144
5 Controversy of Thermodynamics Associated with Surface …
where z is a thermodynamic function of state, which consists of n pairs of thermodynamically conjugate state variables, the i-th pair of state variables being denoted by
x i and y i , and the system is represented in a space with exact dimensions of 2n + 1
(2n state variables plus the state function).
An elementary example of Eq. (5.15) is the case where z is U (internal energy),
n is 2, x 1 is −P (− pressure), y 1 is V (volume), x 2 is T (temperature), and y 2 is S
(entropy), which is expressed by
dU = −PdV + T dS.
(5.16)
In Eq. (5.16), V is conjugate to −P, and S is conjugate to T , implying that the
conjugate state variable to any given state variable is unique. By the way, Eq. (5.16)
does not contain a term of the form x i dy i + y i dx i , in which x i and y i correspond to A
and γ , respectively, in Eq. (5.14). A term such as x i dy i +y i dx i in Eq. (5.14) reduces the
exact number of dimensions by two, because of counting twice the state variables
x i and y i in the term of x i dy i + y i dx i , and thus, the exact number of dimensions
required to represent the system is (2n + 1 − 2), i.e., 2n − 1. In Eq. (5.14), i takes
one specific value since the exact number of dimensions is three for n = 2, implying
that dF
s
= Adγ + γ dA does not match Hermann’s formulation.
Therefore, Bottomley et al. [17] claimed that the derivation of the Shuttleworth
equation is not consistent with Hermann’s formulation of the mathematical structure
of thermodynamics. Furthermore, they have stated that if Hermann’s mathematical
structure of thermodynamics is true, and if Shuttleworth’s equation is a thermodynamic equation, then Shuttleworth’s equation is false. Marichev [23] supported
the argument of Bottomley et al. [17] with a supplement message that the consistency with Hermann’s analysis is necessary, but not sufficient to demonstrate the
consistency of thermodynamic equations since if one or more pairs of conjugate
state variables changed sign (e.g., from +PV to −PV), the function under consideration would change or completely lose its physical sense, even if remaining formally
consistent with the requirements of Hermann’s analysis.
Bottomley et al. [24] mentioned later that the consistency with Hermann’s theory
is a necessary condition. Furthermore, Marichev [23] remarked that the Shuttleworth
equation has not been proven experimentally although the Gokhshtein equation is
confirmed experimentally [13, 15]. On the other hand, Eriksson and Rusanov [25],
and Ibach [26] refuted the argument of Bottomley et al. [17] and asserted the validity
of the Shuttleworth equation from different derivations achieved by them. Moreover,
Ibach [26] mentioned that Hermann’s formal theory concerns the properties of functions defined on the Euclidean space of (independent) Cartesian coordinates, while
F
s , A, and γ do not span a Euclidean space since one variable (F
s ) is the product
of other two variables (A and γ ). The corresponding equation for the free energy of
bulk systems F would be F = f V (V : the volume and f : the volume specific free
energy), where f and V are not conjugate variables as P and V are.
In reply to their refutations, Bottomley et al. [27] argued again that the term of
x i dy i + y i dx i still involves in their different derivations, and the derived Shuttleworth
5 Controversy of Thermodynamics Associated with Surface …
where z is a thermodynamic function of state, which consists of n pairs of thermodynamically conjugate state variables, the i-th pair of state variables being denoted by
x i and y i , and the system is represented in a space with exact dimensions of 2n + 1
(2n state variables plus the state function).
An elementary example of Eq. (5.15) is the case where z is U (internal energy),
n is 2, x 1 is −P (− pressure), y 1 is V (volume), x 2 is T (temperature), and y 2 is S
(entropy), which is expressed by
dU = −PdV + T dS.
(5.16)
In Eq. (5.16), V is conjugate to −P, and S is conjugate to T , implying that the
conjugate state variable to any given state variable is unique. By the way, Eq. (5.16)
does not contain a term of the form x i dy i + y i dx i , in which x i and y i correspond to A
and γ , respectively, in Eq. (5.14). A term such as x i dy i +y i dx i in Eq. (5.14) reduces the
exact number of dimensions by two, because of counting twice the state variables
x i and y i in the term of x i dy i + y i dx i , and thus, the exact number of dimensions
required to represent the system is (2n + 1 − 2), i.e., 2n − 1. In Eq. (5.14), i takes
one specific value since the exact number of dimensions is three for n = 2, implying
that dF
s
= Adγ + γ dA does not match Hermann’s formulation.
Therefore, Bottomley et al. [17] claimed that the derivation of the Shuttleworth
equation is not consistent with Hermann’s formulation of the mathematical structure
of thermodynamics. Furthermore, they have stated that if Hermann’s mathematical
structure of thermodynamics is true, and if Shuttleworth’s equation is a thermodynamic equation, then Shuttleworth’s equation is false. Marichev [23] supported
the argument of Bottomley et al. [17] with a supplement message that the consistency with Hermann’s analysis is necessary, but not sufficient to demonstrate the
consistency of thermodynamic equations since if one or more pairs of conjugate
state variables changed sign (e.g., from +PV to −PV), the function under consideration would change or completely lose its physical sense, even if remaining formally
consistent with the requirements of Hermann’s analysis.
Bottomley et al. [24] mentioned later that the consistency with Hermann’s theory
is a necessary condition. Furthermore, Marichev [23] remarked that the Shuttleworth
equation has not been proven experimentally although the Gokhshtein equation is
confirmed experimentally [13, 15]. On the other hand, Eriksson and Rusanov [25],
and Ibach [26] refuted the argument of Bottomley et al. [17] and asserted the validity
of the Shuttleworth equation from different derivations achieved by them. Moreover,
Ibach [26] mentioned that Hermann’s formal theory concerns the properties of functions defined on the Euclidean space of (independent) Cartesian coordinates, while
F
s , A, and γ do not span a Euclidean space since one variable (F
s ) is the product
of other two variables (A and γ ). The corresponding equation for the free energy of
bulk systems F would be F = f V (V : the volume and f : the volume specific free
energy), where f and V are not conjugate variables as P and V are.
In reply to their refutations, Bottomley et al. [27] argued again that the term of
x i dy i + y i dx i still involves in their different derivations, and the derived Shuttleworth