Transport in degenerate  3He- 4He mixtures in quasi-2D flow channels is discussed. The quasiparticle mean free path combines particle-wall and particle-particle collisions including the interference between them. The temperature, concentration, and polarization dependences of the transport coefficients allow easy extraction of the correlation parameters of random surface roughness from transport data

Jochemsen, Reyer

Meyerovich, A. E.

English

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University of Rhode IslandDigitalCommons@URIPhysics Faculty Publications Physics2002Quasi-2D Flows of 3He-4He MixturesA. E. MeyerovichUniversity of Rhode Island, sfo101@uri.eduReyer JochemsenFollow this and additional works at: https://digitalcommons.uri.edu/phys_facpubsTerms of UseAll rights reserved under copyright.This Article is brought to you for free and open access by the Physics at DigitalCommons@URI. It has been accepted for inclusion in Physics FacultyPublications by an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.Citation/Publisher AttributionMeyerovich, A. E., & Jochemsen, R. (2002). Quasi-2D Flows of 3He-4He Mixtures. J. Low Temp. Phys., 126, 193-197. doi: 10.1023/A:1013751825229Available at: http://dx.doi.org/10.1023/A:1013751825229Quasi-2D Flows of 3 He - 4 He Mixtures Alexander Meyerovich and Reyer J ochemsen * Department of Physics, University of Rhode Island, Kingston, RI 02881, USA * K amerlingh Onnes Laboratory, Leiden University, P. 0. Box 9504, 2300 RA Leiden, The Netherlands Transport in degenerate 3 He - 4 He mixtures in quasi-2D flow channels is discussed. The quasiparticle mean free path combines particle-wall and particle-particle collisions including the interference between them. The tem-perature, concentration, and polarization dependences of the transport coef-ficients allow easy extraction of the correlation parameters of random sur-face roughness from transport data. FAGS numbers: 51.20.+d, 67.60.FP, 67.65.+z. 1. INTRODUCTION Wall scattering is an integral part of almost any physical process in ul-traclean and nanoscale systems. The convoluted nature of surface processes often makes the understanding of surface scattering difficult. In contrast to many other liquids and solids, 3 He allows a direct study of the effects of wall scattering on particle transport over a wide range of parameters.1 Low-temperature 3 He - 4 He mixtures provide even more flexibility by allowing one to scan particle mean free paths and wavelengths by simply changing the temperature, 3 He concentration, and degree of spin polarization. Since the formation of thin 4H e-rich layers near the walls can prevent energy and magnetic accommodation on the walls, one gets experimental access to a unique system with locally specular scattering of 3 He quasiparticles with a practically quadratic energy spectrum. This makes 3 He - 4 He mixtures one of the most convenient tools for studying of the effects of random wall roughness on transport. Below we report theoretical results of a comprehensive joint project on helium transport in ultrathin channels. The experiments are ongoing. 2 A. Meyerovich and R. Jochernsen 2. SURFACE AND BULK RELAXATION We consider degenerate 3 He - 4 He mixtures, T ~ TF, where TF p}/2m* = (37r 2 N 3) 2/ 3 /2m* and PF are the Fermi energy and momentum of 3 He quasiparticles, N 3 is the number of 3 He particles per unit volume, and m* is the effective mass of a single 3 He quasi particle (for values of parameters, see reviews3 ). We are interested in transport through a thin quasi-2D flow channel or film of the (average) thickness L. The random inhomogeneities of the walls are characterized by the correlation function of thickness fluctuations ( (y, z) (for definitions, see Ref. 4). The motion of 3 He quasi particles across the ultra thin films is quantized, where q is the momentum along the film. Such quantization has been ob-served in recent experiments. 5 According to our recent results on transport in quantized systems with both bulk and boundary scattering,4 the effective relaxation time in each of the minibands Ej is given by the equation : where S is the total number of occupied or energetically accessible mini-bands Ej ( q), and Tlb) ( q) is the bulk relaxation time in miniband Ej. The wall-induced transition probability Wjj' ( q, q') between the states Ej ( q) and Ej' ( q') is determined by the Fourier image of the correlation functions of surface inhomogeneities ( ( q - q'), 4T;2 w ( ') 7r IL ·2 ·/2( ( ') jj' q, q = Af2 L6J J q - q . (3) The interference between bulk and boundary scattering, which is one of the most important features of Eq.(2), has been thoroughly analyzed in Ref. 4. The application of these results to helium mixtures is straightforward since in dilute mixtures the bulk relaxation time does not depend on momenta: T(b)(q) =Tb= -- - , sn3 ( 1 ) 2 J 47rm* aT (4) where a r-v -0.9 A is the s-wave scattering length of two 3 He quasiparticles (for consistency, we use the viscous relaxation time). Quasi-2D Flows of 3 He- 4 He Mixtures Since extensive transport measurements in quantized films are not fea-sible at present, one should be more interested in quasiclassical transport in slightly thicker, but still ultrathin quasi-2D systems. Then the analog of Eq.(2) is _1 __ --1:_ (i + J W(p,p') dp' ) T(eff)(p) - Tb (E(p') - µ) 2 /n2 + 1/4TC (27rn)3 ' (5) with 47r ( fR ) 2 ( 2 ) w (p, p') = L nM p;p~ exp - (q - q') R 2 /2n2 ' (6) where we assume that the wall inhomogeneities are Gaussian with the am-plitude (height) and correlation radius (size) f and R respectively. Eq.(5) can be drastically simplified in two limiting cases when the bulk mean free path is either much larger or much smaller than the wall-restricted mean free path for ballistic particles. At ultralow temperatures, (7) (Xis the molar concentration of 3 He in the mixture), the bulk mean free path is larger than the wall-driven one, and the bulk and surface scattering processes are independent from each other. Then the effective relaxation obeys Matthiessen's rule, (8) The express10n for the wall contribution T w is especially simple for atomic- or nano-scale roughness in the low-concentration limit (9) when 1 2 f 2R2N3TF - = 47r -Tw L n. (10) Numerically, Eqs.(7) - (10) correspond at SVP to with T in Kelvin and R, f, and L in metres. A. Meyerovich and R. Jochernsen At higher temperatures or 3 He concentrations, when the bulk mean free path is much shorter than the wall-driven contribution, the effective relax-ation time is a convoluted combination of the bulk and surface scattering parameters. This strong interference between bulk and surface scattering is explained by the fact that the particles return to the wall not after a reflec-tion from the opposite wall, but after a series of bulk scattering processes. The value of the effective relaxation time at (TF /T) 2 ~ ( R/ a ) 2 X 413 is (12) Numerically, this equation corresponds to 1 1 £2 -- = - + 4.5 · 1015 - X 213T s-1 . T(eff) Tb L (13) 3. SPIN-POLARIZED MIXTURES Spin polarization of 3 He - 4 He mixtures provides a useful experimen-tal tool. The polarization results in change in particle wavelengths and bulk mean free paths.3 In addition, in spin-polarized mixtures, spin-up and spin-down quasiparticles have different dynamic and kinetic properties. The polarization dependence of the bulk viscous relaxation times of the spin-up and spin-down particles is given by the equation n3 ( 1 ) 2 5 c+ N ± T± = 2Irm* aT 5 - 3 (cf'-/c+)2/3 cf'-' C± = N3 (14) where C± are the "concentrations" of spin-up and spin-down particles. The above results can be easily expanded to polarized mixtures. Eqs.(7) - (11) for the limit of large bulk mean free path yield 1 2 ( )s/3 £2 R2 N3 TF - + 47r 2q T. T± L IL (15) 1 £2 R2 - + 3.16. 1034-- (2q)8 / 3 x~/3 s-1 . T± L (16) In the opposite case of not very low temperatures (or large-scale inho-mogeneities) and moderate polarizations, (17) Quasi-2D Flows of 3 He- 4 He Mixtures the relaxation time becomes (18) (19) (20) All the polarization dependence is contained in the factors q±, Eq.(20). Interestingly, the wall contributions to the inverse relaxation time 1/Tleff), Eq.(18), go to zero when the mixture becomes completely polar-ized, q± ( c_ ----+ 0) ----+ 0. In reality, the wall contribution does not disappear, at least for spin-ups: at high enough polarization one always leaves the hy-drodynamic regime, Eq.(17), and gets into the opposite, ballistic regime, Eq.(15). Note, that q_ goes to zero at c_ ----+ 0 much faster than q+. At high spin polarization one can observe a peculiar "mixed" low-temperature regime between the limiting cases (15) and (18) when c_ is so small that the majority spins (spin-ups) are in the ballistic regime (15) while the spin-downs are in the hydrodynamic regime (18) with a relatively small bulk mean free path. 4. CONCLUSIONS In summary, we present a theory of wall effects in transport of superfiuid 3 He - 4 He mixtures in narrow quasi-2D flow channels with random rough walls. The low-temperature scattering of 3 He quasi particles from the walls is almost completely specular locally and is a sensitive probe of the correlation function of surface inhomogeneities. Temperature, concentration, and spin polarization dependences of the transport coefficients allow one to monitor the system in a wide range of bulk mean free paths and particle wavelengths. The results are especially simple and easily distinguishable in the lim-iting cases of ballistic and hydrodynamic regimes (Eqs. (10), (12)), when the bulk collisions are either less or more effective than the wall scattering. The condition (7) that separates the ballistic and hydrodynamic limits is determined solely by the temperature and the correlation radius of surface inhomogeneities. High polarization can result in a peculiar regime in which the spin-up quasiparticles are ballistic while the spin-downs remain hydrody-namic and the wall contributions from spin components of the quasiparticle system are distinctly different from each other. The above results can be applied to pure liquid 3 He or metal films A. Meyerovich and R. Jochernsen with some caution. The main constraints are the requirement of quasielastic scattering by the walls and the use of constant bulk relaxation time. Experiments on thermal conductivity and viscosity of helium in ultra-thin channels are currently underway at Kamerlingh Onnes Laboratory. One of the goals is to extract parameters of wall roughness directly from the trans-port data. The wall contribution becomes noticeable at temperatures below 10 mK. At even lower temperatures it may be possible to observe roughness-driven localization. Measurements are being done in arrays of micro- and nanochannel glass capillaries. The material is fabricated at Na val Research Laboratory6 by a glass fiber drawing process that can provide channel glass arrays having channel diameters ranging from about 100 nm to 10 µ. The experimental cell design and nanochannel glass fabrication were discussed at QFS2001. 2 Additional experimental details will be published later. More detailed theory will appear in Ref. 7. ACKNOWLEDGMENTS The authors acknowledge support by NSF grants INT-9724 728 and DMR-0077266 (A.M.) and Stichting FOM with funding by NWO (R.J.). The authors are grateful to Dr. B. Justus (Naval Research Laboratory) for providing nanochannel glass samples. REFERENCES 1. H. Smith, Long Mean Free Paths in Quantum Fluids, in: Progr. Low Temp. Phys., ed. D. F. Brewer, 11, 75 (1987). 2. A. Meyerovich, I. Ponomarev, W. van Loo, R. Jochemsen, B. Justus, and P. Falkenstein, 3 He Transport in Nanochannel Glass Capillaries, P23.14, Int. Symposium on Quantum Fluids and Solids QFS2001 (Konstanz, July 2001), p.37. 3. A. E. Meyerovich, Spin-Polarized Phases of 3 He, in: Helium Three, eds. W. P. Halperin, L. P. Pitaevskii (North Holland, Amsterdam, 1991), p.p.757-879; Spin-Polarized Phases of 3 He - 4 He Solutions, in: Progr. Low Temp. Phys., ed. D. F. Brewer, 11, 1 - 75 (1987), and J. Owers-Bradley, Rep. Prag. Phys., 60, 1173 (1997). 4. A. E. Meyerovich,, and A. Stepaniants, Phys.Rev. B 60, 9129 (1999); J. Phys.: Cond. Matter 13, 5575 (2000). 5. S. L. Phillipson, A. M. Guenault, S. N. Fisher, G. R. Pickett, and P. J. Y. Thibault, Nature 395, 578 (1998) and P. A. Reeves, A. M. Guenault, S. N. Fisher, G. R. Pickett, and G. Tvalashvili, Physica B 284-288 319 (2000). 6. R. J. Tonucci, B. L. Justus, A. J. Campillo, and C. E. Ford, Science 258, 783 (1992). 7. A. E. Meyerovich, J. Low Temp. Phys. 124 No. 3/4 (2001), in print 

Quasi-2D Flows of \u3csup\u3e3\u3c/sup\u3eHe-\u3csup\u3e4\u3c/sup\u3eHe Mixtures

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Quasi-2D Flows of \u3csup\u3e3\u3c/sup\u3eHe-\u3csup\u3e4\u3c/sup\u3eHe Mixtures

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