6120
J. Phys. Chem. 1993,97, 612M123
Compression of Liquids and Solids: Universal Aspects I. C. Sanchez,* J. Cho, and W.-J. Cben Chemical Engineering Department, University of Texas, Austin, Texas 78712 Received: March 19, 1993; In Final Form: April 26, 1993
It is demonstrated that the compression response of a liquid (including water) or solid to hydrostatic pressure satisfies a temperaturepressure superposition principle. A dimensionless pressure variable is used to superpose compression data as a function of temperature into a universal curve. The observation that the pressure coefficient of the adiabatic bulk modulus B1= 5 for many ionic compounds, metals, and semimetals is consistent with the internal energy being nearly harmonic in density around its minimum. Using the classical virial theorem, it is proposed that B1 has a theoretical lower bound of 1113.
Bridgman's classic papers1 earlier this century provide the foundation for understanding pressure effects on material prop erties. A considerable amount of work has followed, new techniques, such as Brillouin scattering,2continueto be developed to investigate material physical propertiesunder pressure. What seemsto be have been overlooked until now is that a corresponding states principle is satisfied, adiabatic compression yields fundamental information on the density dependence of the internal energy Uand the potential energy @, and isothermal compression yields fundamental information on the density dependence of the Helmholtz energy A. What we will show is that the compression can be represented as a functionof a single dimensionlesspressure variable that takes into account both temperature and pressure. Even water, which has an anomalous compressibility, satisfies this universal feature of compression. The compressional strain B is defined as
POLYSTYRENE
-0.02
-
=so'Bo + BIP + zB2P+ ...
1 1 = -p + zB1p' - 5i(2# -
B g 2 ) p 3+ ... (lb) where V is the volume at pressure P, VO is the volume at zero pressure, B is a bulk modulus, and p P/Bo is a dimensionless pressure and BOis the zero-pressurebulk modulus. This pressure variable p plays a key role in all that follows. The compression can be carried out isothermally or adiabatically (isentropically):
.=e
isothermal modulus; c = ln(V/Vo)+ P E PIBO Badiabatic modulus; c = In( V/V0),; p EPIBo (2) A typical example of the isothermal compression behavior of a polymer over a 150 OC temperature range is shown in Figure 1. These data for polystyrene (PS) were obtained on a Gnomix PVT apparatus that has been described in detail elsewhere.3Since the glass transition temperature is 100 OC,some of these data are in the glass and some are in the liquid. Note that the compression of the glass is about a factor of 2 less than the liquid at 200 MPa (2000 bar). The same data shown in Figure 2 are plotted as a functionof the dimensionless pressure defined in eq 2. Note that all data fall onto a single curve. This result implies that compression is a function of p only. In turn, this requires that the first few coefficients in the expansion for the compressive strain, eq 1b, be largely temperature independent;the dominant temperature dependence must be in BO, the zero-pressure, isothermal bulk modulus.
0 0
8
0 0
m
-
e
o ID0
8
e
0
-
8
In
(V/V,)
o -
4.04
.
0
0
"
0
I
-
D
"
0
l 0
-
4.06
8 ' O
8 '
30.1 C 81.6C
8 8
0
m
I1
-0'08--
-0.10
dP 1
P ,,
fi
0
--!
8
150.0C 179.6C I
so
-
L
8 I
8
. . . .
! . . . . 150! .
100
. . . 200
Pressure (MPs)
Figure 1. Isothermal compression data for a high molecular weight polystyrene below and above the glass transition temperatureof 100 OC. The same data are replotted in Figure 2.
Bridgman' studied the isothermal compression of water as a function of temperature up to pressures of 1200 MPa. These data for the compression and zero-pressure modulus are conveniently tabulated. In Figure 3 the compression data are shown as a function of P/Bo, and in Figure 4 the isothermal modulus is pointed against P/Bo. Below the PS and water results are discussed in more detail. The response of a liquid or solid to pressure is directly related to the internal energy Uby the adiabatic thermodynamicrelation
-P = aulav),
(3)
Without loss of generality, U can be expressed as in terms of a dimensionless energy u:
where UO> 0, p = Vo/Vis a dimensionless relative density, and V , , as before, is the volume at zero pressure. Thus, from eq 3
All condensed phases (solids and liquids) are thermodynamically
0022-3654/93/2097-6120$04.00/0 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 6121
Letters
Water
POLYSTYRENE 0.00 .b a
-0.02
*
i
30.1 C 81.6C
8.
-
150.0C
-
179.6C
%,
-
4.04
In (VIVO )
BIB ,
,
*k
-0.06
-
0.
I
m
I
I
n
3
-
4.08
I
-0.101 ' 0.00
.
.
'
8
0.05
'
' .
'
'
'
'
'
'
'
'
0.15
0.10
'
PIB
0.20
PIB,
Figure 2. Compression data in Figure 1 are plotted against PI&. The BovaluesinGPaare3.21 (30.1 0C),2.74(81.60C), 1.31 (150°C),and 1.14 (179.6 "C). Moduli are determined from low-pressure data (P< 50 MPa). Equation 17 provides an excellent fit to these data with E1 = 12.2. Water
,
Figure 4. Reduced isothermal modulus for wate# plotted against PI&. It illustratesthe near-linear behavior of the modulus with pressure and the correspondingstatesrelationship. The straightline has a slope of 6.1.
Thus, eq 5 can be rewritten as
P / B o p = p2u'(p,S)/u,
(9) This is the adiabatic equation of state. It has an interesting consequence: If the dimensionless internal energy function u(p,S) can be expressed as u [ p ( S ) ]so , that u only depends implicitly on entropy through VO,then the dimensionless pressure pis a function of p only. Note that the adiabatic compressive strain c is given by c = -In p . The inverse statement is that c is uniquely defined by p. For a given material, all adiabatic compressions would follow the same universal curve determined by the inverse of p2u'(p)/u2. Weemphasize that this universalitywould onlyobtain if u(P,S) u[p(S)I. Although u(p,S) is unknown, we can formally carry out the requisite inversion if u(p,S) is analytic around its minimum at p = 1, i.e.,
-
In (VIVO
Using the relation aB/aP)s = 8 In B/a In p ) ~the , modulus, given by eq 7, can be expanded in a series at zero pressure (p = 1): .
e...3 0.0
011
0:Z
0:s
0;s
0;4
0;6
PIB
Figure 3. Isothermal compression data for water up to pressures of 1.2 GPa and from 0 to 80 "(2.4 The EOvalues in GPa are 19.4 (0 "C), 22.5 (20 "C), 32 (40"C), 35 (60 "C), and 31 (80 "C). Equation 17 provides an excellent fit to these data with E1 = 6.1.
= u1 = 0;
u"(1S)
u2
>0
B, = 5 + uJu2 B@2 = -[6 -k Us/uz
metastable or stable at zero pressure which requires that
u'(1,S)
where
(6)
i.e., at zero pressure the internal energy of a condensed phase passes through a minimum. The adiabatic bulk modulus B is defined by
+
- %/u21
Substitution of (12) into eq l b yields a series expansion for c:
with the first few coefficients given by
c1= - 1 At zero pressure, p = 1 and ul = 0 so thatS
(12)
c, = 5
+ us/u2
6122 The Journal of Physical Chemistry. Vol. 97, No. 23, 1993
Note that the c, are dimensionless parameters that in general depend on entropy, or equivalently, temperature. Taking the pressure derivative of the bulk modulus yields the following differential equation for the internal energy: p2u”’+ p [ 5 - (aB/aP)]u”+ [4 - 2 ( a B / d P ) ] ~ ’ =0
(15) This equation can be solved in various approximations;the simplest is to assume that the modulus varies linearly with pressure so that (aB/dP) = Bl. Under this approximation we obtain
= u2(pB’ - 1)/(B1p2) or by using the equation of state, eq 9, we obtain 11’
= -ln(l
(16)
+ Blp)/Bl
(17) Equation 17 has the functional form of the Murnaghan equation.Beginning with the thermodynamic relationship, -P = (dA/ d v ) , where A is the Helmholtz free energy, a completely analogous set of equations can be developed for isothermal an, compression with the substitutions: UO Ao, u -* a, Bn Bn, and Cn Cn. Electron and nuclei interactions are Coulombic in nature. For such a system, the classical virial theorem yields a relationship between internal energy U,potential energy @, and kinetic energy T(U= T+@): t
-.
+
+
+
T = - U + 3PV; @ = 2 U - 3PV (18) These equations are known as theSchottkyrel~tionr.~J~ Defining a dimensionless potential energy function 4 as @ = @04(p,S)= 2U&(p,S), we obtain 3
4 = u -p’= 40 + r#J1(p- 1) + ,(p = U o - T3% (p-l)-u2(l
$2
- 1)2+ ...
+q(p-l)2-... 4%
(19)
Alternatively, the uR/u2derivative ratios can be expressed as a function of $,,/$Iderivative ratios (e.g., u3/u2 = 42/41 - 4/3). The equations developed here for adiabatic and isothermal compression are rigorous results applicable to isotropic solids and liquids. The only assumption made is that the internal energy and Helmholtz energy are analytic around p = 1. However, the materials should be homogeneous and undergo no phase change during the compression. For glasses, we assume that the glass is in quasi-equilibrium. This condition prevails at temperatures below the glass transition region. Above the glass transition temperature similar precautions are required because the glass transition temperaturerises with pressure. In the transition region relaxation effects during the experiment will vitiate the assumg tioin of quasi-equilibriumin the glass or equilibriumin the liquid. For crystalline solids, these results are applicable to crystals with isotropic symmetry.” We have shown that isothermalcompression data for a polymer and water measured over range of temperature and pressure can be superposed into universal curves that are only a function of P/Bo. This is only possible if the first few coefficients c, are temperature independent. The c. are functions of the derivative ratios a,/a2. Thus, our conclusion is that the derivative ratios an/a2 must be largely temperature independent. Temperatureindependent derivative ratios imply that the curvature of the Helmhotz energy is independent of temperature. In Figure 2 we have superimposed both liquid and glass compression data. The implication is that both glass and liquid data follow the same universal compression curve. However, even at 200 MPa, P/Bo for polymeric glasses is only about 0.1. Nearly all PVT polymer data in the literature are restricted to pressures below 200 MPa.12 There is a real need to extend the
Letters pressure range for polymeric glasses to ascertain whether glass compression differs from liquid compression. Our own belief is that glass and liquid will continue to superimpose out to higher pressures. It is worth mentioning that various existing empirical and semiempirical equations of state,2.*,13-16such as the well-known Tait equation: can be cast in forms that are consistent with our observation of temperature-pressure superposition. Equation 17, which assumes linearity of the bulk modulus, is a one-parameter equation for the compressive strain valid for adiabatic (Bl) or isothermal (B1) compression. This equation provides an excellent fit to the compression data for both PS and water with B1= 12.2 for PS and B1= 6.1 for water. As is apparent from Figure 4, the modulus varies quite linearly for water over most of the pressure range. This near linear behavior has also been observed in organic liquids.17 We should also mention that BOfor water is anomalous. It passes through a maximum at about 50 OC. A significant result that we have obtained is an expression for the first pressure derivative of the adiabatic bulk modulus, B1 = 5+u3/u2= 11/3 +42/41. Theratiou3/~2isafirst-ordermeasure of asymmetry of the internal energy minimum. If u3 = 0, u(p,S) is to first order quadratic around the minimum, i.e., a harmonic potential. We should only expect that situation if atoms or molecules are very tightly bound so that displacements around the minimum are equally difficult in compression or dilation. The strongest chemical forces are ionic. Remarkably, B1 data reported for seven ionic compounds7 (NaCl, KCl, MgO, Al203, RbI, LiF, and NaF) average 5.0 f 0.4! For eight metals and semimetals (Al, Mg, Cd, Fe, Na, K, Ge, and Si) the average is 4.7 & 0.9.7 Metals for which Bl < 5 (compression easier than dilation) may reflect the phenomenon of “core softening”l8 that is well-documented in liquid metals.19 For organics, with weak attractive van der Waals forces, we expect a much less symmetric internal energy with u3 > 0 (compression more difficult than dilation). For organic liquids Bl 10, which is consistent with this view. The second density derivative of the potential energy 42 changes sign at u3/u2 = -4/3 or 431 = 11/3. The lowest reported value of Bl that we are aware of is 3.6 f 0.1 for sodium.’ We propose that 1 1/3 may be a theoretical lower bound for B1 for condensed matter. The above results for Bl give us great confidence in our analysis of compression. Finally, we should mention that the results presented here for compression are also applicable to dilation by application of negative pressures.20 No additional information is obtainable in dilation experiments,but the observed universal features seen in compression should also be seen in dilation.
-
Acknowledgment. Financial support for this research was provided by AFOSR and NSF. We thank UT colleagues Dr. G. Brannock and Prof. D. T. Blackstock for helpful comments and suggestions and Prof. P. G. Debenedettiof Princeton for pointing out to us the core softening phenomenon. References and Notes (1) Bridgman, P. W. CollecfedExperimentalPapers; Harvard University Press: Cambridge, 1964; Vols. 1-7. (2) Oliver, W. F.; Herbst, C. A,; Lindsay, S. M.; Wolf, G. H. Phys. Rev. Lett. 1991, 67, 2795. (3) Zoller, P.; Bolli, P.; Pahud, V.; Ackermann, H. Reu. Sci. Instrum. 1976, 47, 948. (4) Bridgman, P. W. Proc. Am. Acad. Arts Sct. 1912, 48, 309. (5) Actually, all that is required is that the vapor pressure be much less than the bulk modulus at the same temperature. Liquids in their normal liquid range satisfy this requirement, but it is not satisfied as the gas-liquid critical point is approached. The latter situation can be treated by carrying out the expansionsat finite prmure [see: Sanchez, I. C.;Cho, J.; Chen, W.-J. Macromolecules, in press]. (6) Murnaghan, F. D. Proc. Natl. Acad. Sci. U.S.A. 1944, 30, 244. (7) Anderson, 0. L. J. Phys. Chem. Solids 1966,27, 547. (8) Macdonald, J. R. Reo. Mod. Phys. 1969, 41, 316.
Letters
The Journal of Physical Chemistry, Vol. 97, No. 23, 1993 6123
(1 1) Even for anisotropic crystals, methods for determining the bulk modulus have been developed. See: ref 7 and Thurston, R. N.; Shapiro, M. J. J. Acowt. Soc. Am. 1967, 47, 158. (12) Rodgers, P. A. J. Appl. Polym. Sci. 1993, 48, 1061. 113) Birch. F. J. Geoahvs. Res. 1978. 83. 1257. (14j Vinet; P.;Ferrahe;J.; Smith, J. R.;Rose, J. H. J. Phys. C 1986,19,
Holzapfel, W. B. Europhys. Lett. 1991, 16, 67. Jeanloz, R.; Godwal, B. K.; Meade, C. Nature 1991, 349,687. Houck, J. C. J. Res. Narl. Bur. Stand., Sect. A 1974, 78,617. Debenedetti, P.G.; Raghavan, V. S.;Borick, S . S . J . Phys. Chem. 1991, 95, 4540 and references therein. (19) Hoshino, K.;Leung, C. H.; McLaughlin, I. L.; Rahman, S.M.M.; Young, W. H. J. Phys. F: Mer. Phys. 1987, 17,787. (20) Green, J. L.; Durbin, D. J.; Wolf, G. H.; Angel, C. A. Science 1990,
L467.
249, 649.
(9) Schottky, W. Phys. 2.1920, 21,232. (10) Hirschfelder, J. 0.;Curtis&C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wilcy: New York, 1954; Chapter 4.
(15) (16) (17) (18)
