Why Pore Width of Nanoporous Carbon Materials Determines the

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Why Pore Width of Nanoporous Carbon Materials Determines the Preferred Solvated States of Alkaline Cations: A Density Functional Theory Calculation Study Takashi Yumura, Marie Ishikura, and Koki Urita J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b03568 • Publication Date (Web): 12 Aug 2019 Downloaded from pubs.acs.org on August 13, 2019

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The Journal of Physical Chemistry

Why Pore Width of Nanoporous Carbon Materials Determines the Preferred Solvated States of Alkaline Cations: A Density Functional Theory Calculation Study

Takashi Yumura,1, * Marie Ishikura1 and Koki Urita2

1Faculty

of Materials Science and Engineering, Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto, 606-8585, Japan 2Graduate School of Engineering, Nagasaki University, 1-14 Bunkyo-machi, Nagasaki 852-8521, Japan

*To whom all correspondence should be addressed. E-mail; [email protected]

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The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract:

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Dispersion-corrected DFT calculations were performed to investigate

energetically stable structures of a Li+ cation solvated by some PC molecules inside an AB-stacked graphene bilayer (2G) with a variant interlayer spacing (L ranging from 5 to 8 Å), denoted by PCn−Li+@2G(L), where n is the number of PC molecules (1 ≤ n ≤ 3). The 2G(L) structures model slit-type pore geometries of microporous carbons, where the interlayer spacing corresponds to their pore width.

DFT calculations found L− and

n−dependences of stabilized PCn−Li+@2G(L) structures due to different degrees of CH− and cation− interactions between a graphene bilayer host and a PCn−Li+ guest. Since the interlayer space of 2G(L) at L = 6 Å is the smallest limit that can accommodate a single PC molecule, the restricted interlayer space can be utilized to selectively stabilize a PC1−Li+ structure due to attractive cation− interactions and negligible CH− interactions.

Within a larger interlayer space of graphene bilayers (L

≥ 7 Å), attractive cation− and CH− interactions between the host and guests are significant for stabilizing inner PCn−Li+ structures (n ≥ 3).

Similar structural

preferences can be found in the solvated state of Na+ cations within the interlayer space of 2G(L) at L = 6 Å.

By using the distance of a Li+ cation from a carbon layer in the

PCn−Li+@2G(L) structures compared with that in the optimized PC4−Li+ structure on a graphene monolayer (1G) as a model of external and macroporous carbons, the capacitances of PCn−Li+@2G(L) structures relative to that of PC4−Li+ structure on 1G were approximately estimated.

The estimated capacitance of the PC1−Li+ structure

stabilized inside 2G(L) at L = 6 Å is almost twice as large as that in PC4−Li+ on the 1G structure.

Accordingly, the DFT calculations revealed that the confinement of

PCn−Li+ structures within microporous carbons whose pore size is around 6 Å significantly enhances the capacitance relative to those in external or macroporous carbons due to the selective formation of the significantly desolated Li+ cations.


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Introduction Electric double-layer supercapacitors (EDLCs), which consist of positive and negative electrodes together with electrolytes,1-3 are promising candidates for devices for energy storage and power management.

Porous carbons, which have high specific

surface area, were mainly utilized as the electrode materials.4-11

In EDLCs, electrical

charge is stored by the presence of ions near the surface of the electrodes due to electrostatic interactions.

The charge storage on the surface of EDLCs results in the

generation of moderate energy.

A model of electric double-layer (EDL) was originally

proposed by Helmholz who assumed that a single layer of counterions adsorbed on a flat charged surface.12,13

Then, the capacitance (C) normalized by the accessible

electrode surface area (A) is given in the following equation,

C ∕ A = 0r ∕ d

[1]

where 0 and r are the dielectric constants of the vacuum and electrolyte, respectively, and d is the double-layer thickness.

Eq. (1) suggests that the capacitance is inversely

proportional to the double-layer thickness.

As is well known, porous carbons have

curved and highly nanoporous carbon networks, consisting of macropores (diameter > 50 nm), mesopores (diameter: 2−50 nm), and micropores (diameter ≤ 2 nm).3

To

model the double-layer formed in nanoporous carbons, the Helmholz model has been modified by assuming that carbons have slit-type pore geometries.

As a result, various

models, such as double-cylinder and slit-type capacitor models, have been proposed.14-20



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These models can suggest that the capacitance is linked to the pore size of carbons and an effective radius of ions.14-20 　 Although the importance of the carbon pore size in the capacitance was expected, indispensable prerequisite for porous carbon electrodes is that their pores can accommodate electrolytes.

In this point mesoporous carbons have enough space, and

thus they have been believed to be ideal for EDLCs.

However, recent studies

suggested the accessibility of restricted space of micropores to partially or fully desolvated ions.21-27

More interestingly, an anomalous increase of the capacitances of

microporous carbons was observed compared with those in mesoporous carbons.

To

understand the anomalous capacitance increase in microporous carbons, molecular dynamics (MD) simulations have been performed.28-37 investigated

the

state

of

ions

of

an

ionic

For example, MD simulations liquid

electrolyte

(1-butyl-3-

methylimidazolium (BMI+) hexafluorophosphate (PF6−)) in microporous carbons.28 As a result, Ref. 28, suggested that desolvated ions are formed inside microporous carbons, leading to the capacitance enhancement. Similar desolvation of ions was observed by Urita et al., who used LiClO4/PC (PC = propylene carbonate) as a different electrolyte inside carbon electrodes with various micro/mesoporous ratios.26,27

Note that their carbon materials contain slit-type

micropores whose sizes are less than 0.82 nm with a maximum distribution of approximately 0.6 nm.

7Li-NMR

measurements clarified the solvation number of PC

molecules for Li+ cations inside the activated carbons.

In fact, Li+ cation are solvated

by 1~2 PC molecules inside microporous carbons, whereas the solvation number of PC molecules for Li+ cations on the external surface or in the macropores is approximately 4 ACS Paragon Plus Environment

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4.

The results indicate that desolvation of Li+ cations occurs in the restricted space of

micropores.

Furthermore, Ref. 26 reported that the presence of the significantly

desolvated Li+ cations in micropores results in the capacitance enhancement, because the Li+ cations are expected to more closely approach carbon surfaces. Despite the interesting experimental reports, questions arise as to why desolvated Li+ ions can exist inside micropores as well as how the pore size affects energetically preferable solvation states of inner Li+ cations, and their double-layer thicknesses, being a key parameter to determine the capacitance.

To answer these

questions, the current study performed quantum chemistry calculations based on density functional theory (DFT) instead of MD simulations.

Advantage of the use of DFT

calculations is that one can directly obtain geometrical information on PC-solvated Li+ cations inside nanometer-scaled carbon cavities, especially the distance of Li+ cation from a carbon surface, as well as key interactions in determining the inner solvated structures.

Information obtained from DFT calculations would be important to

understand the capacitance enhancement in microporous carbons, experimentally obtained in Ref. 26.

For the above purpose, the current study used graphene bilayers

with a variant interlayer spacing to model slit-type pore geometries of microporous carbons.

The interlayer spacing in the graphene bilayer models corresponds to the slit-

pore width of microporous carbons.

In DFT calculations, we will focus on roles of the

interlayer spacing in energetically preferable solvation states of a Li+ cation within a graphene bilayer, as well as the distance of a Li+ cation from a graphene layer, which have not been well discussed in previous experimental studies and MD-based computational studies. 5 ACS Paragon Plus Environment


Method of Calculation

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We employed dispersion-corrected DFT calculations

implemented in Gaussian 09 code38 to understand energetically preferable solvated states of an alkaline (Li+ or Na+) cation within slit-type pore geometries of microporous carbons.

To correctly obtain their structural features, where long-range host-guest

interactions occur, we utilized the dispersion-corrected GGA B97 function (B97D) in our DFT calculations.39,40 According to Ref. 41, B97D calculations can yield potential energy surfaces in benzene dimers and between methane and benzene, consistent with those obtained from couple cluster using perturbative triple excitations (CCSD(T)) that are accurate for evaluating dispersion interactions.

Note that dispersion-corrected DFT

calculations successfully revealed key interactions in determining the structures for conjugated guest molecules inside carbon nanotubes.42-46 With respect to models for microporous carbons, Seaton first used parallel-sided slit bounded by semi-infinite slabs of graphite,47 because transmission electron microscopy (TEM) found that carbon pores are approximately slit-shaped.

We also

found slit-shaped carbon pores in the TEM image of pitch-based activated carbons, as shown in Supporting information (Figure S1(a)).

After the first modelling of

microporous carbons proposed by Seaton, slit-shaped porous carbons modeled by infinite- or finite graphene bilayers have been used up to now.48,49

Other porous

carbon models that represents microporous and mesoporous were proposed by Biggs,50 which called solid-like models.

Although various porosity was involved in the solid-

like models, they also consist of parallel finite-sized graphene domains,50 being similar to the standard slit-pore models.47-49

More realistic models for porous carbons were 6

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generated by reverse Monte Carlo (RMC), hybrid RMC, and quenched molecular dynamics (MD) simulations.51,52

A new model obtained from the quenched MD

simulation at 1200 °C has small graphene ribbons composed of two or more parallel graphene fragments, but it does not contain stacking due to neglecting long-range dispersion force in the simulation.51 As models for slit-shaped microporous carbons usable for DFT calculations to find out enegeticall preferable PC-solvated state of inner Li+ cations, the current study used graphene bilayers, as shown in Figure 1.

The graphene bilayer models are

categorized as standard slit-pore models. 　 Detailed discussion on how microporous carbons can be modeled by graphene bilayers is given in Supporting Information A, containing Figure S1(a)−1(d).

According to the experimental data obtained in Ref. 26,

we first constructed a graphene monolayer by 2.2 nm-size C150H30, and the C150H30 structure was fully optimized, as shown in Figure 1.

By using the optimized C150H30

structure (1G), a slit-type pore model of microporous carbons was constructed from two C150H30 clusters stacked in an AB manner with a certain interlayer spacing (2G(L)). Here, L represents the interlayer spacing defined in Figure 1, which is varied from 5 to 8 Å.

The interlayer spacing in the graphene bilayer models represents the slit-pore width

of microporous carbons.

Then, a PC-solvated Li+ or Na+ cation is positioned in the

interlayer space of the graphene bilayer to construct the initial geometries (abbreviated by PCn−Li+@2G(L), where n = 1−3, and L = 5−8 Å).

The PCn−Li+ or PCn−Na+

structures inside the 2G(L) structure were optimized except for 2G(L), whose atomic positions were fixed. 　 Due to limited computational resources, we used the 6-31G** basis set.53-55

Accordingly, the optimization of the graphene bilayer containing one Li+ 7 ACS Paragon Plus Environment


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ion solvated by some PC molecules involved up to 400 atoms containing 9,562 primitive Gaussians.

Results and Discussion PC4−Li+ Structure on Graphene Monolayer

Prior

to

discussing

the

PCn−Li+@2G(L) structures, let us investigate how a PC4−Li+ structure interacts with a graphene monolayer (1G).

Figure 2 shows the optimized geometry for a PC4−Li+

structure on 1G, whose key parameters are given in Table 1.

As shown in Figure 2 and

Table 1, the Li+ ion attached by four PC molecules is separated by 3.46 Å from 1G.

In

addition, we found that some H atoms of PC molecules point toward the carbon surface, whose distances of an H atom from 1G are tabulated in Table 1.

As shown in Table 1,

the shortest distance of a H atom from 1G is 2.29 Å for the fully solvated Li+ structure. The geometrical information indicates that cation− and CH− interactions can operate in the optimized structure. To obtain a better understanding of both interactions, we investigated the potential energy surfaces (PES) of a single PC molecule or Li+ cation approaching the graphene monolayer (1G), as shown in Figure S2 or S3 (Supporting Information). Figure S2 shows that PES of the PC molecule approach to 1G has one local minimum. In this local minimum, the distance of a PC H atom from 1G (D) is 2.27 Å, that is the equilibrium distance (De).

The local minimum is stabilized by 11.4 kcal/mol

compared with the dissociation limit, whose energy value is given by Emin. Table S1.

See also

Further approach of the PC molecule to 1G from D = 2.27 Å increases the

total energy.

At D = 1.86 Å (D0), the total energy becomes positive, indicating that 8 ACS Paragon Plus Environment

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repulsive interactions operate at D < 1.86 Å.

As seen from Figure 1, substantial

stabilization energies were found, when the D values range from 2.0 to 3.5 Å.

A

similar PES can be found in the interaction between a single Li+ cation and 1G (Figure S3(i-b)).

From Figure S3(i-b), the three key parameters (De = 1.88 Å, D0 = 1.01 Å,

and Emin = −60.7 kcal/mol) were obtained, which are also tabulated in Table S2. Considering the key geometrical parameters in Table 1, as well as Tables S1 and S2, we found that distances between a PC H atom and the carbon surface in the PC4−Li+ structure on 1G are similar to the De value in a single PC molecule on 1G (Figure S2 and Table S1).

The similarity suggests that CH− interactions mainly stabilize the

PC4−Li+ structure on a graphene monolayer rather than Li+ cation− interaction.56

As

shown in Figure 2, three PC molecules are arranged parallel to the carbon surface.

The

parallel PC arrangement can be verified from angles between the perpendicular line of the Li+ cation to 1G and the Li-O(PC) bonds (), because their  values range from 74.5 to 88.8 degrees. In the parallel arranged PC molecules, most H atoms separate from the carbon surface by a distance around 2.3 Å, as shown in Table 1.

The separation ranges

are close to the equilibrium distance (De) in the PES of the PC–1G system.

Thus, the

three parallel arranged PC molecules experience attractive CH− interactions from the graphene monolayer.

However, the other PC molecule, tilting from the perpendicular

line ( = 150.1 degree), is far from the carbon surface, and thus it cannot interact with 1G.

PC-Solvated States of a Cation within a Graphene Bilayer (PCn−Li+@2G(L)) This section will focus on energetically preferable PC solvated states of a Li+ 9 ACS Paragon Plus Environment


cation confined by the interlayer space of a graphene bilayer (PCn−Li+@2G(L)). Figure 3 displays optimized geometries for PCn−Li+@2G(L), where n increases from 1 to 3, and L increases from 5 to 8 Å. listed in Table 1.

Key parameters in the optimized structures are

As shown in Figure 3, a single Li+ cation directly binds to a carbon

surface in the optimized PCn−Li+@2G(L) structures, except for the optimized structures at L = 5 Å.

The direct Li+ cation binding, indicating the presence of cation−

interactions, is characteristic in the PCn−Li+@2G(L) structures, because it cannot be seen in the PC4−Li+ structure on 1G (Figure 2).

Another interesting finding from

Figure 3 is that the arrangement of PC molecules within a graphene bilayer varies depending on the interlayer spacing (L).

The variety of arrangement of inner PC

molecules can be seen from the  values in Table 1.

Within a smaller interlayer space

of 2G, PC molecules are arranged parallel to the carbon surfaces ( ≈ 90 degree), while within a larger interlayer space, they are tilted from the perpendicular line of 2G ( being substantially deviated from 90 degree).

Similar tilted PC molecules can also be

reported on the basis of the electron radial distribution function and reverse Monte Carlo analyses by Fukano et al.57 The  angle changes can allow PC molecules to interact with both carbon surfaces through CH− interactions.

In fact, Table 1 shows that lots of H atoms are

close to carbon surfaces, and thus significant CH− interactions can operate in the PCn−Li+@2G(L) structures.

Considering Table 1 and Figure S2, larger number of H

atoms of PC molecules experience repulsive CH− interactions in both PC1−Li+ and PC2−Li+ structures within 2G(L) at L = 5 Å.

On the other hand, attractive CH−

interactions can operate between a PCn−Li structure and 2G(L), when L is larger than 7 10 ACS Paragon Plus Environment

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Å.

Within the interlayer space of 2G(L) at L = 6 Å, there are two types of H atom that

experiences either repulsive or attractive CH− interactions from a carbon surface. Both interactions are largely canceled out to result in negligible CH− interactions, as will be discussed below.

The above geometrical information indicates that the

magnitude of CH− interactions in PCn−Li+@2G(L) can depend on the interlayer spacing.

From a different view of the optimized structures, we noticed that the number

of H atoms pointing to a carbon surface in PCn−Li+@2G(L) is also sensitive to the number of PC molecules (n). can also depend on n.

This indicates that the magnitude of CH− interactions

When the number of PC molecules decreases, weaker CH−

interactions are expected from Table 1.

The CH− interactions between the graphene

bilayer host and a guest induce to change the solvated guest structures.

The change of

the solvated guest structure results in destabilization in PCn−Li+@2G(L), which is a factor in determining the energetics. By using the geometrical information of PCn−Li+@2G(L), let us discuss their capacitance.

Eq. (1) suggests that the capacitance is inversely proportional to the

double-layer thickness (d), that is the distance of a Li+ cation from a graphene layer in PCn−Li+@2G(L).

The d values in PCn−Li+@2G(L) are shown in Figure 3 and Table 1.

Figure 3 shows that smaller d values can be found in a smaller number of PC molecules. This result indicates that a partially desolvated Li+ cation can more easily approach a graphene layer.

In particular, the PC1−Li+@2G(L) structures commonly have

relatively small d values of approximately 1.96 Å, except for L = 5.

When the number

of contained PC molecules are 2 or 3, larger d values were obtained.

In fact, the d

values range from 2.03 to 2.49 Å in n = 2 and from 2.36 to 2.54 Å in n = 3. 11 ACS Paragon Plus Environment


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By comparing in the d values between PCn−Li+@2G(L) and PC4−Li+ on 1G, we can approximately estimate the capacitance of PCn−Li+@2G(L) relative to that in PC4−Li+ on 1G.

Based on Eq.(1), the relative capacitances, obtained by the d value in

PC4−Li+ on 1G divided by that in PCn−Li+@2G(L), are tabulated in Table 1.

Note that

the 1G model can be viewed as a slit-type pore geometry of carbons whose interlayer space is large enough to allow a PC4−Li+ structure to interact with a single surface, unaffected by the presence of the other surface.

Accordingly, the 1G model

corresponds to macroporous carbons, as well as external carbons.

Note that Ref. 26

suggests that the solvation number of PC molecules for Li+ cations on the external surface or in the macropore is 3.6−4.4.

Table 1 shows that the relative capacitance of

PCn−Li+@2G(L) is very sensitive to the number of PC molecules (n) and the interlayer spacing (L).

In particular, the largest capacitance was found in PC1−Li+@2G(L),

where L is larger than 6 Å; the value is almost twice as large as that in the PC4−Li+ structure on 1G.

The capacitances of PCn−Li+@2G(L) decrease as the number of PC

molecules attached to the Li+ ion increases.

The DFT findings indicate that the

presence of a significantly desolvated Li+ cation within the graphene bilayers is important to enhance the capacitance due to the shortening of distance of the Li+ ion from a carbon surface.

Energetics in the Solvated State of a Cation within a Graphene Bilayer discuss the stability of PCn−Li+@2G(L) structures.

Let us

Their stabilized energy (Estabilize) is

defined as follows:


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Estabilize = Etotal(PCn−Li+@2G(L)) – Etotal(2G(L)) – Etotal(PCn−Li+)

[2]

Here, Etotal(PCn−Li+@2G(L)), Etotal(2G(L)), and Etotal(PCn−Li+) represent the total energies of PCn−Li+@2G(L), 2G(L), and PCn−Li+ structures, respectively.

Basis set

superposition errors (BSSEs)53 were corrected in the Estabilize values, whose values are listed in Table 2.

Negative Estabilize values indicate that PCn−Li+@2G(L) structures are

energetically stable relative to the dissociation limit toward a PCn−Li+ structure and 2G(L). Based on Table 2, Figure S4 (Supporting Information) and Figure 4 plot the Estabilize values of PCn−Li+@2G(L) as a function of L.

As shown in Figure S4 and

Table 2, we found positive Estabilize values at L = 5 Å and negative values at L larger than 6 Å.

The results indicate that energetically stable PCn−Li+ structures can exist within

the graphene bilayer whose interlayer spacing is greater than 6 Å.

Here, we focus on

energetically preferable PCn−Li+ structures within a graphene bilayer, whose Establize values ranging from −100 to 25 kcal/mol are displayed in Figure 4 .

From Figure 4 we

see different degree of stability of PCn−Li+ structures inside 2G(L) depending on the interlayer spacing (L).

In fact, at L = 6 Å, the Estabilize value in PC1−Li+@2G(L)

structure is more significant than those of PC2−Li+@2G(L) and PC3−Li+@2G(L), whose energy differences are 20.7 and 42.4 kcal/mol, respectively.

This result suggests that

strong confinement of the interlayer space of 2G(L) at L = 6 Å induces to selectively stabilize a PC1−Li+ structure rather than PC2−Li+ and PC3−Li+ structures.

Within

2G(L) at L ≥ 7 Å, the large interlayer space can accommodate PCn−Li+ structures with n ≤ 3.

In fact, we see from Figure 4 n-independent Estabilize values in PCn−Li+@2G(L) at 13 ACS Paragon Plus Environment


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L = 7 and 8 Å: Estabilize values at L = 7 and 8 Å are approximately −75 and −72 kcal/mol, respectively.

The above DFT results indicate that the interlayer spacing of graphene

bilayers plays an important role in determining energetically preferable PC solvated states of an inner Li+ cation.

Key Factors in Determining the Stability of PCn−Li+@2G(L) Structures To gain a deeper understanding of the stability of PCn−Li+@2G(L) structures, let us discuss the origin of the stabilization energy (Estabilize).　 As mentioned above, CH− guest-host interactions (E(CH−) and cation− interactions (E(Li+−) occur in PCn−M+@2G(L) structures.

To estimate the two types of interaction energy in the

PCn−M+@2G(L) structures, PCn@2G(L) and Li+@2G(L) were considered.

The

PCn@2G(L) and Li+@2G(L) structures were constructed by removing the Li+ cation and PCn moiety from an optimized PCn−Li+@2G(L) geometry, respectively.

Then, the

E(CH− and E(Li+− values were calculated in the following definition.

E(Li+− = Etotal(Li+@2G(L)) − Etotal(Li+) − Etotal(2G(L))

[3]

E(CH− = Etotal(PCn@2G(L)) – n × Etotal(PC) − Etotal(2G(L))

[4]

where Etotal(PCn@2G(L)), Etotal(Li+@2G(L)), Etotal(PC), and Etotal(Li+) are the total energies of the PCn@2G(L) structure, the Li+@2G(L) structure, the optimized PC structure, and the Li+ cation, respectively.

The previous section already found that a

PCn−Li+ guest structure is substantially deformed by the confinement of a graphene 14 ACS Paragon Plus Environment

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bilayer host.

Accordingly, the energy required by the guest deformation from the

optimize structure of PCn−Li+ itself is a key factor in the stability.

The definition of

the deformation energy (E(deform) is given as follows:

E(deform = Etotal(PCn−Li+ guest) − Etotal(optimized PCn−Li+ structure)

[5]

where Etotal(PCn−Li+ guest) is the total energy of a PCn−Li+ guest taken from an optimized PCn@2G(L) structure, and Etotal(optimized PCn−Li+ structure) is that of the optimized PCn−M+ structure.

Considering the above discussion, we partitioned the

Estabilize values into the three energy components. 　 In addition of the Estabilize values, Table 2 tabulates the three components of the calculated energy values and their summation Esum (= E(CH−  E(Li+−  E(deform).

As shown in Table 2, Esum

values are usually larger than the corresponding Estabilize values by ~20 kcal/mol, indicating that the energy decomposition analyses always overstabilize the interaction energies in PCn−Li+@2G(L).

However, trends in the Esum values are similar to those

in the Estabilize values; therefore, we can roughly discuss the origin of the stabilization of PCn−Li+@2G(L) structures by the quantitative energy decomposition analyses. Table 2 shows variety of E(CH− and E(Li+− values in PCn−Li+@2G(L) structures.

In terms of E(CH− components in PCn−Li+@2G(L), positive and

negative values indicate repulsive and attractive CH− interactions operated between PC molecules and the graphene bilayer, respectively.

We can see from Table 2

positive E(CH− values at L = 5 Å, negligibly negative values at L = 6 Å, and substantially negative values at L ≥ 7 Å.

The variety of the CH− interactions



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in PCn−Li+@2G(L) structures can be understood from different confinement effects of a single PC molecule in a variant interlayer space of 2G(L), as shown in Figure S5 & S6 (Supporting Information).

Figure S6 shows parabola-type PESs for a vertical change

of position of an inner PC molecule within 2G(L) where L ranges from 5 to 8 Å: there is one local minimum at each potential energy surface, and their stabilization energies depend on the interlayer spacing, as shown in Table S3.

In fact, the CH− interactions

between the PC molecule and 2G at L = 5 Å are repulsive, those in 2G at L = 6 Å are negligibly attractive, and those in 2G at L = 7 or 8 Å are substantially attractive.

This

indicates that the medium interlayer space of 2G(L) at L = 6 Å is the smallest limit to accommodate a single PC molecule. In terms of E(Li+− components in PCn−Li+@2G(L), Table 2 always show negative values, indicating attractive cation− interactions operated between Li+ cation and graphene bilayer in PCn−Li+@2G(L) structures.

The magnitudes of cation−

interactions in PCn−Li+@2G(L) structures are more significant than that in the Li+ cation on 1G (Figure S3), being also due to the nanospace confinement effects. detailed discussion in Supporting Information (Figure S7 & S8).

See

When the Li+ cation

vertically moves within 2G(L) from its optimal position in Figure S7, double-well PESs appear in Figure S8, being in contrast to PES of that on 1G (Figure S3). Figure S8, each double-well PES has two local minima.

As shown in

Their stabilization energies,

tabulated in Table S4, are more significant than that in the Li+ cation on 1G.

More

interestingly, the stabilization energies in a single Li+ cation inside 2G(L), due to attractive cation− interactions, are sensitive to the interlayer spacing (L).

In fact,

more significant attractive cation− interactions between the Li+ cation and the 16 ACS Paragon Plus Environment

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graphene bilayer can be found at smaller interlayer spacing.

Similar L-dependent

cation− interactions are also found in PCn−Li+@2G(L) structures, as shown in Table 2. To easily look at what factors determine the stability of PCn−Li+@2G(L), the three types of energy values are displayed in Figure 5 as a function of the interlayer spacing (L).

From Figure 5, less significant values were found in the deformation

energies (Edeform).

In contrast, we found that cation− interactions are always a key in

determining the stability of PCn−Li+@2G(L), since their E(Li+− values, ranging from −52.0 to −82.0 kcal/mol in Figure 5 and Table 2, are relatively significant. A more striking finding in Figure 5 is L-dependent CH− interactions in PCn−Li+@2G(L): significantly positive E(CH− values at L = 5 Å, negligibly negative values at L = 6 Å, and substantially negative values at L = 7 and 8 Å.

When the interlayer spacing of

PCn−Li+@2G(L) is 5 Å, repulsive CH− interactions dominate attractive cation− interactions.

The repulsive CH− interactions cannot allow PCn−Li+ structures to be

accommodated within a too small interlayer space of the graphene bilayer.

At L equal

to and greater than 6 Å, attractive CH− interactions occur, whose magnitude is maximized at L = 7 Å.

The attractive CH− interactions are also sensitive to the

number of PC molecules (n); more significant CH− interactions were found in a larger solvation number of PC molecules for a Li+ cation within 2G(L).

On the other hand,

the dependence of cation− interactions on the number of PC molecules is reversed; stronger cation− interactions in the Li+ cation were found in a smaller solvation number of PC molecules for an inner Li+ cation. Accordingly, the magnitude relationships between CH− interactions and cation− interactions change, depending on the number of PC molecules as well as the 17 ACS Paragon Plus Environment


interlayer spacing.

Page 18 of 41

Within the interlayer space of 2G(L) at L = 6 Å, attractive CH−

interactions are negligible, and thus only cation− interactions are important to determine the stability of inner PCn−Li+

　

structures, independent of the solvation

number of PC molecules for a Li+ cation. 　At L ≥ 7 Å, whose interlay space is larger, CH− interactions become substantially attractive.

When the number of PC molecules

is 2 and 3, the magnitudes of CH− and cation− interactions are comparable, and thus the two interactions are main factor in determining energetically preferable solvated structures.

While at n = 1, the magnitude of the CH− interactions is less significant

than that of the cation− interactions.

As a result, the energy-decomposition analyses

demonstrated that balance between the CH− and cation− interactions is important to stabilize PCn−Li+ structures inside a graphene bilayer.

PCn−Na+@2G(L) structures

Although it is now well known that lithium

resources are limited, the increasing importance of EDLCs as well as lithium-ion batteries require abundant lithium.

As a result there are trends of replacing lithium

with sodium due to the abundance of sodium resources.

In accordance with the trend,

we suppose that a sodium cation solvated by PC molecules inside a graphene bilayer can be an alternative candidate for combination of electrodes and electrolytes in EDLC. In this direction, we investigated energetically preferable PC solvation states of a Na+ cation within 2G(L), whose optimized structures are given in Figures S9 and S10, as well as Table S5 (Supporting Information).

Similar to the Li cases, we obtained

Estabilize values for the PCn−Na+@2G(L) structures in Table S6 and Figure S11 (Supporting Information), and partitioned them the three energy components (Table S6 18 ACS Paragon Plus Environment

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and Figure S12).

Figures S11 and S12 show similar trends in the energetics between

the PCn−Na+@2G(L) and PCn−Li+@2G(L) structures.

Briefly, preferences of

PC1−Na+ structures can be seen inside 2G(L) at L = 6 Å rather than PC2−Na+ and PC3−Na+ structures, as shown in Figure S11.

When L is 7 or 8 Å, all PCn−Na+

structures (n ≤ 3) are allowed to be accommodated within a graphene bilayer. The stability

of

PCn−Na+@2G(L)

structures

comes

CH− interactions and cation− interactions.

from

the

balance

between

Our DFT calculations found that

confinement of the graphene bilayer at an interlayer spacing of approximately 6 Å can stabilize a PC1−Na+ structure, similar to the Li cases.

Linking DFT findings to Experimental Reports Our DFT calculations found important roles of the interlayer spacing in determining energetically preferable solvated states of a Li+ cation within a graphene bilayer (2G(L)).

PCn−Li+ structures (1 ≤ n ≤ 3) are stabilized within 2G(L) at L ≥ 7 Å

due to the presence of larger interlayer space.

Then, both cation− and CH− 

interactions contribute to the stabilization of PCn−Li+ structures within the graphene bilayers.

In contrast, 2G(L) at L = 5 Å cannot accommodate even the smallest

PCn−Li+ structure (i.e. n = 1) due to too small interlayer space.

2G(L) at L = 6 Å has a

medium interlayer space, being the smallest limit for the encapsulation of a single PC molecule.

Thus the restricted interlayer space of 2G(L) at L = 6 Å can be utilized to

selectively stabilize PC1−Li+ structure.

The selective stabilization of significantly

desolvated Li+ cations within 2G(L) at L = 6 Å comes from only cation− interactions. In this structure, negligible CH− interactions occur. 19 ACS Paragon Plus Environment


Page 20 of 41

The strong cation− interactions result in the shortening of the distance of a Li+ cation from the carbon surface (1.96 Å).

The shorter distance in PC1−Li+@2G(L) at L

= 6 Å is a strong contrast to that in the PC4−Li+ structure on the graphene monolayer (3.46 Å) where CH− interactions are important.

Note that the graphene monolayer

(1G) models external carbons and macroporous carbons.

Since PC1−Li+@2G(L) at L

= 6 Å has a quite shorter double-layer thickness than PC4−Li+ on 1G, PC1−Li+@2G(L) at L = 6 Å exhibits the capacitance being almost twice as large as that in the fully solvated Li+ cation on a carbon surface.

Therefore, the capacitance enhancement

comes from the selective stabilization of the significantly desolvated Li+ cations by utilizing the restricted interlayer space of 2G(L) at L = 6 Å. Our DFT results can help us to understand the capacitance enhancement reported by Urita et al. who used as EDLC electrode materials activated carbons containing micropores in LiClO4/PC solution.26

In Ref. 26, the porous carbons with

the average pore size of 0.6 nm were synthesized by carefully selecting the carbon precursor and activation condition.

The pore size distributions were obtained by

combined experimental approaches such as TEM measurement, N2 adsorption isotherms, and XRD analyses.

Considering the above DFT calculations, a slit-type

pore geometry of microporous carbons whose pore size is 0.6 nm, modeled by 2G(L) at L = 6 Å, utilized the restricted space to selectively stabilize a significantly desolvated Li+ cation that can more closely approach the carbon surface. 　 Therefore, our DFT findings can explain the capacitance enhancement reported in Ref. 26.

Note that

activated carbons usually have slit-type micropores between neighboring units of micrographites consisting of a few graphene layers, as shown in Figures S1(a) and 20 ACS Paragon Plus Environment

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S1(b).

Thus, micropores are completely different from the interlayer space in

graphene multilayers in terms of the size, as discussed in Supporting information A. There is not an experimental approach to control the pore size of micropores in activated carbons, being similar to the case of graphene multilayers,59-63 and therefore it is difficult to obtain activated carbons like showing quite sharp pore size distribution. However, it is evident that one can obtain activated carbons whose average pore size is around 0.6 nm.26

Our DFT calculations found the optimal pore size of microporous

carbons for enhancing the capacitance in activated carbons.

These computational

findings can provide a nice hint to enhance the capacitance in activated carbon-based electrode materials, which stimulates experimentalists to increase the amount of micropores whose sizes are around 0.6 nm.

Conclusions Dispersion-corrected DFT calculations were employed to investigate energetically preferred structures of a Li+ cation solvated by some PC molecules inside a graphene bilayer (2G) with a certain interlayer spacing (L), which are denoted by PCn−Li+@2G(L), where n is the number of PC molecules. ranges from 5 to 8 Å, and n is up to 3.

Here, we considered that L

The 2G(L) structures model slit-type pore

geometries of microporous carbons, and then its interlayer spacing corresponds to slitpore width of microporous carbons.

The DFT calculations found that preferred

PCn−Li+@2G(L) structures depend on the interlayer spacing of the graphene bilayer (L) and the number of PC molecules (n).

Within 2G(L) at L ≥ 7 Å, PCn−Li+ structures

where n is up to 3 are allowed to be accommodated inside its larger interlayer space. Then, both cation− and CH−  interactions are important to determine inner PCn−Li+ 21 ACS Paragon Plus Environment


structures.

Page 22 of 41

Within 2G(L) at L = 6 Å, a significantly desolvated Li+ cation (PC1−Li+

structure) is selectively stabilized by utilizing the nanospace confinement, because the interlayer space is the smallest limit for the encapsulation of a single PC molecule.

In

this structure, only cation− interactions are important to stabilize the solvated state of a Li+ cation, where negligible CH− interactions occur. The significant attractive cation− interactions in PC1−Li+@2G(L) shorten the distance of a Li+ cation from a carbon layer (1.96 Å), corresponding to the double-layer thickness.

The distance is smaller than those in a fully PC-solvated Li+ cation on a

graphene monolayer, which is models of external or macroporous carbons.

Since the

double-layer thickness is inversly proportional to the capacitance, we estimated the capacitance of PCn−Li+@2G(L) structures relative to that of PC4−Li+ on 1G.

As a

result, we found that PC1−Li+@2G(L) at L = 6 Å has the capacitance that is almost twice as large as that in PC4−Li+ on the 1G structure.

Since significantly desolated Li+

cations are selectively stabilized within the interlayer space of 2G(L) at L = 6 Å, DFT calculations revealed that slit-type pore geometries of microporous carbons whose pore size is around 6 Å, modeled by 2G(L) at L = 6 Å, can utilized the restricted interlayer space to selectively stabilize significantly desolvated Li+ cations (PC1−Li+ structures), which can significantly enhance the capacitance relative to those in external or macroporous carbons.

Acknowledgments: This project was partially supported from a Grant-in-Aid for Scientific Research (C) (No. 18K04864) from JSPS for T. Y., and from KAKENHI Grant (No. 16H05967) from JSPS for K. U. 22 ACS Paragon Plus Environment

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Supporting Information Available:

A transmission electron microscopy (TEM)

image of pitch-based activated carbons involving microporese, and modelling microporous carbons (Figure S1); Potential energy surface (PES) for PC molecule approaching 1G (Figure S2 and Table S1); PES for an alkaline cation (Li+ or Na+) approaching 1G (Figure S3 and Table S2); Estabilize values in PCn−Na+@2G(L) as a function of the interlayer spacing, ranging (L) from −100 to 275 kcal/mol (Figure S4); PES for PC molecule approaching 2G (Figures S5 & S6, and Table S3); Optimized

geometries of alkaline cation on a graphene monolayer and PES for an alkaline cation (Li+ or Na+) approaching 1G (Figures S7 & S8, and Table S4); Optimized

PCn−Na+@2G(L) structures together with the optimized PC4−Na+ structure on 1G (Figures S9−S12 and Table S5); full list in Refs. 4, 38, 57, 59, 60, and 63. materials are available free of charge via the internet at http://pubs.acs.org.


These


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(55) Clark, T.; Chandrasekhar, J.; Spitznagel, G. W.; Schleyer, P. v. R. Efficient Diffuse Function-Augmented Basis Sets for Anion Calculations. III. The 3-21+G Basis Set for First-Row Elements, Li–F. J. Comp. Chem. 1983, 4, 294–301. (56) This can be also confirmed by lifting of the Li+ cation up from the graphene surface、whose distance is larger than the De value of 1.88 Å in Figure S3. (57) Fukano, M.; Fujimori, T.; Ségalin, J.; Iwama, E.; Taberna, F.-L.; Iiyama, T.; Ohba, T.; Kanoh, H.; Gogotsi, Y.; Simon, P.; et al. Vertically Oriented Propylene Carbonate Molecules and Tetraethyl Ammonium Ions in Carbon Slit Pores. J. Phys. Chem. C 2013, 117, 5752. (58) Boys, S. F.; Bernardi, F. The Calculation of Small Molecular Interactions by the Differences of Separate Total Energies. Some Procedures with Reduced Errors. Mol. Phys. 1970, 19, 553–566. (59) Radha, B.; Esfandiar, A.; Wang, F. C.; Rooney, A. P.; Gopinadhan, K.; Keerthi, A.; Mishchenko, A.; Janardanan, A.; Blake, P.; Fumagalli, L.; et al. Molecular Transport through Capillatires Made with Atomic-Scale Precision. Nature 2016, 538, 222-228. (60) Chen, L.; Shi, G.; Shen, J.; Peng, B.; Zhang, B.; Wang, Y.; Bian, F.; Wang, J.; Li, D.; Qian, Z.; Xu, G.; et al. Ion Sieving in Graphene Oxide Membranes via Cationic Control of Interlayer Spacing. Nature 2017, 550, 380-383. (61) Esfandiar, A.; Radha, B.; Wang, F. C.; Yang, Q.; Hu, S.; Garaj, S.; Nair, R. R.; Geim, A.; Gopinadhan, K. Size Effect in Ion Transport through Angstrom-Scale Slits. Science 2017, 358, 511-513.



(62) Gopinadhan, K.; Hu, S.; Esfandiar, A.; Lozada-Hidalgo, M.; Wang, F. C.; Yang, Q.; Tyurnina, A. V.; Keerthi, A.; Radha, B.; Geim, A. K. Complete Steric Exclusion of Ions, and Proton Transfer through Confined Monolayer Water. Science 2019, 363, 145148. (63) Abraham, J.; Vasu, K. S.; Williams, C. D.; Gopinadhan, K.; Su, Y.; Cherian, C. T.; Dix, J.; Prestat, E.; Haigh, S. J.; Grigorieva, I. V.; et al. Tunable Sieving of Ions Using Graphene Oxide Membranes. Nature Nanotech. 2017, 12, 546–550.


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Figure Captions Figure 1

Finite-size graphene monolayer (1G), and graphene bilayer (2G), where

two 1G layers are stacked in an ABAB fashion with a certain interlayer spacing (L). The graphene bilayer, denoted by 2G(L), models slit-type microporous carbons.

The

interlayer spacing in the graphene bilayer models corresponds to the pore width of microporous carbons.　Detailed discussion on modeling slit-type microporous carbons can be seen in Supporting Information A.

Figure 2 (1G).

Optimized structures for PC4−Li+ structure on the graphene monolayer

Blue, black, red, and white indicate lithium, carbon, oxygen, and hydrogen

atoms, respectively. given in Å.

Figure 3

The optimized distance of the Li+ cation from a graphene sheet is

Detailed information on key geometrical parameters is given in Table 1.

Optimized structures for PCn−Li+　 structures within the graphene bilayer

with a certain interlayer spacing (L), denoted by PCn−Li+@2G(L).

Blue, black, red,

and white indicate lithium, carbon, oxygen, and hydrogen atoms, respectively. 　 The optimized distance of the Li+ cation from a graphene sheet is given in Å. Detailed information on key geometrical parameters are given in Table 1.

Figure 4

Estabilize values of PCn−Li+@2G(L) as a function of the interlayer spacing

(L), ranging from −100 to 25 kcal/mol.

The black line; Estabilize value change in n = 1,

the red line; that in n = 2, and the green line; that in n = 3.


Detailed Estabilize values are


given in Table 2. from

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Figure S4 displays a similar graph of the Estabilize values ranging

−100 to 275 kcal/mol.

Figure 5

Energy decomposition analyses on Estabilize values.

The values are

partitioned into CH− interactions (E(CH− cation− interactions (E(Li+−), and the energy required by the deformation of a PCn−Li+ structure induced by encapsulation into the interlayer space of the graphene bilayer (Edeform).

The energy values of the

three components are given as a function of the interlayer spacing. E(CH−, the red line; E(Li+−, and the green line; Edeform.


The black line;

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Table 1 Key parameters of optimized structures for PC4−Li+ structure on a graphene monolayer and PCn−Li+@2G(L)a. Structures db H(PC)– C(Graphene) distances c Ce d PC4−Li+ on 1G 3.46 2.29, 2.97, 2.31, 2.31, 2.36, 2.36, 2.37 74.5, 79.1, 88.8, − 150.1 PC1−Li+@2G(L = 5 Å) 2.44 1.39, 1.57, 1.57, 1.77, 1.80, 2.46 92.1 1.42 PC1−Li+@2G(L = 6 Å) 1.96 1.72, 1.99,2.11, 2.25, 2.27, 2.45 120.3 1.77 PC1−Li+@2G(L = 7 Å) 1.95 2.06, 2.23, 2.30, 3.00, 3.04, 3.19 137.9 1.77 PC1−Li+@2G(L = 8 Å) 1.94 2.31, 2.64, 2.79, 3.21, 3.47 150.1 1.78 PC2−Li+@2G(L = 5 Å) 2.49 1.30, 1.30, 1.45, 1.46, 1.60, 1.60, 1.75, 90.2, 90.3 1.39 1.75, 2.08, 2.10, 2.15, 2.15 PC2−Li+@2G(L = 6 Å) 2.31 1.66, 1.69, 1.91, 1.99, 2.06, 2.14, 2.19, 106.4, 107.6 1.50 2.26, 2.30, 2.38, 2.54, 2.63 PC2−Li+@2G(L = 7 Å) 2.29 2.07, 2.10, 2.26, 2.28, 2.33, 2.39, 2.87, 112.7, 114.6 1.51 2.97, 3.18, 3.19, 3.31, 3.32 PC2−Li+@2G(L = 8 Å) 2.03 2.39, 2.42, 2.58, 2.60, 2.64, 2.70, 3.15, 125.0, 130.7 1.70 3.19, PC3−Li+@2G(L = 6 Å) 2.36 1.67, 1.68, 1.69, 1.92, 1.92, 1.95,1.95, 103.7, 104.0, 1.47 2.00, 2.08, 2.26, 2.32, 2.32, 2.33, 2.41, 104.9 2.49, 2.49, 2.66, 2.66 PC3−Li+@2G(L = 7 Å) 2.37 2.10, 2.11, 2.21, 2.25, 2.26, 2.26, 2.43, 106.1, 106.7, 1.46 2.59, 2.61, 2.61, 2.61, 2.83, 3.04, 3.08, 108.8 3.13, 3.16, 3.41, 3.42 PC3−Li+@2G(L = 8 Å) 2.54 2.28, 2.38, 2.46, 2.75, 2.77, 2.78, 2.91, 98.9, 102.1, 1.36 2.92, 2.98 3.06, 3.09, 3.34, 3.42, 3.44 160.1 a their optimized geometries are displayed in Figures 2 and 3. b the distance of a Li+ cation from a graphene monolayer, whose unit is in Å. c the distance of a H PC atom from a carbon surface, whose value is less than 3.5 Å. In the distance range, substantial CH- interactions are expected to operate, according to Figure S2. Distances corresponding to the range where repulsive interactions operate between a PC molecule and graphene are underlined. Distances without the underline correspond to the range where attractive interactions operate. See also Figure S2. d the angles between the perpendicular line of the Li+ cation to the surface and the Li−O(PC) bonds e the capacitance of PCn−Li+@2G(L) relative to that in PC4−Li+ on 1G

Table 2 Energetics of the optimized structures for PCn−Li+@2G(L). . Structures Estabilize a Edeform d Esum e E(Li+–) b E(CH–) c + PC4−Li on 1G – – – PC1−Li+@2G(L = 5 Å) 103.0 –82.0 128.0 30.2 76.0 PC1−Li+@2G(L = 6 Å) –44.8 –76.0 –0.10 11.5 –65.0 PC1−Li+@2G(L = 7 Å) –71.8 –73.0 –20.0 3.7 –89.0 PC1−Li+@2G(L = 8 Å) –71.5 –72.0 –18.0 1.8 –88.0 PC2−Li+@2G(L = 5 Å) 258 –82.0 248.0 47.3 213.0 PC2−Li+@2G(L = 6 Å) –24.1 –68.0 –3.4 16.0 –55.0 PC2−Li+@2G(L = 7 Å) –75.5 –63.0 –43.0 4.0 –102.0 PC2−Li+@2G(L = 8 Å) –72.6 –70.0 –36.0 3.7 –102.0 PC3−Li+@2G(L = 6 Å) –2.6 –68.0 –3.0 29.1 –42.0 PC3−Li+@2G(L = 7 Å) –79.2 –62.0 –62.0 10.3 –114.0 PC3−Li+@2G(L = 8 Å) –71.0 –52.0 –52.0 6.8 –100.0 a  in kcal/mol defined in equation [1]. stabilize b (Li+–) in kcal/mol defined in equation [2]. c (CH–) in kcal/mol defined in equation [3]. d  deform in kcal/mol defined in equation [4]. e  + sum is the summation of (Li –), (CH–), and deform, whose unit is in kcal/mol. f  is defined by  –  sum stabilized, whose unit is in kcal/mol.


E f – –27 –20 –17 –17 –45 –31 –27 –29 –39 –35 –29


TOC

Why Pore width of Nanoporous Carbon Materials Determines the Preferred Solvated States of Alkaline Cations: A Density Functional Theory Calculation Study

Takashi Yumura*, Marie Ishikura, and Koki Urita


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interlayer spacing (L)

Figure 1. Finite-size graphene monolayer (1G), and graphene bilayer (2G) where two 1G layers are stacked in an ABAB fashion with a certain interlayer spacing (L). The graphene bilayer, denoted by 2G(L), models slit-type microporous carbons. The interlayer spacing in the graphene bilayer models corresponds to the pore width of microporous carbons. Detailed discussion on modeling slit-type microporous carbons can be seen in Supporting Information A.



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3.46

Figure 2. Optimized structures for a PC4–Li+ structure on the graphene monolayer (1G). Blue, black, red, and white indicate lithium, carbon, oxygen, and hydrogen atoms, respectively. The optimized disatnce of Li+ cation from a graphene sheet is given in Å. Detailed information on key geometrical parameters are given in Table 1.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 L17= 5 Å (n = 1) d = 5 Å (n = 2) 18 19 2.44 2.49 20 21 22 23 L24= 6 Å (n = 1) L = 6 Å (n = 2) L = 6 Å (n = 3) 25 26 27 1.96 2.31 2.36 28 29 30 31 L32= 7 Å (n = 1) L = 7 Å (n = 2) L = 7 Å (n = 3) 33 34 35 1.95 2.29 2.37 36 37 38 39 L40= 8 Å (n = 1) L = 8 Å (n = 2) L = 8 Å (n = 3) 41 42 43 44 1.94 2.03 2.54 45 46 47 48 49 Figure 3. Optimized structures for PCn–Li+ structures within the graphene bilayer with a certain interlayer 50 spacing (L), denoted by PCn–Li+@2G( L) . Blue, black, red, and white indicate lithium, carbon, oxygen, 51 and hydrogen atoms, respectively. The optimized disatnce of Li+ cation from a graphene sheet is given in Å. 52 53Detailed information on key geometrical parameters are given in Table 1. 54 55 56 57 ACS Paragon Plus Environment 58 59 60


25

E stabilize (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

PC 2 –Li+@2G(L)

0 PC 3–Li+@2G(L)

–25 –50

PC1 –Li+@2G(L) –75 –100

5

6

7

8

Interlayer spacing (L) (Å)

Figure 4. Estabilize values of PCn–Li+@2G( L) as a function of the interlayer spacing (L), ranging from –100 to 25 kcal/mol. The black line; Estabilize value change in n = 1, the red line; that in n = 2, and the green line; that in n = 3. Detailed Estabilize values are given in Table 2. Figure S4 displays a similar graph of the Estabilize values ranging from –100 to 275 kcal/mol.


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energy values (kcal/mol)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 (b) PC 2 –Li+@2G(L) (c) PC 3–Li+@2G(L) (a) PC1 –Li+@2G(L) 25 26 250 27 E(CH–π) 28 200 29 30 150 31 32 E(CH–π) 100 33 34 35 50 E deform 36 E deform E deform 37 0 38 39 E(CH–π) E(Li + –π) 40 –50 E(Li + –π) 41 E(Li + –π) 42 –100 43 5 7 6 8 5 7 5 7 6 8 6 8 44 Interlayer spacing (Å) Interlayer spacing (Å) Interlayer spacing (Å) 45 46 47 48 Figure 5. Energy decomposition analyses on Estabilize values. The values are partitined into CH–π 49 interactions (E(CH–π), cation–π interactions (E(Li + –π)), and the energy required by the deformation 50 of a PC –Li+ structure induced by encapsulation into the interlayer space of the graphene bilayer n 51 ). The energy values of the three components are given as a function of the interlayer (E 52 deform spacing. The black line; E(CH–π), the red line; E(cation–π), and the green line; Edeform. 53 54 55 56 57 ACS Paragon Plus Environment 58 59 60

Why Pore Width of Nanoporous Carbon Materials Determines the

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