Incommensurate Chirality Density Wave Transition in a Hybrid Molecular Framework

Using single-crystal x-ray diffraction we characterize the 235 K incommensurate phase transition in the hybrid molecular framework tetraethylammonium silver(I) dicyanoargentate, $[{\mathrm{NEt}}_4]{\mathrm{Ag}}_3(\mathrm{CN}{)}_4$. We demonstrate the transition to involve spontaneous resolution of chiral $[{\mathrm{NEt}}_4{]}^{+}$ conformations, giving rise to a state in which molecular chirality is incommensurately modulated throughout the crystal lattice. We refer to this state as an incommensurate chirality density wave (XDW) phase, which represents a fundamentally new type of chiral symmetry breaking in the solid state. Drawing on parallels to the incommensurate ferroelectric transition of ${\mathrm{NaNO}}_2$, we suggest the XDW state arises through coupling between acoustic (shear) and molecular rotoinversion modes. Such coupling is symmetry forbidden at the Brillouin zone center but symmetry allowed for small but finite modulation vectors $\mathbf{q}=[0,0,q_z{]}^{*}$. The importance of long-wavelength chirality modulations in the physics of this hybrid framework may have implications for the generation of mesoscale chiral textures, as required for advanced photonic materials.

Figure 1

(a) Representations of the $[{\mathrm{NEt}}_4{]}^{+}$ molecular geometry distinguishing left-handed (blue) and right-handed (red) conformations, related by mirror symmetry (vertical line). (b) Structural model for the $[{\mathrm{NEt}}_4{]}^{+}$ site in the ambient-temperature phase of $[{\mathrm{NEt}}_4]{\mathrm{Ag}}_3(\mathrm{CN}{)}_4$ [27]: the molecular cation is disordered over the two equally populated conformations, represented here in blue and red. The two conformations are related by a mirror plane bisecting the N atom (gray). H atoms omitted for clarity. (c) The disordered molecular cations are located within cavities formed by the interpenetrating diamondoid $[{\mathrm{Ag}}_3(\mathrm{CN}{)}_4{]}^{-}$ frameworks (shades of green).

Figure 2

(a) Sections of single-crystal x-ray diffraction patterns of $[{\mathrm{NEt}}_4]{\mathrm{Ag}}_3(\mathrm{CN}{)}_4$ collected at temperatures spanning the incommensurate phase transition. The superlattice reflections associated with incommensurate modulation of the structure are visible as satellites of the parent Bragg reflections. Note the presence at the lowest temperatures of higher-order modulation peaks ($m>1$). The transition is entirely reversible, as is evident from the similarity in diffraction patterns at 240 K prior to and after cooling through $T_c$. (b) The left-hand panel shows the temperature dependence of the XDW order parameter, as determined from the relative intensity of first-order modulation reflections (averaged over six representative reflections, see the Supplemental Material [35]). The right-hand panel shows the value of the modulation wave-vector component $q_z$ as a function of temperature. The nearby commensurate value $q_z=\frac{1}{8}$ is shown as a horizontal line.

Figure 3

Incommensurate modulation in $[{\mathrm{NEt}}_4]{\mathrm{Ag}}_3(\mathrm{CN}{)}_4$ determined using single-crystal x-ray diffraction measurements at 100 K. (a) Displacement amplitude (colored lines) and $[{\mathrm{NEt}}_4{]}^{+}$ chirality as a function of modulation coordinate $t$. Displacements of the framework components are shown in shades of green; those of the central N atom in the $[{\mathrm{NEt}}_4{]}^{+}$ cation are shown in gray; and those of the pendant C atoms of the $[{\mathrm{NEt}}_4{]}^{+}$ cations are shown in red or blue, depending on the particular chiral state present ($+1,-1$) for the corresponding $t$ value. (b) Representation of the framework modulation in the commensurate approximant of the XDW state. Note that the framework displacements are predominantly polarized in the $\mathbf{b}$ direction. (c) Representation of the modulation in $[{\mathrm{NEt}}_4{]}^{+}$ cation positions and chirality states (red or blue) in the same approximant model.