White dwarfs, the most abundant stellar remnants, provide a promising means of probing dark matter interactions, complimentary to terrestrial searches. The scattering of dark matter from stellar constituents leads to gravitational capture, with important observational consequences. In particular, white dwarf heating occurs due to the energy transfer in the dark matter capture and thermalisation processes, and the subsequent annihilation of captured dark matter. We consider the capture of dark matter by scattering on either the ion or the degenerate electron component of white dwarfs. For ions, we account for the stellar structure, the star opacity, realistic nuclear form factors that go beyond the simple Helm approach, and finite temperature effects pertinent to sub-GeV dark matter. Electrons are treated as relativistic, degenerate targets, with Pauli blocking, finite temperature and multiple scattering effects all taken into account. We also estimate the dark matter evaporation rate. The dark matter-nucleon/electron scattering cross sections can be constrained by comparing the heating rate due to dark matter capture with observations of cold white dwarfs in dark matter-rich environments. We apply this technique to observations of old white dwarfs in the globular cluster Messier 4, which we assume to be located in a DM subhalo. For dark matter-nucleon scattering, we find that white dwarfs can probe the sub-GeV mass range inaccessible to direct detection searches, with the low mass reach limited only by evaporation, and can be competitive with direct detection in the $1-10^4$ GeV range. White dwarf limits on dark matter-electron scattering are found to outperform current electron recoil experiments over the full mass range considered, and extend well beyond the $sim 10$ GeV mass regime where the sensitivity of electron recoil experiments is reduced.
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