Supplementary Information
Analytical representation of micropores for predicting gas
adsorption in porous materials
Aaron W Thornton*a, Scott A Furmana, Kate M Nairna, Anita J Hilla, James M Hillb and
Matthew R Hilla,c
Section 1: Heat of adsorption
The heat of adsorption is typically calculated through the thermodynamic equation [1],
  ln p 
h  R
 ,
  1 / T   n
(S1)
where R is the universal gas constant, p is pressure, T is temperature and n is the amount of gas in moles. This equation can
be used in a discrete form for experimental isotherms as a function of n as follows [2,3],
 p T  T 
hn   R ln  2  1 2  ,
 p1  T1T2 
(S2)
where p2 and T2 are chosen to match the amount adsorbed n for a second isotherm with p1 and T1.
Alternatively, S1 can be solved analytically if a function for p with respect to T is defined. First we will demonstrate for
the ideal gas law where pV = nRT. Hence we proceed,
 nRT 
ln p  ln 

 V 
 V (1 / T ) 
  ln 
,
 nR 
(S2)
 ln p
 T
 1 / T 
(S3)
h  RT
,
.
(S4)
The heat of adsorption is negative as this is a exothermic reaction while the absolute value of this quantity is known as the
isosteric heat of adsorption, hence RT. This means that any loss of heat is simply measured by a change in temperature.
From S1 we now calculate the heat of adsorption for a system governed by the equation of state for the adsorbed phase
(Equation 8 in the manuscript),
 nRT 
  PE *

ln p  ln 
exp 
 RT
V








  PE *
 nRT    

 ln  exp 
 ln 

 RT
 V   



1
 PE *  1 
 V  1 

   ln 
   ,
R T 
 nR  T  
(S5)
*
 ln p  PE

T ,
1 / T 
R
(S6)
h   PE *  RT
,
(S7)
hst  PE *  RT
.
(S8)
Therefore the isosteric heat of adsorption according to the proposed equation of state is identical to that used in simulation
[4,5].
Section 2: Partitioning of free volume
Slit-shaped cavity:
0
Vad  2 l
2
P
ad
  d
(S9)
0
0
Vbulk  2 l
2
 1  P
ad
    d
(S10)
0
Cylindrical cavity:
0
Vad  l 2 Pad   d

(S11)
0
0
Vbulk  l 2 1  Pad   d

(S12)
0
Spherical cavity:
0
Vad  4 2 Pad   d

(S13)
0
0
Vbulk  4 2 1  Pad   d

(S14)
0
where l is the arbitrary length of a repeatable section of the cavity.
Section 3: Density profile
To assist in a deeper understanding of the theory underlying the TIMTAM construct, here we present Figures S1a-d. Figure
S1a is a schematic that demonstrates the relationship between potential energy, probability of adsorption, pressure and
temperature. Figure S1b is an example that depicts the potential energy profile calculated within a spherical cavity which
2
represents a zeolite cavity with silicon and oxygen atoms smoothed across the surface. This potential energy profile is fed
into the probability functions predicting the phase likelihood within the cavity at 298 K, Figure S1c. Finally, the gas density
profile is calculated by multiplying the probabilities with their corresponding densities, Figure S1d. This demonstrates the
morphological detail captured by the continuous model.
Potential Energy for Adsorption (kJ/mol)
Fig. S1a Schematic for the TIMTAM approach. Potential energy with surface does not change with temperature, however the probability of
adsorption depends on temperature, since temperature affects the balance of kinetic and adsorption energies. Pressure simply increases the
number of gas molecules in each phase according to the appropriate equation of state.
2
0
-2
-4
-20
-10
0
10
20
Position within cavity (Å)
Fig. S1a Potential energy between a hydrogen molecule and the surface of a spherical cavity with radius of 20 Å. The cavity surface is
composed of oxygen and silicon atoms at densities of 0.09 and 0.04 no./ Å2, respectively.
3
Probability of phase (%)
100
80
60
40
20
0
-20
-10
0
10
20
Position within cavity (Å)
Fig. S1b Probability of phase existing within cavity at ambient conditions (298 K and 1atm) according to Equation (4) of the main article
and the potential energy profile within Fig S1a. Bulk gas phase (green line) dominates in the center of the cavity where potential energy is
close to zero and the adsorbed phase (red line) dominates close to the cavity surface where potential energy is strong.
Gas Density (g/L)
0.15
0.1
0.05
0
-20
-10
0
10
20
Position within cavity (Å)
Fig. S1c Gas density profile (black line), which is the total of both phase densities calculated by multiplying the probability profiles from
Fig S1b with their corresponding gas densities. Gas density for bulk gas phase (green line) and for adsorbed phase (red line) according to
Equations (7) and (8) in main article, respectively.
4
Section 4: TIMTAM representations and parameter values
IRMOF-1
Fig. S2 IRMOF-1 with yellow sphere that closely resembles the TIMTAM cavity representation.
Crystallographic information [6]:
Composition = OZn4(O2C-C6H4-CO2)3
Unit cell volume = 17237 Å3 (Dimensions: 25.83 × 25.83 × 25.83 Å)
Porosity = 0.78
Unit cell free volume = 13444.86 Å3
Density = 0.59 g/cm3
TIMTAM representation:
Cavity shape = Spherical
Cavity diameter = 21.8 Å
Number of cavities per unit cell = 8
Surface Density C = 0.064 No./ Å2
Surface Density O = 0.070 No./ Å2
Surface Density H = 0.032 No./ Å2
Surface Density Zn = 0.021 No./ Å2
5
ZIF-8
Fig. S3 ZIF-8 with yellow spheres that closely resemble the TIMTAM cavity representation.
Crystallographic information [7]:
Typology = SOD
Composition = Zn(mIM)2
Unit cell volume = 4905.2 Å3 (Dimensions: 16.99 × 16.99 × 16.99 Å)
Porosity = 0.48
Unit cell free volume = 2354.5 Å3
Density = 0.916 g/cm3
TIMTAM representation:
Cavity shape = Spherical
Cavity diameter = 18.7 Å
Number of cavities per unit cell = 2
Surface Density C = 0.117 No./ Å2
Surface Density N = 0.058 No./ Å2
Surface Density H = 0.175 No./ Å2
Surface Density Zn = 0.022 No./ Å2
6
ZIF-69
Fig. S4 ZIF-69 with yellow and orange cylinders that closely resemble the TIMTAM cavity representation.
Crystallographic information [9]:
Typology = GME
Composition = Zn(nIM)(cbIM)
Unit cell volume = 11435.7 Å3 (Dimensions: 19.4 × 26.08 × 26.08 Å)
Porosity = 0.56
Unit cell free volume = 6403.99 Å3
Density = 1.295 g/cm3
TIMTAM representation:
Cavity shape 1 = Cylindrical
Cavity diameter, length = 19.5, 19.4 Å
Number of cavities per unit cell = 1
Surface Density C = 0.202 No./ Å2
Surface Density N = 0.101 No./ Å2
Surface Density H = 0.242 No./ Å2
Surface Density O = 0.040 No./ Å2
Surface Density Zn = 0.020 No./ Å2
Surface Density Cl = 0.020 No./ Å2
Cavity shape 2 = Cylindrical
Cavity diameter, length = 12.5, 19.4 Å
Number of cavities per unit cell = 2
Surface Density C = 0.157 No./ Å2
Surface Density N = 0.078 No./ Å2
Surface Density H = 0.188 No./ Å2
Surface Density O = 0.031 No./ Å2
Surface Density Zn = 0.016 No./ Å2
Surface Density Cl = 0.016 No./ Å2
7
ZIF-76
Fig. S5 ZIF-76 with yellow and orange spheres that closely resemble the TIMTAM cavity representation.
Crystallographic information [9]:
Typology = LTA
Composition = Zn(IM)(cbIM)
Unit cell volume = 11651.1 Å3 (Dimensions: 22.67 × 22.67 × 22.67 Å)
Porosity = 0.66
Unit cell free volume = 7689.73 Å3
Density = 0.992 g/cm3
TIMTAM representation:
Cavity shape 1 = Spherical
Cavity diameter = 26 Å
Number of cavities per unit cell = 1
Surface Density C = 0.151 No./ Å2
Surface Density N = 0.060 No./ Å2
Surface Density H = 0.166 No./ Å2
Surface Density Zn = 0.022 No./ Å2
Surface Density Cl = 0.015 No./ Å2
Cavity shape 2 = Spherical
Cavity diameter = 24.8 Å
Number of cavities per unit cell = 1
Surface Density C = 0.083 No./ Å2
Surface Density N = 0.033 No./ Å2
Surface Density H = 0.091 No./ Å2
Surface Density Zn = 0.012 No./ Å2
Surface Density Cl = 0.008 No./ Å2
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