58
4 Combinatorial Studies in the Li–Co–Mn–O System
domain having some disorder with cobalt present in the Mn 2 Li-rich domains and
both lithium and manganese in the cobalt-rich domains. This picture illustrates well
what was seen experimentally where the layered–layered composites were found to
be made up of structures with x = 0.2 and 0.8. As such, β T = 1 in the simulation
would appear to correspond to a temperature where phase separation took place,
below 800
◦ C. Though not shown, these disordered domains were also found for
β T = 1.5 and 2.0 demonstrating that these domains minimize the free energy over
a range of temperatures during cooling. Furthermore, β T = 5 corresponds to 1/5
of the temperature at β T = 1, and can therefore be considered to be below room
temperature. Figure 4.10 (c) shows nearly complete phase separation takes place at
this lower temperature, and only trace amounts of cobalt can be seen in the Mn 2 Li
domains and these regions are bounded by pure cobalt domains. This picture is very
similar to the model made by Bare˜ no [23]. Since this equilibrium state was not seen
experimentally here, it can be concluded that slow kinetics must take over at some
point during cooling, preventing the system from reaching the low energy state. In
terms of the simulation, one would need to consider the activation energy involved in
switching two neighboring atoms in order to simulate the “freezing” of the structure
during cooling.
In order to confirm that equilibrium was reached in Fig. 4.10 (b), the simulation
was repeated, keeping β T fixed at 1 for 100,000 Monte Carlo steps (i.e., 10 times
as many as during the simulated slow cool). Figure 4.10 (d) shows the results of
this simulated anneal and confirms that equilibrium was reached in (b) such that the
disorder in the two domains cannot be eliminated at this temperature. The domains
may also be slightly larger after the anneal: the small cobalt-rich domains in (b)
appear to be replaced by a few larger ones in (d). However, the relatively small array
makes it unwise to make such generalizations.
The results from the Monte Carlo simulation suggest that β T = 1 corresponds
roughly to 900 K (i.e., near to but below 800
◦ C). This implies that the Li–Li nearest
neighbor (NN) interaction is β T k B T = 78 meV. Published ab initio calculations of
LiCoO 2 obtain values of 29 meV as the effective cluster interaction for NN Li–Li
clusters, while 7 and 6 meV are obtained for next NN and next next NN, respectively
[79]. The interaction energy obtained here is larger; this can be partially attributed to
the fact that the only interaction used here is for nearest neighbors. Furthermore, this
result implies that a relatively strong interaction was required in order to obtain the
phase separation seen during cooling in the current study. One more consequence
of this result is that δ, the effective charge fraction of the metal atoms, has a value
of 12 % of that expected from the oxidation number. This low value shows that the
nearest neighbor interactions are much weaker than that expected from Coulombic
interactions assuming all metal–oxygen bonds are purely ionic. It should also be
stressed that strain within the lattice was neglected in the simulation (all nearest
neighbors were exactly one lattice parameter away from each other). The unit cell
volume of LiCoO 2 and Li 2 MnO 3 differ by 3.4 % while those for x = 0.2 and 0.8 differ
by only 1.3 % (using lattice parameters from Ref. [18]). Phase separating over the
entire composition line might require fracturing the crystallites, which may be the
reason why the endpoints found experimentally were nearer to the center of the line.
4 Combinatorial Studies in the Li–Co–Mn–O System
domain having some disorder with cobalt present in the Mn 2 Li-rich domains and
both lithium and manganese in the cobalt-rich domains. This picture illustrates well
what was seen experimentally where the layered–layered composites were found to
be made up of structures with x = 0.2 and 0.8. As such, β T = 1 in the simulation
would appear to correspond to a temperature where phase separation took place,
below 800
◦ C. Though not shown, these disordered domains were also found for
β T = 1.5 and 2.0 demonstrating that these domains minimize the free energy over
a range of temperatures during cooling. Furthermore, β T = 5 corresponds to 1/5
of the temperature at β T = 1, and can therefore be considered to be below room
temperature. Figure 4.10 (c) shows nearly complete phase separation takes place at
this lower temperature, and only trace amounts of cobalt can be seen in the Mn 2 Li
domains and these regions are bounded by pure cobalt domains. This picture is very
similar to the model made by Bare˜ no [23]. Since this equilibrium state was not seen
experimentally here, it can be concluded that slow kinetics must take over at some
point during cooling, preventing the system from reaching the low energy state. In
terms of the simulation, one would need to consider the activation energy involved in
switching two neighboring atoms in order to simulate the “freezing” of the structure
during cooling.
In order to confirm that equilibrium was reached in Fig. 4.10 (b), the simulation
was repeated, keeping β T fixed at 1 for 100,000 Monte Carlo steps (i.e., 10 times
as many as during the simulated slow cool). Figure 4.10 (d) shows the results of
this simulated anneal and confirms that equilibrium was reached in (b) such that the
disorder in the two domains cannot be eliminated at this temperature. The domains
may also be slightly larger after the anneal: the small cobalt-rich domains in (b)
appear to be replaced by a few larger ones in (d). However, the relatively small array
makes it unwise to make such generalizations.
The results from the Monte Carlo simulation suggest that β T = 1 corresponds
roughly to 900 K (i.e., near to but below 800
◦ C). This implies that the Li–Li nearest
neighbor (NN) interaction is β T k B T = 78 meV. Published ab initio calculations of
LiCoO 2 obtain values of 29 meV as the effective cluster interaction for NN Li–Li
clusters, while 7 and 6 meV are obtained for next NN and next next NN, respectively
[79]. The interaction energy obtained here is larger; this can be partially attributed to
the fact that the only interaction used here is for nearest neighbors. Furthermore, this
result implies that a relatively strong interaction was required in order to obtain the
phase separation seen during cooling in the current study. One more consequence
of this result is that δ, the effective charge fraction of the metal atoms, has a value
of 12 % of that expected from the oxidation number. This low value shows that the
nearest neighbor interactions are much weaker than that expected from Coulombic
interactions assuming all metal–oxygen bonds are purely ionic. It should also be
stressed that strain within the lattice was neglected in the simulation (all nearest
neighbors were exactly one lattice parameter away from each other). The unit cell
volume of LiCoO 2 and Li 2 MnO 3 differ by 3.4 % while those for x = 0.2 and 0.8 differ
by only 1.3 % (using lattice parameters from Ref. [18]). Phase separating over the
entire composition line might require fracturing the crystallites, which may be the
reason why the endpoints found experimentally were nearer to the center of the line.