Engineering, 2013, 5, 844-849 Published Online November 2013 (http://www.scirp.org/journal/eng) http://dx.doi.org/10.4236/eng.2013.511102

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

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 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 to the misinterpretation of the laws of . This is because expansivity is usually defined at constant . Negative values for volume expansivity can be shown using Maxwell’s reciprocity relations to lead to nega- tive values for absolute for . For real systems, using Helmholz free 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 at room temperature are provided in Table 1. As can be seen from Table 1, the volume expansivity The volume expansivity of pure substances is defined as for pure substances is usually positive. In some cases it [1]: can be negative. Examples of materials given with nega- 1 V tive values for volume expansivity in the literature are   (1)  in the temperature range of O - 4 K, honey, VT P mononoclinic Sellenium, Se, Tellerium, Te, quartz , It is a parameter that is used to measure the volume faujasite, cubic Tungstate, ZrW2O8 [2] in the expansivity of pure substances and is defined at constant temperature range of 0.3 - 1050 K. ZerodourTM [3] is a pressure, P. In the field of materials science, the property glass- material that can be controlled to have zero of linear coefficient of thermal expansion is an important or slightly negative thermal coefficient of expansion and consideration in materials selection and design of prod- was developed by Schott Glass Technologies. It consists ucts. This property is used to account for the change in of a 70 - 80 wt% crystalline with high-quartz volume when the temperature of the material is changed. structure. The rest of the material is a glassy phase. The The linear coefficient of thermal expansion is defined as: negative thermal expansion coefficient of the glassy phase and the positive thermal expansion coefficient of ll l   f 0  (2) the crystalline phase are expected to cancel out each lT T lT T other leading to a zero thermal coefficient material. Ze- 00f 0 rodurTM has been used as the substrate on the For isotropic materials,  = 3. Instruments such as Hubble telescope and the Chandra X-ray telescope. A dilatometers, XRD, X-ray diffraction can be used to dense, optically transparent and zero-thermal expansion measure the thermal expansion coefficient. Typical val- material is necessary in these applications since any ues of volume expansivity for selected isotropic materials changes in dimensions as a result of the changes in the

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Table 1. Volume expansivity of selected materials at room known large distance away from a cannon, and noting temperature. the time delay between the light flash from the muzzle , Volume Expansivity and the sound of the discharge. In proposition 50, Book # Material (*10−6·K−1) II of his Principia Newton [6] theorized that the speed of 1 Aluminum 75.0 sound was related to the of the air and can be given by the reciprocal of the . He as- 2 49.8 sumed that the sound wave propagation was an isother- 3 36.0 mal process. He proposed the following expression for the speed of sound; 4 9.0 1 5 1020 36.0 c  (3)  6 51.9 T where the isothermal compressibility is given by; 7 Epoxy 165.0 1 V 8 Nylon 6,6 240  T   (4) VP T 9 Polyethylene 300.0 Newton calculated a value of 979 ft/s from this ex- 10 210 pression to interpret the artillery test results. The value

11 Partially Stabilized ZrO2 31.8 was 15% lower than the gunshot data. He attributed the 12 Soda-Lime Glass 27.0 difference between experiment and theory to existence of dust particles and moisture in the atmosphere. A century 13 Zirconium Tungstate ZrW2O8 −27.0 later Laplace [7] corrected the theory by assuming that 14 Faujasite −12.0 the sound wave was isentropic and not isothermal. He derived the expression used to this day to instruct senior 15 Water (0 - 4 K) negative level students in Gas Dynamics [5] for the speed of 16 Honey negative sound as: 1 temperature in space will make it difficult to focus the c  (5)  telescopes appropriately. Material scientists have devel- s oped ceramic materials based on zirconium phos- 1 V phate, NZP that have a near-zero-thermal-expansion co-  s   (6) VP efficient. s The occurrence of negative values of volume expan- By the demise of Napaleon the great the relation be- sivity ab initio, is a violation of second law of thermo- tween propagation of sound in gas was better understood. dyna ddUQPV d mics according to some investi- gators such as Stepanov [4]. They propose that the first 3. Theoretical Analysis law of thermodynamics be changed from Clapeyron Equation [1] can be derived to obtain the lines ddUQPV d to dd UQPV d in order to work of demarcation of phase from phase, liquid with materials with zero or negative coefficient of ther- phase from vapor phase and solid phase from vapor mal expansion. phase in a Pressure-Temperature diagram for a pure sub- In a famous problem such as the development of the stance. This can be done by considering a point in the theory of the velocity of sound such a change from de- demarcation line and a small segment in the demarcation fining a isothermal compressibility to isentropic com- line. At the point the free energy of the solid and liquid pressibility brought the experimental observations closer phases can be equated to each other at equilibrium. The to theory [5-7]. The proposal from this study is in part change during melting can be related to the en- motivated by the work of Laplace (see 2. Historical Note tropy change of melting at a certain temperature of phase below). change. Along the segment the change in free energy dGs l 2. Historical Note and dG in the solid and liquid phases at equilibrium can be equated to each other. The combined two laws of By 17th century it was realized that sound propagates thermodynamics are applied and an expression for dP/dT through air at some finite velocity. Artillery tests have can be obtained. Sometimes when ideal gas can be as- indicated that the speed of sound was approximately summed, this expression can be integrated to obtain use- 1140 ft/s. These tests were performed by standing a ful expressions that are used in undergraduate thermody-

Open Access ENG 846 K. R. SHARMA namics instruction. A similar analysis is considered here modynamics, using Helmholz free energy. At the point and segment of dddddddGTSVPTSSTVPST  (14) the phase demarcation line of solid and liquid in a Pres- sure-Temperature diagram of a pure substance; It may be deduced from Equation (14) that; s LsL G A AAA,d d (7)  V P T UTSUTSTSSSLL, SL U SL (8) (14,15) G S Applying the first and second law of thermodynamics T P to change in Helmholz free energy; G ddA UTS d  PVST d d (9) S (16) T P For reversible changes at equilibrium; ddA PVS sat  ST s ddd sat  A L  PV L sat  ST L d sat The reciprocity relation can be used to obtain the cor- responding Maxwell relation. The order of differentiation sat SL SL dVSU of the state property does not matter as long as the prop- SL  SL (10) dT PTP erty is an analytic function of the two variables. Thus, For physical changes Equation (10) can be seen to be 22GG  (17) always positive. This is because of the lowering of pres- PT TP sure as solid becomes liquid and the Combining Equation (15)-(16) with Equation (17); change is positive resulting in a net positive sign in the RHS, right hand side of Equation (10). No V S  (18) was assumed. Only the first two   TPP T was used and reversible changes were assumed in order to obtain Equation (10). Thus reports of materials with Thus Equation (18) can be derived. Combining Equa- negative thermal expansion coefficient is inconsistent tion (1) and Equation (18); with Equation (10). S V   (19) What may be happening is chemical changes. Strong P hydrogen bonded water in 0 - 4 0K shrinks upon heating T st due to chemical changes. This cannot be interpreted us- For a reversible process, the combined statement of 1 nd ing laws that are developed to describe physical changes. and 2 laws [1] can be written as: In the example of faujasite may be lattice structure dddH  TS VP (20) changes take place upon heating. What do you do? At constant temperature for reversible process for real For ideal gas, it is shown below that the volume ex- substances; pansivity can be related to the reciprocal of absolute tem- perature. Per the third law of thermodynamics the lowest HS  TV (21) achievable temperature is 0 0K. Hence volume expansiv-   PPTT  ity is always positive for physical changes. From Equa- tion (1); Combining Equation (21) and Equation (19); 1 H 1 V T 1 (22) V    VP T VT P From [8], Or 11H V S   (23)   (11)  TVTP T TPP T Equation (8) can be seen to be the case as follows. The For ideal gases Equation (23) would revert to the free energy G of pure substances are defined as: volume expansivity,  would equal the reciprocal of ab- solute temperature. This would mean that  can never be GHTS (12) negative as temperature is always positive as stated by where, H is the enthalpy (J/mole), S is the the third law of thermodynamics. So materials with (J/mole/K) negative values for  ab initio are in violation of the st nd ddGHTSHTSS d d dT (13) combined statement of the 1 and 2 laws of thermody- namics. Negative are not possible for vibra- Combining Equation (13) with the First Law of Ther- 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 1 V CCPV  CPT temperatures for there is no upper bound on their ener- s   (34) VT VP V gies. Nuclear spin orientation in a magnetic field is S needed for negative temperatures [9]. This is not appli- For substances with negative coefficient of thermal cable for engineering applications. Enthalpy variation expansion under the proposed definition of isentropic with pressure is weak and small for real substances. This volume expansivity, s does not violate the laws of ther- has to be large to obtain a negative quantity in Equation modynamics quid pro quo. (23). Considering the thermal expansion process for pure substances in general and materials with negative coeffi- 4. Proposed Isentropic Expansivity cient of thermal expansion in particular, the process is not isobaric. Pressure can be shown to be related to the Along similar lines to the improvement given by Laplace square of the velocity of the molecules. to the theory of the speed of sound as developed by Newton (as discussed in Section 2 Historical Note) an 5. Measurements of Volume Expansivity Not isentropic volume expansivity is proposed. Isobaric 1 V s   (24) The process of measurement of volume expansivity can- VT S not be isobaric in practice. When materials expand the Using the rules of partial differential for three vari- root mean square velocity of the molecules increases. For ables, any function f in variables (x,y,z) it can be seen the materials with negative coefficient of thermal coeffi- that; cient the velocity of molecules are expected to decrease. In either case, forcing such a process as isobaric is not a f ff y   (25) good representation of theory with experiments. Such x xyx z yzx  processes can even be reversible or isentropic. Experi- Thus, ments can be conducted in a reversible manner and the energy may be supplied or may be removed as the case VVVP  (26) may be. Hence it is proposed to define volume expansiv- TTPTs PTS ity at constant entropy. This can keep the quantity per se Let, from violation the laws of thermodynamics. P 6. Significance of Treatment of Materials    (27) T S with Nte Plugging Equation (27) into Equation (26); Recently, Miller et al. [10] presented a review article on 1 V materials that were observed to exhibit negative thermal s PT (28) expansion. Most materials demonstrate an expansion VT S upon heating. Few materials are known to contract. At constant pressure, These materials are expected to exhibit a NTE, negative ddH  UPV d (30) thermal expansion coefficient. These materials include complex metal , polymers and zeolites as shown in Equation (30) comes from H = U + PV the definition o Table 2. These can be used to design composited with specific enthalpy in terms of specific internal energy, U, zero coefficient of thermal expansion. When the matrix pressure and volume, P and V respectively. Equation (30) has a positive thermal expansion coefficient and the filler can be written for ideal gas as: material has a negative thermal expansion coefficient the

CCPVdd TPV (31) net expansion coefficient of the composite can be dialed in to zero. They explore supramolecular mechanisms for It can be realized from Equation (31) that; exhibition of NTE. Examples of materials where reports

V CCPV indicate NTE with the references are as follows; V  (32)  P A careful study of these materials under isentropic TPP heating is suggested. The volume expansivity then will Plugging Equation (32) into Equation (30); be within the predictions of the laws of thermodynamics. Further studies on chemical changes on heating for these V CCPV sV CPT (33) materials are suggested. Strong hydrogen bonded sys- TP S tems may also be considered as strongly interacting sys- Or, tems.

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Table 2. Materials that possess NTE. thermal compressibility at a given pressure and tempera- ture of the material. # Material Journal Reference Chemical changes have to be delineated from physical ZrW O , Zirconium 1.0 2 8 Acta Crystallography [11] changes when heating the material. Tungstate (cubic lattice)

2.0 (ZrO)2VP2O7 US Patent [12] 8. Acknowledgements HfW O , Halfnium 3.0 2 8 J. Appl. Phys. [13] Acknowledgements are extended to my undergraduate Tungstate (orthorhombic) students in advanced standing in my CHEG 2043 Ther- ZrMo O , Zirconium 4.0 2 8 Chem. Mater. [14] molybdate, (cubic) modynamics and CHEG 3053 Thermodynamics-II cour- ses at Prairie View A & M University, Prairie View, TX Silicalite-1 & 5.0 Mater. Res. Bulletin [15] Zirconium Silicalite-1 for valuable discussions on Clapeyron equation during office hours. 6.0 CuScO2 (delafossite structure) Chem. Mater. [16]

7.0 Polydiacetylene J of Polym. Sci. [17] REFERENCES 8.0 Graphite Composites Proc. of Royal Society [18] [1] J. M. Smith, H. C. Van Ness and M. M. Abbott, “Intro- duction to Chemical Engineering Thermodynamics,” 7th 7. Conclusions Edition, McGraw Hill Professional, New York, 2005, Most materials expand upon heating. Some materials [2] M. B. Jakubinek, C. A. Whitman and M. A. White, “Negative Thermal Expansion Materials: Thermal Prop- have been reported as NTE, negative thermal expansion erties and Implications for Composite Materials,” Journal coefficient materials. Very little thermodynamic analysis of Thermal Analysis and Calorimetry, Vol. 99, No. 1, pp. has been done on these materials. In this study, Helmholz 165-172. free energy change during melting or solidication was [3] D. R. Askleland and P. P. Phule, “The Science and Engi- undertaken. Equation (10) was derived for expansivity at neering of Materials,” PWS-Kent, Boston. equilibrium. [4] L. A. Stepanov, “Thermodynamics of Substances with For physical changes, Equation (10) can be seen to be Negative Thermal Expansion Coefficient,” Computer always positive. This is because the lowering of pressure Modelling & New Technologies, Vol. 4, No. 2, 2000, pp. as solid becomes liquid and the internal energy change is 72-74. positive resulting in a net positive sign in the RHS, right [5] J. D. Anderson, “Modern Compressible Flow with His- torical Perspective,” Third Edition, McGraw Hill Profes- hand side of Equation (10). No ideal gas law was as- sional, New York, 2003. sumed. Only the first two laws of thermodynamics were [6] Sir Isaac Newton, “Philosophiae Naturalis Principia used and reversible changes were assumed in order to Mathematica,” 1687. obtain Equation (10). Thus reports of materials with ne- [7] P. S. M. de Laplace, “Sur la Vitesse du son dans L’aire et gative thermal expansion coefficient are inconsistent Dan L’eau,” Annales de Chimie et de Physique,1816. with Equation (10). [8] K. R. Sharma, “Polymer Thermodynamics: Blends, Co- For ideal gases, Equation (23) would revert to the polymers and Reversible Copolymerization,” CRC Press, volume expansivity,  would equal the reciprocal of ab- Taylor Francis Group, Bacon Raton, 2010. solute temperature. This would mean that  can never be [9] C. Kittel and H. Kroemer, “Thermal Physics,” 2nd Edi- negative as temperature is always positive as stated by tion, Freeman & Co., New York, 1980. the third law of thermodynamics. So materials are with [10] W. Miller, C. W. Smith, D. S. Mackenzie and K. E. Ev- negative values for  ab initio are in violation of the ans, “Negative Thermal Expansion: A Review,” Journal combined statement of the 1st and 2nd laws of thermody- of Material Science, Vol. 44, 2009, pp. 5441-5451. namics. http://dx.doi.org/10.1007/s10853-009-3692-4 Along similar lines to the improvement given by [11] J. S. O. Evans, W. I. F. Davis and A. W. Sleight, “Struc- Laplace to the theory of the speed of sound as developed tural Investigation of the Negative-Thermal-Expansion Material ZrW2O8,” Acta Crystallographica B, Vol. 55, by Newton (as discussed in Section 2 Historical Note) an No. 3, 1999, pp. 333-340. isentropic volume expansivity is proposed by Equation [12] A. W. Sleight, “Negative Thermal Expansion Material,” (24). This can be calculated using Equation (28) from US Patent 5,322,559, 1994. isobaric expansivity, isothermal compressibility and a [13] J. D. Jorgensen, Z. Hu and S. Short, “Pressure-Induced parameter  that is a measure of isentropic change of Cubic to Orthorhombic Phase Transformation in the pressure with temperature. Equation (34) can be used to Negative Thermal Expansion Material HfW2O8,” Journal obtain the isentropic expansivity in terms of capaci- of Applied Physics, Vol. 89, No. 6, 2001, pp. 3184-3188. ties at constant volume and constant pressure and iso- [14] C. Lind, A. P. Wilkinson, Z. Hu, S. Short and J. D.

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Jorgensen, “Synthesis and Properties of the Negative in CuScO2”, Chemistry of Materials, Vol. 14, No. 6, 2002, Thernal Expansion Material Cubic ZrMo2O8,” Chemistry pp. 2602-2606. http://dx.doi.org/10.1021/cm011633v of Materials, Vol. 10, No. 9, 1998, pp. 2335-2337. [17] R. H. Baughman and E. A. Turi, “Negative Thermal Ex- http://dx.doi.org/10.1021/cm980438m pansion of a Polydiacetyelene Single Crystal”, Journal of [15] D. S. Bhange and Veda, “Negative Thermal Expansion in Polymer Science Part B: Polymer Physics, Vol. 11, 1973, Silicalite-1 and Zirconium Silicalite-1 having MFI Struc- pp. 2453-2466. ture,” Materials Research Bulletin, Vol. 41, No. 7, pp. [18] P. Rupnowski, M. Gentz, J. K. Sutter and A. M. Kumosa, 1392-1402. “An Evaluation of Elastic Properties and Coefficients of http://dx.doi.org/10.1016/j.materresbull.2005.12.002 Thermal Expansion of Graphite from Macro- [16] J. Li, A. Yokochi, T. G. Amos and A. W. Sleight, “Strong scopicc Composite Input Data,” Proceedings of Royal Negative Thermal Expansion along the O-Cu-O Linkage Society A, Vol. 461, 2005, pp. 347-369.

−1 Nomenclature s Isentropic Volume Expansivity (K ) ε Elongational Strain A Helmholz free energy (J/mole)  Isothermal Compressibility (ms2/mole) c Speed of Sound (m/s) T  Isentropic Compressibility (ms2/mole) C at Constant Pressure (J/kg/K) s P  molar (mole/m3) Cv Heat Capacity at Constant Volume (J/Kg/K) G (J/mole) Subscripts H Specific Enthalpy (J/mole) l length of the box (m) 0 Intial state m mass of molecule (kg) f Final state P Pressure (Nm−2) T isothermal S Specific Entropy (J/mole/K) P isobaric T Temperature (K) S isentropic 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) −1 P Isobaric Volume Expansivity (K )

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