It's Not Reusable Because It's Broken

In a \(Li-FeS_2\) battery at room operating temperature,  at the cathode upon first discharge

\(FeS_2 {_{(lattice)}}+2Li^{+}+2e^{-}\rightarrow Li_2FeS_2 {_{(amporhous)}}\) ---(1)

and then a second discharge equation,

\(Li_2FeS_2+2Li^{+}+2e^{-}\leftrightarrow Fe+2Li_2S\) --- (2)

On recharge, equation (2) reverses,

\(Fe+2Li_2S-2e^{-}\leftrightarrow Li_2FeS_2+2Li^{+}\) --- (3)

then,

\(Li_2FeS_2-xe^{-}\leftrightarrow Li_{2-x}FeS_2+xLi^{+}\) --- (4)

but on further recharge,

\(Li_{2-x}FeS_2-(2-x)e^{-}\rightarrow (2-x)Li^{+}+FeS_n+(2-n)S\) ---(5)

where \(x\ge0.8\)

With these background materials...

This is \(Li_2\) when two unpaired orbits, each from one \(Li\), pairs up,

This is the \(FeS_2\) lattice, where each member has six neighbors and each lattice unit is attached to another at two points,

At the cathode, the \(Li\) with an unpaired orbit pairs up another \(Li\) to form \(Li_2\) and attaches itself to the lattice of \(FeS_2\).  This is an ionic bond.

\(Li_2\) replaces one \(Fe\) atom around \(S\) and the replaced \(Fe\) is reduced.

Under lower temperature \(21^{o}C\) to \(30^{o}C\), the lattice starts to crack. This is the first discharge when a new battery come into use. Upon a second discharge via equation (2), hole forms in the lattice with the formation of \(Fe\) and \(Li_2S\).  At this point if a recharge is attempted, the amporhous \(Li_2FeS_2\) ⁽¹⁾ reforms within the hole in the lattice.  Upon further recharge, we obtain \(Li\), but we do not return to the ionic lattice of \(FeS_2\), instead we obtain, \(FeS_n\) and free elemental \(S\). The presence of elemental \(S\) causes havoc at the anode.

Structurally the cathode has disintegrated.

Recharging is not a problem at \(400^oC\) with a salt electrolyte, the \(FeS_2\)  lattice (melting point \(1,177-1,188^oC\) is reformed upon full recharge.  Heat increases the orbital radii of the members of the lattice \(Fe\) and \(S\), and makes the \(FeS_2\) lattice more malleable but not melting it completely.  At this temperature the \(FeS_2\) lattice can accommodate \(Li_2\).

Increasing the distance between the ionic elements can also be done using a polar solvent infused into the lattice.  This solvent should not dissolve the compound but space out the members of the lattice as when the
temperature is at \(400^{o}C\).  \(FeS_2\) is not soluble in water, but water introduced into the lattice when the crystal forms might space themselves between the lattice members and make the crystal soft.  Otherwise a more polar solvent has to be found.  Since, Pyrite is soluble in acids, a good candidate is \(HCl\), other acids are physically too big.  \(FeS_2\) doped with \(HCl\).

\(CuS_2-FeS_2\) has wider lattice spacing \(\sqrt{3}.(3.966\times10^{-10})\) m, compared to \(FeS_2\) of  \(\sqrt{3}.(3.38\times10^{-10})\) m.  And with Arsenic \(As\) in the mix,  \(\sqrt{3}.(3.8488\times10^{-10})\) m.⁽²⁾

If the lattice is intact after the first discharge via equation (1).  The \(Li_2\) member are simply hanging in the lattice of \(FeS_2\) slightly expanded to accommodate the new member.  And subsequent discharge,

\(Li_2FeS_2-2e^{-}\leftrightarrow FeS_2+2Li^{+}\)

pucks the \(Li_2\) as \(2Li^{+}\) after removing the two electrons, from the lattice.

The crux of the problem is to keep the lattice structure of \(FeS_2\) intact during the charge and discharge cycle. A backing lattice that holds onto the \(FeS_2\) lattice and stretches it to accommodate \(Li_2\) but
itself does not part-take in the redox reactions can be the solution.  Stacked layers of the two lattices will then build the bulk of the cathode.  The backing lattice need not stretch the \(FeS_2\) lattice if a single layer of the bounded lattice (\(FeS_2\)+backing) are use.  In this case, the expose surface of the \(FeS_2\) lattice allows \(Li_2\) to bond freely.  A transport layer that allows \(Li^{+}\) to move freely can be sandwiched between two bounded lattices to build bulk.  Many binary metal chalcogenides (compounds with \(S\), \(Se\) or \(Te\)) have the \(FeS_2\) structure, as do oxides like \(CdO_2\), \(\alpha-K_2O\), \(\beta-Na_2O\)⁽³⁾. And "Late" transition metal disulfides (Mn, Fe, Co, Ni) almost always adopt the pyrite or the related marcasite motif.   There are many material options for a backing lacttice, insulated from the redox reactions by a few layers of \(FeS_2\) to provide structural integrity.

Note: It is likely that instead of \(Li^{+}\) we have \(LiH^{+}\).  Since, \(H\) after loosing its electron sit inside \(Li\), \(Li^{+}\) and \(LiH^{+}\) is likely to be of comparable in size.

There is no reversing entropy but entropy can be preserved.

References:

(1) In situ Fe K-EDGE X-ray absorption fine structure of a pyrite electrode in a Li/Polyethylene oxide (LiClO₄)/FeS₂ battery environment.  Dana Totir, In Tae Bae, Yining Hu, Mark R. Antonio and Daniel A. Scherson.  Proceeding Symposium Lithium Battery CONF-961040--23

(2)Schmid-Beurmann P, Lottermoser W Physics and Chemistry of Minerals 19 (1993) 571-577
57Fe-Moessbauer spectra, electronic and crystal structure of members of the CuS2-FeS2
solid solution series.

One set of data from reference (2), (http://rruff.geo.arizona.edu/AMS/AMC_text_files/20081_amc.txt) repeated below, suggests, an atomic center to center of \(\sqrt{3}.(6.25\times10^{-10})\)m.

Pyrite
Oftedal I
Zeitschrift fur Physikalische Chemie 135 (1928) 291-299
Uber die Kristallstrukturen der verbindungen RuS2, OsS2, MnTe2 und AuSb2.
Mit einem Anhang uber die Gitterkonstant von Pyrit
_database_code_amcsd 0017728
5.414 5.414 5.414 90 90 90 Pa3
atom    x    y    z
Fe      0    0    0
S    .625 .625 .625

(3)"Ceramic Materials: Science and Engineering" By C. Barry Carter, M. Grant Norton publisher: Springer Science & Business Media, 4 Jan 2013.