Duality of liquids.  
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PMID: 23851971 Owner: NLM Status: InDataReview 
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Liquids flow, and in this sense are close to gases. At the same time, interactions in liquids are strong as in solids. The combination of these two properties is believed to be the ultimate obstacle to constructing a general theory of liquids. Here, we adopt a new approach: instead of focusing on the problem of strong interactions, we zero in on the relative contributions of vibrational and diffusional motion. We show that liquid energy and specific heat are given, to a very good approximation, by their vibrational contributions as in solids over almost entire range of relaxation time in which liquids exist as such, and demonstrate that this result is consistent with liquid entropy exceeding solid entropy. Our analysis therefore reveals an interesting duality of liquids not hitherto known: they are close to solids from the thermodynamic perspective and to flowing gases. We discuss several implications of this result. 
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K Trachenko; V V Brazhkin 
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Type: Journal Article 
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Title: Scientific reports Volume: 3 ISSN: 20452322 ISO Abbreviation: Sci Rep Publication Date: 2013 Jul 
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Created Date: 20130715 Completed Date:  Revised Date:  
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Nlm Unique ID: 101563288 Medline TA: Sci Rep Country: England 
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Languages: eng Pagination: 2188 Citation Subset: IM 
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School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK. 
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Journal Information Journal ID (nlmta): Sci Rep Journal ID (isoabbrev): Sci Rep ISSN: 20452322 Publisher: Nature Publishing Group 
Article Information Download PDF Copyright © 2013, Macmillan Publishers Limited. All rights reserved openaccess: Received Day: 04 Month: 04 Year: 2013 Accepted Day: 26 Month: 06 Year: 2013 Electronic publication date: Day: 15 Month: 07 Year: 2013 collection publication date: Year: 2013 Volume: 3Elocation ID: 2188 PubMed Id: 23851971 ID: 3711051 Publisher Item Identifier: srep02188 DOI: 10.1038/srep02188 
Duality of liquids  
K. Trachenkoa1  
V. V. Brazhkin2  
1School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK 

2Institute for High Pressure Physics, RAS, 142190, Moscow, Russia 

ak.trachenko@qmul.ac.uk 
The development of basic theories of solids involved a number of important discoveries that date back to over 100 years. This was preceded by the development of the theories of gases. These developments form the basis for current understanding of most essential properties of these two basic states of matter^{1}. The third state of matter, the liquid state, remains poorly understood in comparison. A testament to this comes from a surprising fact that even recent textbooks dedicated to liquids do not discuss most basic liquid properties such as specific heat^{2}. Surprising though it may seem to a scientist outside the area, this fact has long been appreciated by those teaching the subject. In an amusing story about student teaching experience, Granato recalls his persistent fear of a potential student question about liquid specific heat^{3}. Noting that such a question was never asked over many years by a total of 10,000 students, Granato observes that this possibly reflects an important deficiency of our standard teaching method that fails to mention unsolved problems in physics, both in lectures and textbooks.
Liquids flow, and share this property with gases. At the same time, interactions in a liquid are strong, and are similar to those in solids. This presents a fundamental difficulty in calculating liquid energy in general form. Indeed, strong interactions, combined with systemspecific form of interactions, imply that the energy is strongly systemdependent, ostensibly precluding the calculation of energy in general form, contrary to solids or gases^{1}.
Strong interactions are successfully treated in solids in the phonon approach, but this approach has long been thought to be inapplicable to liquids where atomic displacements are large. Stated differently, the “small parameter” in the theory of solids are atomic displacements, and a harmonic contribution to the energy, the phonon energy, is often a good approximation. The small parameter in gases are weak interatomic interactions. On the other hand, liquids have none of these because interactions are strong and displacements are large. The absence of a small parameter was, in Landau view, the fundamental property of liquids that ultimately precluded the construction of a theory of liquids at the same level existing for solids or gases^{1}.
Here, we propose that reformulating the problem and exploring it in the new formulation provides an important way in. Instead of starting at the level of strong interactions at the atomistic scale, we focus on the atomic trajectories that result from these interactions. This reason is not unrelated to the one stated by Landau above, but operates at a different level.
We have recently proposed^{4} that from the point of atomic dynamics, solids and gases are pure states of matter in the sense that dynamics in solids is purely oscillatory and dynamics in gases is purely ballistic and collisional. The dynamics of a liquid, on the other hand, is not pure but mixed: it involves both oscillations and ballistic motions, and the relative contributions of the two types of motion change in response to external parameters, temperature and pressure. Physically, the different behaviors arise because in solids the kinetic energy of particles, K, is much smaller than the energy barriers between various potential minima, U: K ≪ U. In gases, it is the other way around: K ≫ U. In liquids, the mixed nature of dynamics originates because none of these strong inequalities apply.
Notably, the solidlike oscillatory component of liquid dynamics originates from large energy barriers preventing local atomic jumps and diffusion processes. Large energy barriers, in turn, are set by strong interactions  this is how interaction strength, emphasized by Landau, enters our approach.
In our approach to liquid thermodynamics, we therefore focus on atomic trajectories and relative weights of solidlike oscillatory and gaslike diffusional motions. In this approach, we present a proof that liquid energy and heat capacity are given, to a very good approximation, by their vibrational contributions as in solids over almost entire range of relaxation times in which liquids exist as such. We show that this result is consistent with liquid entropy exceeding solid entropy. Our analysis therefore shows that while liquids are close to gases from the point of view of flow, they are also close to solids from the thermodynamic perspective. This reveals a new property of liquids not hitherto known: their duality.
We note that the possibility of oscillatory motion contributing to liquid heat capacity has been contemplated before. This was done on the basis of empirical observation that experimental specific heat of some monatomic liquids (e.g. liquid metals) around the melting point is close to the DulongPetit value^{5}^{, 6}^{, 7}. Our new result here is the rigorous proof that liquid heat capacity is given by the vibrational motion over almost entire range of liquid relaxation time. This is an important advance in view of the absence of a theory of liquids^{3}.
Our first step is to calculate the liquid energy by evaluating relative contributions of oscillatory and ballistic diffusional motions. The evaluation can be done using the concept of liquid relaxation time proposed by Frenkel^{8}. Atoms or molecules in a liquid are not fixed, but rearrange in space due to thermally activated processes, giving liquid flow. Each flow event is a jump of an atom from its surrounding cage, accompanied by largescale rearrangement of the cage atoms. We call this process a local relaxation event (LRE). Frenkel introduced liquid relaxation time τ as the time between LREs at one point in space in a liquid^{8}, and showed that τ is related to liquid viscosity η via the Maxwell relationship η = G_{∞}τ, where G_{∞} is the instantaneous shear modulus.
The concept of τ has been widely used since to discuss liquid dynamics and its changes with temperature^{9}^{, 10}. In this picture, liquid dynamics acquires a simple description: a particle spends time τ oscillating inside the cage before jumping to a nearby quasiequilibrium site. The range of τ is bound by two important values. At low temperature, τ increases until it reaches the value at which the liquid stops flowing at the experimental time scale. This corresponds to τ ≈ 10^{3} s and the liquidglass transition^{9}^{, 10}. At high temperature, τ approaches its minimal value given by Debye vibration period, τ_{D} ≈ 0.1 ps, when the time between the jumps becomes comparable to the shortest vibrational period. As a result, τ varies by about 16 orders of magnitude in which the liquid state of flow can be measured.
There are two contributions to liquid energy, E_{l}: E_{l} = E_{vib} + E_{dif}, where E_{vib} and E_{dif} are the energies of vibrations and diffusion, respectively. E_{dif} includes the kinetic energy of jumping atoms as well as the energy of their interaction with other atoms during the local jump events. E_{vib} has two contributions predicted by Frenkel^{8}: one longitudinal mode and two transverse modes with frequency . The prediction was made on the basis of observation that at times shorter than τ, a liquid is essentially a frozen solid with all three vibrational modes, including transverse ones. On the other hand, a liquid flows and yields to shear stress at times longer than τ, and therefore does not support transverse modes at frequency .
Since Frenkel's prediction, the ability of liquids to support solidlike modes with wavelengths extending to the shortest distance comparable to interatomic separations has been confirmed experimentally^{11}^{, 12}^{, 13}^{, 14}^{, 15}^{, 16}^{, 17}. Notably, most of this experimental evidence is fairly recent, and has started to come to the fore only when powerful synchrotron radiation sources started to be deployed, some 50–60 years after Frenkel's prediction. This longlived absence of experimental data about propagating collective excitations in liquids may have contributed to their poor understanding from the theoretical point of view.
We note here that contrary to sometimes expressed views^{18}^{, 19}, collective excitations in disordered systems such as glasses and liquids do not decay or get damped, and are propagating up to the shortest wavelengths. Indeed, disordered systems support nondecaying solutions, eigenstates, of the secular equation involving the force matrix constructed from the disordered structure (this structure is static in a glass whereas in a liquid it is static at times smaller than τ as discussed by Frenkel). The energy of the disordered system is then equal to the energy of nondecaying eigenstate collective excitations. Harmonic (planewave) excitations, including those measured by the experimental probes, naturally decay in disordered systems, yet importantly these are clearly seen in liquids as solidlike quasilinear dispersion curves up to the shortest wavelengths^{11}^{, 12}^{, 13}^{, 14}^{, 15}^{, 16}^{, 17}. This includes transverse waves with the shortest wavelengths comparable to interatomic separations that are seen in even lowviscous liquids such as liquid Na^{15}, Ga^{16}, water^{17} and so on, consistent with Frenkel's prediction. A detailed discussion of this point is forthcoming. Below, we will make use of the fact that the experimental solidlike quasilinear dispersion curves in liquids imply that their vibrational density of states can be approximated by the quadratic form g(ω) ∝ ω^{2} to the same extent as in solids.
Lets now consider the regime where LREs take place rarely compared to the short period of vibrations:
Let us divide time τ into m time periods of duration τ_{D} each, so that . Then, E_{dif}, averaged over time τ, , is
We now consider Eqs. (7,8) in harmonic and anharmonic cases. In the harmonic case, Eqs. (7,8) give the energy and specific heat of a liquid as 3NT and 3, respectively, i.e. the same as in a harmonic solid:
In the anharmonic case, Eqs. (7) and Eqs. (8) still hold, but the equality of liquid and solid energies and specific heats holds only approximately because anharmonicity affects the vibrational energy and c_{v} of a liquid and a solid in a different way. In particular, c_{v} is modified by the intrinsic anharmonicity related to softening of vibrational frequencies at constant volume, and can be approximately written as c_{v} = 3(1 + αT), where α is the coefficient of thermal expansion^{20}^{, 21}. α is generally larger in liquids compared to solids, resulting in larger liquid c_{v} compared to solid c_{v}, albeit the term αT is usually small compared to 1.
The primary evidence supporting Eqs. (8,10) and our theory comes from experimental specific heat of liquids. Early measurements were done for liquid metals, and indicated that their specific heat is very close to 3, the DulongPetit value^{5}^{, 6}^{, 7}. This takes place close the melting point where Eq. (1) applies and which, according to our theory, gives Eqs. (8,10) and c_{v,l} ≈ 3. As experimental techniques advanced and gave access to high pressure and temperature, specific heats of many noble, molecular and network liquids were measured in detail in a wide range of parameters including in the supercritical region^{22}. Similarly to liquid metals, the experimental c_{v} of these liquids was found to be very close to 3 at low temperature where Eq. (1) applies (see Ref. ^{24} for a compilation of the NIST and other data of c_{v} for 21 liquids of different types). This universal behavior provides strong support to our theory.
We note that on temperature increase when condition τ starts to approach τ_{D} and Eq. (1) no longer applies, experimental c_{v,l} starts to decrease from about 3 at the melting point to 2 at high temperature^{6}^{, 7}^{, 22}. We have recently provided a quantitative description of this effect on the basis of the phonon theory of liquid thermodynamics for many different liquids^{23}^{, 24}. In this theory, the reduction of heat capacity is due to the progressive loss of transverse modes with frequency . On further temperature increase when c_{v} decreases from 2 to its idealgas value of , another mechanism kicks in: the disappearance of the remaining longitudinal mode with the wavelength smaller than the meanfree path of particles^{25}. The two mechanisms naturally give a crossover of c_{v} at c_{v} = 2 that we recently discovered^{25}, the crossover that corresponds to the Frenkel line where many liquid properties change^{4}^{, 26}.
Interestingly, although Eq. (6), combined with Eq. (1), implies that the energy and c_{v} of a liquid are entirely vibrational as in a solid, this does not apply to entropy: the diffusional component to entropy is substantial, and can not be neglected (here and below we imply the equilibrium state, the condition for which t ≫ τ, where t is observation time).
Indeed, if Z_{vib} and Z_{dif} are the contributions to the partition sum from vibrations and diffusion, respectively, the total partition sum of the liquid is Z = Z_{vib} · Z_{dif}. Then, the liquid energy is (here and below, the derivatives are taken at constant volume). Next, from Eq. (6) also implies , where, for brevity, we dropped the subscript referring to the average. Therefore, the smallness of diffusional energy, , gives
Physically, the inequality of liquid and solid entropies, S_{l} ≠ S_{s}, is related to the fact that the entropy measures the total phase space available to the system, which is larger in the liquid due to the diffusional component present in Eq. (13). The diffusional component, ln Z_{dif}, although large, is slowly varying with temperature according to Eq. (11), resulting in a small contribution to c_{v} (see Eqs. (14) and (15)) and giving c_{v,l} = c_{v,s}. On the other hand, the energy corresponds to the instantaneous state of the system (or averaged over τ), and is not related to exploring the phase space. Consequently, E_{l} = E_{vib}, yielding Eq. (11) and the smallness of diffusional contribution to c_{v} despite S_{l} ≠ S_{s}.
On the basis of results in the previous chapter, we conclude that two important properties of a liquid, energy and specific heat, are essentially vibrational, as they are in a solid, provided . For practical purposes, this takes place for . Perhaps not widely recognized, the condition τ ≈ 10τ_{D} holds even for lowviscous liquids such a liquid monatomic metals (Hg, Na, Rb and so on) and noble liquids such as Ar near their melting points^{22}^{, 23}^{, 24}, let alone for more viscous liquids such as roomtemperature olive or motor oil, honey and so on.
Notably, the condition corresponds to almost the entire range of τ at which liquids exist. This fact was not fully appreciated in earlier theoretical work on liquids. Indeed, on lowering the temperature, τ increases from its smallest limiting value of τ = τ_{D} ≈ 0.1 ps to τ ≈ 10^{3} s where, by definition, a liquid forms a glass at the glass transition temperature. Here, τ changes by 16 orders of magnitude. Consequently, the condition , Eq. (1), or , applies in the range 10^{3}–10^{−12} s, spanning 15 orders of magnitude of τ. This constitutes almost entire range of τ where liquids exist as such.
That important elements of liquid thermodynamics can be understood on the basis of thermodynamics of solids is a result hitherto not anticipated. Indeed, starting from earlier proposals^{1}^{, 2}^{, 8}, existing theories approach liquids as interacting gases and consequently attempt to calculate liquid energy as an integral of the product of interatomic interactions and correlation functions (see below). The interactions and correlations are often complex and are not generally known, except in the simplest liquids such as Ar. On the other hand, our theory circumvents this problem because the problem is reduced to calculating the vibrational energy only as in solids.
We now make several further observations regarding the implications of our theory. First, our approach provides general insights into the problem of liquidglass transition, and implies that the theory of solids is a good starting point for discussing important aspects of liquid thermodynamics. This is a useful result in the area where the calculation of thermodynamic properties of viscous liquids approaching glass transition has been a longstanding problem^{9}^{, 10}.
Second, we can revisit the longstanding and puzzling question of why Phillips constraint theory of glasses^{27} works. Proposed over 30 years ago, the theory is based on the idea that a shortranged interatomic bond in a glass can be viewed as a rigid mechanical constraint. Equating the number of constraints and degrees of freedom gives the average coordination number 〈r〉 = 2.4 at which, the theory proposes, the glassforming ability is optimized. The constraint theory has since been used to explain other properties of glass transition^{28}. Importantly, the constraint theory derives its predictions from counting the bonds (constraints) and 〈r〉 in the solid glass, but subsequently uses 〈r〉 to predict the properties measured in the liquid state above the liquidglass transition. This is truly surprising: indeed, in the liquid state above T_{g}, bonds are not intact because atoms rearrange on the experimental time scale, and therefore can no longer be viewed as rigid constraints. This poses an intriguing question of why the constraint theory works.
Our approach to liquids readily answers this question on general grounds: the measured properties operate in the regime where Eq. (1) applies. Indeed, glassforming melts such as silicates, chalcogenides and others are measured at temperatures where they are viscous enough to satisfy Eq. (1)^{28}. Consequently, our theory predicts that if a measured property is related to system energy or specific heat, this property in the liquid state is, to a very good approximation, equal to that in the solid glass, and can therefore be predicted from solidstate properties.
Finally, our approach relates to previous theories of liquids in another interesting way. When interactions and correlations are pairwise only as in simple noble liquids such as liquid Ar, previous theories attempt to calculate c_{v,l} as
There has been an extensive work on calculating g(r) and related probability functions in liquids, involving Gaussian and other approximations (see, e.g. Ref. ^{29}). Using Eq. (16) and our solidlike result for c_{v,l}, (c_{v,l} = 3 in the harmonic case) gives an important way to validate these approximations. In other words, our result opens a new avenue to elucidate the structure of liquids, a developing area with long history^{2}.
In summary, we observe that liquids flow, making them remarkably distinct from solids and close to gases. However, from the point of view of thermodynamics, liquid energy and specific heat are, to a very good approximation, equal to those in solids for relaxation times spanning 15 orders of magnitude, or almost entire range in which the liquids exist as such. In this sense, liquids show an interesting duality of physical properties not hitherto anticipated.
K. T. and V. V. Brazhkin have contributed equally to this work.
K. Trachenko thanks EPSRC for financial support.
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