Why ice floats on water

Solid State Sciences 5 (2003) 683–693 www.elsevier.com/locate/ssscie

Why ice floats on water Sten Andersson a,∗ , B.W. Ninham b,c a Sandviks Forskningsanstalt, S Långgatan 27, 380 74 Löttorp, Sweden b Department of Chemistry and CSGI, University of Florence, via della Lastruccia 3, 50019 Sesto Fiorentino (Firenze), Italy c Department of Applied Mathematics RSPhysSE, Australian National University, Canberra A.C.T. 0200, Australia

Received 29 October 2002; received in revised form 26 February 2003; accepted 28 February 2003

Abstract Why ice floats on water is an unresolved question that underlines the special properties of water. We argue that it is resolved by taking a different approach to structure than is usual, the “exponential scale” description. The exponential scale is placed on a firm foundation based on underlying quantum mechanics. The approach includes and is more general than conventional approaches to molecular and chemical interactions, based on separate and different kinds of “bonds”. These are based on perturbation theories applied to few-atom systems. The present approach to the many-body problem is in the same spirit as, but more general than Lifshitz theory of intermolecular interactions.  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Ice; Water structure; Quantum tunneling; Hydrogen bonding; Chemical bonds

1. Introduction While the modern age rejoices in the discovery of the big bang, and eagerly awaits the speculations of the biologists on the origins of life, some more prosaic matters have defied explanation. Among these are two apparently simple phenomena: why bubble coalescence in water is inhibited above 0.1 M salt [1], and why ice floats on water. These are not so mundane as they seem. For without the latter the astronomers and the biologists would have remained in their past and future state, as archaebacteria dependent on hydrogen sulphide, in a hot watery brimstone cave. We here discuss and resolve the problem of why ice floats on water. It is clear that ice and water are quantum gels. It is equally clear that the problem cannot be resolved in terms of the usual statistical mechanics, and two- and three-body molecular interaction potentials derived from quantum mechanical perturbation theory. Otherwise the problem would have been solved long since. It is an inherently many-body problem. * Corresponding author. Phone: +46-485-2643.

E-mail address: [email protected] (B.W. Ninham).

We first make some preliminary observations. The only theory of melting is embodied in the Lindeman criterion that dates from 1914. It asserts that a solid melts when the kinetic motions of molecule in the self consistent containing cell of its surrounding neighbors are such that the mean square displacement of the molecule reaches about 1/10 of its lattice spacing. That is so and has been established for molecular potentials ranging from short range attractive van der Waals plus hard core potentials, all the way to an electron gas immersed in a neutralizing continuum of positive charge. The Coulomb potential here is repulsive, non-additive, and long ranged. The reason for this apparent universality seems clear. For example imagine the case of a (spherical) molecule with its electron cloud spread out in a Gaussian of width the size of the molecule. Then by the Rayleigh criterion for resolvability [9, p. 157] the two molecules are indistinguishable when the profiles overlap around 20%. At that point given by the Lindeman criterion it is impossible to determine to which cell a molecule belongs, the system is no longer ordered, and diffusion creating holes becomes possible. Normally because of these vibrations about the ordered state, we expect that the liquid will be less dense than the solid. But the physically sensible Lindeman criterion in fact says nothing about the relative density of the resulting liquid.

1293-2558/03/$ – see front matter  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. doi:10.1016/S1293-2558(03)00092-X

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Two things are clear: I Ice (tridymite) is lighter than water. II Cristobalite (the stable form of SiO2 at that temperature, 1100 centigrade) is lighter than its melt. (We remark that were it not so the earth’s crust would be different, and our existence would again be in doubt.) To understand this most unusual behavior we take the cubic cristobalite structure and the hexagonal tridymite type structure to be similar in properties, as we deal with typically intrinsic properties. (See however Appendix A.5.) Diamond has the cristobalite structure with the sp3 electrons in the oxygen positions. Tridymite is another form of SiO2 , quartz is a third. Diamond also takes the tridymite structure—lonsdalite. The density of liquid diamond is, we believe not known. Ice and cristobalite are chemically totally different. The physical behavior must be explainable by special features in the intrinsic molecular structure—shape—that they have in common. This intrinsic molecular shape which reflects the many-body interactions is all that they can have in common. With this as background we ask what is the driving force behind the phenomenon? What happens before melting occurs? What is the real nature of the structure? We look for some important clues on these unusual structures that lead to reduced density for the solid. Several hints are these: The diamond structure has poor packing. But it is very stable (hardness) and is the best heat conductor of all, even better than copper. Ice is a normal heat conductor. The conductivity reduces suddenly by a factor of four on melting. We know nothing about heat conduction in cristobalite. But there is a hint that bears on our argument—gemologists distinguish glass imitations from crystalline gems by licking— glass feels warmer due to its smaller thermal conductivity [2].

Fig. 1. View of oxygen–silicon–oxygen arrangement in cristobalite SiO2 . Or carbon–sp3 electrons–carbon in diamond. Carbon or silicon are the bigger spheres. Coordinate units, (x, y, z) are arbitrary and can be scaled to actual atomic dimensions.

Fig. 2. Vibration mode solid SiO2 .

Eq. (2.1) gives Fig. 1: e−[(x−2)

2 +(y−2)2 +(z−2)2 ]

+ e−[(x+2)

2 +(y−6)2 +(z−6)2 ]

In Fig. 1 is depicted the structure of cristobalite (SiO2 ), and also carbon diamond. In all the descriptions of structures and characterization of their motions below, we use the GD or exponential scale mathematics [3–7]. This representation gives an excellent description of molecular crystals and of biomacromolecules. Its formal link and connection to the more conventional quantum mechanical description is not trivial and is developed explicitly in Appendix A.

2 +(y+2)2 +(z+2)2 ]

+ e−[(x−6)

2 +(y+2)2 +(z−6)2 ]

+ e−[(x−6) +(y−6) +(z+2) ] 1
2 2 2 2 2 2 + e−[(x) +(y) +(z) ] + e−[(x−4) +(y−4) +(z) ] 4 2

2

+ e−[(x−4) 2. Intrinsic structures

+ e−[(x+2) 2

2 +(y)2 +(z−4)2 ]



1 . (2.1) 10 We postulate a vibration mode for the cristobalite SiO2 lattice depicted in Fig. 2 that strains the lattice. It is represented explicitly by Eq. (2.2). Due to the directed momentum, the lattice is expanded. Such collective modes would also explain how are phonons conducted in diamond. + e−[(x)

2 +(y−4)2 +(z−4)2 ]

e−[(x+4) +(y) ] + e−[(x−4) +(y) 1 1 2 2 + e−[(x) +(y) ] = . 4 10 2

2

2

=

2]

(2.2)

S. Andersson, B.W. Ninham / Solid State Sciences 5 (2003) 683–693

685

(a)

Fig. 3. Vibration model in liquid (rocking or semi-rotation) SiO2 . Ionic attraction is high enough to keep a dynamic structure. Melting is a slow procedure in SiO2 , extending over a wide temperature region.

With this expansion of the lattice, rocking or semi-rotation modes, previously forbidden, become allowed. Some of the direct momentum that expands the lattice is now dissipated into these previously forbidden modes. It can be taken up by the new modes as more freedom for their existence is created. We then allow these new vibration modes in solid cristobalite SiO2 depicted in Fig. 3. Directed momentum between atoms is lost, the crystals melt, and density is increased with the new momentum.

(b) Fig. 4. (a) Crystallographic position of hydrogen, 50% occupancy. (b) Crystallographic position of hydrogen, 50% occupancy.

The hindered motion can only take place via the quantum mechanical effect. For a good description see [8]. We exhibit this effect using our GD mathematics in Figs. 5 and 6. e−[(x−2) +(y−2) +(z−2) ] + e−[(x−3) 3 2 3 2 3 2 1 + e−[(x− 4 ) +(y− 4 ) +(z− 4 ) ] 2 2

2

2

3. The ice lattice

+ e−[(x− 4 )

With this established we consider next the ice lattice. Below is the crystallographic picture of ice with hydrogen ‘perfectly’ disordered, exactly 50%, on the two positions as shown in Fig. 4a and b. From left to right in Fig. 4a there are oxygen, hydrogen and a lone pair of electrons attached to its oxygen. And in Fig. 4b there is shown the inversion. If we placed the hydrogen in the middle and take away the lone pair we would have the cristobalite structure. These representations are combined in Eq. (3.1):

+ e−[(x−4)

17 2 17 2 2 +(y− 17 4 ) +(z− 4 ) ]

e−[(x+4) +(y) ] + e−[(x−4) +(y) ] 1 1 1 2 2 2 2 + e−[(x±2) +(y) ] + e−[(x±1) +(y) ] = . (3.1) 4 3 10 We would infer from this that the ice structure is crystallographically highly disordered. We say ice is not disordered. To overcome this apparent contradiction, we assume again a directed vibrational mode as in cristobalite in the crystal of ice. But as there are electrons involved we need to quantize the entire many-body system to get a correct picture. To do this we recall first the usual description of the electron structure of ammonia. Its vibrational mode is responsible for the umbrella like inversion of the molecule. 2

2

2 +(y−3)2 +(z−3)2 ]

2

2

+ e−[(x)

2 +(y−4)2 +(z)2 ]

2 +(y−4)2 +(z−4)2 ]

+ e−[(x−4) 

=

1 . 10

2 +(y)2 +(z−4)2 ]

(3.2)

We now assume an umbrella like motion for the lattice of ice in analogy with the ammonia molecule. Again as the motion involves electrons we describe the ice structure with hydrogen and lone pair motions as a 3D quantum mechanical effect. Eqs. (3.3) and (3.4) for the extreme parts of ice are as in Fig. 7. e−[(x+4) +(y) +(z) ] + e−[(x−4) +(y) 1 2 2 2 + e−[(x−2) +(y) +(z) ] 4 1 1 2 2 2 + e−[(x+1) +(y) +(z) ] = , 3 10

2 +(z)2 ]

e−[(x+4) +(y) +(z) ] + e−[(x−4) +(y) 1 2 2 2 + e−[(x+2) +(y) +(z) ] 4 1 1 2 2 2 + e−[(x−1) +(y) +(z) ] = . 3 10

2 +(z)2 ]

2

2

2

2

2

2

2

2

(3.3)

(3.4)

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Fig. 5. Two NH3 molecules with attached lone pairs, in opposite orientations.

Fig. 6. The ammonia inversion. The umbrella like motion in the tunneling structure of the ammonia molecule. Eq. (3.2) characterizes this tunneling phenomenon.

Eqs. (3.3) and (3.4) are now put together in Eq. (3.5) to give the tunneling shape between two oxygens in ice as shown in Fig. 8.

Fig. 7. Two parts of the ice structure, two oxygens with attached lone pairs, in opposite orientations. The smaller spheres are the hydrogens. This is described as the crystallographic structure with 50% occupancy of hydrogens.

e−[(x+4) +(y) +(z) ] + e−[(x−4) +(y) +(z) ]  1 1 −[(x−2)2+(y)2 +(z)2 ] 1 −[(x+1)2+(y)2 +(z)2 ] e + + e 2 4 3 2

2

+

2

2

2

2

1 −[(x+2)2+(y)2 +(z)2 ] e 4

 1 1 −[(x−1)2+(y)2 +(z)2 ] = . + e 3 10

(3.5)

This umbrella like motion gives a vibrational mode like in crystalline cristobalite SiO2 . That again strains the lattice around and gives rise to the lower density of ice. At melting vibration modes like rocking, rotation and tumbling reduce the strain in the lattice, and density increases.

Fig. 8. The inversion-like motion in the tunneling structure of ice.

S. Andersson, B.W. Ninham / Solid State Sciences 5 (2003) 683–693

4. Conclusion The representation of Fig. 8 shows a “merging” of the hydrogen with the lone pairs, and so allows the onset of rocking-rotation modes (rather reminiscent of a skipping rope). The existence of such modes that accommodate the density difference between ice and water would have another consequence. For as temperature increases, some of the “skipping ropes”, progressively more excited, and at random, introduce more disorder. They interfere with each other, the merged “bonds” break, and an additional rotation of coupled pairs of oxygen molecules emerges a new mode. The merging of hydrogens with lone pairs are given a quantum mechanical basis in Appendix A. At first sight, although this description resolves the apparent logical contradiction above, that ice would be disordered, it conflicts with the usual picture we have involving lone pairs and isolated hydrogens. But that picture derives from quantum mechanical perturbation theory applied to only two molecules. The distinction made between different kinds of “bonds”, ionic, van der Waals dispersion, hydrogen bonding, pi electron bonds and so on is defective as shown in Appendix A. Water and ice are bulk quantum many-body systems which the standard approach does not accommodate.

Acknowledgement The authors are grateful to Mathis Bostrom for a critical reading of the manuscript and for assistance in preparation and submission.

Appendix A. The physical meaning of exponential scale: energy and shape: general comments The equations used in the text to visualize structures are an application of a more general “exponential scale” or “multiplicative geometry” developed by one of us and colleagues [3–6]. Many examples illustrated therein show that this mathematical characterization provides an excellent representation of even extremely complex molecular structures. Even electron densities deduced from X-ray analysis seem to be matched precisely with this simple mathematics [4, pp. 81–83]. But the necessary connection between the underlying quantum mechanical basis of structure, the physics and energetics of this approach, has not previously been spelt out. The connection is not trivial. Here we sketch how that connection is made. In at least in some special cases like the representations used here, and in, e.g., [9], the arguments to establish linkage to quantum mechanics and quantum electrodynamics will be outlined. We claim that the shapes generated for different structures do actually accommodate the molecular forces implicitly.

687

This so whether or not one wishes to characterize the forces as ionic, dispersion, dipolar, covalent or H= bonding, pi electron forces or any other divisions into categories of forces with which we are familiar. The formalism needed to explain and make the connections is in part contained in [9]. This work extended the general theory of molecular interactions to include molecular size. This book [9] is long out of print so that it will be necessary to rehearse the general formalism in some detail below. Before so doing we need to digress to make some remarks on recent developments in the general theory of molecular forces and establish a corollary. A.1. The correspondence of quantum electrodynamic theories of forces and semi-classical theory at zero temperature It has emerged from recent work on molecular and colloid interactions [10,11] that: (1) The Debye–Huckel theory and its extensions in statistical mechanics even including asymmetric electrolytes; (2) The theories of interfacial tension; (3) The DLVO theory of colloidal particle interactions, are fundamentally flawed. To illustrate, the latter theory separates forces between particles (or molecules) into two components: electrostatic (double layer) forces, and dispersion forces. The first are treated in a non-linear theory (Poisson–Boltzmann equation or its extensions). The dispersion forces are handled by the Lifschitz theory, derived originally from an apparently rigorous complete solution of the many-body problem. These theories are fundamentally wrong in that they are inconsistent with thermodynamics (Gibbs adsorption isotherm) and also violate the gauge condition on the electromagnetic field [10,11]. The error lies in the ansatz that separates the electrostatics and the quantum mechanical fluctuation (dispersion) forces. The one, electrostatics, is handled by a non-linear theory; the dispersion forces in a linear theory. When the theory is done consistently, specific ion or Hofmeister effects so important in biology, due to previously neglected dispersion forces acting on ions, all come out predictively. Activity coefficients of electrolytes, interfacial tensions and interaction free energies all emerge correctly in a consistent theory. The new theory includes molecular size. Now the point of all this is to establish a corollary. Lifschitz theory of (physical) interactions whose extensions [10,11] include conducting media, electrolytes, retardation and temperature. It includes all fluctuation correlation frequencies. Through its reliance on measured dielectric susceptibilities. It apparently represented a complete solution of the many-body problem for physical, not chemical, interactions. It used the full apparatus of quantum electrodynamics However it turns out to be EXACTLY equivalent to semi-

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classical theory. That is, it can be derived simply using Maxwell’s equations plus the Planck hypothesis on quantization of the electromagnetic field [9–11]. (The complete QED description should include more than this. The error lies in a subtle point in the original development of Dzyaloshinski, Lifschitz and Pitaevski [12], whereby a non-linear coupling constant integration in Dyson’s integral equation for the non-linear dielectric susceptibility is linearized.) From our point of view it is sufficient that the entire edifice collapses to the statement that as presently used: QED ≡ semi-classical theory at zero temperature. From this approach one can derive trivially the famous Casimir interaction; the London; and Casimir–Polder (retarded, large distance) interaction between atoms; the Keesom interaction; the Debye induced dipole–permanent dipole force; ion fluctuation forces; the resonance interaction– real photon transfer between excited state–ground state; the Forster interaction; and so on as very special limiting cases [9–12]. Text book high order quantum mechanical perturbation theories to describe such forces are carried out at zero temperature. The semi-classical theory that gives out and embraces all such forces is more general in that it includes temperature, and many-body effects. Parenthetically, it turns out [11,13] that the long range Casimir–Polder, resonance, and Forster interactions between molecules turn out to be completely wrong at ANY finite, non-zero temperature! These things have now been corrected [11,12]. (Another point in passing is that, just as for the DLVO theory and for non-zero temperature, in the presence of a plasma, i.e., an electron gas, however dilute this is, excited state interactions again take a completely different form at large distances. It is not permissible to separate electron transfer from photon transfer.) Yet another point coupled to these points is that it also emerges that the whole notion and theory of pH and buffers is also badly deficient! This, and specific ion effects, done correctly in a modified emerging non-linear theory are all connected. (See last reference in [11].) These problems are directly relevant to the present case, and in both inorganic and organic structural chemistry. The concepts and standard language of different kinds of “bonds” derives from the same kinds of flawed ansatz, that separate of forces: ionic, dispersion, etc. Further the use of quantum mechanical perturbation theory for two and few body systems in what is really a many-body system, is restricted to zero temperature only. A.2. Molecular size. Summary The theories of interactions mentioned treat atoms or molecules as points. The interactions diverge at zero separation, so workers in the liquid, solution, colloid sciences invoke hard cores, soft cores or some other artificial cutoffs to get around this. Unlike in solid state, these parameters— like “hydration”—are artificial fit parameters that vary from

situation to situation. (This is because as explained above the basic theories have been deficient.) In developing a general theory of dispersion forces, Mahanty and one of us introduced molecular size [9]. This is defined formally in terms of the complete quantum mechanical k- and frequency-dependent polarisability tensor of an atom in its nth state. It can then be shown that to a very good approximation, this can be approximated by, e.g., a Gaussian atomic form factor in real space with a range of the order of the size of the atom convoluted with by the normal frequency dependent response function of a point particle. The Gaussian form is not crucial and any peaked function consistent the quantum state of the atom or molecule will do. There then developed a theory of dispersion self-energy of an atomic system. This is the precise analogue of the Born self-energy of an ion, except that the size is defined in terms of the quantum state of the molecule. The Born and dispersion self-energies in vacuum are of the same order of magnitude. For, say, a hydrogen atom, the dispersion selfenergy comes out to be the binding energy, but with the opposite sign. For non-polar molecules the dispersion selfenergy gives the energy of transfer from gas to solution, or from one liquid to another, i.e., solubility. The theory can be extended to include background electrons, temperature and so on. It can be used to derive a very respectable estimate of the Lamb shift [9]. It can be written down for interactions in a condensed medium. Just as the Debye–Huckel theory of electrolytes comes from the change in self-energy of an ion due to other ions, so too does the London force emerge as the change in self-energy due to other atoms surrounding it. “Bonds” But the difference now is that the London dispersion force between two molecules or atoms does not diverge on contact. Instead, when two atoms come together, the energy of interaction is the same as, or very close to the covalent binding energy of a molecule that they form. As soon therefore as an atomic form factor is introduced properly into the formalism for energy that involves the polarisability tensor, here modelled as a Gaussian, the distinction between different kinds of “bonds” disappears. The energies of interaction, or bonding come out quantitatively about right. A.3. Characterization of molecular size. Formal theory We shall need to develop these concepts formally. For more details see Ref. [9]. The classical theory of dispersion forces discussed above is based on the concept of the interaction of point molecules with the radiation field. This introduces divergences in the theory. Examples are the divergence in the dispersion self energy of a molecule, and that in the Lifshitz theory between two dielectric slabs at zero separation. In practice the size of the molecules will have significant effects at separations

S. Andersson, B.W. Ninham / Solid State Sciences 5 (2003) 683–693

comparable to that size. There has been a great deal of work in this area. Consider first for illustration a single atom interacting with the radiation field. Qualitatively, the effect of the size of the atom appears when we consider the polarization induced on it due to an electric field in the form of a plane wave. If the wave number is larger than the reciprocal of the linear size of the atom, different parts of the atom will experience different fields (in fact opposing fields), and the net polarization developed will be small. We may expect it to tend to zero as the wave number k → ∞. For a point atom, on the other hand, no such effect can be expected. Quantitatively, this can be expressed in terms of a k-dependent polarisability tensor αn (k, ω), which gives the polarization of the atom in its nth state due to an incident field of the form E(r, t) = E(k, ω)ei(k.r−ωt ),

(A.1)

the amplitude of the polarization being,   Pn (k, ω) = αn (k, ω)E(k, ω) .

(A.2)

The evaluation of αn (k, ω) can be done using linear response theory as in Ref. [9]. For a single electron atom, for instance, the polarization operator in terms of the electron coordinate u measured from the nucleus which is at R, is given by P = −eu.

(A.3)

The perturbation Hamiltonian due to the electric field of Eq. (A.1) is given approximately by taking the electric field at the middle of the dipole, and multiplying with the dipole moment,   HI (t) ≡ eu · E(k, ω)eik.u/2 ei(k.R−ωt ) . (A.4) A more exact treatment is given in Ref. ([9], Appendix B). The polarization developed on the atom in the state |n due to this perturbation is obtained from linear response theory as t 
 
1 n| p t − t  , HI t  |n dt  , Pn (t) = (A.5) i h¯ −∞

where p(t − t  ) is the polarization operator in the Heisenberg representation, p(t) = ei(H0 t /h¯ ) (p)e−i(H0 t /h¯ ) .

(A.6)

H0 being the unperturbed Hamiltonian. Using Eqs. (A.3) and (A.4) in Eq. (A.5) and doing the time integration, after some simplifications we get [9]   Pn (t) = αn (k, ω)E(k, ω) ei(k.r−ωt ), (A.7) where the polarisability tensor αn (k, ω) in the nth state of the atom is given by e2  {n|u|m m|ueik.u/2 |n } αn (k, ω) = − (ωnm + ω) h¯ m

{m|u|n n|ueik.u/2 |m } + . (A.8) (ωnm − ω)

689

Here n|u|m m|ueik.u/2 |n stands for the dyadic formed out of the two matrix elements and En − Em . ωnm = h¯ Eq. (A.7) leads immediately to the result that the polarization due to an arbitrary field E(r, ω), which can always be expressed as a linear combination of E(k, ω)ei(k.r) , will be given by  Pn (ω) = d3 k αn (k, ω)E(k, ω)eik.R  = d3 r αn (r − R; ω)E(r, ω), (A.9) where αn (r, ω) is the Fourier transform of α(k, ω). The quantity αn (r − R; ω)E(r, ω) can thus be interpreted as a polarization density pn (r, ω) whose integral over all space gives the total polarization developed by the atom. Form the explicit form of αn (k, ω) in Eq. (A.8) we note that its k-dependence arises through the factor m|ueik.u/2 |n  ≡ −(i∇k ) d3 u Φm (u)Φn (u)eik.u/2.

(A.10)

For k → ∞ his must tend to zero. Since the atomic wave function Φn (u) is peaked with a finite range, one expects the above matrix element to be peaked around k = 0 with a range that is of the order of the reciprocal of the range of Φn (u), i.e., the size of the atom in the nth state. This property must persist in αn (k, ω), whence αn (r − R; ω) will be peaked function with the peak around R and a range of the order of the atomic size. Because of the peaked nature of αn (r − R; ω), we can also write Pn (r, ω) ≡ αn (r − R; ω)E(r, ω) ∼ = αn (r − R; ω)E(R, ω).

(A.11)

For a point atom, the term in [eik.u/2 ] does not occur in the matrix element, so that αn (k, ω) is independent of k. This leads to αn (r − R; ω) becoming a delta function α(ω)δ(r − R). Thus we arrive at the conclusion that the size of an atomic system, i.e., an atom or a molecule can be characterized in terms of a polarisability tensor αn (r − R; ω) which leads to a polarization density according to Eq. (A.11) having a range of the order of the size of the system. The same kind of reasoning can be used to establish that there will be a size distribution associated with each of the other dipole moments in an external field [9]. We consider only the dipole polarisability here. It dominates dispersion interactions. A.3.1. The dispersion self-energy of the atomic system and interaction energies We now obtain an expression for the dispersion selfenergy of the atomic system, which is the change in the zero point energy of the electromagnetic field due to its coupling

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S. Andersson, B.W. Ninham / Solid State Sciences 5 (2003) 683–693

with the atomic system. We start with the field equations in time independent (Fourier transform) form. In Lorentz gauge, the equation for the vector potential can be written as



ω ω2 2 p(r, ω) ∇ + 2 A(r, ω) = −4πi (A.12) c c and the electric field is given by

ic ω2 E(r, ω) = − A + ∇(∇.A) . ω c2

∞ (A.13)

The relationship between E(r, ω) and p(r, ω) as already been indicated. Identifying p(r, ω) in Eq. (A.12) with pn (r, ω) of Eq. (A.11), we can write Eq. (A.12) in the form of an integral equation

 
  3 
 A(r, ω) = −4π G r − r ; ω α r − R; ω d r ×

ω2 A(R; ω) + ∇(∇.A) r=R , (A.14) c2

where the dyadic Green function G(r − r  ; ω) is  
 1 eik.(r−r ) G r − r ; ω = I 3 (A.15) d3 k. 2π (ω2 /c2 ) − k 2 From Eqs. (A.13) and (A.14) we get a secular equation for the perturbed frequencies of the field which is   I + 4πG(R, R; ω) = 0, (A.16) where 2


  ω G r, r ; ω = I + (∇r ∇r ) c2 

 × G r − r  ; ω α r  − r  ; ω d3 r  

 ≡ G r, r  ; ω α r  , r  ; ω d3 r  . (A.17) Here (∇r ∇r ) is the dyadic operator formed out of the gradient and the second form is transparent. As in Ref. [9] we can write the dispersion self-energy of the atomic system in the form of a contour integral  h¯ ω d  1 ln I + 4πG(R, R; ω) dω ES = 2πi 2 dω ∞   h¯ dξ lnI + 4πG(R, R; iξ ). = (A.18) 2π 0

The transformation of the contour integral to the last form as a real integral over imaginary frequencies is standard [9, 10]. (The formalism can be extended to include temperature effects. The energy becomes a free energy and the integral is replaced by a sum over imaginary frequencies 2πnkT /h¯ , n = 0, 1, 2, . . ., ∞.) Notice that from the definition of G in Eq. (A.17) that the elements of G(R, R; ω) = lim G(r, R; ω) r→R

are not likely to diverge, although those of G(R, R; ω) obtained from Eqs. (A.15) and (A.17) do diverge. The latter singularity is responsible for the infinite self-energies of point molecules. The finite size of the atomic system occurring in the function α(r − R; ω) is responsible for the finite value of its self-energy. To leading order in the polarisability we get ES = 2h¯

dξ Tr G(R, R; iξ ).

(A.19)

0

Further computation requires knowledge of the explicit forms of α(r − R; ω), which in principle can be evaluated from the wave functions of the atomic system as discussed above. To bring out the physical concepts involved, we assume that the atomic system is isotropic and that α(r − R; ω) has the form α(r, ω) ≈ I α(ω)f (r),

(A.20)

where f (r) is a peaked function. We further assume, for illustration, that f is a Gaussian with width a, f (r) =

1 π 3/2 a 3

e−r

2 /a 2

,

(A.21)

where a is the size of the atomic system. With this choice we obtain [9] Tr G(R, R; iξ )  3 α(iξ ) d k [3ξ 2 /c2 + k 2 ] exp(−k 2 a 2 /4) = 3 (2π) [ξ 2 /c2 + k 2 ] α(iξ ) 4α(iξ ) = 3/2 3 + π a πa 3

 2 2 1 aξ aξ aξ −(aξ/2c)2 × √ − e erfc . 2c 2c π 2c (A.22) Here erfc is the complementary error function, defined by 2 erfc(z) = √ π

∞

e−t dt. 2

z

When this expression is substituted into (A.19) the main contribution to the ξ -integration will come from the characteristic adsorption frequencies of the atomic system that determine the poles of α(ω). For the corresponding value of ξ , (aξ/2c) ∼ = (a/2λch ), where λch is the wavelength of a characteristic adsorption line. This ratio will generally be negligible, since a would be of the order of a few Angstroms at most while λch is much larger. Hence for all practical purposes for determination of the self-energy of an atomic system we can use the approximate formulae, α(iξ ) Tr G(R, R; iξ ) ∼ = 3/2 3 π a

(A.23)

S. Andersson, B.W. Ninham / Solid State Sciences 5 (2003) 683–693

and ES ∼ =

h¯ π 3/2 a 3

∞ dξ α(iξ ).

(A.24)

−∞

This simplified example brings out the main point that the size (and shape) of the atomic system enters into the expression for its dispersion self-energy in a very direct way. The dispersion self-energy can be expected to depend on the inverse cube of the size of the system. In free space the energy does not depend on R, the location of the atomic system. One point we notice is that Eq. (A.23) arises only from the non-retarded form (c → ∞) of G(r − r  ; ω) so that for the purposes of estimating the self-energy, the nonretarded Green function only need be used. For the simple classical form α(iξ ) = e2 /m(ω02 + ξ 2 ). √ Eq. (A.24) reduces to ES ≈ h¯ e2 / π a 3 mω0 . If we make the identification h¯ ω0 ≈ EG = e2 /2a0 , where EG is the ground state energy for hydrogen, and take a = a0 , the √ Bohr radius for√a hydrogen atom, we see that ES = (2/ π )(e2 /a0 ) = 4/ π × Rydberg; i.e., the self-energy of an atom is of the same order of magnitude as its binding energy but of opposite sign. The numerical factor is not important and arises from the crude choice of form factor. The same approach can be used to calculate a very respectable estimate of the Lamb shift [9], and it is easy to extend the formalism to include temperature as in Lifschitz theory. Interaction energies and “bonds” It is straightforward to extend this formalism to two or more interacting molecules [9]. We omit details and summarize some results. Now, if molecular size is included as above then for example the London dispersion interaction potential between two atoms no longer diverges on contact. Rather in the non-retarded region we have V (R) = VLondon(R)F (R),

(A.25)

where the London potential is 3h¯ VLondon(R) = − πR 6

∞ dξ α1 (iξ )α2 (iξ )

(A.26)

0

and F (R) → 1 at large distances and for R → 0, 2 3 8 R F (R) → , 9π a1 a2

lim V (R) = −

R→0

3π 2 a13 a23

(A.27)

∞ dξ α1 (iξ )α2 (iξ ).

At “contact”, R = (R1 − R2 ) = 0, the two Gaussian form factors overlap with the centers separated by a distance a1 +a2 . Two “separated” atoms now merge into a continuous electron cloud as they should. A.4. The exponential scale and quantum descriptions With these arguments let us now go back to the equations used in the main text to model the shape of molecules, separate or “bound”. With the (modified) Gaussian equations of our ice–water problem, and a simple scaling of the (x, y, z) distance coordinates, these can be interpreted as a good approximation to the rigorous quantum mechanical form factor of the molecules–atom–lone pair polarisability tensor system. The constant on the right side which is a normalization constraint that determines how far bonds can distort is evidently equivalent to the well-known sum rules for oscillator strengths of a coupled atomic–molecular system. (In our case it is a section of the entire many-body system.) So if one wished to get the energy of the system one would simply multiply by the (frequency dependent) polarisability and integrate the equation convoluted with the Green function of the system as in the formalism above. For a system with given lattice symmetry without or defects, restricted or directed random walks, the appropriate Green functions are known and occupy a vast literature, e.g., the books on Random and Restricted Walks of B.D. Hughes (Oxford). (One could also include temperature.) The equivalence of quantum mechanics and the exponential scale now comes within sight. The merging of shapes to give “bonds” in the exponential scale description is real and firmly based. The exponential scale does seem to catch the essence of the problem. Qualitatively, the difference between the exponential scale approach and the full QM treatment is not significant. But in the second case all insight tends to be lost in a welter of calculation. It is similar to the difference between the Born–Oppenheimer approximation where the nuclei are first fixed, and ab initio calculations. For the latter, one of course gets slightly better numerical agreement with laser spectroscopy, but the notion of chirality is often lost. A.5. The HCP vs. FCC structures

so that 8h¯

691

(A.28)

0

It not insignificant that in this limit the interaction energy for two like atoms is of the same order of magnitude as the binding energy of a molecule formed from the two of them. We will discuss this point further below.

There is in fact more in the exponential scale, which we now interpret as including atomic size and symmetry and self interactions. The binding energy of rare gas crystals is a case in point. In both, the number of nearest neighbors is the same. So any differences in energy must come from the long range dispersion part. For point molecules the dispersion energies are the same within computational limits. For a while it was thought that the structure sensitivity energy difference might be a result of multipolar interactions rather than the dipole–dipole interactions. But if the finite

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size of the molecules is taken into account as in the above formalism—equivalent to the exponential scale—the problem is resolved. The differences in energy arise due to the spread of the molecular polarisability. This was shown in Refs. [9, pp. 129–131,13]. A.6. Quantum bound states: interpretation of exponential scale In 1894 P.N. Lebedev wrote “. . . Of special interest and difficulty is the process that takes place in a physical body when many molecules interact simultaneously, the oscillations of the latter being interdependent owing to their proximity. If the solution of this problem ever becomes possible we shall be able to calculate in advance the values of the intermolecular forces due to molecular inter-radiation, deduce the laws of their temperature dependence, solve the fundamental problem of molecular physics whether all the so-called “molecular forces” are confined to the already known mechanical action of light radiation, to electromagnetic forces, or whether some forces of hitherto unknown origin are involved . . .”. Lebedev, a friend of J. Clerk Maxwell, discovered radiation pressure. It is about as succinct a statement of the problem as one could wish. It is especially fitting that Lebedev’s speculations and grand vision concerning the electromagnetic origins of molecular forces should have been confirmed by the Russians in the dramatic simultaneous advance, in theory by Feynman [14a] and Lifschitz [14b], and in experiment by Deryaguin and Abrikosova [15]. Deryaguin led the dominating school of colloid and surface science (along with the Dutch school under J.Th.G. Overbeek) for over half a century. He was Lebedev’s son-inlaw. The theory went further when Dzyaloshinski, Lifshitz and Pitaevski [14] solved the problem of interactions across an intervening medium. This was a tour de force in the then, in vogue, quantum electrodynamics. We have discussed how this theory included all the previous theories of interactions between molecules based on zero temperature quantum mechanical perturbation theories. They all dealt with point molecules which leads to infinite self-energies and divergences on contact which are removed once the theory is extended to include the notion of molecular size. The analysis above outlines how to do this, notwithstanding that the explicit example given is restricted to simple spherically symmetric molecules, dipolar dispersion forces with a special Gaussian form factor chosen for illustration only. In principle exact the Lifschitz theory failed to do as much as was expected because of the restriction to a linear coupling constant integration. That is natural enough, since there is no theory of the dielectric susceptibility in general. In the same way there is no theory of ionization equilibrium,

either in vacuum, or in solution beyond the statement of a “chemical” reaction. The theory collapses to the statement that the forces calculated are equivalent to the application of semi-classical theory—Maxwell’s equations plus the Planck hypothesis for the free energy of quantized allowed modes. The theory combined with double layer theory underlies all of solution theory electrochemistry, and physical forces between molecules or colloid particles. It is inconsistent with the Gibbs adsorption isotherm and the gauge condition on the electromagnetic field. We have seen how its deficiencies can be removed [10,11]. Nonetheless Lifschitz theory was a key advance in that it relied on measured or measurable frequency dependent dielectric susceptibilities of the interacting molecular systems. That is important because it means that in principle, and in fact in practice, all many-body forces are included. It is pointless to add up, as used to be done, pairwise London, Keesom, Debye . . . forces deduced for dilute media to try to obtain the interactions between extended molecules or colloid particles or biomolecules in media like water. The forces are not additive. In principle accessible via quantum mechanics, they are better obtained from indirectly from measurement of the frequency dependent dielectric responses of the bulk interacting media. Precisely the same deficiencies underlie present applications of quantum perturbation theory to give us the notions of “bonds”, ionic, covalent, hydrogen bonds, van der Waals bonds and so on. In any extended medium or multimolecular system the “bonds” are shared over many molecules in ways inaccessible to even the most powerful computers. The exponential scale approach follows the same principle as Lifshitz theory in relying on measured structure of a molecule in a condensed medium obtained from X-ray diffraction. We have shown how it connects to the underlying fundamental quantum mechanics. It is not limited to a harmonic polarization distribution. Anisotropy can be taken into account as in the equations of this text by addition of linear and cubic terms, and unharmonicity of lattice vibrations by including, e.g., quartic terms. The question that will occur to those of us schooled in the conventional approach is: How can such a description give a discrete eigenvalue, a real chemical bond that we are used to thinking of, one separated from the continuum of (implicit) oscillators used to describe a multiatomic system? The pictures characterize the transition astonishingly well. This problem, like a theory for ionization equilibrium or the dielectric susceptibility poses a fair question. However it seems to do so. The only analytic exactly solved model that bears on the problem is the solution for the eigenvalues of a 1D coupled chain of harmonic oscillators with defects, or variations in the force constant. Depending on the position and strength of these defects a long lived eigenvalue does indeed emerge from the continuum [16]. There is other evidence that this approach to the many-body problem including “bonds”, chemical reactions, is effective. For example with zeolites catalysis, comparison of self energies of a long chain hydrocarbon and its daughter products in

S. Andersson, B.W. Ninham / Solid State Sciences 5 (2003) 683–693

vacuo, and adsorbed in a zeolite predicts the measured enthalpies and mechanism of the catalytic process [17]. This would be impossible in a quantum chemistry approach. Likewise for catalytic reactions of molecules like CO2 , H2 , N2 [14]. See also Ref. [12]. With these preliminaries established we return to the main text.

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