#
Overview of Low-Temperature Heat Capacity Data for Zn_{2}(C_{8}H_{4}O_{4})_{2}^{.}C_{6}H_{12}N_{2} and the Salam Hypothesis

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}(C

_{8}H

_{4}O

_{4})

_{2}

^{.}C

_{6}H

_{12}N

_{2}(Zn-DMOF). In Zn-DMOF, left-twisted D

_{3}(S) and right-twisted D

_{3}(R) DABCO molecules (C

_{6}H

_{12}N

_{2}) can transform into each other by tunneling to form a racemate. Termination of tunneling leads to a phase transition in the subsystem of twisted molecules. It is suggested that Zn-DMOF may be considered a model system to study the mechanisms of phase transitions belonging to the same type as hypothetical Salam phase transitions.

## 1. Introduction

_{6}H

_{12}N

_{2}) in the metal-organic framework Zn

_{2}(C

_{8}H

_{4}O

_{4})

_{2}

^{.}C

_{6}H

_{12}N

_{2}(Zn-DMOF) and that the mechanism of Salam phase transitions remains possible. In Zn-DMOF, the enantiomers are represented by left- and right-twisted DABCO molecules, which transform into each other as a result of tunneling.

## 2. Structure of DABCO Molecule in Zn_{2}(C_{8}H_{4}O_{4})_{2}·C_{6}H_{12}N_{2}

_{3h}and D

_{3}point group symmetries, depending on intermolecular interactions. Also, a quasi-D

_{3h}form of DABCO is possible due to strong vibrations of the molecule around the C

_{3}axis. The molecules with the D

_{3}symmetry, which can be left-twisted D

_{3}(S) or right-twisted D

_{3}(R), are considered to be chiral isomers (enantiomers) [4].

_{8}H

_{4}O

_{4}]

^{2−}(BDC

^{2−}) which are linked to {Zn

_{2}} pairs by carboxylate anions. The vertical edges are formed by DABCO molecules (linkers), the point symmetry of which does not contain a 4-fold rotational symmetry axis (Figure 1). This is the reason why DABCO molecules are orientationally disordered; moreover, D

_{3}(S) and D

_{3}(R) forms can transform into each other (by activation or tunneling) [4,6]. Calorimetry, nuclear magnetic resonance, and X-ray structural analysis data provide evidence of the presence of phase transitions in Zn-DMOF at ~14, ~60, and ~130 K [7,8,9,10,11,12].

## 3. Mobility of DABCO Molecules in Zn_{2}(C_{8}H_{4}O_{4})_{2}·C_{6}H_{12}N_{2}

^{2}

^{−}anions and DABCO molecules are involved in activation mobility. According to the nuclear magnetic resonance (NMR) studies of the activation mobility of BDC

^{2}

^{−}anions, the [C

_{6}H

_{4}] groups of BDC

^{2}

^{−}anions rotate about the C

_{2}axis through an angle of 180° (flipping) [13,14,15]. No effect of BDC

^{2}

^{−}flipping on the mobility of DABCO in Zn-DMOF was discovered [13,16,17].

^{1}H NMR T

_{1}(T)), D

_{3}(S) and D

_{3}(R) forms of DABCO can make up a racemic mixture, and their mirror symmetry may be broken during the phase transition at ~60 K [6,11,12]. The time decay of nuclear magnetic moments (

**M**) of hydrogen atoms in DABCO was analyzed to find the distribution of DABCO molecules over different states. Above ~165 K, the time decay of

**M**is a single exponential function characterized by a single value T

_{1}. In this case, DABCO molecules with D

_{3}and D

_{3h}symmetries reorient similarly, their proton spins constitute a single spin system, the activation barrier is equal to ~4 kJ/mol. Between 165 and 60 K, the time decay of

**M**is a biexponential function containing two values T

_{1}, each corresponding to a certain fraction of nuclear spins in

**M**. The ratio of these fractions is estimated to be

**⅓:⅔**. The

**⅓·M**fraction corresponds to

^{1}H spins of DABCO molecules of the D

_{3h}symmetry, the mobility of which is characterized by a short value T

_{1SH}. The

**⅔·M**fraction corresponds to

^{1}H spins of the sum of S- and R-forms of DABCO. In this case, these forms are indistinguishable due to tunneling transitions, so the above fraction (

**⅔·M**) represents the racemic state of DABCO molecules and is characterized by a single value T

_{1L}of a larger magnitude. During the phase transition at 60 K and down to 25 K, the behavior of T

_{1L}is interpreted as the termination of tunneling between energy degenerate quantum states of R- and S- forms of DABCO, and their fractions in

**M**remain equal to each other (

**⅓:⅓**). Below 25 K, the decay

**M**is nonexponential and can be conventionally characterized by three values T

_{1}. So, the phase transition at ~14 K is associated with the redistribution of DABCO molecules over different energy states characterized by contributions

**¼·M**,

**¼·M**, and

**½·M**and the appearance of a chiral polarized state [11].

^{1}H NMR T

_{1}(T) data testify that phase transitions are associated with the mobility of DABCO molecules. The analysis of the function

**M**provides quantitative data on the distribution of DABCO molecules over different states at various temperatures. However, it is still unclear how these states are structurally realized in Zn-DMOF. Low-temperature heat capacity data for Zn-DMOF may be used to clarify this problem.

## 4. Low-Temperature Heat Capacity in Zn_{2}(C_{8}H_{4}O_{4})_{2}·C_{6}H_{12}N_{2}

_{p}= C

_{p}− C

_{p}

^{L}, where C

_{p}is the heat capacity of the substance and C

_{p}

^{L}is the regular part of heat capacity "in the absence of phase transitions". The entropy of the phase transitions is shown in Table 2.

^{4}He affect the states of D

_{3}(S) and D

_{3}(R) forms of DABCO (phase transitions ~14 and ~60 K) and do not affect the ordering and disordering of BDC

^{2−}anions during the phase transition at 130 K. This result can be explained by the fact that the structure of DABCO is flexible [23,24] as compared to that of BDC

^{2−}anions and can therefore be deformed in the presence of adjoining

^{4}He atoms, whereas the structure of BDC

^{2−}anions remains unchanged.

**a**and

**b**axes (dL/LdT(

**a,b**) = −9.59·10

^{−}

^{6}K

^{−}

^{1}) and shrinks along the

**c**axis (dL/LdT(

**c**) = 12.2·10

^{−}

^{6}K

^{−}

^{1}) as the temperature decreases to ~130 K and below ~130 K |dL/LdT(

**c**)| > |dL/LdT(

**a,b**)|(dL/LdT is the coefficient of thermal expansion) [9]. Hence, the interactions in the -Zn-DABCO-Zn- chain directed along the

**c**axis are assumingly stronger than BDC

^{2}

^{−}-[Zn

_{2}]

^{4+}-BDC

^{2}

^{−}- interactions in the

**ab**plane, which determines a one-dimensional elastic continuum for the behavior of the heat capacity. The phase transition at ~130 K was interpreted as an order-disorder phase transition associated with a change in the relative spatial arrangement of BDC

^{2}

^{−}anions, while the DABCO molecules preserve their activation mobility and remain disordered [9].

_{v}is expressed in terms of two relationships, C

_{v}/(3

**N**k) and T/T

_{m}, where

**N**is the number of repeated vibrating units, T

_{m}= hν

_{m}/k, h is the Planck constant, k is the Boltzmann constant, and ν

_{m}is the maximum frequency of stretching vibrations in the chain. The repeated vibrating unit along the

**c**axis in Zn-DMOF consists of two Zn atoms and one DABCO molecule ({Zn

_{2}DABCO}) [5].

_{p}obtained as functions of temperature in [8] were represented on a log-log plot and fitted by best tabulated values C

_{v}/(3

**N**k) for each T/T

_{m}assuming that C

_{p}−C

_{v}is small [27] (Figure 3). As a result, it was found that the vibrating chain is formed by ~38–39 {Zn

_{2}DABCO} units above 130 K, by ~30 units at 60–130 K, and by ~12 units at 14–60 K (Table 3). Below 14 K, the heat capacity obeys the ~T

^{3}law (Figure 3) to indicate that interchain interactions become stronger and the lattice vibrational modes become three-dimensional [8,25].

_{m}~ 1250 cm

^{−1}and ν

_{m}~ 765 cm

^{−1}fall into the region of stretching vibrations of DABCO, and ν

_{m}~ 285 cm

^{−1}fall into the region of Zn-N and Zn-Zn stretchings (Table 3) [28]. Thus, the obtained values ν

_{m}correspond to the stretchings in the chains, in accordance with the model [27].

_{p}for

**N**{Zn

_{2}DABCO} units correlate with fractions (

**M**) in different phases of Zn-DMOF, if

**N**and

**M**values above 130 K are taken as a unit (Table 3). The obtained quantitative agreement between NMR data and the analysis of heat capacity suggests the following conclusions. Above 130 K, the chains consisting of ~39 {Zn

_{2}DABCO} units contain DABCO molecules with D

_{3}(S), D

_{3}(R), and D

_{3h}symmetries. At 60-130 K, the longest chains (~29 {Zn

_{2}DABCO} units) contain only D

_{3}forms in the racemic state. The vibrations of these chains make the largest contribution to the heat capacity, while the vibrations of the chains consisting of D

_{3h}forms make no contribution practically, due to their shorter size. Finally, below ~60 K there are three types of chains (~12 {Zn

_{2}DABCO} units) of the same length but containing three different DABCO forms (D

_{3}(S), D

_{3}(R), and D

_{3h}). The size of the chains below 14 K cannot be estimated, since the heat capacity is no more linear at these temperatures.

## 5. Heat Capacity Behavior during the Phase Transition at 60 K and the Salam Hypothesis

_{2}DABCO]

^{4+}cation were obtained in [29]. The difference between the energies of mirror isomers is as small as ~5·10

^{−16}kJ/mol (~5.2·10

^{−18}eV) for DABCO and an order of magnitude higher (~5·10

^{−15}kJ/mol or ~5.2·10

^{−17}eV) for the [Zn

_{2}DABCO]

^{4+}cation. Therefore, the contribution of PVED increases in the presence of Zn

^{2+}cations and is determined mainly by the contribution of zinc cations. This contribution increases if Zn

^{2+}cations are replaced by heavier cations Cd

^{2+}and Hg

^{2+}[30]. Hence, it can be assumed that the PVED breaking of mirror symmetry between D

_{3}(S) and D

_{3}(R) forms of DABCO may be caused by their external environment in the Zn-DMOF structure.

_{3}(S) and D

_{3}(R) forms of DABCO, then, according to the Salam hypothesis, the behavior of heat capacity must correspond to the behavior of heat capacity during the superconducting phase transition [1,2].

_{p}= C

_{p}− C

_{p}

^{L}, and the behavior of C

_{p}

^{L}was described using the Tarasov model [25]. Figure 3 shows the obtained ΔC

_{p}values.

_{3}(S) and D

_{3}(R) forms of DABCO as the temperature decreases [11,12]. Based on the hypothesis suggested in [1,2], a study was carried out to verify the compliance of heat capacity ΔC

_{p}to the exponential dependence ~ exp(−Δ/T), гдe Δ = 1.76·T

_{c}(Δ is the energy gap at 0 K). Figure 4 shows ΔC

_{p}as a function of 1/T in the temperature region 15 K < T < 60 K. As can be seen, a good agreement with the exponential law is achieved for the parameter Δ equal to ~56 K (or ~5 ·10

^{−3}eV). [31]. The obtained value Δ turned out to be almost twice as small as expected (~106 K for T

_{c}= 60 K). There is probably some inaccuracy with the parameters determining function C

_{p}

^{L}, which may cause the error of determining the Δ value. However, the detected exponential behavior of ΔC

_{p}below 60 K signifies the presence of a BE condensation. The amplitude of ΔC

_{p}during the phase transition at is 60 K ≈ 10 J/mol/K (Table 1), which corresponds to the thermal energy jump (ΔC

_{p}·T

_{c}) ≈ 600 J/mol (or 6·10

^{−3}eV), which agrees well with Δ. The value of (ΔC

_{p}·T

_{c}) is 10

^{15}times bigger than the PVED (~5.2·10

^{−18}eV) of one DABCO molecule, but it can be explained by the phenomenon of BE condensation [1,2].

_{p}data nor

^{1}H NMR T

_{1}(T) data show any energy difference between D

_{3}(S) and D

_{3}(R) forms of DABCO below 60 K (according to the Salam hypothesis, the ratio between D

_{3}(S) and D

_{3}(R) forms of DABCO should change). Apparently, the energy difference between D

_{3}(S) and D

_{3}(R) forms remains negligible and can be observed only at lower temperatures, when the thermal energy of the crystal approaches zero [6]. Indeed, according to

^{1}H NMR T

_{1}(T) data, the decay M as a function of time shows anomalous behavior below 25 K [12], but it is not manifested in the C

_{p}behavior until the phase transition at ~14 K.

## 6. Conclusions

_{3}(S) and D

_{3}(R) forms of DABCO is estimated to be ~4 kJ/mol [7] and 5 kJ/mol [27] according to NMR data and quantum chemical calculations, respectively. Thus, this barrier is ~40 times smaller than the barrier between L- and D-forms of alanine [30]. The NMR data indicate the presence of tunneling between D

_{3}(S) and D

_{3}(R) forms of DABCO. The tunneling splitting for the DABCO molecule in the free state is estimated to be ~6 cm

^{1−}(~8.6 K) [24], which is comparable to the temperature range of observed phase transitions in Zn-DMOF. The behavior of heat capacity below 60 K corresponds to the heat capacity during the BE condensation. According to the NMR data, still lower temperatures are associated with a redistribution of DABCO with different symmetries over energy states to form a chiral polarized state. In the model system [Zn

_{2}DABCO]

^{4+}, the R-form is most favorable due to the PVED [29,30], but it is currently unclear which symmetry of the chains built of {Zn

_{2}DABCO} units corresponds to the most energetically favorable state. The method of resonant X-ray diffraction with circularly polarized X-rays [33] or optical methods seem to be most preferable for use at low and extra-low temperatures.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The structure of Zn

_{2}(C

_{8}H

_{4}O

_{4})

_{2}·C

_{6}H

_{12}N

_{2}(Zn-DMOF), space group P4/mmm. Positions of carbon atoms in triethylenediamine (DABCO) molecules are disordered [5]. Hydrogen atoms are omitted for clarity. DABCO and BDC

^{2−}structures are shown in the insets. (Compiled from Figure 1 in [10] and Figure 1 in [1]).

**Figure 2.**Tabulated (T/T

_{D}) experimental (+) and calculated (solid lines) dependences of heat capacity (Cp) for Zn-DMOF and one-dimensional elastic continuums (Tarasov model [25]). Vertical dashed lines show the temperatures of phase transitions. Solid blue, green, and red lines corresponds to the Debye temperatures of 1490 K, 2230 K, and 2950 K, respectively (according to the data reported in [11]).

**Figure 3.**Log-log plot of the Zn-DMOF heat capacity versus temperature. Experimental (crosses) and calculated values of heat capacity at 14.7–57.4 K (blue lines), 130.1–72.6 K (green lines), and 299.6–141.6 K (red lines).

**Figure 4.**Temperature dependence of ΔC

_{p}(in gram-atom units) for Zn-DMOF (left) and ΔC

_{p}plotted as a function of 1/T below ~60 K (right). ΔC

_{p}is shown on the logarithmic scale (according to the data from Figures 2 and 4 in [31]).

**Table 1.**ΔC

_{p}(J/mol/K) values at the phase transitions in Zn-DMOF under various pressures of the heat-exchange gas

^{4}He (P·10

^{5},Pa).

P | ~14 K | ~60 K | ~130 K |
---|---|---|---|

0.51 | 6.0 ± 0.4 | 8.0 ± 0.2 | 23.0 ± 0.3 |

1.52 | 5.0 ± 0.4 | 11.0 ± 0.2 | 23.0 ± 0.3 |

**Table 2.**Entropies ΔS/R of the phase transitions in the region of critical temperatures (T

_{c}, K) under various pressures of the heat-exchange gas

^{4}He (P·10

^{5},Pa) in Zn-DMOF. R is the universal gas constant.

T_{c} | ~14 | ~60 | ~130 |
---|---|---|---|

P | ΔS/R | ΔS/R | ΔS/R |

0.51 | 0.42 ± 0.05 | 0.14 ± 0.02 | 0.30 ± 0.04 |

1.52 | 0.28 ± 0.04 | 0.23 ± 0.02 | 0.30 ± 0.04 |

**Table 3.**Calculated parameters for Zn-DMOF. $\stackrel{\xb7}{\mathbf{M}}$ is the nuclear magnetic moment, $\stackrel{\xb7}{\mathbf{N}}$ is the number of {Zn

_{2}DABCO} units normalized with respect to corresponding values above 130 K.

Region of Fit, K | 299.6–141.6 | 130.1–72.6 | 57.4–14.7 |
---|---|---|---|

ν_{m}, cm^{−1} | 1250 | 765 | 285 |

N | ~38.5 | ~28.9 | ~12.0 |

$\stackrel{\xb7}{\mathbf{N}}$ | 1 | ~0.75 | ~0.31 |

$\stackrel{\xb7}{\mathbf{M}}$ | 1 | ~0.67 | ~0.33 |

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**MDPI and ACS Style**

Kozlova, S.; Ryzhikov, M.; Pishchur, D.; Mirzaeva, I.
Overview of Low-Temperature Heat Capacity Data for Zn_{2}(C_{8}H_{4}O_{4})_{2}^{.}C_{6}H_{12}N_{2} and the Salam Hypothesis. *Symmetry* **2019**, *11*, 657.
https://doi.org/10.3390/sym11050657

**AMA Style**

Kozlova S, Ryzhikov M, Pishchur D, Mirzaeva I.
Overview of Low-Temperature Heat Capacity Data for Zn_{2}(C_{8}H_{4}O_{4})_{2}^{.}C_{6}H_{12}N_{2} and the Salam Hypothesis. *Symmetry*. 2019; 11(5):657.
https://doi.org/10.3390/sym11050657

**Chicago/Turabian Style**

Kozlova, Svetlana, Maxim Ryzhikov, Denis Pishchur, and Irina Mirzaeva.
2019. "Overview of Low-Temperature Heat Capacity Data for Zn_{2}(C_{8}H_{4}O_{4})_{2}^{.}C_{6}H_{12}N_{2} and the Salam Hypothesis" *Symmetry* 11, no. 5: 657.
https://doi.org/10.3390/sym11050657