Thermodynamics of condensed matter with strong pressure-energy correlations

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by  Trond S. Ingebrigtsen

Thermodynamics of condensed matter with strong pressure-energy correlations
∗
Trond S. Ingebrigtsen, Lasse Bøhling, Thomas B. Schrøder, and Jeppe C. Dyre
DNRF Centre “Glass and Time”, IMFUFA, Department of Sciences,
Roskilde University, Postbox 260, DK-4000 Roskilde, Denmark
(Dated: February 1, 2012)
WeshowthatforanyliquidorsolidwithstrongcorrelationbetweenitsNVT virialandpotential-
energy equilibrium fluctuations, the temperature is a product of a function of excess entropy per
particleandafunctionofdensity,T =f(s)h(ρ). Thisimpliesthat1)thesystem’sisomorphs(curves
in the phase diagram of invariant structure and dynamics) are described by h(ρ)/T = Const., 2)
the density-scaling exponent is a function of density only, 3) a Gru¨neisen-type equation of state
2
applies for the configurational degrees of freedom. For strongly correlating atomic systems one
1 has h(ρ)=P C ρn/3 in which the only non-zero terms are those appearing in the pair potential
0 expanded as vn(r)n= P v r−n. Molecular dynamics simulations of Lennard-Jones type systems
2 n n
confirm thetheory.

n
a
J Theclassofstronglycorrelatingliquidswasintroduced static correlation functions are isomorph invariant, 4)

1 in Refs. 1 and 2. These liquids are defined by hav- a jump between isomorphic state points takes the sys-
3 ing a correlation coefficient above 0.9 of the constant- teminstantaneouslyto equilibrium. Using reducedunits
volume equilibrium fluctuations of virial W and poten- means measuring length in terms the unit ρ−1/3 where
]
t tialenergyU. TheWU correlationcoefficientvarieswith ρ ≡ N/V is the particle density, and time in units of
of state point, but we found from computer simulations ρ−1/3 m/kBT where m is the average particle mass.
s that a system has either poor WU correlations in the Sincepisomorphs are generally approximate, isomorph
.
t entire phase diagram or is strongly correlating at most properties are likewise rarely rigorously obeyed.
a
of its condensed-phase state points [1–5]. Van der Waals Allthermodynamicquantitiesconsideredbelowareex-
m
and metallic liquids are generally strongly correlating, cess quantities, i.e., in excess of those of an ideal gas at
d- whereas hydrogen-bonded, ionic, and covalently bonded the same density and temperature. Thus S is the excess
n liquids are generally not. The solid phase is at least as entropy (S <0), CV is the excess isochoric specific heat,
o correlating as the liquid phase. Theoretical arguments, p is the excess pressure (i.e., p=W/V), etc.
c numerical evidence, and experiments show that strongly Briefly,the reasonthatS andC areisomorphinvari-
V
[ correlatingliquids aresimpler than liquids ingeneral[1– antsisthefollowing[3]. Theentropyisdeterminedbythe
3 7]. canonicalprobabilities,whichareidenticalforscaledmi-
v Thesimplicityofstronglycorrelatingliquidscompared croconfigurations of two isomorphic state points. From
0 toliquidsingeneral[8]derivesfromthefactthatthefor- Einstein’s formula C = h(∆U)2i/k T2 the isomorph
V B
3 merhave“isomorphs”intheirphasediagram. Twostate invariance of C follows easily by taking the logarithm
V
1
pointswithdensityandtemperature(ρ ,T )and(ρ ,T ) of Eq. (1) and making use of the isomorph invariance of
3 1 1 2 2
aretermedisomorphic[3]ifallpairsofphysicallyrelevant scaled microconfigurationprobabilities.
.
7 microconfigurationsofthestatepointsthattriviallyscale Since S and C are invariant along the same curves
V
10 into one another (i.e., ρ11/3r(i1) =ρ12/3r(i2) for all particles in the phase diagram, CV is a function of S: CV =
1 i), have proportional configurationalBoltzmann factors: φ(S). Thus T(∂S/∂T)V = φ(S) or at constant volume:
dS/φ(S) = dT/T. Integrating this leads to an expres-
:
v sion of the form ψ(S) = ln(T) + k(V), which implies
i
X e−U(r(11),...,r(N1))/kBT1 =C12e−U(r(12),...,r(N2))/kBT2. (1) T = exp[ψ(S)]exp[−k(V)]. The generic version of this
involves only intensive quantities (s≡S/N):
r
a
Only inverse-power law liquids [9] have exact isomorphs
(here C = 1), but as shown in Appendix A of Ref.
12 T = f(s)h(ρ). (2)
3 a system is strongly correlating if and only if it has
isomorphs to a good approximation. For inverse power law interactions (∝ r−n) the entropy
The invariance of the canonical probabilities of scaled is well-known to be a function of ργ/T where γ = n/3:
microconfigurations along an isomorph has several im- S = K(ργ/T). Applying the inverse of the function K,
plications, for instance [1–3]: 1) The excess entropy and shows that these perfectly correlating systems obey Eq.
the isochoric specific heat are isomorph invariants, 2) (2) with h(ρ)=ργ.
thereduced-unitdynamicsisisomorphinvariantforboth The thermodynamic separationidentity Eq. (2) is the
Newtonian and stochastic dynamics, 3) all reduced-unit main result of this paper. We proceed to discuss some
consequences and numerical tests.
1. Density scaling
Since entropy is an isomorph invariant, it follows from
∗Electronicaddress: [email protected] Eq. (2) that the variable characterizing an isomorph
2
may be chosen as h(ρ)/T. In particular, the reduced U = F(S)h(ρ)+k(ρ) leads to Eq. (6), in which γ(ρ) is
relaxationtime τ˜, which is also isomorph invariant, may given by Eq. (5).
be written for some function G 4. The isomorphs of atomic systems
We consider now predictions for systems of “atomic”
particles interacting via pair potentials of the form [15]
h(ρ)
τ˜=G . (3) (where r is the distance between two particles)
(cid:18) T (cid:19)
This is the form of “density scaling” proposed by Alba-
Simionesco et al. in 2004 from different arguments [10]; v(r) = v r−n. (7)
n
at the same time Dreyfus et al., as well as Casalini and Xn
Roland,favoredthemorespecificformτ˜=G(ργ/T)[10].
Isochrones for many supercooled liquids and polymers For simplicity of notation the case of identical particles
follow to a good approximation the latter “power-law is considered, but the arguments generalize trivially to
density scaling” relation [11]. For large density changes, multicomponent systems. Consider the thermal average
however, it was recently shown that the density-scaling hr−ni. Switching to reduced units defined by r˜≡ ρ1/3r,
exponentgenerallyvariesinbothsimulationsandexper- we have hr−ni= hr˜−niρn/3. Since structure is isomorph
iment [12]; these cases conform to the more general Eq. invariant in reduced units, hr˜−ni is an isomorph invari-
(3). ant. Consequently,itisafunctionofanyotherisomorph
2. An expression for the density-scaling exponent invariant, for instance the entropy: hr˜−ni = Gn(S).
Thegeneral,state-pointdependentdensity-scalingexpo- Noting that the average potential energy is a sum of
nent γ is defined [2, 3] by Eq. (7) over all particle pairs, we conclude that (where
H (S)∝v G (S))
n n n
∂lnT ∂lnT
γ ≡ = . (4)
(cid:18)∂lnρ(cid:19)S (cid:18)∂lnρ(cid:19)τ˜ U = Hn(S)ρn/3. (8)
Xn
The physical interpretation of Eq. (4) is the following.
If density is increased by 1%, temperature should be
Taking the derivative of this equation with respect to
increased by γ% for the system to have the same en-
temperature at constant volume leads to
tropy and reduced relaxationtime. Equation (2) implies
dlnT = dlnf(s)+dlnh(ρ); thus along an isomorph –
wheresandτ˜arebothconstant–onehasdlnT =dlnh.
∂U ∂S
Via Eq. (4) this implies = H′(S) ρn/3. (9)
(cid:18)∂T(cid:19) n (cid:18)∂T(cid:19)
V Xn V
dlnh
γ = . (5) The left hand side is T (∂S/∂T) , so Eq. (9) implies
V
dlnρ
In particular, γ depends only on density: γ =γ(ρ) [3].
3. Configurational Gru¨neisen equation of state T = Hn′(S)ρn/3. (10)
The Gru¨neisen equation of state expresses that pressure Xn
equals a density-dependent number times energy plus a
This is consistent with the thermodynamic separation
term that is a function of density only [13]. This equa-
′
identity Eq. (2) only if all the functions H (S) are
tion of state is used routinely for describing, in particu- n
proportional to some function, i.e., if one can write
lar,solids under highpressure. We proceedto showthat
′
H (S)=C φ(S). We identify φ(S) as the function f(s)
stronglycorrelatingsystemsobeytheconfigurationalver- n n
of Eq. (2), which means that
sion of the Gru¨neisen equation of state, which as sug-
gested by Casalini et al. [14] has the density-scaling ex-
ponent as the proportionality constant [3, 4]
h(ρ) = C ρn/3. (11)
n
Xn
W = γ(ρ)U +Φ(ρ). (6)
Thusforstronglycorrelatingatomicliquids,the thermo-
To prove this, note first that (∂U/∂S) =T =f(S)h(ρ) dynamicfunctionh(ρ)hasananalyticalstructure,which
ρ
by integration implies U = F(S)h(ρ) + k(ρ) where isinheritedfromv(r) inthe sensethatthe onlynon-zero
′
F (S) = f(S) (S is the extensive entropy). Since W = terms of h(ρ) are those corresponding to the non-zero
(∂U/∂lnρ) (which follows from the standard identity terms of v(r). Note that not all systems with potentials
S
TdS =dU+pdV),wegetW =F(S)dh/dlnρ+dk/dlnρ. ofthe formEq. (7) are strongly correlatingand that the
SubstitutingintothelatterexpressionF(S)isolatedfrom derivation applies only if this is the case.
3
7.0 ρ = 1.20 (T = 0.43 −> T = 0.80) 99.9% in its entire phase diagram. At low densities
N ρ = 1.60 (T = 1.70 −> T = 9.15) (ρ ≪ 1) the repulsive LJ fluid behaves as an r−6 fluid,
> / ρρ == 24..0000 ((TT == 48.04.00 −−>> TT == 2452.50)) whereas it for ρ ≫ 1 is effectively an r−12 fluid. Thus
12 6.0 ρ = 10.0 (T = 3257 −> T = 17005) the density-scaling exponent γ(ρ) varies from 2 to 4 as
-12σ rijij5.0 Increasin LinearK roegbr-eAssniodne rfsietn Binary dperenvsiiotyusilnycsrteuadseiesd, astmrouncghlylacorgrreerlavtairnigatsiyosntetmhasn. that of
εij g te Lennard-Jones
<j m
Σ< 4i4.0 peratur Since h(ρ)is only definedwithin anoverallmultiplica-
4 e tive constant, one can write for the repulsive LJ fluid
-ρ h(ρ) = αρ4 + (1 − α)ρ2. This leads via Eq. (5) to
3-.014.0 -13.0 -12.0 -11.0 -10.0 -9.0 γ0 =2+2α, implying that
- ρ-2 < 4Σ ε σ 6 r -6 > / N
i<j ij ij ij
FIG. 1: The thermal average of r−12 versus that of −r−6 in
reduced units for a large range of state points of the Kob-
Andersen binary Lennard-Jones liquid simulated with 1000
particles (ε = σ = 1). These quantities correspond
AA AA
to H12(S) and H6(S) in Eq. (8). The theory predicts that
′ ′
H12(S) ∝ H6(S), implying that all data points should fall
onto a common line according to H12(S)=αH6(S)+β.
h(ρ) = (γ /2−1)ρ4+(2−γ /2)ρ2. (13)
0 0
As an illustration we present results from simulations
oftheKob-AndersenbinaryLennard-Jones(KABLJ)liq-
uid [16], which is strongly correlating at its condensed-
phase state points [1–3]. The applicationof the aboveto
′ ′
LJsystemspredictsthatH (S)∝H (S),whereH (S)
12 6 12
is the reduced coordinate averageof the r−12 term of U,
Our simulations identified from the expression γ =
0
etc. Integrating this leads to H12(S)=αH6(S)+β, im- h∆W∆Ui/h(∆U)2i [3] the exponent γ = 3.56 at the
0
plying that if the repulsive term in U is plotted against
state point (ρ,T)=(1,1). Equation (13) with γ =3.56
0
the attractive term in reduced units, all points should
was tested in two different ways. First, we compared at
fall onto a common line. Figure 1 presents data where
each state point along an isomorph the exponent γ(ρ)
density was changed by a factor of eight and tempera-
predicted from Eqs. (5) and (13) with that calculated
ture a factor of 40,000. The data collapse is good but from the fluctuations via γ = h∆W∆Ui/h(∆U)2i (right
not exact, which reminds us that the relations derived
panel of Fig. 2). The left panel presents a second test
are approximate.
of Eq. (13) by showing results from simulating five tem-
The theory implies a simple mathematical description
peratures at ρ = 1, plotting for each temperature in-
of the isomorphs in the (ρ,T) phase diagram. From the
stantaneousvaluesofthe potentialenergyversusthe po-
factthatthepotentialenergycontainsonlyr−12 andr−6
tential energy of the same microconfigurations scaled to
terms, it follows that h(ρ) = Aρ4−Bρ2. Consequently,
three other densities (ρ = 0.5,1.6,2.0). The theory be-
LJ isomorphs are given by
hindtheobservedstraightlinesisthefollowing. Consider
two isomorphic state points (ρ ,T ) and (ρ,T) and sup-
0 0
Aρ4−Bρ2 pose each temperature is changed a little, keeping both
= Const. (12) densities constant. If the two new state points are also
T
isomorphic, the entropy change is the same for both:
The invariance of the Boltzmann statistical weights of dU0/T0 = dU/T. This implies dU/dU0 = T/T0, i.e.,
scaledmicroconfigurationsimpliesthatanisomorphcan- (∂U/∂U0)ρ0,ρ = T/T0. Since h(ρ)/T is constant along
notcrosstheliquid-solidcoexistencecurve. Inparticular, an isomorph, this implies (∂U/∂U ) = h(ρ)/h(ρ ).
0 ρ0,ρ 0
thecoexistencecurveisitselfpredictedtobeanisomorph Integrating this at constant ρ and ρ leads to U =
0
[3], which was recently confirmed by simulations of gen- [h(ρ)/h(ρ )]U + φ(ρ ,ρ). In our case ρ = 1 and
0 0 0 0
eralized LJ liquids [4, 17]. Consequently the coexistence h(ρ ) = 1. Thus plotting U versus U is predicted to
0 0
line is given by Eq. (12). This validates a recent conjec- result in straight lines with slope h(ρ) (yellow asterices
ture of Khrapak and Morfill [18]. in the left panel of Fig. 2). The scaled state points are
5. Predictions for the repulsive Lennear-Jones fluid isomorphic to the original ρ = 1 state points, with tem-
As a final illustration we consider the “repulsive”single- peraturesgivenbyT =T h(ρ). Viathe“directisomorph
0
componentLJfluiddefinedbythepairpotential(r−12+ check”[3]thisimpliesthatthescaledmicroconfigurations
r−6)/2, a system with WU correlation coefficient above form elongated ovals with slope h(ρ).
4
Insummary,we haveshownthat forstronglycorrelat-
ing liquids or solids, temperature separates into a func-
50 4 tion of entropy times a function of density. For these
systems the energy scale is consequently determined by
40 densityalone. Itisanopenquestionwhether,conversely,
Slope = 13.4 3.5
the thermodynamic separation identity Eq. (2) implies
ρ = 2.00
Prediction
30 that the system in question is strongly correlating.
Simulation
ρ)U( Repulsive Lennard-Jones 3γ
20
ρ = 1.60 Slope = 5.68
2.5
10
ρ = 1.00
ρ = 0.50
0 2
3.25 3.5 3.75 4 4.25 4.5 0.1 ρ 1 10
U(1.00)
FIG. 2: “Multiple direct isomorph check” applied to simula-
tions of N = 1000 particles of the repulsive LJ fluid defined
by the pair potential (r−12 +r−6)/2. The left panel shows
the potential energies of pairs of microconfigurations, where
thepotential energy of a given microconfiguration at density
1.0 is denoted U(1.00) and that of the same microconfigura-
tion scaled to density ρ is denoted U(ρ) (ρ = 0.5;1.6;2.0).
This was done for T = 0.6;0.8;1.0;1.2;1.4. The black
lines are the predictions (see the text) with slopes deter-
mined via Eq. (13) from the fluctuations calculated at the
state point (ρ,T) = (1,1) marked by an arrow. The right
panelshowsthedensity-scalingexponentforeachstatepoint
along an isomorph predicted from Eqs. (5) and (13) (full
curve) and the exponent calculated via the fluctuation for- The centre for viscous liquid dynamics “Glass and
mula γ = h∆W∆Ui/h(∆U)2i [3] (red crosses). The arrow Time” is sponsored by the Danish National Research
marks thestate point (ρ,T)=(1,1).
Foundation (DNRF).
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