussion:
H
≈
6
̊
A,
≈
9
̊
A, and
>
12
̊
A.
For small layer distances, potentials are similar to the harmonic oscillators:
ε
z
(
H
)
≈
1
2
mω
2
z
2
,
(S2)
where
ε
z
=
ε
vdW
−
ε
00
, and
ω
is the frequency. Substituting in Eq. 8, the partition functions
are:
q
z
(
L
)
≈
1
Λ
∫
+
H/
2
−
H/
2
exp
(
−
mω
2
z
2
2
k
B
T
)
d
z
=
1
Λ
√
2
πk
B
T
mω
2
·
erf
(
H
2
)
,
(S3)
where “erf” is the error function. With
H
increasing,
q
z
increases.
The partition functions reach maxima around 9
̊
A. This can be understood with a “three-
stage model” shown in Fig. S5. The
L
1
parts describe the two potential wells, and the
L
2
describes the middle between the two potential wells. By substituting these values in Eq. 8,
the partition function is:
q
z
=
1
Λ
[
L
1
+
L
2
exp
(
−
ε
2
k
B
T
)]
.
(S4)
3
At around 9
̊
A,
ε
2
is small while
L
2
is large, leading to maxima around these layer distances.
6
7
8
9
10
11
12
13
14
15
16
Layer Distance (A)
0
4
8
12
16
20
251
K
270
K
290
K
310
K
329
K
q
z
Figure S4: The
z
-direction partition functions of the adsorbed phase at different tempera-
tures given by the results in Fig. 5 and Eq. 8.
Distance to the Middle Plane
ε
z
L
2
L
1
ε
1
= 0
ε
2
Figure S5: The three stage model used for thermodynamics calculation.
The three-stage model is still useful for layer distances larger than 12
̊
A. Here,
ε
2
re-
mains at relatively high values, while the length of the middle barrier
L
2
continues to in-
crease. Therefore, the partition functions slightly increase, as a large
ε
2
leads to a small
exp(
−
ε
2
/k
B
T
).
The three-stage model helps explain the trends of
u
z
and
c
z
in Fig. 7. The corresponding
internal energy and heat capacity are:
u
z
=
ε
2
L
1
L
2
exp
(
ε
2
k
B
T
)
+ 1
+ 0
.
5
k
B
T
;
(S5)
4
c
z
=
ε
2
2
k
B
T
2
·
exp
(
−
ε
2
k
B
T
)
[
L
1
L
2
exp
(
−
ε
2
k
B
T
)
+ 1
]
2
·
L
1
L
2
+ 0
.
5
k
B
.
(S6)
Thermal Occupancy of the Pores
5
10
15
20
25
30
Surface Distance (Å)
CNS-201
250 K
270 K
290 K
310 K
330 K
MSC-30
250 K
270 K
290 K
310 K
330 K
Distribution of the Adsorbed Gas (Å
-1
)
(b)
5
10
15
20
25
30
Surface Distance (Å)
(a)
310 K
(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
270 K
Distribution of the Adsorbed Gas (Å
-1
)
0
0.1
0.2
0.3
Figure S6: The distribution of the adsorbed gas, which corresponds to the
n
(
H
) in Eq. 12 over
the total adsorption amount
n
. These distributions can also be given by
n
(
H
)
/n
=
K
(
H
)
/K
.
The black dash lines are the surface area distribution functions shown in Fig. 1 as reference.
Figure S6 presents the thermal occupancy of pores with different sizes, following the dis-
tribution of Fig. 1. This thermal occupancy was obtained from Eq. 13 with the partition
function of Eq. 8. For CNS-201, the peaks of the surface area distribution are around 10.6
̊
A
and 12.1
̊
A. Following to Fig. 7, the Kr surface dynamics causes a change with temperature of
the effective adsorption energy of approximately 0
.
4
k
B
and 0
.
62
k
B
. With increasing temper-
5
ature, the gas distribution shifts slightly from the first to the second peak, while the larger
pores also attract more gas. These larger pores have higher heat capacities and internal
energies, leading to a temperature dependence of the internal energies.
For MSC-30, the transfer of the gas molecules to larger pores has a larger effect on thermal
averages. Despite their lower total population, the energetically favorable sites smaller than
10.7
̊
A adsorb more gas than the sites around 12.2
̊
A. With increasing temperature, there is
a decrease in the number of gas molecules in these energetically favorable sites, while large
pores attract more Kr atoms. Because of this transfer to larger sites with temperature,
MSC-30 has a large slope of 2
.
01
k
B
in Fig. 3.
Results with Small Pores
The NLDFT method of determining pore sizes from BET data is limited in detecting ener-
getically favorable pores smaller than 8.0
̊
A, since the method works best for the detection
of larger pores at lower temperatures. However, by adding a few small pores to the two
materials, the temperature dependence of adsorption energy is changed. The surface area
distributions of added pores are shown in Fig. S7a with dash lines. These distributions are
Gaussian functions with standard deviations of 0.2
̊
A. The mean values of these two Gaussian
functions are 7.3
̊
A for CNS-201 and 7.0
̊
A for MSC-30. The total added surface areas are
both 100 m
2
·
g
−
1
, which is about 10.4% and 4.2% of the original NLDFT surface areas of
CNS-201 and MSC-30.
The adsorption energies of the new surface area distributions are shown in Fig. S7b. From
250 K to 330 K, the adsorption energies change from -0.166 eV to -0.137 eV for CNS-201 and
from -0.139 eV to -0.112 eV for MSC-30. The slopes of these new fittings are 4.32
k
B
and
3.89
k
B
, which are now close to the experimental analysis with Henry’s law. Fig. S8 shows
the gas distribution with the assumption of some pores of smaller size. Compared with the
results in Fig. S6, these small pores are extremely favorable for the gas. With increasing
6
temperature, however, the gas molecules accumulated in these small pores transfer to larger
pores.
7
250
260
270
280
290
300
310
320
330
T
(K)
-0.22
-0.20
-0.18
-0.16
-0.14
-0.12
-0.10
-22
-20
-18
-16
-14
-12
-10
Δu
st
(eV)
Δu
st
(kJ·mol
-1
)
5
10
15
20
25
30
35
0
100
200
300
400
500
600
Surface Area Density
(m
2
·g
-1
·Å
-1
)
Surface Distance (Å)
MSC-30
CNS-201
(a)
(b)
CNS-201
CNS-201
MSC-30
MSC-30
Figure S7: The impact of small-size pores on adsorption energies. (a) The dash lines are
surface area distributions of added pores with small layer distances. The solid lines are the
surface area distribution shown in Fig. 1. (b) Isosteric adsorption energies with the new
modified surface area distribution are shown with the dash lines. Other results are the same
as Fig. 3.
8
5
10
15
20
25
30
Surface Distance (Å)
CNS-201
250 K
270 K
290 K
310 K
330 K
MSC-30
250 K
270 K
290 K
310 K
330 K
Distribution of the Adsorbed Gas (Å
-1
)
(b)
5
10
15
20
25
30
Surface Distance (Å)
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
Distribution of the Adsorbed Gas (Å
-1
)
0
0.2
0.4
0.6
Surface Area Density
(m
2
·g
-1
·Å
-1
)
Surface Area Density
(m
2
·g
-1
·Å
-1
)
0
200
400
600
800
1000
1200
0
200
400
600
800
0.8
330 K
0.7
0.8
1400
1600
Figure S8: The distribution of the adsorbed gas with the new modified surface area distri-
bution for (a) CNS-201 and (b) MSC-30. Similar to Fig. S6, the black dash lines are the
surface area distribution functions as a reference.
9