Equivalent Definitions of the Mie-Grüneisen Form
aa r X i v : . [ phy s i c s . g e n - ph ] O c t Equivalent Definitions of the Mie-Gr¨uneisen Form
Kirill A. Velizhanin a) and Joshua D. Coe b) Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545 (Dated: 3 December 2020)
We define the Mie-Gr¨uneisen form in five different ways, then demonstrate their equivalence.PACS numbers: Valid PACS appear hereKeywords: equation of state, EOS, Mie-Gr¨uneisen, thermodynamics
I. INTRODUCTION
Equations of state (EOS) often are described as beingof Mie-Gr¨uneisen (MG) form, or type. This is usuallymeant to convey that the pressure relative to that of somereference curve (subscripted ‘0’) can be derived from theenergy relative to that curve via P ( V, T ) − P ( V ) = Γ( V ) V [ E ( V, T ) − E ( V )] , (1)where the Gr¨uneisen parameter Γ is a function of volumeonly. MG EOS appear frequently in high-pressure and-temperature physics. They are particularly useful whenapplied to materials for which a single locus (such as theprincipal Hugoniot or isentrope) is well-characterized ex-perimentally, with far fewer or lower-quality data else-where. This is true of many materials with shock datarecorded in the LASL and LLNL compendia, as well asmost high explosive (HE) product mixtures. Proceduresfor extracting the cold compression curve from shock datausing Eq. (1) appeared very early in the history of shockphysics, and fits of the Jones-Wilkins-Lee (JWL) form to cylinder expansion data can be made thermodynami-cally complete on the same basis. While the MG form often is defined through Eq. (1), additional statements (discussed below) have been de-rived as consequences. Aside from the question as towhether (1) is actually overdetermined ( i.e. , redundant),many discussions implicitly suggest its primacy by theorder in which alternative expressions are presented, asif they were merely its consequences.Our purpose here is to clarify the precise relationshipbetween various formulae that have appeared in discus-sions of the MG form. Specifically, we will define theform in five different ways that do not include (1), thendemonstrate their complete equivalence. Each pairwiserelationship is one of mutual entailment, meaning thatany single definition is sufficient to prove the others andthat primacy cannot be assigned to any individual. Be-cause no one definition is independent or unique, its em-phasis in a particular context is purely a matter of con-venience. Some of these results are well-known and have a) Electronic mail: [email protected] b) Electronic mail: [email protected] appeared elsewhere (at most depth in Ref. 10), but weare aware of no previous attempts to so explicitly makethese connections clear.The following section introduces the definitions, theirequivalence is proven in Sec. III, and in Section IV wediscuss some implications.
II. DEFINITIONS
The following five definitions of the MG form are equiv-alent, meaning that any one serves as the necessary andsufficient condition for any other. Not every definition isexpressed in a single equation, so definitions will be dis-tinguished from equation numbers by a prepended ‘
Definition
The first definition is a slight generalization of (1), P ( V, T ) − P ( V ) = Γ V [ E ( V, T ) − E ( V )] , (2)where subscripts have the same meaning as before and Γis the Gr¨uneisen parameter defined byΓ = V (cid:18) ∂P∂E (cid:19) V = − VT (cid:18) ∂T∂V (cid:19) S . (3)Unlike (1), viz. , that it is a function only of V ). Definition
In general, the Gr¨uneisen parameter as defined byEq. (3) is an arbitrary function of the thermodynamicstate. In the MG form, it is a function of volume only,Γ = Γ( V ) . (4) Definition
The entropy, S = f ( x ) , (5)is a function only of the scaled temperature, x = T / Θ( V ). The scaling factor Θ( V ) is usually a charac-teristic temperature such as that of Debye or Einstein and is, at most, a function of V . Definition
The heat capacity at constant volume, C V = (cid:18) ∂E∂T (cid:19) V = T (cid:18) ∂S∂T (cid:19) V , (6)is a function only of the entropy, C V = C V ( S ) . (7) Definition
The temperature- and volume-dependencies of theHelmholtz free energy can be decomposed as A ( V, T ) = φ ( V ) + Θ( V ) f ( x ) , (8)where f ( x ) and φ ( V ) are single-argument functions thatare otherwise arbitrary, and the same comments apply to x = T / Θ( V ) as in (5). The first term on the right handside is an internal energy as well as a free energy. III. EQUIVALENCE
Each of the following subsections demonstrates equiv-alence of a pair of definitions drawn from the five givenabove. Specifically, we consider four different pairs indi-cated by the arrows in Figure 1. This diagram is com-plete in the sense that any definition follows from anyother, either directly ( e.g. , → e.g. , → → e.g. , → conceptual significance to A. ←→ Differentiation of Eq. (2) with respect to T at constant V produces (cid:18) ∂P∂T (cid:19) V = (cid:18) ∂ Γ ∂T (cid:19) V [ E ( T, V ) − E ( V )] V + Γ V C V . (9)The left-hand-side (lhs) of (9) can also be evaluated usingthe chain rule in combination with Eqs. (3) and (6), (cid:18) ∂P∂T (cid:19) V = (cid:18) ∂P∂E (cid:19) V (cid:18) ∂E∂T (cid:19) V = Γ V C V . (10) FIG. 1. Five definitions of the Mie-Gr¨uneisen form, with ar-rows indicating demonstrations of equivalence provided in thetext. Γ satisfies (3), and x = T / Θ( V ). Equality of Eqs. (9) and (10) requires that (cid:0) ∂ Γ ∂T (cid:1) V = 0,establishing that T atconstant V under the assumption that Γ /V is a functionof V only (definition P = Γ V E + C ( V ) . (11)Taking C ( V ) = P ( V ) − Γ( V ) V E ( V ) for some referencecurve { P ( V ) , E ( V ) } produces Eq. (2). Definition B. ←→ We first demonstrate → Γ =
V αB T C V , (12)where α is the volumetric expansion coefficient, α = 1 V (cid:18) ∂V∂T (cid:19) P , (13)and B T is the isothermal bulk modulus, B T = − V (cid:18) ∂P∂V (cid:19) T . (14)Based on definition C V = T (cid:18) ∂S∂T (cid:19) V = x dSdx , (15)and the numerator is V αB T = V (cid:18) ∂P∂T (cid:19) V = V (cid:18) ∂S∂V (cid:19) T = V dSdx (cid:18) ∂x∂V (cid:19) T = − V T Θ dSdx d Θ dV = − x dSdx d ln Θ d ln V . (16)The first of the equalities in (16) follows from the cyclicrule for partial differentiation (cid:18) ∂x∂y (cid:19) z (cid:18) ∂z∂x (cid:19) y (cid:18) ∂z∂x (cid:19) y = − , (17)and the second from a Maxwell relation in the Helmholtzrepresentation. Substitution of (15) and (16) into (12)gives Γ = − d ln Θ d ln V , (18)a function only of V . This argument is very similar tothat of Wallace, and is sufficient to prove that → V ) = − VT (cid:18) ∂T∂V (cid:19) S = VT (cid:18) ∂T∂S (cid:19) V (cid:18) ∂S∂V (cid:19) T , (19)or as a partial differential equation for the entropy, V Γ( V ) (cid:18) ∂S∂V (cid:19) T − T (cid:18) ∂S∂T (cid:19) V = 0 . (20)Equation (20) can be solved by the method of charac-teristics, where the characteristics satisfy the followingsystem of ordinary differential equations parameterizedby t : dVdt = V Γ( V ) (21) dTdt = − T (22) dSdt = 0 . (23)The last of these demonstrates that the characteristiccurves are isentropic, whereas the first two can be com-bined to eliminate tdTT + Γ( V ) V dV = 0 , (24)and then straightforwardly integrated via separation ofvariables, yielding T e R VV V ′ ) V ′ dV ′ = C. (25)The solution to (20) is the union of all such curves (con-veniently labeled by the integration constant C ) and en-tropy can thus be expressed as S = f ( C ) = f (cid:18) T e R VV V ′ ) V ′ dV ′ (cid:19) (26)for arbitrary function f . This expression is simply Eq.(5) with Θ( V ) = e − R VV V ′ ) V ′ dV ′ , thereby proving → C. ←→ The equality of these statements was shown originallyby Davis and discussed in more detail by Menikoff. It is based on the invariance of mixed third derivativesof E ( S, V ) to the order in which they are taken ( i.e. , ahigher-order analogue of the Maxwell relations). In thiscase, ∂∂V ∂ E∂S = ∂ ∂S ∂E∂V , (27)which can be reexpressed as − Γ V TC V (cid:18) ∂ Γ ∂S (cid:19) V = VC V (cid:18) ∂C V ∂V (cid:19) S . (28)The lhs vanishes if Γ is a function only of V , requiringthat C V be a function only of S and vice-versa. D. ←→ That → (cid:0) ∂A∂T (cid:1) V = − S . Conversely, integration of S = f ( T / Θ( V ))with respect to T along some path of constant volumegives A ( T, V ) − A ( V ) = θ ( V ) g ( T / Θ( V )) , (29)where dg ( x ) dx = − f ( x ). The function g ( x ) is known upto an integration constant. Convenient choices for thisconstant include g (0) = 0 and θ ( V ) g (0) = E ZP , where E ZP is the vibrational zero-point energy (zpe). The firstresults in A ( V, T ) | T =0 = A ( V ) equivalent to the zero-temperature isotherm, the second in its being the coldinternal energy without zpe. The latter is often calcu-lated using density functional theory, and is sometimesreferred to as the cold curve. IV. COMMENTS • Definitions ↔ • It is sometimes convenient to generalize Eq. (1)to Γ as a function of variables in addition to vol-ume, such as internal energy or temperature. Thisposes no difficulty so long as it is understood thatthe new Γ(
V, E ) or Γ(
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