Polarized 3He : dendritic melting

MAI 1986 No 5 Tome 47 LE JOURNAL DE J. PHYSIQUE Physique 47 (1986) 723-725 Marl 1986, 723 Classification Physics Abstracts 68.70 - 67.50 POLARIZED 3He : DENDRITIC MELTING PUECH, G. BONFAIT and L. CASTAING B. Centre de Recherches sur les Très Basses Températures, C.N.R.S., BP 166 X, F-38042 Grenoble Cedex, France (Reçu le 16 dgcembre 1985, accepté le 17 fgvrier 2986) Résumé Abstract Nous montrons, par une analyse similaire à celle de Mullins et Sekerka, que l’interface de fusion d’un solide 3He polarisé est en général instable. De notre analyse nous tirons un ordre de grandeur du rayon de courbure à l’extrémité des dendrites de liquide. Nous montrons qu’il existe une gamme de vitesses d’interface pour laquelle ce rayon est supérieur à la longueur de diffusion de l’aimantation dans le solide. We show, by a Mullins and Sekerka like analysis, that melting polarized solid 3He gives generally an unstable liquid-solid interface. From our analysis we obtain an order of magnitude estimation of the radius of curvature at the tip of the liquid dendrites. We show that, in some range of interface velocities, this radius is larger than the diffusion length in the solid. 1. Introduction tip of the dendrites. Comparing it to the diffulength in the solid gives the minimum interface velocity for which the m mL hypothesis is valid. the sion The experiments of Bonfait et al (1) and Dutta et al (2) recently raised the hope of obtaining non trivial knowledge of polarized liquid 3He. Both groups obtained a polarized liquid-solid in equilibrium by decompression of a low temperature, high field polarized solid (3). problem is then to determine the effective field H* corresponding to the measured average magnetization. The Bonfait et al (1) have suggested that meltin such conditions might be dendritic. The hypothesis was then made of a fine division of the solid, yielding that the measured magnetization m is also that of the solid at the interface m ms This interpretation was hardly in accord with the low magnetization data, obtained at the end of the relaxation (1). But it is even more clearly in disagreement with the results of Stony Brook (2). The remark then has been made (4) that, if one assumes that the interface during melting has a radius of curvature larger than the diffusion length in the solid, it is the liquid magnetization at the interface, mL, which is equal to the measured magnetization m (m mL hypothesis). mL ing = = = XLH* = xSH*. = In this paper, by a linear stability analysis similar to the Mullins-Sekerka (5) one, we show that liquid dendrites really occur in these melting conditions. We further obtain an order of magnitude estimate for the radius of curvature at = 2. The Planar Instability As we consider only times shorter than the spin-lattice relaxation time Tl, our situation is that of a mixture of plus and minus spins. The pro- blem is thus very similar to the Mullins and Sekerka (5) one which has been deeply investigated (see for instance the review by Langer (6)). There are however important differences in our case. First we melting and not growing of the solid, and we do no neglect, as is often done, the diffu- two consider second sion in the solid phase. For the particular system (liquid-solid 3He), the temperature of interest here and the pressure be considered as uniform in the sample. Indeed, remarked in reference (1), it is the phonons of the solid and the quasi particles of the liquid which govern the exchange of atoms (and thus of magnetization) between the two phases. It is thus their common temperature which is important thermodynamically. The phonon diffusion in the solid is very rapid and it results in heat conductivity which is larger in the solid than in the liquid above 30 mK. On the liquid side the heat and spin diffusions are of the same order of magnitude but, as we shall see, the instability length is much shorter than the diffusion length in the liquid, and the above remarks show that can as Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004705072300 724 of the solid make the temperainterface. high conductivity the uniform ture at the Note that consider a plane interface moving in the solid direction. The steady state situation is then for the magnetization, in the frame of the unperturbed interface : in the liquid (z 0) Let us - - in the solid (z > slopes velocity of the interface and Di the magnediffusivity in the phase i (i S, L). We assume here pure diffusive transport of magnetization in the solid, a simplifying assumption which could be wrong for high magnetizations,and/or low temperatization = same Taking We shall now the lo- to where a is the liquid-solid surface tension. With a system of units where m is the relative magnetization and H is expressed in Kelvins, § equals the molar volume divided by the perfect gas constant (1). (7). bation of the interface XS and XL refer here Finally, compared to the planar situation liquid pressure, the changes in chemipotential per unit volume are : with the cal ami/aH. 0) V is the tures cal consider an harmonic pertur- Eq. (9) becomes, into account Equations (10) first order : to (15), to position : and search for the relation between y and k imposed the conservation laws. Associated with this deformation is a perturbation of the magnetization on both sides : by Eq. (18) then gives : - 3. General Discussion cases magnetization conservation given by the equation : The We recall that where we are interested in the in both media is » discussing the (8) we distinguish and thus qs and equations : qL are the positive solutions of the case where y « Dsk hold and the fully Due to qi using Eq. (7) and the quasi stationary close to the instability thresunstable case where y » value of the For two cases : , Dsk2. inequalities (A) we can write : We have here neglected the difference in densities between liquid and solid, as usual (8). The magnetization conservation across the interface then leads to the equation : at where the quantities are taken/at z V + E. v is the instantaneous velocity of the interface. Up to the first order in E : = = reawhere taking into account inequalities (A) we have neglected sonable values of DSk One can see that it is also necompared to in the cases where this gligible compared to last term is important. Here : DLXLk. that same nar imposes the condition correspond to the field H*+h where H* refers to the pla- The local equilibrium at the interface and mL effective ms interface and h = hoeYte1kx : XS/XL VXL and XS/V 725 Approximate f, . values -. time the dendrite needs to progress a diffusion On times shorter than this the denlength is drite is static, which is coherent only if the evolution time of the curvature, due to the surface is also large. tension, are I DS/V2. " (p/ia)(DS/v2), Thus p must be of the condition (A) as : above 60 mK. there is no The first consequence of instability for (12) : order Qa° and we can write Eq. (23) is that with V. s 10-4 mIs, having the radius of Under this critical velocity, the instability indeed occurs. In order to estimate its growing we have to take the opposite limit : y > In such Dsk . a case Eq. (20) becomes : time * larger which is the condition for curvature of the interface than the diffusion length in the solid. we examine whether condibe considered as fulfilled in the actual experiments (ref. (1) and (2)). This would demonstrate the applicability of the m mL hypothesis. we have strongly focused on melting of polariwe think that the situation we have discusszed ed in the present paper can be found in classical systems. It could correspond to ordinary solid dendrites in a case where the heat diffusivity in the liquid is several orders of magnitude smaller than in the solid. We think that liquid dendrites could even be observed at slow enough melting with high enough impurities concentration in the solid. In another paper tion (B) can = We have WhileHe, neglected The most unstable responds perturbation thus cor- to : Let us end this section by a physical discussion which will be helpful shortly. As is always the case, the growing time of the unstable mode comes from the competition of two characteristic is the growing time in times. One, of order the absence of surface tension, where any mode is DS/V , unstable. The second comes from the stabilizing effect of the surface tension : due to it, the deformation creates a field undulation which tends to relax by magnetization diffusive currents, mainly in the This second time, thus proportional to is of order (a DL liquid. )-1 DS /V2Q a k. 4. Conclusion REFERENCES (1) G. B. Bonfait, L. Puech, A.S. Greenberg, Castaing, D. Thoulouze, Phys. Rev. G. Eska, Lett. 53 - (1984) 1092. (2) A. Dutta, C.N. Archie, Phys. Rev. Lett. 55 (1985) 2949. (3) B. Castaing, P. Nozières, J. Physique-Lettres 40 (1979) 275. (4) G. Bonfait, B. Castaing, A. Schuhl, M. Chapellier J. Physique-Lettres 46 (1985) 1073. (5) W.W. Mullins, R.F. Sekerka, J. Appl. Phys. 35 (1964) 444. (6) J.S. Langer, Rev. Mod. Phys. 52 (1980) 1. (7) B. Castaing, Proceeding of L.T. 17, Physica 126B (1984) 212. (8) Some recent papers take it into account. See - - that with 10-4 m/s is well under Vo and gives a growing time for the most unstable mode of approximately 50 ms. It is thus almost certain that As a conclusion let us note (M. S-MOL ) = 0.3, V -= in current polarized 3He faced with the problem of fully developed dendrites and particularly to the relation between their tip velocity and their radius of curvature. We cannot solve completely this problem which is always open for ordinary dendrites (6). dendritic melting occurs experiments. We are thus (9) - C3-55. (10) We shall however remark that this radius p cannot be much larger than t. : on such a quasi planar interface, instabilities of wavelength A would develop. On the other hand, as the shape of the dendrite is probably close to a paraboloid, the characteristic length for the variation of the curvature is also p at the tip. We have ding section, a thus, as competition the end of the precebetween two times. The at for instance B. Caroli, C. Caroli, C. Misbah, B. Roulet, J. Physique-Lettres 46 (1985) 1657. M.G. Richards, Adv. Magn. Resonance 5 (1971) 305 ; A. Landesman, J. Physique C3 31 (1970) J. Wilks, The Properties of Liquid Helium.(Clarendon, Oxford 1967). (11) This is estimated and Solid from the temperature of the roughening transition (E. Rolley, S. Balibar private communication). See also : S. Balibar, B. Castaing, Surface Science Reports 5 (1985) - 87. (12) In fact V > Vo corresponds inequality (A). namely qL ~ lity at large V But taking DLk2/V, remains. one can to breaking of opposite limit, the the see that the stabi-