Engineering, 2013, 5, 844-849
Published Online November 2013 (http://www.scirp.org/journal/eng)
http://dx.doi.org/10.4236/eng.2013.511102
Open Access ENG
On Thermodynamic Analysis of Substances with Negative
Coefficient of Thermal Expansion
Kal Renganathan Sharma
Lone Star College-North Harris, Houston, USA
Received November 27, 2012; revised January 8, 2013; accepted January 15, 2013
Copyright © 2013 Kal Renganathan Sharma. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
There are reports in the literature on the discovery of novel materials that were observed to shrink upon heating. Treat-
ment of these materials in the same manner as the materials with positive coefficient of thermal expansion can lead to
the misinterpretation of the laws of thermodynamics. This is because volume expansivity is usually defined at constant
pressure. Negative values for volume expansivity can be shown using Maxwell’s reciprocity relations to lead to nega-
tive values for absolute temperature for ideal gas. For real systems, using Helmholz free energy analysis at equilibrium
an expression for the volume expansivity was derived. It can be seen that this expression would be always positive for
real physical changes, either heating or cooling. Isentropic volume expansivity is proposed as better suited for analysis
of materials with negative thermal expansion, NTE and composites used in space such as Hubble telescope and Chandra
telescope with zero coefficient of thermal expansion. This kind of a switch from isobaric to isentropic has precedence in
the history of development of speed of sound.
Keywords: Negative Thermal Expansion; NTE Materials; Helmholz Free Energy; Second Law of Thermodynamics;
Speed of Sound; Volume Expansivity
1. Introduction
The volume expansivity of pure substances is defined as
[1]:
1
P
V
VT






(1)
It is a parameter that is used to measure the volume
expansivity of pure substances and is defined at constant
pressure, P. In the field of materials science, the property
of linear coefficient of thermal expansion is an important
consideration in materials selection and design of prod-
ucts. This property is used to account for the change in
volume when the temperature of the material is changed.
The linear coefficient of thermal expansion is defined as:

0
0
00
f
f
ll l
lT T
lT T











(2)
For isotropic materials,
 = 3  . Instruments such as
dilatometers, XRD, X-ray diffraction can be used to
measure the thermal expansion coefficient. Typical val-
ues of volume expansivity for selected isotropic materials
at room temperature are provided in Table 1.
As can be seen from Table 1, the volume expansivity
for pure substances is usually positive. In some cases it
can be negative. Examples of materials given with nega-
tive values for volume expansivity in the literature are
water in the temperature range of O - 4 K, honey,
mononoclinic Sellenium, Se, Tellerium, Te, quartz glass,
faujasite, cubic Zirconium Tungstate, ZrW2O8 [2] in the
temperature range of 0.3 - 1050 K. ZerodourTM [3] is a
glass-ceramic material that can be controlled to have zero
or slightly negative thermal coefficient of expansion and
was developed by Schott Glass Technologies. It consists
of a 70 - 80 wt% crystalline phase with high-quartz
structure. The rest of the material is a glassy phase. The
negative thermal expansion coefficient of the glassy
phase and the positive thermal expansion coefficient of
the crystalline phase are expected to cancel out each
other leading to a zero thermal coefficient material. Ze-
rodurTM has been used as the mirror substrate on the
Hubble telescope and the Chandra X-ray telescope. A
dense, optically transparent and zero-thermal expansion
material is necessary in these applications since any
changes in dimensions as a result of the changes in the

K. R. SHARMA 845
Table 1. Volume expansivity of selected materials at room
temperature.
# Material
 , Volume Expansivity
(*10−6·K−1)
1 Aluminum 75.0
2 Copper 49.8
3 Iron 36.0
4 Silicon 9.0
5 1020 Steel 36.0
6 Stainless Steel 51.9
7 Epoxy 165.0
8 Nylon 6,6 240
9 Polyethylene 300.0
10 Polystyrene 210
11 Partially Stabilized ZrO2 31.8
12 Soda-Lime Glass 27.0
13 Zirconium Tungstate ZrW2O8 −27.0
14 Faujasite −12.0
15 Water (0 - 4 K) negative
16 Honey negative
temperature in space will make it difficult to focus the
telescopes appropriately. Material scientists have devel-
oped ceramic materials based on sodium zirconium phos-
phate, NZP that have a near-zero-thermal-expansion co-
efficient.
The occurrence of negative values of volume expan-
sivity ab initio, is a violation of second law of thermo-
dyna mics according to some investi-
gators such as Stepanov [4]. They propose that the first
law of thermodynamics be changed from
to in order to work
with materials with zero or negative coefficient of ther-
mal expansion.
dd dUQPV
dd dUQ PV dd dUQPV
In a famous problem such as the development of the
theory of the velocity of sound such a change from de-
fining a isothermal compressibility to isentropic com-
pressibility brought the experimental observations closer
to theory [5-7]. The proposal from this study is in part
motivated by the work of Laplace (see 2. Historical Note
below).
2. Historical Note
By 17th century it was realized that sound propagates
through air at some finite velocity. Artillery tests have
indicated that the speed of sound was approximately
1140 ft/s. These tests were performed by standing a
known large distance away from a cannon, and noting
the time delay between the light flash from the muzzle
and the sound of the discharge. In proposition 50, Book
II of his Principia Newton [6] theorized that the speed of
sound was related to the elasticity of the air and can be
given by the reciprocal of the compressibility. He as-
sumed that the sound wave propagation was an isother-
mal process. He proposed the following expression for
the speed of sound;
1
T
c


 (3)
where the isothermal compressibility is given by;
1
T
T
V
VP



 


(4)
Newton calculated a value of 979 ft/s from this ex-
pression to interpret the artillery test results. The value
was 15% lower than the gunshot data. He attributed the
difference between experiment and theory to existence of
dust particles and moisture in the atmosphere. A century
later Laplace [7] corrected the theory by assuming that
the sound wave was isentropic and not isothermal. He
derived the expression used to this day to instruct senior
level students in Gas Dynamics [5] for the speed of
sound as:
1
s
c


 (5)
1
s
s
V
VP



 


(6)
By the demise of Napaleon the great the relation be-
tween propagation of sound in gas was better understood.
3. Theoretical Analysis
Clapeyron Equation [1] can be derived to obtain the lines
of demarcation of solid phase from liquid phase, liquid
phase from vapor phase and solid phase from vapor
phase in a Pressure-Temperature diagram for a pure sub-
stance. This can be done by considering a point in the
demarcation line and a small segment in the demarcation
line. At the point the free energy of the solid and liquid
phases can be equated to each other at equilibrium. The
enthalpy change during melting can be related to the en-
tropy change of melting at a certain temperature of phase
change. Along the segment the change in free energy dGs
and dGl in the solid and liquid phases at equilibrium can
be equated to each other. The combined two laws of
thermodynamics are applied and an expression for dP/dT
can be obtained. Sometimes when ideal gas can be as-
summed, this expression can be integrated to obtain use-
ful expressions that are used in undergraduate thermody-
Open Access ENG

K. R. SHARMA 846
namics instruction. A similar analysis is considered here
using Helmholz free energy. At the point and segment of
the phase demarcation line of solid and liquid in a Pres-
sure-Temperature diagram of a pure substance;
,d d
s
LsL
A
AAA (7)
,
SS LLSLSL
UTSUTSTSU (8)
Applying the first and second law of thermodynamics
to change in Helmholz free energy;
dddd d
A
UTS PVST  (9)
For reversible changes at equilibrium;
dd dddd
Ssats satLLsatLsat
A
PV STAPV ST
d
d
sat SL SL
SL SL
VSU
TPTP


 
 
 (10)
For physical changes Equation (10) can be seen to be
always positive. This is because of the lowering of pres-
sure as solid becomes liquid and the internal energy
change is positive resulting in a net positive sign in the
RHS, right hand side of Equation (10). No ideal gas law
was assumed. Only the first two laws of thermodynamics
was used and reversible changes were assumed in order
to obtain Equation (10). Thus reports of materials with
negative thermal expansion coefficient is inconsistent
with Equation (10).
What may be happening is chemical changes. Strong
hydrogen bonded water in 0 - 4 0K shrinks upon heating
due to chemical changes. This cannot be interpreted us-
ing laws that are developed to describe physical changes.
In the example of faujasite may be lattice structure
changes take place upon heating. What do you do?
For ideal gas, it is shown below that the volume ex-
pansivity can be related to the reciprocal of absolute tem-
perature. Per the third law of thermodynamics the lowest
achievable temperature is 0 0K. Hence volume expansiv-
ity is always positive for physical changes. From Equa-
tion (1);
1
P
V
VVT






From Maxwell Relations [8],
P
T
V
TP

 

 

 
S
T
(11)
Equation (8) can be seen to be the case as follows. The
free energy G of pure substances are defined as:
GHTS (12)
where, H is the enthalpy (J/mole), S is the entropy
(J/mole/K)

dddd dGHTSHTSS (13)
Combining Equation (13) with the First Law of Ther-
modynamics,
dddddddGTSVPTSSTVPST

 (14)
It may be deduced from Equation (14) that;
T
P
GV
P
GS
T












(14,15)
P
GS
T





 (16)
The reciprocity relation can be used to obtain the cor-
responding Maxwell relation. The order of differentiation
of the state property does not matter as long as the prop-
erty is an analytic function of the two variables. Thus,
22
G
PT TP

G


(17)
Combining Equation (15)-(16) with Equation (17);
P
T
V
TP

 

 

 
S
(18)
Thus Equation (18) can be derived. Combining Equa-
tion (1) and Equation (18);
T
S
VP






(19)
For a reversible process, the combined statement of 1st
and 2nd laws [1] can be written as:
ddd
H
TS VP

 (20)
At constant temperature for reversible process for real
substances;
TT
HS
T
PP

 

 

 
V
(21)
Combining Equation (21) and Equation (19);
11
T
HT
VP



 


 (22)
Or
11
T
H
TVT P



 


(23)
For ideal gases Equation (23) would revert to the
volume expansivity,
 would equal the reciprocal of ab-
solute temperature. This would mean that  can never be
negative as temperature is always positive as stated by
the third law of thermodynamics. So materials with
negative values for
 ab initio are in violation of the
combined statement of the 1st and 2nd laws of thermody-
namics. Negative temperatures are not possible for vibra-
tional and rotational degrees of freedom. A freely mov-
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K. R. SHARMA 847
ing particle or a harmonic oscillator cannot have negative
temperatures for there is no upper bound on their ener-
gies. Nuclear spin orientation in a magnetic field is
needed for negative temperatures [9]. This is not appli-
cable for engineering applications. Enthalpy variation
with pressure is weak and small for real substances. This
has to be large to obtain a negative quantity in Equation
(23).
4. Proposed Isentropic Expansivity
Along similar lines to the improvement given by Laplace
to the theory of the speed of sound as developed by
Newton (as discussed in Section 2 Historical Note) an
isentropic volume expansivity is proposed.
1
s
S
V
VT






(24)
Using the rules of partial differential for three vari-
ables, any function f in variables (x,y,z) it can be seen
that;
z
yz
x
f
ffy
x
xyx
 
 

  

 
 (25)
Thus,
s
PT
VVVP
TT PT






S
(26)
Let,
S
P
T






(27)
Plugging Equation (27) into Equation (26);
1
s
P
S
V
VT T







 (28)
At constant pressure,
dd d
H
UPV

 (30)
Equation (30) comes from H = U + PV the definition o
specific enthalpy in terms of specific internal energy, U,
pressure and volume, P and V respectively. Equation (30)
can be written for ideal gas as:

dd
PV
CC TPV (31)
It can be realized from Equation (31) that;

PV
P
P
CC
VV
TP







 (32)
Plugging Equation (32) into Equation (30);

PV
s
PT
S
CC
V
V
TP
C





 


 (33)
Or,

 1PV
P
T
s
S
CC C
V
VTVP V





 


 (34)
For substances with negative coefficient of thermal
expansion under the proposed definition of isentropic
volume expansivity, 
s does not violate the laws of ther-
modynamics quid pro quo.
Considering the thermal expansion process for pure
substances in general and materials with negative coeffi-
cient of thermal expansion in particular, the process is
not isobaric. Pressure can be shown to be related to the
square of the velocity of the molecules.
5. Measurements of Volume Expansivity Not
Isobaric
The process of measurement of volume expansivity can-
not be isobaric in practice. When materials expand the
root mean square velocity of the molecules increases. For
the materials with negative coefficient of thermal coeffi-
cient the velocity of molecules are expected to decrease.
In either case, forcing such a process as isobaric is not a
good representation of theory with experiments. Such
processes can even be reversible or isentropic. Experi-
ments can be conducted in a reversible manner and the
energy may be supplied or may be removed as the case
may be. Hence it is proposed to define volume expansiv-
ity at constant entropy. This can keep the quantity per se
from violation the laws of thermodynamics.
6. Significance of Treatment of Materials
with Nte
Recently, Miller et al. [10] presented a review article on
materials that were observed to exhibit negative thermal
expansion. Most materials demonstrate an expansion
upon heating. Few materials are known to contract.
These materials are expected to exhibit a NTE, negative
thermal expansion coefficient. These materials include
complex metal oxides, polymers and zeolites as shown in
Table 2. These can be used to design composited with
zero coefficient of thermal expansion. When the matrix
has a positive thermal expansion coefficient and the filler
material has a negative thermal expansion coefficient the
net expansion coefficient of the composite can be dialed
in to zero. They explore supramolecular mechanisms for
exhibition of NTE. Examples of materials where reports
indicate NTE with the references are as follows;
A careful study of these materials under isentropic
heating is suggested. The volume expansivity then will
be within the predictions of the laws of thermodynamics.
Further studies on chemical changes on heating for these
materials are suggested. Strong hydrogen bonded sys-
tems may also be considered as strongly interacting sys-
tems.
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K. R. SHARMA 848
Table 2. Materials that possess NTE.
# Material Journal Reference
1.0 ZrW2O8, Zirconium
Tungstate (cubic lattice) Acta Crystallography[11]
2.0 (ZrO)2VP2O7 US Patent [12]
3.0 HfW2O8, Halfnium
Tungstate (orthorhombic) J. Appl. Phys. [13]
4.0 ZrMo2O8, Zirconium
molybdate, (cubic) Chem. Mater. [14]
5.0 Silicalite-1 &
Zirconium Silicalite-1 Mater. Res. Bulletin[15]
6.0 CuScO2 (delafossite structure) Chem. Mater. [16]
7.0 Polydiacetylene Crystal J of Polym. Sci. [17]
8.0 Graphite Fiber Composites Proc. of Royal Society[18]
7. Conclusions
Most materials expand upon heating. Some materials
have been reported as NTE, negative thermal expansion
coefficient materials. Very little thermodynamic analysis
has been done on these materials. In this study, Helmholz
free energy change during melting or solidication was
undertaken. Equation (10) was derived for expansivity at
equilibrium.
For physical changes, Equation (10) can be seen to be
always positive. This is because the lowering of pressure
as solid becomes liquid and the internal energy change is
positive resulting in a net positive sign in the RHS, right
hand side of Equation (10). No ideal gas law was as-
sumed. Only the first two laws of thermodynamics were
used and reversible changes were assumed in order to
obtain Equation (10). Thus reports of materials with ne-
gative thermal expansion coefficient are inconsistent
with Equation (10).
For ideal gases, Equation (23) would revert to the
volume expansivity,
 would equal the reciprocal of ab-
solute temperature. This would mean that  can never be
negative as temperature is always positive as stated by
the third law of thermodynamics. So materials are with
negative values for
 ab initio are in violation of the
combined statement of the 1st and 2nd laws of thermody-
namics.
Along similar lines to the improvement given by
Laplace to the theory of the speed of sound as developed
by Newton (as discussed in Section 2 Historical Note) an
isentropic volume expansivity is proposed by Equation
(24). This can be calculated using Equation (28) from
isobaric expansivity, isothermal compressibility and a
parameter  that is a measure of isentropic change of
pressure with temperature. Equation (34) can be used to
obtain the isentropic expansivity in terms of heat capaci-
ties at constant volume and constant pressure and iso-
thermal compressibility at a given pressure and tempera-
ture of the material.
Chemical changes have to be delineated from physical
changes when heating the material.
8. Acknowledgements
Acknowledgements are extended to my undergraduate
students in advanced standing in my CHEG 2043 Ther-
modynamics and CHEG 3053 Thermodynamics-II cour-
ses at Prairie View A & M University, Prairie View, TX
for valuable discussions on Clapeyron equation during
office hours.
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Open Access ENG
849
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Nomenclature
A Helmholz free energy (J/mole)
c Speed of Sound (m/s)
CP Heat Capacity at Constant Pressure (J/kg/K)
Cv Heat Capacity at Constant Volume (J/Kg/K)
G Gibbs Free Energy (J/mole)
H Specific Enthalpy (J/mole)
l length of the box (m)
m mass of molecule (kg)
P Pressure (Nm−2)
S Specific Entropy (J/mole/K)
T Temperature (K)
U Internal Energy (J/mole)
v velocity of molecule (m/s)
V Molar Volume (m3/mole)
Greek
α Linear coefficient of thermal expansion (K−1)

P Isobaric Volume Expansivity (K−1)

s Isentropic Volume Expansivity (K−1)
ε Elongational Strain

T Isothermal Compressibility (ms2/mole)

s Isentropic Compressibility (ms2/mole)
 molar density (mole/m3)
Subscripts
0 Intial state
f Final state
T isothermal
P isobaric
S isentropic