Origin of the exceptional negative thermal expansion in metal-organic framework-5 ${\text{Zn}}_4\text{O}{(1,4-\text{benzenedicarboxylate})}_3$

Metal organic framework-5 (MOF-5) was recently suggested to possess an exceptionally large negative thermal-expansion coefficient. Our direct experimental measurement of the thermal expansion of MOF-5 using neutron powder diffraction, in the temperature range of 4 to 600 K, shows that the linear thermal-expansion coefficient is $\approx -16\times {10}^{-6}{\text{K}}^{-1}$. To understand the origin of this large negative thermal-expansion behavior, we performed first-principles lattice dynamics calculations. The calculated thermal-expansion coefficients within quasiharmonic approximation agree well with the experimental data. We found that almost all low-frequency lattice vibrational modes (below $\sim 23\text{meV}$) involve the motion of the benzene rings and the ${\text{ZnO}}_4$ clusters as rigid units and the carboxyl groups as bridges. These so-called “rigid-unit modes” exhibit various degrees of phonon softening (i.e., the vibrational energy decreases with contracting crystal lattice) and thus are directly responsible for the large negative thermal expansion in MOF-5. Initial efforts were made to observe the phonon softening experimentally.

Figure 1

(Color online) [100] (left) and [110] (right) views of the crystal structure of MOF-5, which consists of BDC linkers connecting ${\text{ZnO}}_4$ clusters located at the tetrahedral site of an fcc lattice.

Figure 2

(Color online) (a) The experimental (dots) and calculated (lines) temperature dependence of the lattice constant $a$ of MOF-5. (b) The experimental (dots) and calculated (lines) linear thermal-expansion coefficient of MOF-5. Note that the experimental data points of $a$ have very small error bar $\left(<0.01\%\right)$ while the errors associated with the derived data points of $\alpha$ are large $\left(\sim 10\%\right)$. This is due to the limited number of $a$ data points used in the numerical interpolation and differentiation. Computationally, LDA underestimates the lattice constant while GGA gives overestimated result as generally expected. Nevertheless, both LDA and GGA generate nearly the same linear thermal-expansion coefficients, which agree reasonably well with the experimental data.

Figure 3

(Color online) According to the “rigid-unit modes” mechanism, the ${\text{ZnO}}_4$ tetrahedra and the benzene rings serve as the rigid units while the carboxyl groups serve as the bridge. The large amplitude transverse vibration of the bridge leads to negative thermal expansion.

Figure 4

Calculated phonon frequency at $\Gamma$ as a function of lattice constant $a$. Only the low-energy portion is shown. Note that nearly all optical-phonon modes below $180{\text{cm}}^{-1}$ exhibit phonon softening to various extent. In contrast, phonon modes with higher frequencies (not shown here) exhibit normal behavior, i.e., the phonon frequencies increase with contracting lattice. Note that $1\text{meV}\cong 8.07{\text{cm}}^{-1}$.

Figure 5

(Color online) Schematics of several representative low-frequency phonon modes. Corresponding phonon energies from LDA calculation at 0 K are also shown. Note that all the phonon modes below 23 meV originate from various combinations of local motions of the bridging carboxyl groups.