3.2. Calculating the Second Moment Plateau Values
The van Vleck formula can be used to calculate NMR second moments in rigid lattices of solid-state materials with well-known molecular and crystal structures [
39,
40]. A theoretical description of the NMR second moment is given in
Appendix A. Using the above-mentioned formula, we calculated the
1H and
19F second moments of the rigid lattices of the three crystal structures of ELM-11. Second-moment reductions were also calculated by taking into account the anisotropy parameter [
40,
41] associated with the isotropic reorientation of BF
4− and the torsional flipping of the 4,4′-bipyridine linkers. The second moments in the rigid lattices determined for the
1H and
19F nuclei are summarized in
Table A1 and
Table A2 in
Appendix A, while
Table 1 and
Table 2 show the evaluated reductions in the
1H and
19F second moments.
The reduction in the 1H second moment, , which is the sum of , , and , is about (0.4–2) × 10−8 T2, which indicates a low contribution to the total magnetic dipolar relaxation rate. On the other hand, the reduction in the 19F second moment, , which is the sum of , , , and , ranged between 21 × 10−8 and 24 × 10−8 T2. In particular, the isotropic reorientation of BF4− effectively modulates the F-F and F-B vectors, leading to a large reduction in the second moment, which suggests that spin–lattice relaxation is expected to be effective through a mechanism involving fluctuations in magnetic dipolar interactions that act on 19F nuclei and control the 1H spin–lattice relaxation rate through cross-relaxation between the 1H and 19F spin systems.
3.3. Temperature Dependence of T1 in the Closed form of ELM-11
Figure 2a shows the temperature dependence of
1H
T1 in
1. Below 250 K,
T1 was almost constant, at 520 μs; it decreased above 250 K and then increased to 499 μs at 360 K after exhibiting a minimum value of 492 μs at 323 K.
T1 only changed by 30 μs in this region, which is only a 5.8% change compared to the original value of 520 μs. If the
T1 minimum is caused by the thermal motions of BF
4- and/or 4,4′-bipyridine, then the apparent activation energy (0.5 kJ mol
−1) is much smaller than the reported
Ea values for the isotropic rotation of BF
4− (10–26 kJ mol
−1) [
40,
42,
43,
44] and/or the torsional flipping of 4,4′-bipyridine (~10 kJ mol
−1) [
45,
46].
ELM-11 contains paramagnetic Cu
2+ (
S = 1/2) ions and four kinds of NMR-active nucleus:
1H (
I = 1/2),
19F (
S = 1/2),
10B (
S = 3), and
11B (
S = 3/2). In this case, the nuclear spin systems relax through two mechanisms: paramagnetic and dipolar relaxation. In general, relaxation times through paramagnetic ions are one or two orders of magnitude shorter than the relaxation times of diamagnetic substances. According to the multi-paramagnetic-center model, which is preferred for paramagnetic materials with dense paramagnetic-centers, the paramagnetic relaxation rate (
R1p) is given by [
47,
48].
where
and
D is the efficiency of direct relaxation and the diffusion coefficient for spin diffusion, respectively, and
Np is the number of paramagnetic centers per unit volume of the sample. In the powder sample,
is represented by
where
γS and
γI are the gyromagnetic ratios of the electron spin and resonant nuclei, respectively,
S is the spin of the paramagnetic ion,
τe is the correlation time for the
z-component of the paramagnetic spin (longitudinal relaxation time for the electron spin), and
ωI is the resonance frequency of a resonant nucleus. According to Bloembergen [
49],
D =
a2/50T
2, where
a is the average
1H-
1H distance (0.551 nm for
1) and
T2 is the
1H spin–spin relaxation time (average of experimental values; ~22 μs). As a result,
D = 2.87 × 10
−16 m
2 s
−1 for
1. This is reasonable because it is of the same order of magnitude as the
D value (6.25 × 10
−16 m
2 s
−1) for the high spin state of [Fe(ptz)
6](BF
4)
2 (ptz = 1-
n-propyl-1
H-tetrazole) [
35]. Furthermore, we evaluated
Np as 1.91 × 10
27 m
−3 for the body-centered lattice formed by the Cu
2+ ions in
1. Thus,
R1p depends strongly on
τe.
On the other hand, the dipolar relaxation rate (
R1d) is mainly controlled by fluctuations in the magnetic dipolar interactions among the
1H (
I = 1/2),
19F (
S = 1/2),
10B (
S = 3), and
11B (
S = 3/2) spins. In such a multi-spin system, cross relaxation between the
1H,
19F,
10B, and
11B nuclei are taken into account [
40]. Here, assuming that both the
1H and
19F nuclei dominantly contribute to cross relaxation because of their large gyromagnetic ratios, the actual relaxation rates are given by the eigenvalues of the relaxation matrix
R [
43,
44,
50,
51,
52]:
In general, these relaxation rates lead to the non-exponential recovery of magnetization: however, the
1H magnetization recovers exponentially in ELM-11. In this context, as mentioned in
Appendix B, we can regard
RHH,
RFF ≈
RFH,
RHF; hence one of the two eigenvalues is almost zero. The observed relaxation rate then takes the following form
where
RHH and
RFF are diagonal elements of the relaxation matrix
R. In this case,
RHH and
RFF are given by [
39,
40]:
The analytical formulas for
and
are given by [
40,
50]:
Assuming that a thermal activation process is responsible for the fluctuation in the internuclear vector, the temperature dependence of
τi (
i = H, F) is given by the Arrhenius equation, as follows
where
Ea,i (
i = H, F) is the activation energy for BF
4− and 4,4′-bipyridine. Consequently, we analyzed the temperature dependence of
1H
T1 using the sum of the contributions from both paramagnetic relaxation (
R1p) and dipolar relaxation (
R1d):
The experimental data were fitted to Equation (8), the results of which are shown in
Figure 2a,b. The
R1p component was optimized at
τe = 1.22 × 10
−11 s, resulting in a
T1p value more than one order of magnitude smaller than
T1d. The evaluated
τe value is reasonable because typical
τe values for paramagnetic metal ions range between 10
−8 s and 10
−12 s [
53]; it is also sufficiently fast to average out the width of the
1H resonance line due to
1H-electron dipolar interactions. As described below, the average value of
1H
T2 is about 22 μs, which corresponds to a full width at half maximum (FWHM) of 15 kHz, where FWHM = 1/π
T2. This value is much narrower than the linewidth (~500 kHz) caused by the average local magnetic field between interlayer Cu-H pairs.
Table 3 summarizes the activation parameters and
and
values for the isotropic rotation of BF
4− and the torsional flipping of 4,4′-bipyridine. The
and
values determined from the optimization of
R1d are in good agreement with those calculated assuming an isotropic BF
4− reorientation and the torsional flipping of 4,4′-bipyridine. This observation suggests that the
T1 minimum observed at 323 K is mainly caused by averaging the
19F-
19F and
19F-
11B magnetic dipolar interaction by isotropic BF
4− reorientation. On the other hand, the small dips observed at 200 and 250 K are attributed to the averaging of the
1H-
1H and
1H-
19F magnetic dipolar interactions by the torsional flipping of the 4,4′-bipyridine as well as isotropic BF
4− reorientation. That is, the
1H-
1H and
1H-
19F magnetic dipolar interactions contribute less to the total
T1 compared to the
19F-
19F and
19F-
11B magnetic dipolar interactions; hence, the calculated
T1 curve is less sensitive to the 4,4′-bipyridine activation parameters. Therefore, in order to improve the reliability of the optimization results and to guarantee that the parameters have physical meaning, we assumed an
Ea value for the torsional flipping of the 4,4′-bipyridine. In fact, Moreau et al. reported that the torsional barrier for phenylene rings within linkers in a series of isoreticular octacarboxylate MOFs depended on the steric hindrance around the linkers, as well as the electronic structure of the framework [
54]. Furthermore, Inukai et al. reported that in [{Zn(5-nitroisophthalate)
x (5-methoxyisophthalate)
1−x (deuterated 4,4′-bipyridyl)} (DMF·MeOH)]
n, a kind of flexible PCP referred to as “CID-5/6”, the energy barrier for the rotation of the pyridyl ring depended on the steric hindrance around the linkers: the
Ea values for the 4-site and 2-site flip rotations are 20 and 25 kJ mol
−1 for CID-5/6 (
x = 0.55), and 32 and 27 kJ mol
−1 for CID-5/6 (
x = 0.37) [
55]. In the latter case, the intermolecular distances between 4,4′-bipyridine linkers in CID-5 and 6 are 4.11 Å and 3.91 Å, whereas it is 6.21 Å in the closed form of ELM-11, which suggests that there is less steric hindrance between the linkers in ELM-11. Therefore, we referred to the
Ea value as reported in the gas phase (4.0 kcal mol
−1) [
45] for simplicity, and then fixed the
Ea value to be close to this value during our
T1 analysis.
As a result, the
Ea value (32 kJ mol
−1) obtained for the isotropic reorientation of BF
4− is slightly larger than those (10–26 kJ mol
−1) reported in various systems [
40,
42,
43,
44]. The relatively short Cu-F interatomic distance of 2.404 Å facilitates the formation of a strong hydrogen-bond-like interaction (C-F...M
+ [
56]) between Cu(II) and a F atom in BF
4− (B-F...Cu
2+). As a result, the BF
4− isotropic reorientation in ELM-11 has a large
Ea value.
The gate phenomenon is closely associated with lattice vibration as well as the diffusivity of gas molecules. The rotational flipping of the 4,4′-bipyridine moiety is a type of phonon acoustic lattice-vibration mode of ELM-11. Gas molecules, such as CO2, perturb the rotational motion of the 4,4′-bipyridine moiety through molecular collisions. In particular, the inelastic collisions between gas molecules and the ELM-11 framework is considered to effectively perturb the thermally activated rotational motion of the 4,4′-bipyridine moiety, which then triggers the structural transition for gate opening. Thus, energy-transfer efficiency between the gas molecules and the ELM-11 framework determines the gate-opening pressure.
Furthermore, the torsional flipping and/or rotational motion of the 4,4′-bipyridine moiety also affects the orientational selectivity of the CO2 molecules toward molecular diffusion and arrangement in 1 at the first gate opening. Torsional flipping gives rise to an excluded volume for the pyridyl ring that is larger than the rigid one. This reduces the effective free volume along the b-axis because twisted 4,4′-bipyridine moieties lie along the b-axis. As a result, the accessible space for the CO2 molecules elongates along the b-axis as a prolate spheroid, which not only affects the molecular orientation when CO2 molecules penetrate into the ELM-11 crystal lattice, but also facilitates the alignment of CO2 molecules along the b-axis. In fact, the CO2 molecules are accommodated in the interlayer void spaces formed between the neighboring layered square grids in 2, which results in the alignment of the molecular axes with the b-axis.
3.4. CO2-Uptake Dependence of T1 in ELM-11
Figure 3a,b shows the dependence of
1H
T1 on the amount of CO
2 sorbed into ELM-11 at 273 and 195 K, respectively. The
T1 value was observed to decrease in a stepwise manner at 273 K, from 500 to 455 μs at
P/
P0 = 0.01. On the other hand, the
T1 value decreased in a stepwise manner at 195 K, from 532 to 490 μs at
P/
P0 = 0.01, and then increased again to 529 μs in the 0.2–0.4
P/
P0 range. These observed changes are in good agreement with the stepwise increases in the uptake of CO
2 shown in the sorption isotherms (
Figure 1). The crystal structure of ELM-11 changes through the stepwise sorption of CO
2, resulting in an increase in the interlayer distance. Therefore, this feature suggests that variations in
T1 due to CO
2 sorption are closely related to the structural changes undergone by ELM-11.
Table 4 lists the
T1 values for each ELM-11 structure at 273 and 195 K. The
T1 changes observed between 529 and 455 μs are due to structural changes, and the change in
T1 during a one-step structural change is in the 39–45 μs range.
T1 appears to depend on CO2 uptake, which is ascribable to: (1) an increase in the interlayer distance, and (2) an increase in the chemical pressure due to the impact of CO2 on the molecular motions of BF4− and 4,4′-bipyridine. The change in T1 in 1 in moving from 250 to 323 K is about 32 μs, which is smaller than those observed for the CO2-uptake dependence. Since dominates , a further increase in is required in order to explain the relationship between T1 and CO2 uptake. However, the structure of BF4− is not significantly affected by changes in the crystal structure of ELM-11; consequently, isotropic BF4− reorientation cannot be used to reasonably explain the observed change in T1 due to CO2 sorption.
On the other hand, the increase in the interlayer distance between the stacked two-dimensional [Cu(bpy)
22+]
n sheets increases the unit cell volume and the interlayer Cu-Cu distance; these affect
Np and
τe, which dominate
R1p. Since
R1p depends on
Np2 and
Np4/3 [
47], an increase in the cell volume decreases
Np (see
Table 4), resulting in a decrease in
R1p. In contrast,
τe is affected by interactions between electron spins (dipolar interactions and/or exchange interactions) and, as a first approximation, 1/
τe is proportional to the magnetic dipolar and/or exchange interaction [
53]. The average Cu-Cu distance in a [Cu(bpy)
22+]
n layer is 1.11 nm, whereas the average Cu-Cu distance between layers is 0.9105 nm in
1, 0.9959 nm in
2, and 1.0692 nm in
3. This feature strongly suggests that interlayer spin–spin interactions dominate more than intralayer ones. Since the magnetic dipolar and exchange interactions decay with increasing inter-spin distance,
τe increases with inter-spin distance. Consequently, the change in
T1 due to CO
2 sorption can be examined using
τe as a variable.
Table 4 summarizes the experimental and calculated values of
T1 for each crystal structure.
T1p, calc was calculated using
τe as a variable so as to reproduce
T1p, exp. At 273 K, the experimental value for
1 is somewhat smaller than the calculated one; this difference stems from the contribution of
R1d. Compared to
τe at 195 K, a longer interlayer Cu-Cu distance leads to a longer
τe. Thus, expansion of the unit cell due to CO
2 sorption decreases the spin density, whereas elongation of the interlayer distance increases
τe. These two effects act on
T1p in opposite directions, and in ELM-11 they are balanced and determine the total
T1p of the system. The
T1p of
2 is shorter than that of
1 because the contribution of
τe is rather large. On the other hand, both effects are comparable in
3 and, as a result, its
T1p is almost the same as that of
1.
3.5. Spin–Spin Relaxation Time (T2) in ELM-11
Figure 4a,b shows the dependence of
T2 on the amount of CO
2 sorbed at 273 K and 195 K. At 273 K, ELM-11 shows a stepwise increase in
T2 at
P/
P0 ~0.01, despite a decrease in
T1. On the other hand, ELM-11 shows two stepwise increases in
T2 at 195 K, at
P/
P0 ~0.01 and ~0.3. These changes in
T2 also correspond to the gate sorption of CO
2, as was observed for
T1, which accompanies a structural change in the crystal structure, in particular, an increase in the interlayer distance. The spin system satisfies a condition that
ωHτ >> 1 in these temperature regions, because
T1 ≠
T2 and
T2 <<
T1; hence
T2 is governed by the local magnetic field at the
1H nuclei (1/
T2). The local magnetic field caused by a spin with magnetic moment
μ at a position far from the spin, is given by (
μ0/4π)(
μ/
r3)(3cos
2θ – 1) [
39]. Here,
θ is the angle between the inter-spin vector and the external magnetic field and μ
0 is the magnetic permeability of a vacuum. The
1H,
19F, and electron spins contribute to the local magnetic field in ELM-11.
The magnitude of the local magnetic field is inversely proportional to the cube of the inter-spin distance. The contribution of Cu2+ can be evaluated from the average Cu-H distance between the stacked two-dimensional [Cu(bpy)22+]n sheets, which is 0.7801 nm in 1, 0.8668 nm in 2, and 0.9702 nm in 3. The square of the local magnetic field, , is evaluated using these distances to be 383 × 10−8 T2, 203 × 10−8 T2, and 103 × 10−8 T2, respectively. Consequently, extending the interlayer distance results in a decrease in to 53% in 2, and 27% in 3, of that of 1. Actually, the magnetic moment of Cu2+ is partially averaged out by the fast flip-flopping of the electron spin; hence, the net magnetic moment of Cu2+ reduces to .
The contributions from the
1H and
19F magnetic moments can also be evaluated through the second moments in the rigid lattices (see
Table A1 and
Table A2).
1 and
2 contain two kinds of 4,4′-bipyridine linkers with different conformations, whereas
3 has four kinds of 4,4′-bipyridine linker. The
values for the two conformers of
1 are 7.348 × 10
−8 T
2 and 2.086 × 10
−8 T
2, while in
2 they are 5.683 × 10
−8 T
2 and 1.958 × 10
−8 T
2, and they are 8.454 × 10
−8 T
2, 6.202 × 10
−8 T
2, 6.093 × 10
−8 T
2, and 2.357 × 10
−8 T
2, for the four conformers of
3. In each case, the conformer with the somewhat smaller torsion angle, in which
1H-
1H distances are relatively short, gives a larger
value than that with the larger torsion angle. Furthermore, the values of
of the planar and twisted conformers are similar in each compound, but
decreases in the order:
1 >
2 >
3, which indicates that the intermolecular
1H-
1H dipolar interaction is affected little by the conformation of the 4,4′-bipyridine moiety, but decreases due to the increase in the interlayer distance. On the other hand,
,
,
,
, and
are almost identical in the rigid lattices of the three substances, which suggests that the increase in the interlayer distance affects the intermolecular
1H-
19F dipolar interactions little. Therefore,
1H-
1H dipolar interactions are also considered to be among the factors that affect
T2 through the local magnetic field.
In terms of the structural changes that occur in going from
1 to
2 and then from
2 to
3, increases in the interlayer distance and the conformational changes undergone by the 4,4′-bipyridine linkers decrease both the
1H-electron and
1H-
1H dipolar interactions, i.e., the local magnetic field around the protons, resulting in an increase in
T2. In addition, at 195 K,
T2 is somewhat lower for CO
2 sorption between the first and the second steps. Since no lattice shrinkage was observed by powder XRD to accompany the decrease in interlayer distance during this process, we infer that the decrease in
T2 is not related to a change in interlayer distance (i.e., the
1H-electron distance). In fact, the closest
1H-
1H distance in 4,4′-bipyridine, pairs of which contribute the most to the local magnetic field, changes periodically with torsion angle. The local field is smallest at a twist angle of 90°, in which two pyridine rings are perpendicular to each other, and is largest for the planar structure, with a twist angle of 0° or 180°. Hence, we speculate that the conformational change undergone by the 4,4′-bipyridine linkers is one of the origins of the observed decrease in
T2 between the first and the second CO
2-sorption steps. The CO
2 uptake during the first gate sorption is estimated to be 160 mg g
-1, which corresponds to the sorption of two CO
2 molecules per [Cu(bpy)
2](BF
4)
2 formula unit, after which the CO
2 uptake increases gradually with
P/
P0, to a value of 230 mg g
−1 just prior to the second gate sorption. This uptake corresponds to the sorption of 2.9 CO
2 molecules per ELM-11 formula unit. Furthermore, uptake was observed to increase to 500 mg g
−1 following the second gate sorption, which corresponds to the sorption of 6.2 CO
2 molecules per ELM-11 formula unit. Hiraide et al. reported the crystal structures of
2 and
3, and revealed that the torsion angle around the C-C axis becomes small as the structure transforms from
2 into
3 [
11,
29]. This feature is considered to avoid repulsion between CO
2 and 4,4′-bipyridine, which increases the amount of sorbed CO
2 because the planar 4,4′-bipyridine structure has less free volume around its linkers than the other conformers. In fact, the conformation of the 4,4′-bipyridine linkers reportedly approaches that of the planar conformer by reducing the torsional angles from 0.74° and 70.64° in
2 to 0.14° and 68.74° in ELM-11⸧3CO
2 [
11,
29]. As the molecular structure of 4,4′-bipyridine approaches planarity, the intramolecular
1H-
1H distances (particularly, at the 2,6 and 2’,6’ positions) become shorter, which increases the
1H-
1H magnetic dipolar interactions. This conclusion is also supported by the
values of the 4,4′-bipyridine moieties, which are significantly different for the planar (5.683 × 10
−8 T
2) and twisted (1.958 × 10
−8 T
2) orientations. These
values correspond to
T2 contributions of 13 and 22 μs. Therefore, the increase in the intramolecular
1H-
1H dipolar interaction is regarded as a possible explanation for the decrease in
T2 observed between the first and second sorption steps.