  We present a simple, top-down approach for the calculation of minimum energy
consumption of electrosorptive ion separation using variational form of the
(Gibbs) free energy. We focus and expand on the case of electrostatic
capacitive deionization (CDI), and the theoretical framework is independent of
details of the double-layer charge distribution and is applicable to any
thermodynamically consistent model, such as the Gouy-Chapman-Stern (GCS) and
modified Donnan (mD) models. We demonstrate that, under certain assumptions,
the minimum required electric work energy is indeed equivalent to the free
energy of separation. Using the theory, we define the thermodynamic efficiency
of CDI. We explore the thermodynamic efficiency of current experimental CDI
systems and show that these are currently very low, less than 1% for most
existing systems. We applied this knowledge and constructed and operated a CDI
cell to show that judicious selection of the materials, geometry, and process
parameters can be used to achieve a 9% thermodynamic efficiency (4.6 kT energy
per removed ion). This relatively high value is, to our knowledge, by far the
highest thermodynamic efficiency ever demonstrated for CDI. We hypothesize that
efficiency can be further improved by further reduction of CDI cell series
resistances and optimization of operational parameters.
