Rigid Unit Modes
Encyclopedia
Rigid unit modes represent a class of lattice vibrations or phonons that exist in network materials such as quartz
, cristobalite
or zirconium tungstate
. Network materials can be described as three-dimensional networks of polyhedral groups of atoms such as SiO4 tetrahedra or TiO6 octahedra. A RUM is a lattice vibration in which the polyhedra are able to move, by translation and/or rotation, without distorting. RUMs in crystalline materials are the counterparts of floppy modes in glasses, as introduced by Jim Phillips and Mike Thorpe.
The original work in silicates showed that many of the phase transitions in silicates could be understood in terms of soft modes that are RUMs.
After the original work on displacive phase transitions, the RUM model was also applied to understanding the nature of the disordered high-temperature phases of materials such as cristobalite
, the dynamics and localised structural distortions in zeolites, and negative thermal expansion
.
of the network, an engineering analysis that dates back to James Clerk Maxwell
and which was introduced to amorphous materials by Jim Phillips and Mike Thorpe. If the number of constraints exceeds the number of degrees of freedom, the structure will be rigid. On the other hand, if the number of degrees of freedom exceeds the number of constraints, the structure will be floppy.
For a structure that consists of corner-linked tetrahedra (such as the SiO4 tetrahedra in silica, SiO2) we can count the numbers of constraints and degrees of freedom as follows. For a given tetrahedron, the position of any corner has to have its three spatial coordinates (x,y,z) match the spatial coordinates of the corresponding corner of a linked tetrahedron. Thus each corner has three constraints. These are shared by the two linked tetrahedra, so contribute 1.5 constraints to each tetrahedron. There are 4 corners, so we have a total of 6 constraints per tetrahedron. A rigid three-dimensional object has 6 degrees of freedom, 3 translations and 3 rotations. Thus there is an exact balance between the numbers of constraints and degrees of freedom.
(Note that we can get an identical result by considering the atoms to be the basic units. There are 5 atoms in the structural tetrahedron, but 4 of there are shared by two tetrahedra, so that there are 3 + 4*3/2 = 9 degrees of freedom per tetrahedron. The number of constraints to hold together such a tetrahedron is 9 (4 distances and 5 angles)).
What this balance means is that a structure composed of structural tetrahedra joined at corners is exactly on the border between being rigid and floppy. What appears to happen is that symmetry reduces the number of constraints so that structures such as quartz
and cristobalite
are slightly floppy and thus support some RUMs.
The above analysis can be applied to any network structure composed of polyhedral groups of atoms. One example is the perovskite
family of structures, which consist of corner linked TiO6 or ZrO6 octahedra. A simple counting analysis would in fact suggest that such structures are rigid, but in the ideal cubic phase symmetry allows some degree of flexibility. Zirconium tungstate
, the archetypal material showing negative thermal expansion
, contains ZrO6 octahedra and WO4 tetrahedra, with one of the corners of each WO4 tetrahedra having no linkage. The counting analysis shows that, like silica, zirconium tungstate has an exact balance of the numbers of constraints and degrees of freedom, and further analysis has shown the existence of RUMs in this material.
Quartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
, cristobalite
Cristobalite
The mineral cristobalite is a high-temperature polymorph of silica, meaning that it has the same chemical formula, SiO2, but a distinct crystal structure. Both quartz and cristobalite are polymorphs with all the members of the quartz group which also include coesite, tridymite and stishovite...
or zirconium tungstate
Zirconium tungstate
Zirconium tungstate is a metal oxide with unusual properties. The phase formed at ambient pressure by reaction of ZrO2 and WO3 is a metastable cubic phase, which has negative thermal expansion characteristics, namely it shrinks over a wide range of temperatures when heated...
. Network materials can be described as three-dimensional networks of polyhedral groups of atoms such as SiO4 tetrahedra or TiO6 octahedra. A RUM is a lattice vibration in which the polyhedra are able to move, by translation and/or rotation, without distorting. RUMs in crystalline materials are the counterparts of floppy modes in glasses, as introduced by Jim Phillips and Mike Thorpe.
The interest in rigid unit modes
The idea of rigid unit modes was developed for crystalline materials to enable an understanding of the origin of displacive phase transitions in materials such as silicates, which can be described as infinite three-dimensional networks of corner-lined SiO4 and AlO4 tetrahedra. The idea was that rigid unit modes could act as the soft modes for displacive phase transitions.The original work in silicates showed that many of the phase transitions in silicates could be understood in terms of soft modes that are RUMs.
After the original work on displacive phase transitions, the RUM model was also applied to understanding the nature of the disordered high-temperature phases of materials such as cristobalite
Cristobalite
The mineral cristobalite is a high-temperature polymorph of silica, meaning that it has the same chemical formula, SiO2, but a distinct crystal structure. Both quartz and cristobalite are polymorphs with all the members of the quartz group which also include coesite, tridymite and stishovite...
, the dynamics and localised structural distortions in zeolites, and negative thermal expansion
Negative thermal expansion
Negative Thermal Expansion is a physicochemical process in which some materials contract upon heating rather than expanding as most materials do. Materials which undergo this unusual process have a range of potential engineering, photonic, electronic, and structural applications...
.
Why rigid unit modes can exist
The simplest way to understand the origin of RUMs is to consider the balance between the numbers of constraints and degrees of freedomDegrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
of the network, an engineering analysis that dates back to James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
and which was introduced to amorphous materials by Jim Phillips and Mike Thorpe. If the number of constraints exceeds the number of degrees of freedom, the structure will be rigid. On the other hand, if the number of degrees of freedom exceeds the number of constraints, the structure will be floppy.
For a structure that consists of corner-linked tetrahedra (such as the SiO4 tetrahedra in silica, SiO2) we can count the numbers of constraints and degrees of freedom as follows. For a given tetrahedron, the position of any corner has to have its three spatial coordinates (x,y,z) match the spatial coordinates of the corresponding corner of a linked tetrahedron. Thus each corner has three constraints. These are shared by the two linked tetrahedra, so contribute 1.5 constraints to each tetrahedron. There are 4 corners, so we have a total of 6 constraints per tetrahedron. A rigid three-dimensional object has 6 degrees of freedom, 3 translations and 3 rotations. Thus there is an exact balance between the numbers of constraints and degrees of freedom.
(Note that we can get an identical result by considering the atoms to be the basic units. There are 5 atoms in the structural tetrahedron, but 4 of there are shared by two tetrahedra, so that there are 3 + 4*3/2 = 9 degrees of freedom per tetrahedron. The number of constraints to hold together such a tetrahedron is 9 (4 distances and 5 angles)).
What this balance means is that a structure composed of structural tetrahedra joined at corners is exactly on the border between being rigid and floppy. What appears to happen is that symmetry reduces the number of constraints so that structures such as quartz
Quartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
and cristobalite
Cristobalite
The mineral cristobalite is a high-temperature polymorph of silica, meaning that it has the same chemical formula, SiO2, but a distinct crystal structure. Both quartz and cristobalite are polymorphs with all the members of the quartz group which also include coesite, tridymite and stishovite...
are slightly floppy and thus support some RUMs.
The above analysis can be applied to any network structure composed of polyhedral groups of atoms. One example is the perovskite
Perovskite
A perovskite structure is any material with the same type of crystal structure as calcium titanium oxide , known as the perovskite structure, or XIIA2+VIB4+X2−3 with the oxygen in the face centers. Perovskites take their name from this compound, which was first discovered in the Ural mountains of...
family of structures, which consist of corner linked TiO6 or ZrO6 octahedra. A simple counting analysis would in fact suggest that such structures are rigid, but in the ideal cubic phase symmetry allows some degree of flexibility. Zirconium tungstate
Zirconium tungstate
Zirconium tungstate is a metal oxide with unusual properties. The phase formed at ambient pressure by reaction of ZrO2 and WO3 is a metastable cubic phase, which has negative thermal expansion characteristics, namely it shrinks over a wide range of temperatures when heated...
, the archetypal material showing negative thermal expansion
Negative thermal expansion
Negative Thermal Expansion is a physicochemical process in which some materials contract upon heating rather than expanding as most materials do. Materials which undergo this unusual process have a range of potential engineering, photonic, electronic, and structural applications...
, contains ZrO6 octahedra and WO4 tetrahedra, with one of the corners of each WO4 tetrahedra having no linkage. The counting analysis shows that, like silica, zirconium tungstate has an exact balance of the numbers of constraints and degrees of freedom, and further analysis has shown the existence of RUMs in this material.