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Impurity Effects in Thermal Regelation Soft Matter View Article Online PAPER View Journal | View Issue Impurity effects in thermal regelation a ab Cite this: Soft Matter, 2020, Navaneeth K. Marath and J. S. Wettlaufer * 16, 5886 When a particle is placed in a material with a lower bulk melting temperature, intermolecular forces can lead to the existence of a ‘‘premelted’’ liquid film of the lower melting temperature material. Despite the system being below the melting temperatures of both solids, the liquid film is a consequence of thermodynamic equilibrium, controlled by intermolecular, ionic and other interactions. An imposed Received 30th March 2020, temperature gradient drives the translation of the particle by a process of melting and refreezing known as Accepted 17th May 2020 ‘‘thermal regelation’’. We calculate the rate of regelation of spherical particles surrounded by premelted DOI: 10.1039/d0sm00558d films that contain ionic impurities. The impurities enhance the rate of motion thereby influencing the dynamics of single particles and distributions of particles, which we describe in addition to the rsc.li/soft-matter-journal consequences in natural and technological settings. Creative Commons Attribution-NonCommercial 3.0 Unported Licence. 1 Introduction translational velocity and displacement of the particle, when it is subjected to a temperature gradient. Lastly, we develop Premelted liquid films can separate the surface of a solid from the theory to understand the combined effects of thermal a foreign substrate at temperatures below the solid’s bulk regelation and diffusion on the particle motion. In Section 3, melting temperature. The solid melts against the substrate in we describe the displacement of particles of different sizes order to minimize the free energy of the solid–liquid–substrate and impurity concentrations for various temperature gradients. system.1 Thorough reviews1–3 discuss the theory of premelting The examples we provide are motivated by the dating of ice and its consequences in a swath of natural and technological cores used in paleoclimate research and the manipulation of This article is licensed under a settings. One of the consequences of premelting is the transla- composite materials, as described in Section 4. We conclude tion of a foreign particle through a solid when subjected to a in Section 5. temperature gradient. In this process of thermal regelation the Open Access Article. Published on 21 May 2020. Downloaded 9/24/2021 3:11:03 PM. solid melts at the warmer side of the particle and the fluid flows to the colder side and refreezes thereby facilitating the transla- 2 Theory tion. Evidently Gilpin4 first modeled thermal regelation, which We analyse the translation of a spherical particle of radius R that was later discussed in the context of interfacial melting5 and is surrounded by a solid, which premelts against the particle frost heave.6 In most settings impurities are present in the forming a thin film of liquid of thickness d, as shown in Fig. 1(b). films surrounding particles, but the rate of regelation has only At temperatures below the solid’s bulk melting temperature, Tm, been calculated for particles surrounded by pure films. The the film’s thickness depends on the nature of the intermolecular effect of impurities on the premelting of ice has been calculated and coulombic forces operative in the system and is a function of for a wide range of planar substrates.7 By combining the key the impurity concentration and the degree of undercooling.7,8 ingredients of pure and impure systems,5–11 we analyze the For sufficiently large concentrations,1,7,8 the thickness of the effects of impurities on the rate of thermal regelation of premelted film on a flat substrate (R - N) is dominated by spherical particles and discuss their implications for environ- colligative effects and is given by mental and engineering problems, by treating specific materials, in particular ice. R T 2N d ¼ g m i: (1) In Section 2, we give the theory for thermal regelation rlqmDT of spherical particles embedded in a solid. First, we describe how impurities in the premelted films that surround the Here Rg is the universal gas constant, rl is the molar density particles control the film thickness. Then, we calculate the of the liquid, DT = Tm À T is the undercooling, with T is the temperature of the solid/liquid/substrate system, qm is a Nordic Institute for Theoretical Physics, Stockholm, Sweden. the latent heat of melting per mole of the solid and Ni is the E-mail: [email protected] number of moles of impurity per unit surface area of the b Yale University, New Haven, Connecticut, USA substrate. The concentrations and the temperatures in which 5886 | Soft Matter, 2020, 16, 5886--5891 This journal is © The Royal Society of Chemistry 2020 View Article Online Paper Soft Matter ‘‘thermodynamic buoyancy’’ force13 and it pushes the particle in the direction of the gradient-towards the warmer side of the particle. As the particle moves, the thickness of the film must obey eqn (1), with larger thicknesses at larger temperatures. Therefore, the solid melts on the warm side, and the melt flows through the film to the colder side where it refreezes. The flow is driven by a thermomolecular pressure gradient,13 or tem- perature gradient induced liquid pressure gradient, and it will exert a hydrodynamic force on the particle governed by lubrica- tion theory.14 At a polar angle y (Fig. 1(b)) the volume flux from lubrication theory is equated to the particle motion as pd3 sin y dp ÀpðR sin yÞ2U ¼ l; (3) 6m dy where U is the magnitude of the translational velocity of the particle and m is the viscosity of the liquid. Substituting the thickness from eqn (1) into eqn (3) and integrating the latter with respect to y we obtain the liquid 6 pressure.Ð Hence, the hydrodynamic force is given by Fig. 1 (a) Shows particles embedded in a solid subjected to a temperature Fl ¼ SplndA, and integration yields gradient of T and (b) shows a zoomed view of the particle that is r 4 enclosed by the box in (a). The radius and the translational velocity of 8pmR Fl ¼ 2 U; (4) the particle are denoted by R and U, respectively. The premelted film that d0 Creative Commons Attribution-NonCommercial 3.0 Unported Licence. separates the solid from the particle is of thickness d. where d0 is the thickness of the film at the equator of the particle, when its axis is parallel to the temperature gradient. this linear colligative-type relationship between d, Ni and DT The expression for the hydrodynamic force is the same as that holds, depends on the nature of the materials in the system.7 derived for the case of a premelted film without impurities;5,6 Firstly, for spheres of radius R,solongasd { R, the curvature however, here the relationship between d0 and the undercool- of the solid–liquid interface is approximately the same as that of ing DT is given by eqn (1). Equating the hydrodynamic and the particle–liquid interface. Secondly, for sufficiently large thermodynamic buoyancy forces we obtain the translational impurity concentrations and particles, then the Gibbs–Thomson velocity of the particle as This article is licensed under a 8 effect does not control the film thickness. This requires that 3 rsqmrTd0 DT 4 2Tmgsl/Rrsqm, so that for micron sized particles in ice, U ¼ ; (5) À2 6mRTm where gsl =0.033Jm is the solid–liquid interfacial free energy, Open Access Article. Published on 21 May 2020. Downloaded 9/24/2021 3:11:03 PM. E 6 À3 Tm =273.15K,andrsqm 334 Â 10 Jm , and thus showing that the particle translates parallel to the temperature DT 4 0.05 K. Therefore, unless the system is within 50 mK of gradient. the bulk melting point, eqn (1) determines the film thickness for high concentrations and particles larger than about a micron, 2.2 Particle displacement by thermal regelation which is the situation we treat here. Substituting the thickness from eqn (1) into eqn (5) we can rewrite the latter as 2.1 Rate of thermal regelation ÀÁ dz A Uz ¼ p ¼ ÀÁ3 ; (6) The pressure in the solid ( ps) will be different from that in the p t 3 d A1 À A2zp liquid ( pl) and the difference is calculated using the Gibbs– Duhem equation12 and is given by p À p E r q DT/T . This s l s m m where zp is the position of the particle measured parallel to the pressure difference creates a force acting on the particle temperature gradient, with zp =0att =0,A1 = rlqm(Tm À Tint)/ given by T , A = r q |rT|/T , A = r q |rT|(R T N )3/6mRT , and T ð ð m 2 l m m 3 s m g m i m int rsqmDT rsqmrT msLhrTi is the temperature at the initial position of the particle. Fp À ndA ¼ dV ¼ ; (2) S Tm V Tm Tm Integration of eqn (6) leads to a quartic equation in zp whose solution is where n is the unit vector normal to the surface (S) of the ÀÁ 4 1=4 particle–film system, V is the volume, ms is the mass of the A1 À A1 À 4A2A3t zp ¼ ; (7) displaced host solid, L is the latent heat of melting per unit A2 mass of the solid and hrTi is the temperature gradient aver- which gives the net displacement of the particle at time t. When aged over V. We assume that the materials have the same z { A /A , then the displacement grows linearly with time, viz., thermal conductivities and the temperature gradient is a con- p 1 2 A3 stant vector, which implies hrTi = rT.
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