Structural Dynamics of An ELM-11 Framework Transformation Accompanied with Double-Step CO₂ Gate sorption: An NMR Spin Relaxation Study

Abstract

[Cu(4,4′-bipyridine)₂(BF₄)₂] (ELM-11), an elastic layer-structured MOF (metal-organic framework), is expected to be a sophisticated CO₂ reservoir candidate because of its high capacity and recovery efficiency for CO₂ sorption. While ELM-11 shows a unique double-step gate sorption for CO₂ gas, the dynamics of the structural transition have not yet been clarified. In this study, the dynamics of the 4,4′-bipyridine linkers and the BF₄⁻ anions were studied by determining ¹H spin-lattice relaxation times (T₁). The ELM-11 structural transition accompanying CO₂ sorption was also examined through the CO₂ uptake dependence of the ¹H spin–spin relaxation time (T₂), in addition to T₁. In its closed form, the temperature dependence of the ¹H T₁ of ELM-11 was analyzed by considering the contributions of both paramagnetic and dipolar relaxations, which revealed the isotropic reorientation of BF₄⁻ and the torsional flipping of the 4,4′-bipyridine moieties. The resultant activation energy of 32 kJ mol⁻¹ for the isotropic BF₄⁻ reorientation is suggestive of strong (B-F...Cu²⁺) interactions between Cu(II) and the F atoms in BF₄⁻. Furthermore, the CO₂ uptake dependence of T₁ was found to be dominated by competition between the increase in the longitudinal relaxation time of the electron spins and the decrease in the spin density in the unit cell.

1. Introduction

Porous metal-organic frameworks (PMOFs) and porous coordination polymers (PCP), which exhibit dynamic structural transitions attributed to soft interactions in their crystal structures, are expected to have sorption properties that are different to those of traditional porous materials [1,2,3]. One of the most interesting phenomena in a flexible MOF is its guest-induced structural transition, which typically occurs at a threshold gas pressure and leads to an abrupt increase in the sorption isotherm, a phenomenon referred to as “breathing” and “gate sorption” [4,5,6,7,8,9,10]. The breathing of MIL-53 involves micropore filling accompanied by structural shrinkage and swelling, with volume expansion [7,8]. The gate sorption of a layer-structured MOF is accompanied by an abrupt increase and decrease in the sorbed quantity at a definite pressure, with almost no sorption below the threshold pressure [4,5,6,11,12,13,14,15]. Such guest-induced framework transitions have also been studied using theoretical and computational methods [8,16,17,18]. With such novel properties, these materials are expected to be developed into a unique class of material for gas separation and molecular sensing technologies [4,19,20,21,22,23].

The gate sorption of [Cu(4,4′-bipyridine)₂(BF₄)₂] (ELM-11), an elastic layer-structured MOF, is a representative example of novel sorption behavior. This material shows unique sorption isotherms for CO₂, N₂, and CH₄ through the expansive modulation of its layer structure [4,24,25,26,27,28,29]. ELM-11 also exhibits a better capacity and recovery efficiency for CO₂ sorption compared to other nanoporous materials [9,26]. Layer stacking is stabilized by soft interactions, such as π–π interactions and H...F hydrogen bonds [11,24]. ELM-11 shows a double-step gate sorption for CO₂ gas [28]. The first gate-opening occurs at a relative pressure (P/P₀) of 0.003 at 195 K, accompanied with a 28% increase in the interlayer distance, while the second sorption occurs at P/P₀ = 0.3 with a 56% expansion from the initial interlayer distance. More detailed structural analyses provided its fine structures before and after CO₂ sorption [29].

The crystal structures of ELM-11 with different CO₂ uptake levels were studied by Hiraide et al in the 195–298 K temperature range using in situ synchrotron X-ray powder diffractometry. For example, the unit cell of 1, the closed form of ELM-11 before CO₂ sorption, is monoclinic (space group C2/c, No. 15) with lattice constants: a = 1.24227(8) nm, b = 1.11618(6) nm, c = 1.61420(11) nm, β = 100.534(4)° at 273 K, and includes four formula units (Z = 4) [12]. By encapsulating two CO₂ molecules per [Cu(bpy)₂(BF₄)₂] (bpy = 4,4′-bipyridine) monomer unit at 273 K, the closed form of ELM-11 transforms into ELM-11⸧2CO₂ (2), which corresponds to the first gate-opening process. In this transformation, the unit cell expands along its a- and c-axes: a = 1.36851(6) nm, b = 1.10446(3) nm, c = 1.87175(6) nm, β = 95.687(3)°, although the crystal system and the space group are the same as those of the closed form. The CO₂ molecules penetrate through 1D channels composed of stacked square grids after expansion, and are then accommodated into the interlayer void spaces formed between the neighboring layered square grids through extension of the interlayer distance. The second gate-opening process accompanies the structural transition into ELM-11⸧6CO₂ (3), in which six CO₂ molecules are encapsulated in the [Cu(bpy)₂(BF₄)₂] monomer unit at 195 K. In this form, the unit cell is triclinic (space group P1, No. 1), with lattice constants: a = 1.10894(7) nm, b = 1.11193(5) nm, c = 1.43930(9) nm, α = 86.608(6)°, β = 75.513(5)°, and γ = 86.791(9)°, and includes two formula units (Z = 2) [29]. The lattice volume with four included formula units expands from 2.2005(3) nm³ for the closed form at 273 K to 3.4274(4) nm³ for ELM-11⸧6CO₂ via 2.8157(2) nm³ for ELM-11⸧2CO₂. Similar to the increase in the interlayer distance, the lattice volume is also 28% larger following the first step, and 56% larger following the second, compared with that of the closed form. Understanding the dynamic structure of its component moieties, such as the 4,4′-bipyridine linkers and BF₄⁻ anions, is necessary in order to clarify the transition mechanism of ELM-11 that accompanies gate sorption from a microscopic viewpoint. In addition, magnetic spin interactions between the paramagnetic spins will play important roles that induce the structural phase transition.

Electron-paramagnetic resonance (EPR) spectroscopy is a standard analytical procedure used to examine the local structures and spin–spin interactions in paramagnetic MOFs. The first EPR and ¹¹B magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectra were acquired by Jiang et al, and revealed the reversible structural changes that occur during the adsorption and desorption of probe molecules (CH₃OH and CH₃CN) [30]. Furthermore, Kultaeva et al. studied the formation and transformation mechanism of ELM-11 using powder and single-crystal EPR spectroscopy [31]. Based on the principal value of the g tensor and its anisotropy, they found that the cupric ions have elongated octahedral coordination symmetries and different axial ligands in the as-synthesized and activated forms of both [Cu(bpy)₂(CH₃OH)₂](BF₄)₂ and [Cu(bpy)₂(CH₃CN)₂](BF₄)₂.

Recently, the nuclear spin–lattice relaxation rates in paramagnetic substances have attracted much attention due to interest in distance-geometry [32], MRI-relaxation-agents [33], and quantum-computation [34] applications. The molecular motions, phase transitions, and inter-spin interactions in paramagnetic materials have been discussed through ¹H spin-lattice relaxation times (T₁) [35,36,37,38]. Therefore, ¹H nuclear magnetic relaxation in ELM-11 is expected to provide useful information about the structural changes and spin–spin interactions that accompany CO₂ gate sorption.

In this study, we investigated the dynamic behavior of the 4,4′-bipyridine linkers and the BF₄⁻ anions in the closed form of ELM-11 by determining its temperature-dependent ¹H spin-lattice relaxation times (T₁), after which the structural transition of ELM-11 accompanying CO₂ sorption was examined by the CO₂ uptake dependence of the ¹H spin–spin relaxation time (T₂) as well as T₁. Finally, the structural change due to CO₂ sorption was examined in terms of magnetic dipolar interactions between nuclear spins and between paramagnetic spins.

2. Experimental

ELM-11 was prepared according to the reported method [27]. After pretreatment under vacuum (<0.1 Pa) at 373 K for 10 h, CO₂ sorption isotherms were obtained volumetrically at 273 and 195 K using BELSORP Mini II (MicrotracBEL Corp., Osaka, Japan) instruments. The CO₂ gas was 99.9999% pure.

The NMR sample was prepared as follows: a 300 mg sample of ELM-11 powder was introduced into a glass NMR tube (ϕ 10 mm) and maintained under vacuum at 373 K for 10 h. CO₂ gas was loaded into the tube at 273 or 195 K and adjusted to the appropriate pressure. The tube was sealed with a valve and then inserted into the NMR spectrometer, with the temperature controlled at 273 or 195 K.

A JNM-MU25 pulse NMR spectrometer (JEOL, Akishima, Tokyo, Japan) with a ¹H resonance frequency of 25 MHz (0.5872 T, permanent magnet) was used to measure ¹H relaxation times. T₁ values were measured with the inversion recovery method using a radio-frequency pulse width of 2 μs, a repetition time of 2 ms, and 50 datapoints with a sampling interval of 30 μs. T₂ values were measured with the solid-echo method using a radio-frequency pulse width of 2 μs, a repetition time of 2 ms, and 500 datapoints with a sampling interval of 0.2 μs.

3. Results and Discussion

3.1. CO₂ Sorption Isotherms

CO₂ sorption isotherms for ELM-11 at 273 and 195 K are shown in Figure 1. The CO₂ sorption isotherm of ELM-11 to P/P₀ ~0.03 at 273 K reveals a vertical uptake at P/P₀ ~0.01 (Figure 1a), which corresponds to gate opening, as previously reported [4,24,26,27]. Another steep increase in sorption is seen at 195 K at P/P₀ ~0.3, as shown in Figure 1b. Similar double-step sorption isotherms have previously been reported [13,14,28,29]. Detailed structural analyses showed that ELM-11 absorbs two CO₂ molecules per Cu atom to form 2, with a 28% expansion in the interlayer distance at the first step at 273 K, and absorbs four more CO₂ molecules per Cu atom to form 3, with a 56% expanded layer structure compared to the initial structure at 195 K [28,29].

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 BF₄⁻ 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. ¹H second moments (in 10⁻⁸ T²) for all ELM-11 substances.

Interaction	Rigid Lattice	Averaged Value
		bpy Flip	BF4− Rotation	bpy Flip + BF4− Rotation
1
bpy 1 (0.09°)1
$M_{2,intra}^{HH}$	7.348	7.348	7.348	7.348
$M_{2,inter}^{HH}$	2.865	1.327	2.865	1.327
$M_{2,inter}^{HF}$	1.217	1.217	0.692	0.692
total	11.43	9.892	10.905	9.367
bpy 2 (54.6°)1
$M_{2,intra}^{HH}$	2.086	1.634	2.086	1.634
$M_{2,inter}^{HH}$	2.829	2.264	2.829	2.264
$M_{2,inter}^{HF}$	0.883	0.443	0.499	0.317
total	5.798	4.341	5.414	4.215
2
bpy 1 (0.74°) 1
$M_{2,intra}^{HH}$	5.683	5.683	5.683	5.683
$M_{2,inter}^{HH}$	1.267	1.252	1.267	1.252
$M_{2,inter}^{HF}$	1.301	1.301	0.637	0.637
total	8.251	8.236	7.587	7.572
bpy 2 (70.64°)1
$M_{2,intra}^{HH}$	1.958	1.735	1.958	1.735
$M_{2,inter}^{HH}$	1.278	1.152	1.278	1.152
$M_{2,inter}^{HF}$	0.808	0.435	0.576	0.250
total	4.044	3.322	3.812	3.137
3
bpy 1 (0.52°) 1
$M_{2,intra}^{HH}$	8.454	8.454	8.454	8.454
$M_{2,inter}^{HH}$	0.664	0.572	0.664	0.572
$M_{2,inter}^{HF}$	0.712	0.712	0.370	0.370
total	9.830	9.738	9.488	9.396
bpy 2 (14.98°) 1
$M_{2,intra}^{HH}$	6.202	5.045	6.202	5.045
$M_{2,inter}^{HH}$	0.690	0.670	0.690	0.670
$M_{2,inter}^{HF}$	0.612	0.612	0.368	0.368
total	7.504	6.327	7.260	6.083
bpy 1’ (17.52°) 1
$M_{2,intra}^{HH}$	6.093	4.756	6.093	4.756
$M_{2,inter}^{HH}$	0.725	0.540	0.725	0.540
$M_{2,inter}^{HF}$	0.682	0.682	0.373	0.373
total	7.500	5.978	7.191	5.669
bpy 2’ (49.46°) 1
$M_{2,intra}^{HH}$	2.357	1.770	2.357	1.770
$M_{2,inter}^{HH}$	0.586	0.319	0.586	0.319
$M_{2,inter}^{HF}$	0.507	0.146	0.299	0.079
total	3.450	2.235	3.242	2.168

¹ The number in parentheses is the torsion angle of the 4,4′-bipyridine linker around C-C axis.

and

Table A2. ¹⁹F second moments (in 10⁻⁸ T²) for all ELM-11 substances.

Interaction	Rigid Lattice	Averaged Value
		bpy Flip	BF4− Rotation	bpy Flip + BF4− Rotation
1
$M_{2,intra}^{FF}$	6.507	6.507	0	0
$M_{2,inter}^{FF}$	6.872	6.872	2.369	2.369
$M_{2,inter}^{FH}$	5.337	4.220	3.028	2.563
$M_{2,intra}^{F10B}$	0.757	0.757	0	0
$M_{2,intra}^{F11B}$	8.660	8.660	0	0
total	28.133	27.016	5.397	4.932
2
$M_{2,intra}^{FF}$	6.337	6.337	0	0
$M_{2,inter}^{FF}$	6.485	6.485	0.089	0.089
$M_{2,inter}^{FH}$	5.358	5.088	3.080	2.622
$M_{2,intra}^{F10B}$	0.729	0.729	0	0
$M_{2,intra}^{F11B}$	8.343	8.343	0	0
total	27.252	26.982	3.169	2.711
3
$M_{2,intra}^{FF}$	5.478	5.478	0	0
$M_{2,inter}^{FF}$	2.811	2.811	0.040	0.040
$M_{2,inter}^{FH}$	6.302	2.486	3.537	1.308
$M_{2,intra}^{F10B}$	0.628	0.628	0	0
$M_{2,intra}^{F11B}$	7.189	7.189	0	0
total	22.408	18.592	3.577	1.348

in Appendix A, while

Table 1. Reductions in the ¹H second moments (in 10⁻⁸ T²) in ELM-11.

Interaction	Motional Mode
	bpy Flip	BF4− Rotation	bpy Flip + BF4− Rotation
1
bpy 1
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0	0	0
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	1.538	0	1.538
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0	0.525	0.525
total	1.538	0.525	2.063
bpy 2
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0.452	0	0.452
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.565	0	0.565
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0.44	0.384	0.566
total	1.457	0.384	1.583
2
bpy 1
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0	0	0
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.015	0	0.015
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0	0.664	0.664
total	0.015	0.664	0.679
bpy 2
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0.223	0	0.223
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.126	0	0.126
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0.373	0.232	0.558
total	0.722	0.232	0.907
3
bpy 1
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0	0	0
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.092	0	0.092
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0	0.342	0.342
total	0.092	0.342	0.434
bpy 2
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	1.157	0	1.157
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.02	0	0.02
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0	0.244	0.244
total	1.177	0.244	1.421
bpy 1’
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	1.337	0	1.337
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.185	0	0.185
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0	0.309	0.309
total	1.522	0.309	1.831
bpy 2’
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0.587	0	0.587
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.267	0	0.267
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0.361	0.208	0.428
total	1.215	0.208	1.282
bpy 1
$\mathrm{\Delta }{M}_{2,intra}^{HH}$	0	0	0
$\mathrm{\Delta }{M}_{2,inter}^{HH}$	0.092	0	0.092
$\mathrm{\Delta }{M}_{2,inter}^{HF}$	0	0.342	0.342
total	0.092	0.342	0.434

and

Table 2. Reductions in the ¹⁹F second moments (in 10⁻⁸ T²) in ELM-11.

Interaction	Motional Mode
	bpy Flip	BF4− Rotation	bpy Flip + BF4− Rotation
1
$\mathrm{\Delta }{M}_{2,intra}^{FF}$	0	6.507	6.507
$\mathrm{\Delta }{M}_{2,inter}^{FF}$	0	4.503	4.503
$\mathrm{\Delta }{M}_{2,inter}^{FH}$	1.117	2.309	2.774
$\mathrm{\Delta }{M}_{2,intra}^{F10B}$	0	0.757	0.757
$\mathrm{\Delta }{M}_{2,intra}^{F11B}$	0	8.66	8.660
total	1.117	22.736	23.201
2
$\mathrm{\Delta }{M}_{2,intra}^{FF}$	0	6.337	6.337
$\mathrm{\Delta }{M}_{2,inter}^{FF}$	0	6.396	6.396
$\mathrm{\Delta }{M}_{2,inter}^{FH}$	0.270	2.278	2.736
$\mathrm{\Delta }{M}_{2,intra}^{F10B}$	0	0.729	0.729
$\mathrm{\Delta }{M}_{2,intra}^{F11B}$	0	8.343	8.343
total	0.270	24.083	24.541
3
$\mathrm{\Delta }{M}_{2,intra}^{FF}$	0	5.478	5.478
$\mathrm{\Delta }{M}_{2,inter}^{FF}$	0	2.771	2.771
$\mathrm{\Delta }{M}_{2,inter}^{FH}$	3.816	2.765	4.994
$\mathrm{\Delta }{M}_{2,intra}^{F10B}$	0	0.628	0.628
$\mathrm{\Delta }{M}_{2,intra}^{F11B}$	0	7.189	7.189
total	3.816	18.831	21.06

show the evaluated reductions in the ¹H and ¹⁹F second moments.

The reduction in the ¹H second moment, $\mathrm{\Delta }{M}_2^H$, which is the sum of $\mathrm{\Delta }{M}_{2, intra}^{HH}$, $\mathrm{\Delta }{M}_{2, inter}^{HH}$, and $\mathrm{\Delta }{M}_{2, inter}^{HF}$, 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, $\mathrm{\Delta }{M}_2^F$, which is the sum of $\mathrm{\Delta }{M}_{2, intra}^{FF}$, $\mathrm{\Delta }{M}_{2, inter}^{FF}$, $\mathrm{\Delta }{M}_{2, inter}^{FH}$, and $\mathrm{\Delta }{M}_{2, intra}^{FB}$, 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 BF₄^- and/or 4,4′-bipyridine, then the apparent activation energy (0.5 kJ mol⁻¹) is much smaller than the reported E_a values for the isotropic rotation of BF₄⁻ (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 Cu²⁺ (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 (R_1p) is given by [47,48].

$$R_{1p}=2\overline{C}{N}_p^2+50{\left(\overline{C}D\right)}^{1/2}{N}_p^{4/3},$$

(1)

where $\overline{C}$ and D is the efficiency of direct relaxation and the diffusion coefficient for spin diffusion, respectively, and N_p is the number of paramagnetic centers per unit volume of the sample. In the powder sample, $\overline{C}$ is represented by

$$\overline{C}=\frac{2}{5}{\left(\frac{{\mu }_0}{4\pi }\right)}^{2}S\left(S+1\right){\gamma }_S^2{\gamma }_I^2{ħ}^2\frac{{\tau }_e}{1+{\omega }_I^2{\tau }_e^2},$$

(2)

where γ_S and γ_I are the gyromagnetic ratios of the electron spin and resonant nuclei, respectively, S is the spin of the paramagnetic ion, τ_e is the correlation time for the z-component of the paramagnetic spin (longitudinal relaxation time for the electron spin), and ω_I is the resonance frequency of a resonant nucleus. According to Bloembergen [49], D = a²/50T², 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 m² s⁻¹ for 1. This is reasonable because it is of the same order of magnitude as the D value (6.25 × 10⁻¹⁶ m² s⁻¹) for the high spin state of [Fe(ptz)₆](BF₄)₂ (ptz = 1-n-propyl-1H-tetrazole) [35]. Furthermore, we evaluated N_p as 1.91 × 10²⁷ m⁻³ for the body-centered lattice formed by the Cu²⁺ ions in 1. Thus, R_1p depends strongly on τ_e.

On the other hand, the dipolar relaxation rate (R_1d) 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]:

$$\mathit{R}=\left[\begin{array}{cc}{R}_{HH}& R_{HF}\\ R_{FH}& R_{FF}\end{array}\right].$$

(3)

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 R_HH, R_FF ≈ R_FH, R_HF; hence one of the two eigenvalues is almost zero. The observed relaxation rate then takes the following form

$${\left(R_{1d}\right)}_{HH}={\left(R_{1d}\right)}_{FF}=R_{HH}+R_{FF},$$

(4)

where R_HH and R_FF are diagonal elements of the relaxation matrix R. In this case, R_HH and R_FF are given by [39,40]:

$$R_{HH}=\frac{2}{3}{\gamma }_H^2\mathrm{\Delta }{M}_2^{HH}{g}_1\left({\omega }_H,{\tau }_H\right)+\frac{1}{2}{\gamma }_H^2\mathrm{\Delta }{M}_2^{HF}{g}_2\left({\omega }_H,{\omega }_F,{\tau }_H\right),$$

(5a)

$$\begin{array}{cc}\hfill R_{FF}=& \frac{2}{3}{\gamma }_H^2\mathrm{\Delta }{M}_2^{FF}{g}_1\left({\omega }_F,{\tau }_F\right)+\frac{1}{2}{\gamma }_F^2\mathrm{\Delta }{M}_2^{FH}{g}_2\left({\omega }_F,{\omega }_H,{\tau }_F\right)+\frac{1}{2}{\gamma }_F^2\mathrm{\Delta }{M}_2^{F10B}{g}_2\left({\omega }_F,{\omega }_{10B},{\tau }_F\right)\hfill \\ & +\frac{1}{2}{\gamma }_F^2\mathrm{\Delta }{M}_2^{F11B}{g}_2\left({\omega }_F,{\omega }_{11B},{\tau }_F\right)\hfill \end{array}$$

(5b)

The analytical formulas for $g_1\left({\omega }_i,{\tau }_i\right)$ and $g_2\left({\omega }_i,{\omega }_j,{\tau }_i\right)$ are given by [40,50]:

$$g_1\left({\omega }_i,{\tau }_i\right)=\frac{{\tau }_i}{1+{\omega }_i^2{\tau }_i^2}+\frac{4{\tau }_i}{1+4{\omega }_i^2{\tau }_i^2},$$

(6a)

$$g_2\left({\omega }_i,{\omega }_j,{\tau }_i\right)=\frac{{\tau }_i}{1+{\left({\omega }_i-{\omega }_j\right)}^2{\tau }_i^2}+\frac{3{\tau }_i}{1+{\omega }_i^2{\tau }_i^2}+\frac{6{\tau }_i}{1+{\left({\omega }_i+{\omega }_j\right)}^2{\tau }_i^2}.$$

(6b)

Assuming that a thermal activation process is responsible for the fluctuation in the internuclear vector, the temperature dependence of τ_i (i = H, F) is given by the Arrhenius equation, as follows

$${\tau }_i={\tau }_{0,i}\exp \left(E_{a,i}/RT\right),$$

(7)

where E_a,i (i = H, F) is the activation energy for BF₄⁻ and 4,4′-bipyridine. Consequently, we analyzed the temperature dependence of ¹H T₁ using the sum of the contributions from both paramagnetic relaxation (R_1p) and dipolar relaxation (R_1d):

$$R_{1H}=1/T_{1H}=R_{1p}+R_{1d}.$$

(8)

The experimental data were fitted to Equation (8), the results of which are shown in Figure 2a,b. The R_1p component was optimized at τ_e = 1.22 × 10⁻¹¹ s, resulting in a T_1p value more than one order of magnitude smaller than T_1d. The evaluated τ_e value is reasonable because typical τ_e 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/πT₂. This value is much narrower than the linewidth (~500 kHz) caused by the average local magnetic field between interlayer Cu-H pairs.

Table 3. Activation parameters for the isotropic reorientation of BF₄⁻ and the torsional flipping of 4,4′-bipyridine in 1 determined from dipolar relaxation data.

Parameter	Expt.	Calc.
1H interaction
τH,0/s	1.0 × 10−12	-----
Ea(H)/kJ mol−1	18	-----
ΔM2HH/10−8 T2	1.28	1.28
ΔM2HF/10−8 T2	0.55	0.55
19F interaction
τF,0/s	4.0 × 10−14	-----
Ea(F)/kJ mol−1	32	-----
ΔM2FF/10−8 T2	10	11.0
ΔM2FH/10−8 T2	1.6	2.77
ΔM2F11B/10−8 T2	7.0	8.66
ΔM2F10B/10−8 T2	0.61	0.76

summarizes the activation parameters and $\mathrm{\Delta }{M}_2^{ii}$ and $\mathrm{\Delta }{M}_2^{ij}$ values for the isotropic rotation of BF₄⁻ and the torsional flipping of 4,4′-bipyridine. The $\mathrm{\Delta }{M}_2^{ii}$ and $\mathrm{\Delta }{M}_2^{ij}$ values determined from the optimization of R_1d are in good agreement with those calculated assuming an isotropic BF₄⁻ 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 BF₄⁻ 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 BF₄⁻ 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 E_a 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 E_a 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 E_a value as reported in the gas phase (4.0 kcal mol⁻¹) [45] for simplicity, and then fixed the E_a value to be close to this value during our T₁ analysis.

As a result, the E_a value (32 kJ mol⁻¹) obtained for the isotropic reorientation of BF₄⁻ 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 BF₄⁻ (B-F...Cu²⁺). As a result, the BF₄⁻ isotropic reorientation in ELM-11 has a large E_a 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 CO₂ 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 CO₂ shown in the sorption isotherms (Figure 1). The crystal structure of ELM-11 changes through the stepwise sorption of CO₂, resulting in an increase in the interlayer distance. Therefore, this feature suggests that variations in T₁ due to CO₂ sorption are closely related to the structural changes undergone by ELM-11.

Table 4. ¹H T₁ via paramagnetic centers, electron longitudinal relaxation times, diffusion coefficients for spin diffusion, spin densities, and average interlayer Cu-Cu distances in ELM-11.

Parameter	1		2		3
T/K	273	195	273	195	195
T1p, exp/μs	500	532	455	490	529
T1p, calc/μs	523		449	490	528
τe/s	1.22 × 10−11		3.08 × 10−11	2.59 × 10−11	4.05 × 10−11
D/m2s−1	2.87 × 10−16		2.87 × 10−16		2.87 × 10−16
Np/m−3	1.91 × 1027		1.51 × 1027		1.21 × 1027
rave.(Cu-Cu)/nm	0.9105		0.9959		1.0692

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 $\mathrm{\Delta }{M}_2^{F11B}$ dominates $\mathrm{\Delta }{M}_2$, a further increase in $\mathrm{\Delta }{M}_2^{F11B}$ 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 N_p and τ_e, which dominate R_1p. Since R_1p depends on N_p² and N_p^4/3 [47], an increase in the cell volume decreases N_p (see Table 4), resulting in a decrease in R_1p. In contrast, τ_e is affected by interactions between electron spins (dipolar interactions and/or exchange interactions) and, as a first approximation, 1/τ_e 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, τ_e increases with inter-spin distance. Consequently, the change in T₁ due to CO₂ sorption can be examined using τ_e as a variable.

Table 4 summarizes the experimental and calculated values of T₁ for each crystal structure. T_{1p, calc} was calculated using τ_e as a variable so as to reproduce T_{1p, exp}. At 273 K, the experimental value for 1 is somewhat smaller than the calculated one; this difference stems from the contribution of R_1d. Compared to τ_e at 195 K, a longer interlayer Cu-Cu distance leads to a longer τ_e. Thus, expansion of the unit cell due to CO₂ sorption decreases the spin density, whereas elongation of the interlayer distance increases τ_e. These two effects act on T_1p in opposite directions, and in ELM-11 they are balanced and determine the total T_1p of the system. The T_1p of 2 is shorter than that of 1 because the contribution of τ_e is rather large. On the other hand, both effects are comparable in 3 and, as a result, its T_1p 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 CO₂ 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 CO₂, 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 ω_Hτ >> 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$\propto ⟨B_{\mathrm{loc}}^2⟩$). 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)(3cos²θ – 1) [39]. Here, θ is the angle between the inter-spin vector and the external magnetic field and μ₀ 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, $B_{\mathrm{loc}}^2$, 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 $B_{\mathrm{loc}}^2$ 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 $B_{\mathrm{loc}}^2$ to $⟨B_{\mathrm{loc}}^2⟩$.

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 $M_{2,intra}^{HH}$ values for the two conformers of 1 are 7.348 × 10⁻⁸ T² and 2.086 × 10⁻⁸ T², while in 2 they are 5.683 × 10⁻⁸ T² and 1.958 × 10⁻⁸ T², and they are 8.454 × 10⁻⁸ T², 6.202 × 10⁻⁸ T², 6.093 × 10⁻⁸ T², and 2.357 × 10⁻⁸ T², 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 $M_{2,intra}^{HH}$ value than that with the larger torsion angle. Furthermore, the values of $M_{2,inter}^{HH}$ of the planar and twisted conformers are similar in each compound, but $M_{2,inter}^{HH}$ 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, $M_{2,intra}^{FF}$, $M_{2,inter}^{FF}$, $M_{2,inter}^{FH}$, $M_{2,intra}^{F10B}$, and $M_{2,intra}^{F11B}$ 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 CO₂ 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 CO₂ uptake during the first gate sorption is estimated to be 160 mg g^-1, which corresponds to the sorption of two CO₂ molecules per [Cu(bpy)₂](BF₄)₂ formula unit, after which the CO₂ 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 CO₂ 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 CO₂ 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 CO₂ and 4,4′-bipyridine, which increases the amount of sorbed CO₂ 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⸧3CO₂ [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 $M_{2,intra}^{HH}$ values of the 4,4′-bipyridine moieties, which are significantly different for the planar (5.683 × 10⁻⁸ T²) and twisted (1.958 × 10⁻⁸ T²) orientations. These $M_{2,intra}^{HH}$ 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.

4. Conclusions

We calculated the ¹H and ¹⁹F second moments in the rigid lattices of the three crystal structures of ELM-11, and the reductions in the second moments due to both isotropic BF₄⁻ reorientation and the torsional flipping of the 4,4′-bipyridine linkers. ¹H second-moment reductions of (0.4–2) × 10⁻⁸ T² were determined, indicative of a low contribution to the total magnetic dipolar relaxation rate. On the other hand, reductions of (21–24) × 10⁻⁸ T² were determined for the ¹⁹F second moment. These large reductions suggested that ¹H spin–lattice relaxation effectively takes place through fluctuations in the magnetic dipolar interactions that act on ¹⁹F nuclei through cross-relaxation between the ¹H and ¹⁹F spin systems.

The temperature dependence of ¹H T₁ in the closed form of ELM-11 was analyzed using the sum of the contributions from both paramagnetic relaxation (R_1p) and dipolar relaxation (R_1d). We found that R_1p makes a dominant contribution to the total ¹H spin–lattice relaxation rate, but the T₁ minimum observed at 323 K is mainly due to the averaging of ¹⁹F-¹⁹F and ¹⁹F-¹¹B magnetic dipolar interactions through isotropic BF₄⁻ reorientation. The large E_a value (32 kJ mol⁻¹) obtained for the isotropic BF₄⁻ reorientation supports the formation of a strong hydrogen-bond-like interaction (B-F...Cu²⁺) between Cu(II) and a F atom in BF₄^-. We also discussed the role that torsional flipping of the 4,4′-bipyridine moiety plays in relation to the gate-opening phenomenon, as well as the orientational selectivity of the CO₂ molecules in relation to their diffusion and arrangement in the lattice.

The dependence of T₁ on CO₂ uptake is the result of a corresponding increase in the interlayer distance. The increase in the unit cell volume due to CO₂ sorption led to a decrease in spin density, whereas an increase in the interlayer distance resulted in an increase in the longitudinal relaxation time of the electron spins (τ_e). These two effects, which act on T_1p in opposite directions, balance each other and control the T₁ value.

The local magnetic field at the ¹H nuclei governs the T₂ value, and a decrease in the local magnetic field increases the T₂ value. The local magnetic field associated with the net magnetic moments of Cu²⁺ and the intermolecular ¹H dipolar interaction decreases with increasing interlayer distance in ELM-11, leading to an increase in T₂. Furthermore, the conformational change in the 4,4′-bipyridine unit, from the twisted form to the planar form, enables the intramolecular ¹H dipolar interaction to increase, which shortens T₂.