On Thermodynamic Analysis of Substances with Negative Coefficient of Thermal Expansion

doi:10.4236/eng.2013.511102

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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

Email: [email protected]

Received November 27, 2012; revised January 8, 2013; accepted January 15, 2013

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)

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:



ll l

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;





 (3)

where the isothermal compressibility is given by;







 





(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:





 (5)







 





(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

LsL

AAA (7)

SS LLSLSL

UTSUTSTSU (8)

Applying the first and second law of thermodynamics

to change in Helmholz free energy;

dddd d

UTS PVST  (9)

For reversible changes at equilibrium;

dd dddd

Ssats satLLsatLsat

PV STAPV ST

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);

VVT













From Maxwell Relations [8],



 



 



 

(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;

























(14,15)











 (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,

PT TP



G





(17)

Combining Equation (15)-(16) with Equation (17);



 



 



 

(18)

Thus Equation (18) can be derived. Combining Equa-

tion (1) and Equation (18);













(19)

For a reversible process, the combined statement of 1st

and 2nd laws [1] can be written as:

ddd

TS VP



 (20)

At constant temperature for reversible process for real

substances;



 



 



 

V

(21)

Combining Equation (21) and Equation (19);







 





 (22)

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-

Open Access ENG

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.













(24)

Using the rules of partial differential for three vari-

ables, any function f in variables (x,y,z) it can be seen

that;

ffy

xyx





 





 



 

 (25)

Thus,

VVVP

TT PT













(26)

Let,













(27)

Plugging Equation (27) into Equation (26);

VT T















 (28)

At constant pressure,

dd d

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:



CC TPV

(31)

It can be realized from Equation (31) that;

















 (32)

Plugging Equation (32) into Equation (30);













 





 (33)

Or,





1PV

CC C

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.

Open Access ENG

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.

REFERENCES

[1] J. M. Smith, H. C. Van Ness and M. M. Abbott, “Intro-

duction to Chemical Engineering Thermodynamics,” 7th

Edition, McGraw Hill Professional, New York, 2005,

[2] M. B. Jakubinek, C. A. Whitman and M. A. White,

“Negative Thermal Expansion Materials: Thermal Prop-

erties and Implications for Composite Materials,” Journal

of Thermal Analysis and Calorimetry, Vol. 99, No. 1, pp.

165-172.

[3] D. R. Askleland and P. P. Phule, “The Science and Engi-

neering of Materials,” PWS-Kent, Boston.

[4] L. A. Stepanov, “Thermodynamics of Substances with

Negative Thermal Expansion Coefficient,” Computer

Modelling & New Technologies, Vol. 4, No. 2, 2000, pp.

72-74.

[5] J. D. Anderson, “Modern Compressible Flow with His-

torical Perspective,” Third Edition, McGraw Hill Profes-

sional, New York, 2003.

[6] Sir Isaac Newton, “Philosophiae Naturalis Principia

Mathematica,” 1687.

[7] P. S. M. de Laplace, “Sur la Vitesse du son dans L’aire et

Dan L’eau,” Annales de Chimie et de Physique,1816.

[8] K. R. Sharma, “Polymer Thermodynamics: Blends, Co-

polymers and Reversible Copolymerization,” CRC Press,

Taylor Francis Group, Bacon Raton, 2010.

[9] C. Kittel and H. Kroemer, “Thermal Physics,” 2nd Edi-

tion, Freeman & Co., New York, 1980.

[10] W. Miller, C. W. Smith, D. S. Mackenzie and K. E. Ev-

ans, “Negative Thermal Expansion: A Review,” Journal

of Material Science, Vol. 44, 2009, pp. 5441-5451.

http://dx.doi.org/10.1007/s10853-009-3692-4

[11] J. S. O. Evans, W. I. F. Davis and A. W. Sleight, “Struc-

tural Investigation of the Negative-Thermal-Expansion

Material ZrW2O8,” Acta Crystallographica B, Vol. 55,

No. 3, 1999, pp. 333-340.

[12] A. W. Sleight, “Negative Thermal Expansion Material,”

US Patent 5,322,559, 1994.

[13] J. D. Jorgensen, Z. Hu and S. Short, “Pressure-Induced

Cubic to Orthorhombic Phase Transformation in the

Negative Thermal Expansion Material HfW2O8,” Journal

of Applied Physics, Vol. 89, No. 6, 2001, pp. 3184-3188.

[14] C. Lind, A. P. Wilkinson, Z. Hu, S. Short and J. D.

Open Access ENG

K. R. SHARMA

Open Access ENG

849

Jorgensen, “Synthesis and Properties of the Negative

Thernal Expansion Material Cubic ZrMo2O8,” Chemistry

of Materials, Vol. 10, No. 9, 1998, pp. 2335-2337.

http://dx.doi.org/10.1021/cm980438m

[15] D. S. Bhange and Veda, “Negative Thermal Expansion in

Silicalite-1 and Zirconium Silicalite-1 having MFI Struc-

ture,” Materials Research Bulletin, Vol. 41, No. 7, pp.

1392-1402.

http://dx.doi.org/10.1016/j.materresbull.2005.12.002

[16] J. Li, A. Yokochi, T. G. Amos and A. W. Sleight, “Strong

Negative Thermal Expansion along the O-Cu-O Linkage

in CuScO2”, Chemistry of Materials, Vol. 14, No. 6, 2002,

pp. 2602-2606. http://dx.doi.org/10.1021/cm011633v

[17] R. H. Baughman and E. A. Turi, “Negative Thermal Ex-

pansion of a Polydiacetyelene Single Crystal”, Journal of

Polymer Science Part B: Polymer Physics, Vol. 11, 1973,

pp. 2453-2466.

[18] P. Rupnowski, M. Gentz, J. K. Sutter and A. M. Kumosa,

“An Evaluation of Elastic Properties and Coefficients of

Thermal Expansion of Graphite Fibers from Macro-

scopicc Composite Input Data,” Proceedings of Royal

Society A, Vol. 461, 2005, pp. 347-369.

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