Comparing isotropic solids and fluids at either imposed volume or pressure, we investigate various correlations of the instantaneous pressure and its ideal and excess contributions. Focusing on the compression modulus K, it is emphasized that the stress fluctuation representation of the elastic moduli may be obtained directly (without a microscopic displacement field) by comparing the stress fluctuations in conjugated ensembles. This is made manifest by computing the Rowlinson stress fluctuation expression Krow of the compression modulus for NPT-ensembles. It is shown theoretically and numerically that Krow|P = Pid(2 − Pid/K) with Pid being the ideal pressure contribution.
INTRODUCTION
Among the fundamental properties of any equilibrium system are its elastic moduli characterizing the fluctuations of its extensive and/or conjugated intensive variables.1–7 The isothermal compression modulus K of an isotropic solid or fluid may thus be obtained in the NPT-ensemble at imposed particle number N, pressure P, and temperature T from the fluctuations
with kB being Boltzmann's constant. Equivalently, K may be obtained in a canonical NVT-ensemble using Rowlinson's stress fluctuation relation4,8,9
with β = 1/kBT being the inverse temperature,
with Pid being the ideal pressure contribution. We demonstrate first Eq. (3) by considering theoretically the fluctuations of the instantaneous normal pressure
BACKGROUND
As discussed in the literature,2,8,10 a simple average
where K = −V∂P/∂V2 has been used. For
i.e., the compression modulus K may be obtained from the difference of the pressure fluctuations in both ensembles. Interestingly, the numerically more convenient Rowlinson expression Krow for NVT-ensembles can be derived from Eq. (5)9 without using a microscopic displacement field (only possible for solids)5 and avoiding the volume rescaling trick used originally for liquids.4
MC-GAUGE
There is a considerable freedom for defining the instantaneous pressure
and the instantaneous excess pressure
with
NON-AFFINE CONTRIBUTION
In the following, the concise notation
AFFINE (BORN) CONTRIBUTION
The second moment of any intensive variable computed in an ensemble, where its mean value is imposed, is obtained readily by integration by parts. Using Eq. (7), this shows that
where a prime denotes a derivative with respect to the indicated variable. Albeit the indicated averages are taken over all states s and all volumes
CORRELATIONS AT CONSTANT P
We focus now on stress fluctuations in the NPT-ensemble. By comparing with Eq. (5), one sees that if the Rowlinson formula Krow is applied at imposed P, this must yield
Interestingly, Eq. (10) does not completely vanish for finite T as does the corresponding stress fluctuation expression for the shear modulus G at imposed shear stress τ.9 As a next step, we demonstrate the relations
from which Eq. (3) is then directly obtained by substitution into Eq. (10). Returning to the general transformation relation Eq. (4), we first note that the l.h.s. must vanish if at least one of the observables is a function of
making the steepest-descent approximation
to leading order for V → ∞. This relation implies finally the claimed correlation between ideal and excess pressure fluctuations, Eq. (12), using
SOME ALGORITHMIC DETAILS
The numerical results reported here to check our predictions have been obtained by MC simulation of (i) one-dimensional (1D) nets with permanent cross-links and (ii) two-dimensional (2D) glass-forming liquids. Periodic boundary conditions are used and the pressure P is first imposed using a standard MC barostat.8,9 After equilibrating and sampling in the NPT-ensemble, the volume is fixed,
Compression modulus K computed using the rescaled volume fluctuations Kvol|P (filled spheres), the Rowlinson stress fluctuation formula Krow|V (crosses), the difference between the total pressure fluctuations in both ensembles (squares) and the fluctuations of the inverse volume
Compression modulus K computed using the rescaled volume fluctuations Kvol|P (filled spheres), the Rowlinson stress fluctuation formula Krow|V (crosses), the difference between the total pressure fluctuations in both ensembles (squares) and the fluctuations of the inverse volume
COMPUTATIONAL RESULTS
As shown in Fig. 1, the compression modulus K may be determined using the volume fluctuations in the NPT-ensemble, Eq. (1), or using Rowlinson's stress fluctuation formula, Eq. (2), for the NVT-ensemble. The same values of K are obtained from the Legendre transform for the pressure fluctuations, Eq. (5), and from the ideal pressure fluctuations ηid|P, Eq. (11), which thus confirms both relations. As seen in the inset of Fig. 1, the compression modulus of the 1D nets decreases with δk. Also indicated is the “affine” contribution η|P to K, measuring the mean spring constant ⟨kl⟩ = 1, and the “non-affine” contribution η|V which is seen to increase with δk. The decrease of K is thus due to the increase of the non-affine contribution.
As shown in the inset of Fig. 2, we have also checked the correlations between the ideal and the excess pressure fluctuations ηmix|P. To make both models comparable, the reduced correlation function y = ηmix|P/Pid is traced as a function of the reduced ideal pressure x = Pid/K with K as determined independently above. A perfect data collapse on the prediction y = 1 − x (bold line) is observed for all systems. The main panel of Fig. 2 shows finally the scaling of the Rowlinson formula computed in the NPT-ensemble. As before a scaling collapse of the data is achieved by plotting y = Krow|P/Pid vs. x. The bold line indicates our key prediction, Eq. (3). Interestingly, the latter result does not depend on the MC-gauge which has been used above to simplify the derivation of Eq. (3). Please note that it is not possible to increase x beyond unity for our liquid systems (K ⩾ Pid) and the deviations from the low-temperature plateau y = 2 are thus necessarily small. The additional Pid/K correction has thus been overlooked in our previous publication.9
Characterization of stress fluctuations in the NPT-ensemble. Large spheres refer to 2D pLJ beads for P = 2, all other symbols to 1D nets for different δk and P as indicated. (Main panel) Rescaled Rowlinson formula Krow|P/Pid as a function of the reduced ideal pressure x = Pid/K. The bold line represents our key prediction, Eq. (3), on which all data points collapse. (Inset) Similar scaling for the reduced correlation function ηmix|P/Pid confirming Eq. (12).
Characterization of stress fluctuations in the NPT-ensemble. Large spheres refer to 2D pLJ beads for P = 2, all other symbols to 1D nets for different δk and P as indicated. (Main panel) Rescaled Rowlinson formula Krow|P/Pid as a function of the reduced ideal pressure x = Pid/K. The bold line represents our key prediction, Eq. (3), on which all data points collapse. (Inset) Similar scaling for the reduced correlation function ηmix|P/Pid confirming Eq. (12).
CONCLUSION
Emphasizing the underlying Legendre transform, Eq. (5), of the stress fluctuation formalism, we have investigated here the well-known Rowlinson stress fluctuation expression Krow for the compression modulus, Eq. (2), using deliberately the NPT-ensemble. Correcting several statements made in Ref. 9, it has been demonstrated theoretically and numerically that Eq. (3) holds. The latter result, as the other correlation relations indicated in the paper, may allow to readily calibrate (correctness, convergence, and precision) various barostats commonly used.8
ACKNOWLEDGMENTS
P.P. thanks the Région Alsace and the IRTG Soft Matter and F.W. the DAAD for funding. We are indebted to A. Blumen (Freiburg) and A. Johner (Strasbourg) for helpful discussions.