Supplemental material
A. Derivations of Eqs. (12), (13), (14)
We derive here the statistical thermodynamic expressions of the adsorption stress, Eqs. (12),
(13), and (14), for the simple model of the stacked-layer PCP. The layers are numbered from
–NL/2 to NL/2-1 as shown in Fig. S1(a), where NL is the total number of the layers. For
simplification of the derivation, NL is set to be even number. When NL is odd, the same
adsorption stress can be derived by only changing the numbering of the layers as: –(NL-1)/2
to (NL-1)/2. The zeroth layer is fixed at z = 0. The ith interlayer cell is bounded by the ith and
the (i + 1)th layers (Fig. S1(b)). The lengths of the system along the x and y directions are sx
and sy, respectively (Fig. S1(c)). All the widths of the interlayer cells are assumed to be the
same, and then the length of the system along the z direction, sz, can be expressed with NL
and the interlayer width h:
(S1)
sz  N L h
When the interlayer width expands or shrinks by dsz (= NLdh), the ith layer moves by idh.
Periodic boundary conditions are applied for all the directions.
(a)
number of layers: -NL/2
simulation box

-1
0
1
-h
0
h

NL/2-1
z
z-position: -sz/2
(b)
sz/2
(c)
ith & (i + 1)th layers
ith interlayer cell
sy
sx
Figure S1
Schematic representation of the simple stacked-layer PCP model.
The adsorption stress ads, which is the stress normal to the z direction for expanding the
interlayer width, can be calculated as the derivative of the osmotic free energy OS with
respect to sz.
1
 ads ( s z )  
1  Ω OS 


,
A  s z T ,  , A
(S2)
where A, T, and  are the surface area of the layer (= sxsy), the temperature, and the chemical
potential of the guest, respectively. The osmotic free energy is defined as the sum of the
grand potential of the guest, , the Helmholtz free energy of the host, Fhost, and the PV-term
Ω OS  Ω  F host  PV ,
(S3)
where P is the bulk gas pressure, and V is the system volume (= Asz). Thus, the adsorption
stress can be calculated as
 ads ( s z )  
1  Ω 
1  F host 



 
 P.
A  s z T ,  , A A  s z T ,  , A
(S4)
The first term on the right-hand side is the internal stress, int, applied by the guest particles,
and the second term is identical to the force, Fzss, which works between the layers:

1  F host 
1  U ss 
F ss



  
 z ,
A  sz T ,  , A
A  sz T ,  , A
A
(S5)
where Uss is the layer-layer interaction potential. Namely, the adsorption stress can be
expressed as
Fzss
(S6)
 ads ( s z )   int 
P.
A
In what follows, we derive the internal stress, int. First, the grand potential of the guest
particle can be calculated from the grand partition function :
Ω  kBT ln Ξ ,
(S7)
where kB is Boltzmann constant. Then,  is represented as

 N  3

Ξ T ,  , sz , A   exp 
Z N T , N , sz , A ,
N 0
 kBT  N!
(S8)
where N is the number of guest particles,  is the thermal de Broglie wavelength, and ZN is
the configurational part of the canonical partition function. By using Eqs. (S7) and (S8), int
can be expressed as
1   

A  sz T ,  , A
 int   

kBT 1  Ξ 


A Ξ  sz T ,  , A
kT 1
 B
A Ξ
 N   Z N
 3
  
exp 

N 0 N!
 kBT   sz

(S9)


.
T ,  , A
2
The partial derivative of ZN is given by
Z N


dr N exp  U (r N ) 
sz
s z 


sy / 2
sz / 2
  N sx / 2


dxi 
dyi 
dzi exp  U (r N ) ,


sy / 2
sz / 2
s z i 1  s x / 2



(S10)
where U is the total potential energy of the guest particles,  = 1/kBT, rN represents
3N-dimensional configuration of the guest particles, and xi, yi, and zi denote the configuration
of the ith particle. Applying Leibniz’s rule to Eq. (S10), we obtain
N
sx / 2
sy / 2
sz / 2
Z N

exp  U 
 
dxi 
dyi 
dzi

s
/
2

s
/
2

s
/
2
x
y
z
sz
sz
i 1


U
 
dxi 
dyi 
dzi   
 exp  U .
sx / 2
sy / 2
sz / 2
sz
i 1


N
sx / 2
sy / 2
(S11)
sz / 2
Thus,


Z N
U
  dr N   
 exp  U  .
s z
s z


(S12)
By substituting Eq. (S12) into Eq. (S9), we have
 int  
kBT 1
A Ξ
 N 


3
U
  dr N   
exp 
 exp  U 

sz
N 0 N!
 kBT 




1 U
A sz

1 U ff
1 U fs

,
A sz
A sz
(S13)
where Uff and Ufs are the fluid-fluid and the fluid-solid interaction potential energies.
Equation (S13) indicates that the internal stress can be obtained as the sum of the derivatives
of the potential energies. The first term on the right-hand side of Eq. (S13) can be expressed
as
U ff
s z

ff
1 N L / 21 N k N L / 21 N m rij u (rij )

,
   
2 k   N L / 2 i 1 m   N L / 2 j 1 s z
rij
(S14)
where Nk and Nm indicate the numbers of the particles in the kth and the mth interlayer cells, rij
is the distance between the ith particle in the kth interlayer cell and the jth particle in the mth
interlayer cell, uff is the fluid-fluid interaction potential for a pair of the guest particles. The
distance rij can be calculated as
rij  xij  yij  (zki  kh)  (z mj  mh) ,
2
2
2
(S15)
3
where xij and yij are the x and y components of rij, and zki is the distance between the kth layer
and the ith particle. Because the interlayer width h only depends on sz in Eq. (S15), we have
rij
s z

zij k  m

.
rij N L
(S16)
Substituting Eq. (S16) into Eq. (S14), we obtain
U ff
sz
ff
1 N L / 2 1 N k N L / 2 1 N m zij k  m u (rij )


.
    
2 k   N L / 2 i 1 m   N L / 2 j 1 rij N L
rij
(S17)
Eq. (S17) indicates that the interaction between the guest particles in the different cells (i.e., k
≠ m) only contributes to the adsorption stress. Here, all the fluid cells are statistically
identical because all the interlayer widths are set to be the same. Thus, the total fluid-fluid
interaction Uff can be obtained by multiplying the interactions between the particles in the
interlayer cell of m = 0 and those in the other cells (-NL/2 ≤ k ≤ NL/2-1), with the number of
the interlayer cells, NL, to obtain
U ff
sz
NL

2
ff
k u (rij )
   N  r
k   N L / 2 i 1 j 1 rij
L
ij
N L / 2 1 N k N 0
zij
zij k u ff (rij )

 
 Fzff ,


r
2

r
k   N L / 2 i 1 j 1 ij
ij
N L / 2 1 N k N 0
(S18)
where N0 indicates the number of the particles in the zeroth cell. This is the molecular
expression of the derivative of Uff. Then, the derivative of Ufs is given by
U fs

s z
N L / 2 1 N m
N L / 2 1
zik u fs ( zik )
,
  
zik
m   N L / 2 i 1 k   N L / 2 s z
(S19)
where zik is the distance between the ith particle in the mth interlayer cell and the kth layer, and
ufs is the fluid-solid interaction potential of a guest particle. The derivative of zik is written as
zik kh  (zim  mh) k  m


.
(S20)
s z
N L h
NL
By substituting Eq. (S20) into Eq. (S19), we obtain
U fs

s z
N L / 2 1 N m
k  m u fs ( zik )
.
  
zik
m   N L / 2 i 1 k   N L / 2 N L
NL / 2
(S21)
In a manner similar to Eq. (S17), Eq. (S21) can be rewritten as
N0 Nl / 21
U fs
k u fs ( zik )
 NL  
sz
zik
i 1 k  N L / 2 N L
N L / 21
u fs ( zik )
   k
 Fzfs .
zik
i 1 k  N L / 2
N0
(S22)
4
By substituting Eqs. (S13), (S18), and (S22) into Eq. (S6), the “force” expression of the
adsorption stress is obtained as
 ads 
Fzfs  Fzff  Fzss
(S23)
 P.
A
An alternative expression for the adsorption stress can be derived by scaling the z positions
of the guest particles and the layers by sz according to
z ~
z  z / sz .
(S24)
Thus, Eq. (S11) is rewritten as
sy / 2
sz / 2

Z N
  N sx / 2

  s / 2 dxi  s / 2 dyi  s / 2 dzi exp  U 
y
z
s z
s z  i 1 x


sy / 2
1/ 2

  N N sx / 2
~
s z   s / 2 dxi  s / 2 dyi 1 / 2 dzi exp  U 
x
y
s z 
i 1

N
sx / 2
sy / 2
1/ 2
 
N 
 
 s z    
dxi 
dyi  d~zi exp   U 
sy / 2
1 / 2
 s z
 i 1  s x / 2
N
sx / 2
sy / 2
1/ 2
N 
 sz
d
x
d
y
d~zi exp  U 

i
i




s
/
2

s
/
2

1
/
2
y
s z i 1 x
(S25)
N
sx / 2
sy / 2
1/ 2
N
U
N
 Z N  sz   
dxi 
dyi  d~zi
exp  U .

s
/
2

s
/
2

1
/
2
x
y
sz
sz
i 1
Because the scaling affects U (= Uff + Ufs), we obtain
rij
sz


sz
2
1 2 zij
2
2
2
xij  yij  sz ( ~
zj  ~
zi ) 2  sz ~
zij 
rij
sz rij
zik

sz (~zk  ~zi )  zik .

sz sz
sz
(S26)
Thus, the derivatives of the fluid-fluid and the fluid-solid interactions on the right-hand side
of Eq. (S25) can be obtained as
ff
U ff 1 N L / 2 1 N k N L / 2 1 N m u (rij ) rij

   
sz
2 k   N L / 2 i 1 m   N L / 2 j 1 rij sz
ff
1 1 N L / 2 1 N k N L / 2 1 N m u (rij ) zij
1
 
 Wzzff ,




sz 2 k   N L / 2 i 1 m   N L / 2 j 1 rij
rij
sz
2
(S27)
and
N L / 21 N m N L / 21
U fs
u fs ( zik ) zik
   
sz
zik sz
m N L / 2 i 1 k  N L / 2
1

sz
N L / 21 N m
N L / 21
u fs ( zik )
1
zik  Wzzfs ,



zik
sz
m N L / 2 i 1 k  N L / 2
(S28)
5
where Wzzff and Wzzfs are the fluid-fluid and the fluid-solid virials, respectively, applied in a
direction normal to the layer. By substituting Eqs. (S27) and (S28) into Eq. (S25), we obtain
N
sx / 2
sy / 2
1/ 2
Z N N
W ff  Wzzfs
N
 Z N  sz   
dxi 
dyi  d~zi zz
exp  U 
sx / 2
sy / 2
1 / 2
sz
sz
sz
i 1
N
W ff  Wzzfs
 Z N    dr N zz
exp  U .
sz
sz
(S29)
Finally, we get the “virial” expression of the adsorption stress by substituting Eq. (S29) into
Eq. (S9):
 ads  
N k BT
V

Wzzfs  Wzzff
V

Fzss
P.
A
(S30)
Then, the third alternative expression for the adsorption stress can be derived with the local
density profile of the guest particle, (r). Here, we rewrite Eq. (S18) as
U ff
s z

N L / 2 1
u ff (r0 k ) z0 k
k
d
r
d
r

(
r
)

(
r
)
  0  k 0 k r  r ,
k  N L / 2 2
0k
(S31)
where r0 and rk are the coordinates in the zeroth and the kth cells, r0k is the distance between
the two positions (i.e., r0k = | r0  rk |), and z0k is the z component of r0k. By assuming that the
guest particles have the uniform density (z) along the layer direction, Eq. (S31) can be
rewritten as
U ff
s z


u ff (r0 k ) z0 k 
k
d
z
d
z

(
z
)

(
z
)
d
x
d
y
d
x
d
y
  0  k 0 k  0  0  k  k r  r  , (S32)
k  Nl / 2 2
0k 

N l / 2 1
where x0, y0, and z0 are the x, y, and z components of r0, and xk, yk, and zk are those of rk. By
assuming the 12-6 Lennard-Jones (LJ) potential as the fluid-fluid interaction potential for a
pair of the guest particles, the multiple integration in Eq. (S32) can be calculated as
   ff
2

  r0 k
  12   6 
  dx0  dy0  8ff z  ff    ff  
 z0 k 
 z0 k  
  11   5 
 8ff  ff A ff    ff  ,
 z0 k 
 z0 k  
 24
u ff (r0 k ) z0 k
d
x
d
y
d
x
d
y
 0  0  k  k r  r0k   dx0  dy0  dxk  dyk    ff 2ff
14


   ff

 r0 k



8

z

(S33)
where ff and ff are the LJ parameters of the guest particles. Finally, we can obtain the
fluid-fluid interaction force. By substituting Eq. (S33) into Eq. (S32), we have
6

( k 1) h
k h

 
d
z
d
z
k  8ff  ff A  ( z 0 )  ( z k )
0 0 kh

k  N L / 2 2

N L / 2 1
U ff
s z
 
ff

z
 0 k
11
5

  ff  
  
  . (S34)

 z 0 k  
Then, the derivative of the fluid-solid interaction potential of the Steele 10-4 potential can be
calculated as
U fs

sz
Nk
N L / 2 1
 
k
i 1 k   N L / 2
u fs ( zik )
zik

   k  dz0  8s  fs  fs A  ( z0 )
0

k  NL / 2

N L / 2 1
h
11
5
 

  fs  
fs
  
 ,

 z0  kh 
 z0  kh  
(S35)
where fs and fs are the fluid-solid LJ parameters. By using Eqs. (S6), (S13), (S34), and
(S35), the third expression of the adsorption stress is obtained as
 ads
11
5


  fs  
  fs 


  
 
   k  dz0  8s  fs  fs  ( z0 ) 
0
z  kh 

k  N L / 2

 z0  kh  
 0


11
N L / 21


( k 1) h

k h
  ff 



 
d
z
d
z

8



(
z
)

(
z
)
  ff

0
k
ff ff
0
k


0
kh
2
z

k  N L / 2

 z0 k
 0 k 

N L / 21
h



5





(S36)
ss

Fz
 P.
A
B. Derivation of Eq. (3)
The layer-layer interaction, Eq. (3), is calculated by area integral of the LJ 10-4 potential.
The interaction between the layer and a particle included in the other layer, ul-pl, is expressed
as:
u
l - pl
 2   ss 10   ss  4 
(h)  2s ss  
 
 ,
 5  h 
 h  
2
ss
(S37)
where h is the distance between the layer and the particle. By integrating Eq. (S37) with
respect to the x-y layer directions, interaction between the layer in unit cell and the other layer
is obtained:
u ll (h) 
1 sx / 2 s y / 2
 s u l-pl (h)dxdy .
A  sx / 2  s y / 2
Here, h is independent of x and y directions, thus,
u ll (h) 
sx s y
A
 s u l-pl (h)
 2   ss 10   ss  4 
 2    
 
 
 5  h 
 h  
2
s ss
.
2
ss
7