  Constitutive laws underlie most physical processes in nature. However,
learning such equations in heterogeneous solids (e.g., due to phase separation)
is challenging. One such relationship is between composition and eigenstrain,
which governs the chemo-mechanical expansion in solids. In this work, we
developed a generalizable, physically-constrained image-learning framework to
algorithmically learn the chemo-mechanical constitutive law at the nanoscale
from correlative four-dimensional scanning transmission electron microscopy and
X-ray spectro-ptychography images. We demonstrated this approach on
Li$_X$FePO$_4$, a technologically-relevant battery positive electrode material.
We uncovered the functional form of composition-eigenstrain relation in this
two-phase binary solid across the entire composition range (0 $\leq$ X $\leq$
1), including inside the thermodynamically-unstable miscibility gap. The
learned relation directly validates Vegard's law of linear response at the
nanoscale. Our physics-constrained data-driven approach directly visualizes the
residual strain field (by removing the compositional and coherency strain),
which is otherwise impossible to quantify. Heterogeneities in the residual
strain arise from misfit dislocations and were independently verified by X-ray
diffraction line profile analysis. Our work provides the means to
simultaneously quantify chemical expansion, coherency strain and dislocations
in battery electrodes, which has implications on rate capabilities and
lifetime. Broadly, this work also highlights the potential of integrating
correlative microscopy and image learning for extracting material properties
and physics.

Materials Science