Lack of a name for the Thermodynamic potential wrt the natural variables From the first law of thermodynamics relation and using Legendre Transform of the Internal Energy, we can possibly define 8 Thermodynamic potentials out of which we know 5. I could not find any names of the other 3 and any use thereof. Also, for the microcanonical ensemble, we devise $S=S(E,V,N)$ and precisely $E,V,N$ are the parameters held fixed in the formulation of the ensemble. This property seems to be true for all the 5 known potentials (for instance, in the grand potential, we derive $\phi=\phi(T,V,\mu)$ which are the quantities held fixed in the grand canonical ensemble). So for the other 3 "not known" potentials, can we cook up other ensembles and if there are any, what are there uses? In the image, I have put $K_i$ for ease of referencing while answering.

 A: The correspondence between ensembles and relevant thermodynamic potentials requires a partial correction.
Let me start with a general remark about names. Within thermodynamics, the generic name for a function of state embodying the full information about the system is fundamental equation. Energy as a function of $S,V,N$, is a fundamental equation. Moreover, energy and its Legendre transforms are collectively known as thermodynamic potentials.
However, there are fundamental equations that are not energy or any of its Legendre transforms. This is the case of the entropy as a function of energy $U$, $V$, and $N$ and its Legendre transforms, sometimes called Massieu functions (see for instance H.B.Callen's textbook Thermodynamics and an introduction to thermostatistics). There is no uniform conventional symbol (or specific names) for the Massieu functions, although the chemistry community (IUPAC) and IUPAP chose symbols and name for a couple of them. Sometimes they have been indicated as $S_1$, $S_2$, and so on. However, on the one hand, it is worth recalling that also thermodynamic potentials lack a completely uniform notation. Internal energy is sometimes indicated as $E$, sometimes as $U$. The Helmholtz free energy may be indicated as $F$, $A$, or $\Phi$, to cite the most frequent notations. This status of things requires checking the meaning of symbols carefully in a paper or a book. On the other hand, the Massieu functions can be written in terms of thermodynamic potentials, although as functions of different variables.
Thermodynamic potentials are better suited for discussing experimental data. Entropy-based fundamental equations (the Massieu functions) are the natural environment for statistical mechanics. This should not be a surprise since measurements of energy transfers are typically accessible in the experiments. At the same time, statistical mechanics was built around the concept of counting the number of states.
Therefore we have two tables, one more relevant for statistical mechanics and another more relevant for thermodynamics. I show in the following the most important entries of both. Notice that the natural variables of the usual statistical mechanics ensembles are those pertaining to entropy and Massieu functions. No usual ensemble in statistical mechanics has entropy among its natural variables.
Entropy and Massieu functions




Name/symbol of the fundamental equation
natural variables
ensemble




entropy/$S$
$U,V,N$
microcanonical


Massieu function$^{(*)}$/$J^{(*)}=S_1=S-\frac{U}{T}=-\frac{F}{T}$ (?, after division by $k_B$, it is the Helmholtz free-energy in units of $k_BT$, changed of sign)
$\frac{1}{T},V,N$
canonical


Planck function$^{(*)}$/$Y^{(*)}==S_3=S-\frac{PV}{T}-\frac{U}{T}=-\frac{G}{T}$ (?,after division by $k_B$, it is the Gibbs free-energy in units of $k_BT$, changed of sign)
$\frac{1}{T},\frac{P}{T},N$
isothermal-isobaric


?/$S_4=S-\frac{U}{T}+\frac{\mu N}{T}=\frac{PV}{T}$ (?,after division by $k_B$, it is the grand potential, in units of $k_BT$, changed of sign)
$\frac{1}{T},V,\frac{\mu}{T}$
isothermal-isochoric




$^{(*)}$: according to the IUPAC document Quantities, Units and Symbols in Physical Chemistry (section 2.1) and to the IUPAP-SUNAMCO Red book (section 4.4).
Other Massieu functions and the corresponding ensembles can be conceived, and occasionally have been used in computer simulations.
Thermodynamic potentials




Name/symbol of the fundamental equation
natural variables




internal energy/$U$
$S,V,N$


Helmholtz free energy/$F=U-TS$
$T,V,N$


enthaply/$H=U+PV$
$S,P,N$


Gibbs free energy/$G=U-TS+PV=\mu N$
$T,P,N$




A final comment is about the special cases of the Massieu function obtained as Legendre transform of entropy with respect to all its natural variables:
$$
\zeta=S-\frac{U}{T}-\frac{PV}{T}+\frac{\mu N}{T}
$$
and the thermodynamic potential
$$
Z=U-TS+PV-\mu N.
$$
As a consequence of the homogeneity of degree one of $U$ and $S$, as functions of their own variables, it should be $\zeta = Z=0$. For this reason, somebody called $Z$ the "zero"-thermodynamic potential. However, in real conditions these fundamental equations are not useless. Their theoretical vanishing is just a consequence of the exact homogeneity predicted at the thermodynamic limit. For finite-size systems, the "zero"-potential does not vanish exactly, due to sub-dominant contributions, like the surface terms. Therefore, they can be used as a magnifying glass for surface thermodynamics.
