Generalization of the adsorption process in crystalline porous materials and its application to Metal-Organic Frameworks (MOFs)

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by  Alexander A. Aduenko

Generalization of the adsorption process in crystalline porous materials and its
application to Metal-Organic Frameworks (MOFs)
Jose L. Mendoza-Cortes1,2,3,∗ and Alexander A. Aduenko4
1Department of Chemical & Biomedical Engineering, FAMU-FSU College of Engineering,
Florida State University, Tallahassee FL, 32310, USA
2Scientific Computing Department, Materials Science and Engineering,
High Performance Materials Institute, Florida State University, Tallahassee FL, 32310, USA
3Condensed Matter Theory, National High Magnetic Field Laboratory, Tallahassee FL, 32310, USA
4Department of Control and Applied Mathematics,
Moscow Institute of Physics and Technology, 9 Institutsky lane, Dolgoprudny, 141700, Russia
5
In this paper we present an approach for the generalization of adsorption of light gases in crys-
1
talline porous materials. Our approach allows the determination of gas uptake considering only
0
geometrical constrains of the porous framework and interaction energy of the guest molecule with
2
the framework. The derivation of this general equation for the uptake of any crystalline porous

p framework is presented. Based on this theory, we calculated optimal values for the adsorption en-
e thalpyatdifferenttemperaturesandpressures. Wealsopresenttheuseofthistheorytodetermine
S theoptimallinkerlengthforatopologicalequivalentframeworkseries. Wevalidatethistheoretical
approach by comparing the predicted uptake to experimental values for MOF-5, MOF-14, MOF-
8
1 177, MOF-200, SNU-77H and Li-metalated MOF-177 and MOF-200. We obtained the universal
equation for optimal linker length given a topology of a porous framework. This work applies the

] generalequationtoMetal-OrganicFrameworks(MOFs)butitcanbeusedforothercrystallinema-
i terials such as Covalent-Organic Frameworks (COFs) and Zeolitic imidazolate frameworks (ZIFs).
c
s These results will serve to design new porous materials that exhibit high net storage capacities, in
- particular for molecular hydrogen (H ).
l 2
r
t
m
I. INTRODUCTION sections IIA and IIB, we derive the general equation for
.
t adsorptioninanyframework,givenitsgeometricproper-
a
ties (i.e. void volume, pore volume, surface area and
m
Wepresentanapproachthatallowsthedetermination pore size) and enthalpy of adsorption with the guest
- of gas uptake in crystalline porous materials considering molecule (i.e. interaction energy with the surface). We
d
n only geometrical constraints for the crystal and its inter- also describe an approach to find the effective number
o actionenergywiththeguestmolecule. Thederivationof of layers in the more realistic restricted multilayer the-
c thisgeneralequationfortheuptakeofcrystallineporous ory (RMT). Frameworks with several types of topolo-
[ frameworks was validated with experimentally available gies (cells) are also considered. In subsection IIIA, we

3 data for Metal-Organic Frameworks (MOF). We applied compare our predictions of adsorption isotherm to ex-
v this general equation to calculate the net storage capac- perimental measurements for system temperatures T =
0 ity of molecular hydrogen storage (H2) in MOFs. In this 77,243,273and298 K. In subsection IIIB we derive the
3 work we propose the ideal interaction energy and the optimal enthalpy of adsorption and linker-length/void-
6 ideallinkerlengthforMetal-OrganicFrameworks(MOF) volume for the highest net gas storage capacity between
2
given a topology (or crystalline lattice Figure 1), how- atmospheric pressure and the maximum pressure a tank
.
1 ever, in principle, this approach can be extended to any can achieve.
0 crystallineporousmaterial. Currentlytherearenotsim-
4 ple guidelines for the optimal design of the linker length
1
and the storage capacity of porous frameworks. These II. METHODOLOGY
:
v results will serve to design materials that maximize net
Xi gas storage capacity. A. Restricted multilayer adsorption theory
r This paper is organized into several sections. In sub-
a
The derivation of the general formula for the adsorp- available for our system.
tion process is solely restricted to the topology of the The first set of theories available to estimate the ad-
framework. We define the topology of a framework as sorptionpropertiesofsurfacesweretheLangmuirtheory,
theconnectivityinalldirectionstomakeaperiodicstruc- published in 1916, followed by the Brunauer-Emmett-
ture. AsimpleexampleisshowninFigure1. Thistopol- Teller (BET) theory, published in 1938 [1, 2]. However
ogy is going to determine the number of absorption sites these two theories, though very useful, have limited ap-
plication to crystalline porous materials. The Langmuir
theory assumes that only one layer of sorbate molecules
can be formed on the surface, whereas the BET theory
∗ [email protected] assumesthataninfinitenumberoflayerscanexist. Both
2
BO OB
O O
OBO
OBO
O O
BO OB
BO OB BO OB
O O O O
OBO OBO
OBO OBO
O O O O
BO OB BO OB
BO OB
O O
OBO
OBO
BOO OOB Topological Model Minimum
Porous Material Porous Material of the porous material representation
Figure 1: A Porous Framework in converted into a topological construction, i.e. in only edges and nodes
teractions in the system, which are
1. Those between the surface (n = 0) and first layer
molecules (n = 1), with partition function j
1
2. Those between first layer molecules (n = 1) and sec-
ond layer molecules (n = 2), with partition function j ,
2
and
3. Thosebetweennth(n>1)layermoleculesandnth+1
layer molecules, with partition function j .
∞
This creates a columnar structure of bound molecules.
Additionally, we assume that there are no interactions
between molecules of neighboring columns and the par-
titionfunctionj isequaltoj . Giventhetotalnumber
2 ∞
of molecules, N, then N of them are in the first layer
1
andN =N N areinhigherlayers. Fromthesesetof
n 1
−
assumptions, we can derive the adsorption behavior for
a restricted number of multilayers (n layers).
Using the grand canonical partition function, we have
Figure 2: Restricted number of multilayers without
(cid:88)
(left) and with (right) absorbed gas. Ξ(B,µ,T)= exp[(Nµ)/kT]Q(N,B,T)
N≥0
(cid:88)B (cid:88)N1 (cid:88)N2 N(cid:88)n−1 B!
= ...
assumptionareunphysicalforacrystallineporousmate- (B N )!N !
1 1
rial,whichcanusuallyhostmorethanonelayer,butnot N1=0N2=0N3=0 Nn=0 −
an infinite number. Thus, a more appropriate approach N1!
...
is needed to treat the case of adsorption in crystalline ×(N N )!N !
1 2 2
−
porousmaterials,whichwecallrestrictedmultilayerthe- Nn−1!(cx)N1xN2+...+Nn
ory(RMT)inthiswork. ThederivationoftheLangmuir .
× (N N )!N !
and BET theories can be found in the Suplementary In- n−1− n n
formation (Section VI) or elsewhere [1, 2]. We will show In the preceding equations, µ is chemical potential, T is
belowthattheequationforarestrictednumberofmulti- temperature and k is Boltzmann constant.
layersisamoregeneralequationthatgivestheLangmuir
and the BET theory as solutions. Summing in turn over N , N , ..., N we find
n n−1 1
Restricted Multilayer theory (RMT) For the general
Ξ=[1+cx(1+x+x2+...+xn−1)]B
system presented in Figure 2 we posit that there are B
equivalent sites, and N molecules can be distributed in (cid:20) (cid:18)1 xn(cid:19)(cid:21)B
= 1+cx −
them. We assume that there are only three types of in- 1 x
−
3
where we have defined, elsewhere. However, weshouldnotethatedgeeffectsare
significant only for nets for which the linker length is
c=j1/j∞, x=j∞exp(µ/kT). nearly the same as the diameter of guest gas molecule.
Most known synthesized frameworks have linker lengths
Using this equation in conjunction with the identity, that are significantly larger than the hydrogen molecule
diameter, thus the edge effects should be negligible for
(cid:18) (cid:19)
∂logΞ
N =kT , current synthesized materials.
∂µ
T,B
Finally, we obtain the equation for uptake, which can
we find the equation for the adsorption of a restricted
be applied to any type of framework with only one type
number of multilayers:
of conventional unit cell.
N cx[1 (n+1)xn+nxn+1]
= − . (1)
B (1 x)(1 x+cx cxn+1)
− − −
γµV S N
p
Thisequationismoregeneralbecausethesubstitution m= (n˜). (2)
N πr2 V B
of n = 1 or n = gives the Langmuir and BET equa- A w p
∞
tions, respectively (Sup. Info. VI). Similar results were
S and V in equation (2) are pore surface area and pore
p p
obtained earlier [4, 5], however in these treatments, the
volume, respectively.
topology of the framework, along with other geometrical
constraintsandthespecificgaspropertieswerenottaken
N
into account. We do this in the following section. The occupancy ratio (n˜) is defined using the
B
RMT(1)byassumingthatj andj arelinkedwithen-
1 ∞
thalpiesofadsorptiononthefirst∆H layerandhigher
ads
B. General equation for estimation of the layers ∆H∗ as follows (for more details see Supp. Info
ads
molecular uptake for any framework using restricted VI B).
multilayer theory (RMT)
j =exp(∆H /RT), j =exp(∆H∗ /RT)
1 ads ∞ ads
Here we derive a general equation for the uptake, m,
definedasthetotalmassofgasthatcanbeincorporated
per unit volume of porous framework (in units of g/L). Let us derive a more general equation for frameworks
This equation applies for any framework based on the which have M types of the conventional unit cells with
RMT of adsorption and the geometric constraints of the volumes and surface areas Vi and Si, respectively. We
p p
framework. also introduce factors γ , ..., γ for the fraction of vol-
1 M
Given that the molar mass of hydrogen M = 2.018 ume occupied by every type of the cell.
g/mol, the uptake per V =1L=1027˚A3 equals
M
(cid:88)
γ = γ .
i
N γV N i=1
m=M =M B (n˜),
N V N B
A p A Applyingequation(2)toeverytypeofcellseparately,we
get the total uptake
whereV isthevolumeofapore,B isthenumberofsites
p
perpore, N isanumberofadsorbedhydrogenmolecules
per 1 L of adsorbent and N is the Avogadro constant.
A
The factor γ corresponds to the fraction of free (or void)
volume for the framework, i.e. void volume = γV. n˜ m=(cid:88)M γiµV Spi N(n˜). (3)
is the effective number of layers (Supl. Info. section N πr2 Vi B i
VIII).WealsoconsiderD+δ astheeffectivediameterof i=1 A w p
the absorbed molecule which equals 2r = 2 2.76 ˚Afor
w
·
hydrogen(seeSup. Info.). Usingthismodelitispossible
to estimate the number of adsorption sites B as Thisequationisverygeneral. Infact,itcanbeusedfor
thechallengingproblemofpredictingtheuptakeofmulti-
S variate Metal-Organic Frameworks (MTV-MOFs) where
B = ,
πr2 the pore volume changes due to the different function-
w
alities of its linkers [3], thus generating a supercell of
where S is a surface area of one cell. different types of cells where the uptake is not the aver-
Strictly speaking we should also consider edge effects age,butamorecomplexrelationship. Theapplicationof
when one hydrogen molecule occupies two or more dif- this equation to the different MTV-MOFs is beyond the
ferent sites. These considerations will be accounted for scope of the current work.
4
III. RESULTS AND DISCUSSIONS in [9], thus suggesting that an effective ∆H should be
ads
used rather that the one at low coverage as reported in
experiments (see Sup. Info. IX).
A. Comparison to experiments. T=298 K. We also compare our theory with Grand
Canonical Monte Carlo (GCMC) simulations for Li-
metalated frameworks. Han et al. in [18] considered Li-
T =77 K. In order to validate our results, we com- metalated frameworks with high enthalpy of adsorption
pared our predictions to experimental data for differ- at room temperature. This enthalpy decreases signifi-
ent MOFs at T = 77 K. We took 6 reported frame- cantlywithincreasinguptake. Wetookthischangeinen-
works, namely MOF-5, MOF-177, MOF-14, SNU-77H thalpyintoaccountanddefinedaneffectiveenthalpy(see
and MOF-200 for this comparison. Data over geometric Suppl. Info. IX)foreveryconsideredpressure. Wemade
properties and adsorption enthalpy was taken from pub- an assumption that introducing lithium does not signif-
lisheddata. Weusedtherestrictedmultilayeradsorption icantly influence the geometric properties of the frame-
equation Eq. (3) for the calculations. We present the re- work, as suggested in [18], and therefore we used the
sults we obtained for these framework in Table I. The same geometric parameters of the non-metalated frame-
agreement between experimental and theoretical uptake works. Thetheoreticalresultsforthethreedifferenttem-
is within 4 % at this temperature. In Figure 3, we also peratures T = 243 K, T = 273 K and T = 298 K were
compared the experimental and theoretical full isotherm compared with simulations (see Figure 4). The theoret-
at T = 77 K for MOF-5 and MOF-177. The agreement ical predictions are in good agreement with the GCMC
is fairly good, however there are some discrepancies in resultsintheentirepressurerange. Theaverageabsolute
the pressure range of 5-30 bar. This can be explained deviationfortwocompoundsarereportedintableII;this
by the slight decrease of adsorption enthalpy reported error does not exceed 2.2 %.

30
30
25
L L25
/ /
g g
e,20 e,20
k k
a Approximation from theory a Approximation from theory
upt15 Experimental value from [6] upt15 Experimental value from [6]
s s
es10 es10
c c
x x
E E
5 5
0 0

0 10 20 30 40 50 0 20 40 60
P/P0,bar P/P0,bar
(a)MOF-5,77K (b)MOF-177,77K
Figure 3: Comparison of theory and experiments for hydrogen isotherm in (a) MOF-5 and (b) MOF-177 at T=77 K.
Table I: Experimental and theoretical excess uptake for different MOFs given their geometrical properties. ∆H is
ads
given as the enthalpy of adsorption, a is the characteristic length of the cell in a framework, and S is the surface
A
area.
Compound a, ˚A Void vol., γ n˜ S , m2/g V , cm3/g ∆H , kJ/mol P, bar Exp., g/L Theor., g/L Err.,%
A p ads
MOF-5 12 [8] 0.798 2 2900 [8] 1.04 [8] 4.8 [16] 50 31.0 [9] 31.22 +0.7
MOF-14 7.7, 14 [12] 0.67 [12] 2 2000 [12] 0.71 [13] 7.0 [13] 40 28.1 [13] 27.25 -3.0
SNU-77H 8 [14] 0.69 [14] 2 3900 [11] 1.52 [11] 7.05 [11] 90 47.4 [11] 49.26 +3.9
MOF-200 18 [8] 0.90 [15] 3 6400 [15] 3.59 [15] 3.46 [17] 90 16.5 [15] 17.18 +4.0
MOF-177 8.5 [9] 0.83 [8] 1 4740 [8] 1.59 [11] 4.4 [9] 70 32.0 [9] 31.17 -2.6
B. Optimal parameters of the framework for the drogen as possible at some fixed pressure P (usually 100
highest gas delivery using the RMT bar), but also to deliver the maximum amount of hy-
drogen to the surrounds upon depressurization, which is
knownasthedeliveryamountofthestoragedevice. This
Theultimategoalofmakingporousmaterialsforstor- is defined as the difference in uptake between the atmo-
ing hydrogen is not only to store as much molecular hy- spheric pressure, P (usually 1 bar), and the maximum
0
5
20
20
20
L L L15
/ / /
e,g15 e,g15 e,g
k k k
upta10 ASipmpurolaxtiemd avtaiolune f rforomm t h[1e6o]ry upta10 ASipmpurolaxtiemd avtaiolune f rforomm t h[1e6o]ry upta10 ASipmpurolaxtiemd avtaiolune f rforomm t h[1e6o]ry
s s s
es es es
Exc 5 Exc 5 Exc 5
0 0 0

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
P/P0,bar P/P0,bar P/P0,bar
(a)MOF-200-27Li,243K (b)MOF-200-27Li,273K (c)MOF-200-27Li,298K
35
35 30
30
L30 L L25
/ /25 /
g25 g g
ake,20 Approximation from theory ake,20 Approximation from theory ake,20 Approximation from theory
upt15 Simulated value from [16] upt15 Simulated value from [16] upt15 Simulated value from [16]
xcess10 xcess10 xcess10
E E E
5 5 5
0 0 0

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
P/P0,bar P/P0,bar P/P0,bar
(d)MOF-177-15Li,243K (e)MOF-177-15Li,273K (f)MOF-177-15Li,298K
Figure 4: Theoretical and simulated adsorption isotherms for (a-c) MOF-200-27Li and (d-f) MOF-177-15Li.
Table II: Average absolute deviation of excess uptake in theory is further applied for high temperatures (for in-
g/L for MOF-177-15Li and MOF-200-27Li stance,forT =298K),justbyconsideringthefullchem-
ical potential (Supl. Info. XI). One may use the full
Compound T =243 K T =273 K T =298 K chemicalpotentialforthelowtemperatureregionaswell.
Howeverinthisregiontheequilibriumratioofpara-and
MOF-177-15Li 1.80 (6.0%) 2.12 (7.1%) 1.67 (5.6%)
ortho- hydrogen is far from 3:1. This implies full quan-
MOF-200-27Li 1.02 (5.0%) 1.21 (6.0%) 0.40 (2.0%)
tummechanicalconsiderationsoforthoandpara-hydro-
gen without simplification (see Suppl. Info. XI, XII).
However this does not change the result significantly be-
pressure of the tank, P (typically 100 bar). Therefore, causetherotationaldegreesoffreedomarenotactivated
obtaining the maximum delivery amount is the most im- at low temperature.
portant characteristic of a storage device. We propose
doing this by obtaining an optimal enthalpy of adsorp-
tion ∆Hopt. Porous materials with low ∆H do not According to our theory, geometric properties of a
ads ads framework only define some factors to the uptake. The
store enough gas, even at the maximum pressure P. On
shape of the isotherm is defined by the factor ν =
the other hand, if ∆H is too high the framework re-
ads N/B(n˜)fromtheRMT.Thustogetthehighestdelivery
tains all the adsorbed gas at low pressure, and the gas
amount for fixed number of layers (fixed geometry) one
cannotbeusedeither. Inbothcases,thedeliveryamount
must derive the enthalpy ∆H that leads to the high-
is negligible. ads
est possible difference in the factor ν between pressure
We first derive equations for optimal enthalpy of ad-
P and P.
sorption, ∆Hopt, using RMT. Then, we present an ap- 0
ads
proach of determining the optimal geometric properties
∆ν =[ν(P) ν(P )] max .
0
(i.e. effective linker length) for a given series of frame- − →∆Hads
works.
The function ν(P) is smooth, thus the necessary condi-
Enthalpy of adsorption ∆H . First, we find opti-
ads tion to obtain the maximum is
mal values of ∆H in a low-temperature region (for
ads
instance,forT =77K),whererotationaldegreesoffree- ∂∆ν (cid:0)∆Hopt(cid:1)=0. (4)
dom are not significantly activated. Nevertheless this ∂∆H ads
ads
6
(cid:113)
Forconvenienceweintroducethemultipliermdefinedas b= 1 (n+1)mnβ˜n+nmn+1β˜n+1.
P =mP − 2 2
0
We also define
Particularly for the high-temperature region, (e.g.
β =exp ∆H /(RT) , β =exp ∆H∗ /(RT) ,
1 { ads } 2 { ads } T=298 K), and the associated value of α (5) we get
(cid:16) (cid:16) (cid:17)(cid:17)
β˜1 =αβ1, β˜2 =αβ2, ∆Haodpst =RT lnβ1 =RT ln β˜1T7/2/ α0T07/2 , (7)
where α =5.1 10−6, T =298 K and
0 0
c=β /β , x=αP/P β . ·
1 2 0 2
β˜ =α (T /T)7/2β .
Here,H∗ istheinteractionoffirstlayerwiththesecond 1 0 0 1
ads Thus for the RMT, for example at T = 77 K, P =
layer of adsorbed molecules. 100 bar, n = 3, we have ∆Hopt = 3.81 k0J/mol. On the
ads
other hand, at T = 298 K, P = 100 bar, n = 3, we have
P 0
α= 0 exp(µ/kT) ∆Hopt = 24.5 kJ/mol.
P ads
is a function of temperature. For the low temperature The optimal ∆ν is defined as ν(P) ν(P ) where β˜
0 1
−
region chemical potential is defined by a translational is denoted optimal given by Eq. 7. The optimal ∆ν for
part (µ≈µtr) and n=1ortheLangmuircasecanbeobtainedifβ˜2 =0and
therefore a=b=1 in the previous equations. Similarly,
(cid:18) (cid:19)−3/2
P µkT theoptimal∆νforn= ortheBETcasecanbederived
0
α= kT 2πNA(cid:126)2 . by excluding powers of∞β˜2 higher than first power and
setting therefore a = b = 1. This is another example
Forthehightemperatureregion,whenrotationaldegrees wheretherestrictedmultilayertheoryisthegeneralcase
of freedom are fully activated for the Langmuir and BET theories.
(cid:18) (cid:19)−3/2(cid:18) (cid:19)
P µkT 2T Now, to illustrate the consequences of these equations
0 r
α= . (5)
kT 2πN (cid:126)2 T we first plot optimal enthalpy at different temperatures
A
and different number of layers (n = 2,3,6) in Figure 5
Here µ is molar mass of gas, T is the activation temper- and show some of the values in table III.
r
ature for rotational degrees of freedom, and P =1 bar.
0
Deriving ∂∆ν/∂∆H for the RMT gives the values
ads
foroptimalenthalpy∆Hopt. Thus,theoptimalenthalpy
for the restricted multialdasyer case is given by eq. (6). Table III: Optimal enthalpy ∆H∗ for different
ads
temperature T and number of layers n for P =100 bar
∆Hopt =RT lnβ =RT lnβ˜ /α, (6)
ads 1 1
Number of layers n T =50 K T =77 K T =298 K
where
1 2.02 3.80 24.48
a(1 mβ˜ ) b√m(1 β˜ ) 2 1.99 3.80 24.48
β˜ = − 2 − − 2 ,
1 b√m(1 β˜n) am(1 mnβ˜n) 3 1.99 3.81 24.48
− 2 − − 2 6 1.99 3.81 24.48
(cid:113)
a= 1 (n+1)β˜n+nβ˜n+1,
− 2 2
This table shows the similar performance for different ference can be much more significant when applying the
amountoflayers. ThusfiguresarethesameforRMTfor different theories. The safe way, of course, is to use the
any number of layers. In other words, the plots for n = RMT.
2,3,6looksimilarandweonlyshown=3. Discrepancies Optimal length of the linker and void volume. In the
are noticeable only at very low temperatures and with
previous section we obtained the optimal value of en-
very different number of layers. From (Eq. 6) we find
thalpy which influences the shape of the isotherm. Here,
that the optimal enthalpy increases with temperature.
we present an approach to define optimal linker length
for a series of frameworks. The linker length does not
Finally, we plot the results for the optimized value for
influence the shape of the curve, but it does influence
∆ν = ∆(N/B) which directly influences the (see Fig-
its asymptotic value (when P ). This is because
ure 6). These results show that the difference between
→ ∞
the pore volume V , surface area S and fraction of void
all three cases becomes significant only at low temper- p p
volume γ depend on the linker length.
ature. However, one must be cautious for other gases,
which have higher ∆H∗ than hydrogen and this dif- Togettheoptimallinkerlengthweconsiderthegeneral
ads
7
100 300
7 30
90 290
28
6
280
80 26
T,K 70 5kJ/molT,K270 24kJ/mol
4
260
22
60
3 250
20
50 2 240 18
20 40 60 80 100 20 40 60 80 100
P/P , bar P/P , bar
0 0
(a)Restrictedmultilayercase,50-100K,n=3 (b)Restrictedmultilayercase,240-300K,n=3
Figure 5: Temperature and pressure dependence of optimal enthalpy ∆Hopt for low and high temperature region
ads
and n=3.
0.8
2.5
0.8
2 0.6
0.6
B1.5 B B
/ / /
N N N0.4
Δ Δ0.4 Δ
1
0.2
0.5 0.2
Multilayer Multilayer Multilayer
Restr.multilayer, n=2 Restr.multilayer, n=2 Restr.multilayer, n=2
Monolayer Monolayer Monolayer
0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100
P/P0,bar P/P0,bar P/P0,bar
(a)T=50K (b)T=77K (c)T=298K
N
Figure 6: ∆ for different temperatures
B
equationforuptakeonemoretime(Eq. 3)and,usingone volume ofthe netgrows onlywithlinkerlength, whereas
type of cell (M =1), we obtain (Eq. 2), the cross section of linkers is constant.
γµV S N
p
m= (n˜).
N πr2 V B
A w p
Thus we get
The factor (N/B)(n˜) has been considered in the pre-
vious section. Now we consider the second one V κ
net
γ =1 =1 , (8)
S − V − a2−ξ
p
γ
Vp where κ is some constant depending on the framework.
We also get S /V = c/a, where c is constant for ev-
since all the other terms are constant and do not depend p p
ery framework not changing when enlarging the linkers.
on the geometrical characteristics of the framework. We
Thus,
assume that the framework has one characteristic length
a, which is the length of the linker. We assume that S (cid:16) κ (cid:17) c
p
f(a)=γ = 1 max.
tThoetanlevtocluanmebeisexgpivaenndeads bVasedao3n, aburtedfuorcetdheunviotlucemlle. Vp − a2−ξ a → a
∼
of the framework itself we suggest Vnet a1+ξ, where This gives
∼
ξ indicates how the volume of the net grows with the
growth of a. For instance, if ξ = 0, it means that the a2−ξ =(3 ξ)κ
−
8
and therefore
0.9 −1.4
Initial dataset
1 0.88 Approximation
daTtaheiffγactisorknκocwannafbo=er(ea(at3s−illeyaξs)dtκe)ofi2nn−eeξdl.efnrogmtheoxfptehreimliennk(t9ear)l Voidvolume,γ0000...888.2468 IAnpitpiarol xdiamtaasteiotn log(1γ)−−−11−..286
a or can be measured directly for any framework just 0.78
−2.2
by using geometry. Therefore, we obtain the universal 0.76
5 6 7 8 9 10 11 1.6 1.8 2 2.2 2.4
recipe of best linker length under these assumptions. Vp/Sp˚A log(Vp/Sp)
To show an example, given ξ = 0, for any framework (a)Fitinnormalaxes (b)Linearfitinlogarithmicscale
the best value is
Figure 7: Void volume dependence on linker length.
κ
Notice the logarithmic axes in (b) and its direct
γ =1 =2/3.
− (√3κ)2 correlation
Nowweapplyequation(9)tothetopologicallyequivalent
IR-MOFs, IRMOF-1 to IRMOF-16. This will determine For compounds that are isoreticular to MOF-5 (topol-
the optimal length of the linker for such frameworks and ogy pcu), we found that the optimal effective linker
therefore the optimal void volume. First we determine length is 1.95 ˚A and the enthalpy of adsorption is 3.75
the model parameters κ and ξ. To get this we several kJ/mol at 77 K and 24.5 kJ/mol at 298 K, respectively,
calculations that follow. Rearranging Eq. 8 we obtain which will give a maximum expected delivery amount of
81.8% or 34.65 g/L.
log(1 γ)=logκ (2 ξ)loga.
− − −
Now, using the data over the fraction of void volume γ
V. SUPPORTING INFORMATION.
and the effective linker length V /S for the considered
p p
IRMOFs, we obtain ξ and κ using a linear regression
Full derivations for normal non-interpenetrated and
(Figure 7a). The regression has a high correlation coeffi-
cient,R2 =0.9524showingthesignificanceofthedepen- interpenetrated frameworks, low and high temperature
considerations, analyses of different multilayers and en-
dence of the void volume and the effective linker length.
thalpies of the adsorption, and other comparisons to ex-
Thus for this familyof MOFs, weobtain logκ= 0.015,
− periments. This material is available free of charge on
2 ξ =0.932 and the estimations for the coefficients are
− the Internet at http://pubs.acs.org/.
κ=0.966, ξ =1.068.
Thus we get the optimal void volume γ =0.4824. and
theoptimaleffectivelinkerlength(V /S )=1.95˚A.Note
p p
that the effective linker length is not the same as the
linker length alone. For instance, for a cubic cell, the
effectivelinkerlengthisa3/(6a2)whichis6timessmaller
than the linker length a. The closest one to the reported
figures is IRMOF-5 with γ =0.5 and a =2.05 ˚A, which
correspond to 98.5% of the predicted optimal uptake for
this framework.
Theoptimalquantityforeffectivelinkerlengthisclose
to the value for IRMOF-5 and therefore further im-
provements for the IRMOFs should not consider the
linkerlengthbutinsteadtheenthalpyofadsorption(24.5
kJ/mol, for n = 1 and T = 297 K, given that not more
thanonelayerofhydrogencanbeinsertedinsuchpores).
IV. CONCLUSIONS
We presented a theory for a realistic restricted multi-
layer adsorption of gases in crystalline porous materials.
We applied this theory to estimate molecular hydrogen
adsorption in Metal-Organic Frameworks (MOFs) and
our approximations predict uptakes that agree with ex-
perimental values at 77 K and 298 K. We found that the
results for this theory can differ from the Langmuir and
BET theories at low temperatures.
9
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