Behind the Mpemba paradox

Chang Qing Sun*Nanyang Technological University; Singapore

Mpemba paradox results from5 hydrogen-bond anomalous relaxa-

tion. Heating stretches the O:H nonbondand shortens the H��O bond via Cou-lomb coupling; cooling reverses this pro-cess to emit heat at a rate depending on

10 its initial storage. Skin ultra-low massdensity raises the thermal diffusivity andfavors outward heat flow from the liquid.

As the source and central part of all15 lives, hydrogen bond (O:H��O, the ‘:’ is

the lone pair and the ‘-’ bonding pair ofelectrons) plays key role in DNA and pro-tein folding, drug-target binding, ion-channel activating/deactivating, gene

20 delivering, cancel cell curing, messaging,and signaling, etc. However, response ofhydrogen bond to temperature and coor-dination environment is crucial to theseactivities though mechanisms remain yet

25 unknown. This article shows how the O:H��O bond relaxes in resolving theMpemba paradox based on our extensiveresearch.1

Figure 1 illustrates (a) the potential30 paths and (b) the initial temperature

dependence of the liner velocity ofthe O:H��O bond during thermal

relaxation.2 The hydrogen bond consistsof the intermolecular O:H nonbond(left-hand side of the HC reference ori-gin) and the intramolecular H��O bond(right-hand side) other than either ofthem alone. The O:H��O bond acts asan asymmetric, HC bridged, and coupledoscillator pair with memory, whose coop-erative relaxation stems anomalies ofwater ice such as ice floating, regelation,ice slippery, hydrophobic and toughwater skin, Hofmeister effect, etc.1 Aninterplay of the O:H nonbond (van derWaals-like) interaction, the H��O cova-lent bond interaction, the O��O Cou-lomb repulsion and externally appliedstimulus always dislocate O atoms in thesame direction but by different amountsalong the respective potential paths.3 Thesofter O:H nonbond relaxes more thanthe stiffer H��O does.

Generally, heating stores energy in asubstance by stretching and softening allbonds involved. However, heating storesenergy in water by stretching the O:Hnonbond and simultaneously compressingthe H��O bond (red line linked spheresin Figure 1a are in the hot state) by thejoint interactions. Cooling reverses (blueline linked spheres) this process, analogousto suddenly releasing a coupled, highlydeformed bungee pair, one of which isstretched and the other compressed. Thisprocess emits energy at a rate that dependson the deformation history of the bungeepair (i.e., how much they were stretchedor compressed).4

The O:H��O bond exhibits memorywith thermal momentum during cooling.The linear velocity of the H��O bondDdH/Dt in Figure 1b is the product ofslopes for the known temperature depen-dence of H��O length dH(u) and themeasured initial temperature and timedependence of water temperature u(ui, t)shown in Figure 1c as inset. Because ofthe Coulomb coupling of the O:H andH��O, the velocity Ddx/Dt and energyrate DEx/Dt (x D L for O:H and x D H

for H��O) can be readily derived but herewe show the representative only for sim-plicity. Figure 1b indicates that the ini-tially shorter H��O bond at highertemperature remains highly active com-pared to its behavior otherwise when theymeet on the way to freezing.4

Conversely, molecular undercoordina-tion (with fewer than 4 neighbor as inthe bulk) has the same effect of heatingon O:H��O bond relaxation.5 H��Obond contracts and O:H expands, whichshrinks molecular size and expands theirseparations with an association of polari-zation.6 Water skin forms such a super-solid phase that is elastic, hydrophobic,ice like, and less dense (0.75 gcm¡3).The lower mass density raises the thermaldiffusivity and favors outward heat flowfrom the liquid.

Figure 1c and d show the theoreticalduplication of measured (insets) Mpembaparadox.4 The Mpemba paradox is charac-terized by7: (i) hot water freezes faster thancold water under the same conditions; (ii)the liquid temperature u drops exponen-tially with cooling time (t); (iii) water skinis warmer than sites inside the liquid andthe skin of hotter water is even warmer.

Besides the thermal diffusivity andconvection velocity involved in the non-linear Fourier thermal fluid transportequation, systematic examination of allpossible boundary conditions of a 10-mmlong, one-dimensional tube cell of waterwith a one-millimeter thick skin coolingfrom ui to uf D 0 �C revealed thefollowing:

1. Characterized by the crossing tempera-ture in Figure 1c, the Mpemba effecthappens only in the presence of thesupersolid skin.

2. The Mpemba effect is sensitive to thesource volume, the extent of skinsupersolidity, skin radiation and thedrain temperature uf.

3. The thermal convection has little effecton observations.

Keywords: biomolecules, convention,freezing, hydrogen bond, thermaldiffusivity

*Correspondence to: Chang Qing Sun; Email:Ecqsun@ntu.edu.sgQ1

Submitted: 09/26/2014

Revised: 10/03/2014

Accepted: 10/03/2014

http://dx.doi.org/10.4161/23328940.2014.974441

This is an Open Access article distributed underthe terms of the Creative CommonsAttribution-Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/), whichpermits unrestricted non-commercial use, distri-bution, and reproduction in any medium, pro-vided the original work is properly cited. Themoral rights of the named author(s) have beenasserted.

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130 It is necessary to emphasize that theMpemba effect happens only in circum-stances where the water temperature dropssharply from ui to uf at the source–draininterface. Larger liquid volume may pre-

135 vent this effect by heat-dissipation hinder-ing. Any spatial temperature decaybetween the source and the drain couldprevent the Mpemba effect. Conductingexperiments under identical conditions is

140 necessary to minimize artifacts such asradiation, source/drain volume ratio,exposure area, container material, etc.Therefore, conditions for the Mpembaeffect are indeed very critical, which

145 explains why it is not frequently observed.Therefore, O:H��O bond memory

and water-skin supersolidity determines

the heat ‘emission-conduction-dissipation’dynamics of the Mpemba paradox in the‘source-path-drain’ cycle system and theskin-drain interface must be highlynonadiabatic.

Disclosure of Potential Conflicts of Interest

No potential conflicts of interest weredisclosed.

References

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3. Huang Y, Ma Z, Zhang X, Zhou G, Zhou Y, Sun CQ.Hydrogen-bond asymmetric local potentials in com-pressed ice. J Phys Chem B 2013; 117(43):13639-45;PMID:24090472; http://dx.doi.org/10.1021/jp407836n

4. Zhang X, Huang Y, Ma Z, Zhou Y, Zhou J, ZhengW, Jiang Q, Sun CQ. Hydrogen-bond memory andwater-skin supersolidity resolving Mpemba paradox.Phys Chem Chem Phys, 2014; 16:22995-3002;PMID:25253165; http://dx.doi.org/10.1039/C4CP03669G

5. Zhang X, Huang Y, Ma Z, Zhou Y, Zheng W, ZhouJ, Sun CQ. A common superslid skin covering bothwater and ice. Phys Chem Chem Phys 2014;16:22987-94; PMID:25198167; http://dx.doi.org/10.1039/C1034CP02516D

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Figure 1. (a) O:H��O bond asymmetric, short-range potential paths, being analogous to springs and (b) O:H��O bond memory effect - initial tempera-ture dependence of the H��O bond relaxation velocity at cooling. Numerical duplicates the measured (insets) initial temperature dependence of the(c) u(ui, t) curves

7 and (d) the skin-bulk temperature difference u(ui, t) of water at cooling.2 The dH0 and dL0 in (a) are the respective references at 4�C.

Potentials include the O:H nonbond interaction (vdW of EL »0.1 eV bond energy; left-hand side), the H��O bond exchange interaction (EH » 4.0 eV;right-hand side), and the Coulomb repulsion between electron pairs (paired green dots) on adjacent oxygen atoms. These interactions dislocate O atomsin the same direction by different amounts with respect to the HC origin under excitation. The relaxation proceeds along the O:H��O bond potentialsfrom hotter (red line linked spheres, labeled ‘hot’) to colder state (blue line linked spheres, labeled ‘cold’).

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