|
Full text download---------
Jozsef Garai
1. Introduction
Melting of solids is probably one of the oldest known physical phenomena. However, the question of how and why a crystalline solid melts, and what is the melt itself, still remains something of an enigma. Reaching a critical level of atomic vibration [1-2], the vanishing of the shear modulus of the crystal [3], or developing a critical level of defects [4] have all been proposed as the primary criteria for the occurrence of melting. Each of these is an important feature of the formation of melt; however, none of them give a comprehensive description of melting.
The criteria determining the primarily phases such as solid, liquid, or gas should lay at the most fundamental atomic level. Contrary to liquids, the phases of solid and gas are reasonably well defined at the atomic level. Matter is in the gas phase if the thermal energy of the atoms is higher then the disassociation energy, and sufficient space for the free movements of the atoms is available. Solid and liquid phases are distinct from gas because the energy of the atoms in these phases is lower then the disassociation energy, resulting in the existence of interatomic potential energy (IPE). In solids, atoms have a well defined position and energy is required to change the atomic position; in liquids, atoms do not have position stability and atoms are in a metastable equilibrium. Solids transforms to liquid because the atoms lose their position stability. In order to understand the solid-liquid phase transformation this position instability developed at the formation of melt will be investigated.
2. Developing atomic position instability in the melt
Taking the simplest system, a monoatomic pair, it can be seen that as long as IPE exists between the atoms their position is defined [Fig. 1]. Atoms lose their position stability only when IPE vanishes between them. Based on the nature of the interatomic forces there is no reason to assume that IPE is discontinuous and that this discontinuity could lead to position instability. In this monoatomic pair system liquid phase can not be formed between solid and gas phase since IPE can either exist or not. The same is true for a one dimensional solid, which would transform from solid to gas phase directly. In order to form a gas phase the atoms have to be displaced to a distance where their IPE is practically vanished, otherwise gas phase can not be formed. The distance of separation required for the formation of gas can not be reached inside a solid or a liquid where atoms are quite tightly packed. In solids and liquids the IPE of the neighboring atoms will always overlap. The overlapping of IPE causes position instability above a certain energy level. In a crystalline solid where the same atomic arrangement is repeated these instable regions will form an interconnected metastable atomic network. In this network atoms can be repositioned vibrationally [5]. It is suggested that solids transforms to liquid when the thermal energy of the atoms overcomes the potential barrier of metastability.
In a monoatomic system IPE is the function of the separation of atoms; greater displacement always requires more energy. Therefore repositioning requiring the smallest displacements of the neighboring atoms will be energetically favored. Vibration can reposition atoms, atomic lines and atomic sheets. The smallest displacement per atom is required for the reposition of atomic sheets [Fig. 2]. This also means that the vibrational repositioning of atomic sheets requires the least energy; therefore, this is the most probable way to reposition atoms contrarily to atomic lines or individual atoms. The so-called tunnel model proposed by Barker [6] therefore energetically unfavorable. Based on energy considerations it is proposed that solid is destabilized between atomic plains when melt is formed. This model is consistent with the detected layered ordering of fluids at solid interface [7].
The proposed sheet like atomic structure of the melt is consistent with the observed one surface of the liquid.
Atomic sheets situating on the surface of the solid are in contact with the neighboring atoms only on their inner side. The reposition of an inner atomic sheet requires the displacement of both of the neighboring atomic sheets contrarily to the sheets on the surface. Surface atomic sheets; therefore, will be destabilized at lower energy level. The destabilization of the first sheet on the surface opens the surface for the next atomic sheet and a destabilizing domino effect will start. The first melt will be formed on the surface of the crystal and melting will progressively develop towards to the center of the crystal. Experimental results are consistent with this prediction, since the melting preferentially starts at the surface. If the surface is not exposed and instead coated with metal layer then the crystal would not melt at its melting point but rather at higher temperatures [8].
If this proposed vibrating atomic sheet structure of the melts is correct then these atomic sheets should diffract x-rays and the diffraction pattern should be consistent with the interlayer separation of the sheets. These predictions will be considered in detail.
3. X-ray diffractions of liquids
The diffraction halo of liquids is well known since the early decades of the last century [9]. It is believed that the diffraction is the result of a short range atomic order surrounding any average atom [10]. The local point symmetry of this short range order is not resolved even for monoatomic systems. Recent investigation, scattering of totally internally reflected x-rays, indicated a five-fold local symmetry in liquid lead adjacent to a silicon wall [11].
Based on the analyses of previous works few concerns has been raised against the short range atomic orders of liquids. Eventhough the observed diffraction patterns can be explained fully by a certain short range atomic order, this does not mean that this is the right atomic order. Even in crystals, with a very large number of independent reflections, it is always possible to explain the same kind of scattering with a different structure. The short range atomic orders were detected by the observed diffraction peaks for alcohol as a function of the number of [12] and by the size of the Benzene ring [13]. These short range atomic orders were detected by single diffraction peaks which do not necessarily mean that all of the peak positions are governed by the same physical phenomenon. The detected atomic distributions are not always consistent with thermal expansion. In the case of lithium even the direction of the change is opposite since the number of nearest neighbor increases rather then decreases at melting [14]. These concerns indicate that the short range atomic order might not be the right explanation for all of the diffraction peaks observed in liquids.
In order to gain a deeper insight into the atomic ordering of liquids, the x-ray diffraction patterns of melts formed from solids with highly symmetrical packing arrangements (HSPA) has been analyzed. These crystal structures have been chosen because they have the highest symmetry and thus the simplest atomic arrangements. From the previous investigations [14-22] of melted Li, Na, Mg, Al, Ar, K, Ca, Sc, Ti, V, Cr, Fe Co, Ni, Cu, Zn, Rb, Sr, Zr, Pd, Ag, Cd, Xe, Cs, Ba, La, Ce, Eu, Gd, Tb, Yb, Pt, Au, Tl, Pb, the d-spacing of each peak has been determined. The atomic diameters of the investigated elements strongly correlate to the determined d-spacing of the different diffraction peaks regardless of the original crystal structure [Fig. 3]. This correlation indicates that the d-spacing represents the same atomic ordering in these liquids; the structure of the liquid, therefore, is the same for all of these elements. The ratios of the d-spacing show that the second, third, forth, and fifth peaks are most likely the diffractions of the same plane and that this plane contributes to the diffraction of the first peak as well [Tab. 1]. The ratios of interlayer separation and atomic diameter are different for the first and for the rest of the peaks, averaging 0.825 and 0.916 respectively. These different values along with the Full Length Half Maximum of the peaks indicate that the diffraction peaks represent two different d-spacings [Fig. 4].
4. Vibrating double atomic sheet model
The distance between the densest atomic sheets in fcc and hcp is . This d-spacing of the atomic sheets seems to correspond to the value of 0.825d measured in liquids. If the atomic layers are separated by 0.825d then the vibrational displacement of the atomic sheets is not possible. This constraint indicates that liquids should contain atomic sheets vibrating jointly.
In HSPA the minimum interlayer separation allowing the vibrational reposition of two closed packed atomic sheets is . This separation allows the reposition atoms from one tetrahedron site into another. In an atomic sheet the available tetrahedron sites are not lined up indicating a zigzag movement of the contacting sheets [Fig. 5]. The directional change of the vibration would require a sizeable energy which is not consistent with observations. No extra energy is needed when the atomic sheets repositioned along a straight line. This would require a slightly higher displacement between of the atomic sheets then the 0.866d value. The minimum separation for hcp is , while for fcc is . These values are in reasonable agreement with the measured interlayer separation of 0.916d. The second d-spacing detected in liquids seems to be consistent with the distance separating the vibrating atomic sheets. The proposed structure of liquids is shown in Fig. 6. The d-spacing measured in liquids is closer to the minimum displacement required for hcp vibration indicating that the liquid should have an ABABA... layer structure; however, the possibility of an ABCABCA... layering can not be excluded.
If this twin structure of the vibrating sheets is correct, then the volume increase up to the melting should be the same as the volume increase calculated from the measured interlayer separations of the atomic sheets. The volume increase might also help to determine the layer structure of HSPA liquids.
5. Volume increases up to melting
The volume increase of the elements, in the temperature range of absolute zero and 300K, was calculated from thermal expansion data [23]. Density values measured at 300K and at the melting temperature were used to determine the thermal expansion for this temperature range [24-25]. The total thermal expansion of the liquid from absolute zero to the melting point is given in Table 2. The average volume increase of the bcc, fcc, hcp original crystal structures are 0.096, 0.140, and 0.095 respectively. The weighted average of the three different original structures is 0.113.
Supposing that the measured 0.825d separation of the closest packing sheets expresses the expansion in three dimensions then the volume increase can be calculated as:
Using the average of the ratios of interlayer separation and atomic diameter the volume increase up to the melting is:
This value agrees reasonably well with the measured volume increases at the melting temperatures. The calculated expected volume increase for melts formed from original hcp structure is . The minimum required displacement of was used for the calculation. The expected volume increase for fcc is using the value of . These calculated values are consistent with the measured volume increase of hcp and fcc melts.
The packing density of the bcc structure is 0.68 while 0.74 for closed packing arrangement. The transformation of a bcc structure to a closed packing arrangement would result an 8.1% volume decrease of the original bcc structure. The uniform liquid structure of the melts formed form HSPA indicates that the original bcc structure should transform to a closed packing atomic sheet arrangement. This transformation should decrease the volume of the original bcc structure by 8.1%. The measured volume increases are higher for melts formed form fcc structure in comparison to melts formed from bcc structures. This is consistent with theoretical predictions however the volume difference is smaller then the predicted value.
6. Conclusion
Based on theoretical considerations it has been suggested that metastable double atomic sheets should be developed in solids when melting occurs. The diffraction patterns of liquids formed from elements with highly symmetrical packing arrangements are consistent with this prediction. The volume increase from absolute zero to the melting point calculated from the measured displacements of the atomic sheets is consistent with the measured values. The volume increase of melts formed from bcc structure is smaller then the volume increase of melts formed from fcc supporting that bcc structure transformed to closed packing structure. The difference between the volume increases of these two structures does not reach the theoretical prediction. The accuracy of the available data does not allow determining the precise layered structure of the vibrating sheets.
Investigations were limited to melts formed form HSPA; however, it can be predicted that the metastable equilibrium of vibrating atomic sheets should be universally applicable to any melt formed from crystalline solids. The structure of the vibrating atomic layers in melts formed from lower symmetry crystals is probably more complicated and requires further investigations.
[1] F. A. Lindemann, Phys. Z. 11 (1910) 609.
[2] J.J. Gilvarry, Phys. Rev. 102 (1956) 308.
[3] M.J. Born, J Chem. Phys. 7 (1939) 591.
[4] R.M. Cotterill, J. Cryst. Growth 48 (1980) 582.
[5] The word reposition is used when atoms change their position from one potential well to another while the word displacement is used for atomic position changes in the same potential well.
[6] J.A. Barker, Lattice Theories of the Liquid State, Pergamon Press, Ltd., Oxford (1963)
[7] S.E. Donnelly et. al., Science 296 (2002) 507.
[8] R. W. Cahn, Nature 323 (1986) 668
[9] P. Debye and P. Scherrer, Gottingen Nachrichten 16 (1916)
[10] G.W. Stewart, Phys. Rev. 37 (1931) 9 .
[11] H. Reichert et al., Nature 408 (2000) 839.
[12] G.W. Stewart and R.M. Morrow, Phys. Rev. 30 (1927) 232.
[13] G.W. Stewart, Phys. Rev. 33 (1929) 889.
[14] C. Gamertsfelder, J. Chem. Phys. 9 (1941) 450.
[15] R.W. Harris and G. Clayton, Phys. Rev. 153 (1967) 229.
[16] Y. Washeda, The Structure of Non-Crystalline Materials (McGraw-Hill Inc., 1980) p. 54.
[17] Y. Waseda and S. Tamaki, Phil. Mag. 32 (1975) 273.
[18] L.P. Tarasov and B.E. Warren, J. Chem. Phys. 4 (1936) 236.
[19] C.D. Thomas and N.S. Gingrich, J. Chem. Phys. 6 (1938) 411.
[20] A. Eisenstein and N.S. Gingrich, Phys. Rev 62 (1942) 264.
[21] R.R. Fessler, R. Kaplow, and B.L. Averbach, Phys. Rev. 150 (1966) 34.
[22] N.S. Gingrich, Rev. Mod. Phys. 15 (1943) 90.
[23] G.S. Grigoriev, and E.Z. Meilikhov, Handbook of Physical quantities (CRC Press, Inc., 1997)
[24] D.R. Lide, Editor, CRC Handbook of Chemistry and Physics (CRS Press, 1999)
[25] J. Emsley, The Elements (Oxford University Press, New York, 1998)
Home Page
---------------------Research Statement
---------------------Liquid Structure Melting
|