  Wadsley--Roth crystallographic shear phases form a family of compounds that
have attracted attention due to their excellent performance as lithium-ion
battery electrodes. The complex crystallographic structure of these materials
poses a challenge for first-principles computational modelling and hinders the
understanding of their structural, electronic and dynamic properties. In this
article, we study three different niobium-tungsten oxide crystallographic shear
phases (Nb$_{12}$WO$_{33}$, Nb$_{14}$W$_{3}$O$_{44}$, Nb$_{16}$W$_5$O$_{55}$)
using an enumeration-based approach and first-principles density-functional
theory calculations. We report common principles governing the cation disorder,
lithium insertion mechanism, and electronic structure of these materials.
Tungsten preferentially occupies tetrahedral and block-central sites within the
block-type crystal structures. The lithium insertion proceeds via a three-step
mechanism, associated with an anisotropic evolution of the host lattice. Our
calculations reveal an important connection between long-range and local
structural changes: in the second step of the mechanism, the removal of local
structural distortions leads to the contraction of the lattice along specific
crystallographic directions, buffering the volume expansion of the material.
Niobium-tungsten oxide shear structures host small amounts of localised
electrons during initial lithium insertion due to the confining effect of the
blocks, but quickly become metallic upon further lithiation. We argue that the
combination of local, long-range, and electronic structural evolution over the
course of lithiation is beneficial to the performance of these materials as
battery electrodes. The mechanistic principles we establish arise from the
compound-independent crystallographic shear structure, and are therefore likely
to apply to Ti/Nb oxide or pure Nb oxide shear phases.
