- Email: [email protected]
Onset of superfluidity far from equilibrium: dynamical effects on the correlation length
Physica B 280 (2000) 45}49
Onset of super#uidity far from equilibrium: dynamical e!ects on the correlation length R.V. Duncan *, D.A. Sergatskov , S.T.P. Boyd , T.D. McCarson , A. Babkin , P.K. Day, D. Elliott Department of Physics and Astronomy, University of New Mexico, 800 Yale Blvd. NE, Albuquerque NM 87131-1156, USA Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
Abstract Nonlinear heat conduction has recently been measured near the super#uid transition in pure He at very low heat #ux Q. Since both dynamic e!ects and gravity limit the divergence of the super#uid correlation length near the transition at low-Q, these measurements must be repeated in the microgravity environment in order to observe the dynamic e!ects in isolation. Comparison of the microgravity data to similar data obtained on Earth will provide experimental insight into the e!ect of gravity on this nonlinear conduction region at low heat #ux where theoretical predictions are lacking. While some measurement advantages exist in the microgravity laboratory, it is the study of the direct e!ect of gravity on the nonlinear conduction measurements that motivate the microgravity need. 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Critical phenomena; Heat transport; Microgravity; Nonequilibrium super#uidity
1. Introduction We report on recent studies of dynamic e!ects near the super#uid transition in pure He, which is a continuous phase transition in the static limit. The physics of these dynamic e!ects is the focus of this report, and the experimental design of a prototype instrument for space #ight is discussed in a companion paper [1]. Such e!ects are studied experimentally in this work by placing a heat #ux through the He either along, or in the opposite direction to, the gravitational acceleration g. A quantitative theory for nonlinear heat transport, based on renormalization group techniques, has been developed by Haussmann and Dohm (HD) (Fig. 1) [2,3] for all heat #ux Q and for temperatures ¹'¹ , where ¹ is the static super#uid H H transition temperature under its vapor pressure. This analysis by Haussmann and Dohm, which is based on &Model F' of Halperin et al. [4,5] does not consider
* Corresponding author. E-mail address: [email protected] (R.V. Duncan)
gravitational e!ects. Calculations of gravitational e!ects on the nonlinear heat conduction have been made using mean-"eld theories [6}10], however these e!orts lack a fundamental treatment of the e!ect of #uctuations. These #uctuation e!ects become large when Q becomes very small. It is in this same limit, however, that the e!ects of gravity become most important [2,3,9,10]. Recently, a renormalization theory of nonlinear heat transport under gravity has been developed [11]. While this theory provides predictions over the full nonlinear heat transport region, its range of applicability is limited to Q'70 nW cm\, which is at the upper end of our measurement range. This predicted [2,3] nonlinear region is displayed in Fig. 1, along with recent experimental data [12]. In Fig. 1 t (Q) is de"ned as [¹ (Q)!¹ ]/¹ , where ¹ (Q) H H is de"ned by HD as that temperature where nonlinear e!ects make a 5% correction to the predicted heat transport within linear response region. By convention, t (Q) H and ¹ (Q) are now referred to as t (Q) and ¹ (Q), to H avoid confusion with the true critical temperature ¹ , H which is only de"ned in the static (Q"0) limit.
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R.V. Duncan et al. / Physica B 280 (2000) 45}49
2. Recent results
been observed by Liu and Ahlers [22] and Murphy and Meyer [23], however, the interpretation of the nature of this nonlinear region is not yet determined. These measurements, and a recent Caltech study [24] at lower Q, suggest that the depression of the super#uid transition temperature and the Kapitza resistance singularity may be closely related, as discussed later in this paper. For the intermediate range (70 nW cm\(Q(10 lW cm\) the isothermal nature of the helium just below ¹ (Q) permits the heat capacity of the helium under helium counter#ow to be measured. This pioneering measurement has recently been made by Harter et al., and their initial results are reported here at LT-22 [25]. Such an enhancement of the heat capacity just below ¹ (Q) had been predicted on the basis of numerous theories [26}28]. In the lowest heat #ux region Q(70 nW cm\, where gravitational e!ects dominate in nonlinear heat transport, experiments have only been conducted recently [12]. Experiments have measured these nonlinear e!ects down to 10 nW cm\, and deviations from theory [2,3] at these low values of Q are thought to arise from gravitational in#uences [11,29]. Future measurements, currently in de"nition for the International Space Station, will permit this nonlinear region to be studied in the absence of gravitational bias. Such measurements will also elucidate the fundamental way in which gravitational acceleration in#uences nonlinear thermal conductivity through its e!ect on the correlation length near the super#uid transition, as described below. The intrinsic e!ect of gravity on the nonlinear dynamics near the super#uid transition arises from a combination of gravitational e!ects and dynamical e!ects on the correlation length in the direction of heat transport [2,9}11]. The near-divergence of this correlation length, made possible by the chemical and physical purity of the helium sample, results in the sharp heat capacity singularity that gives the lambda point its name. This singularity, and its rounding by "nite size e!ects, have been extensively studied on orbit by the lambda point experiment and the con"ned helium experiment in the static (Q"0) limit [30,31]. In the dynamics, it is the #uctuations of the super#uid order parameter in normal #uid He that create the divergence of the normal #uid thermal conductivity, which has been studied over the last three decades [32}39]. It is the modi"cation of these #uctuations by the heat #ux that gives rise to the recently observed failure of Fourier's law near the super#uid transition [12]. Furthermore, it is the convolution of these dynamical e!ects with the e!ect of gravity on the correlation length [12], discussed in more detail below, that make a comprehensive experimental study of these
New phenomena have been discovered recently in all three ranges of heat #ux discussed above. In the highest range of heat #ux (Q'10 lW cm\) a new, dissipative phase of heat transport near the super#uid transition has
Critical dynamics in Microgravity (DYNAMX) is part of the &M1' Mission Set, in de"nition and competition for #ight on the International Space Station in year 2004.
Fig. 1. Predictions for the extent of the nonlinear region from Haussmann and Dohm (HD) [2,3], and experimental data from Day et al. [12]. Here t (Q) is the reduced temperature where H counter#ow heat transport fails catastrophically, and t (Q) is the reduced temperature where nonlinear heat transport makes only a 5% correction from linear heat transport, both consistent with HD.
Three distinct ranges of Q are of interest in studies of dynamic e!ects near the super#uid transition. First, for Q(70 nW cm\, the hydrostatic variation in ¹ across H the helium column [13] pushes the system away from criticality faster than does the temperature pro"le that results from Q [2,3]. It is desirable to conduct experiments in this range of Q aboard an orbiting laboratory to remove these gradients in the super#uid transition temperature, making the nonlinear region near the super#uid transition much wider and easier to study [14]. As discussed below, gravity intrinsically a!ects the divergence of the correlation length close to criticality, which should result in dramatic di!erences in the nonlinear conductivity observed on Earth and on orbit. Just the opposite situation exists for Q'70 nW cm\, where the temperature gradient in the middle of the nonlinear region pushes the system further from criticality than does the gravitationally induced gradient in ¹ . Measurements H made by Duncan, Ahlers, and Steinberg (DAS) [15], and recent measurements made by Baddar et al. [16] indicate that, above about 10 lW cm\, the bulk super#uid develops a measurable thermal gradient before counter#ow heat transport fails catastrophically at ¹ (Q). Finally, in the region Q'10 lW cm\, a Gorter}Mellink-like [17] bulk thermal gradient becomes detectable with subnanokelvin resolved thermometry [18}21], making the sharp transition to the thermally resistive state more di$cult to study in this high Q region [16].
R.V. Duncan et al. / Physica B 280 (2000) 45}49
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lower Q the data show a clear, systematic departure from the HD prediction due to the gravitational e!ects discussed above, which have not been considered in the renormalized theory at these low values of Q [2,3]. A clear understanding of the gravitational e!ects on the nonlinear region will not be possible until the data reported in Ref. [12] are repeated on earth-orbit.
3. Depression of the super6uid transition temperature by a heat 6ux
Fig. 2. Recent measurements [12] of ¹ (Q), as de"ned by Day et al. [12], and ¹ (Q), as de"ned by Haussmann and Dohm [2,3]. The fact that these two temperatures do not extrapolate to the same value in the QP0 limit has recently been explained by Haussmann [11].
predicted nonlinear e!ects impossible within the Earthbased laboratory [29]. In zero gravity, as the super#uid transition is approached from the static (Q"0) normal #uid phase by lowering the He temperature at constant pressure, the correlation length diverges as m(t)"m t\J, where M m "3.4;10\ cm, l"0.672, and t"(¹!¹ )/¹ M H H [40]. Under gravity, however, the pressure variation across the correlation length in the direction of gravity creates a gradient in the super#uid transition temperature that limits the divergence of the correlation length along gravity to m +110 lm, corresponding to a min imum closest approach to the transition of t+3;10\ in the bulk, when "nite-size e!ects are not a concern [12,41]. The Q-induced limit to the correlation length divergence at ¹ in the absence of gravity is predicted to H equal m when Q"60 nW cm\ [2,3]. This implies that these convolved gravitational and dynamical e!ects must be studied at the lowest values of Q that are tractable experimentally today [12]. Fig. 2 of Ref. [12] displays the thermal resistivity of He very close to the super#uid transition. These data, which were taken on Earth, show better agreement with the Q"0 theoretical predictions of HD as Q is increased. This is expected, since gravitational e!ects are much less important at higher Q. At
As a rough estimate we do not distinguish between the correlation length above and below the super#uid transition.
The depression of the super#uid transition temperature ¹ (Q) by a heat #ux Q has been measured by many experimentalists [12,15,16]. These measurements have been made using two di!erent procedures. In one type of measurements, the breakdown of counter#ow is detected by observing rapid heating at the heated endplate of the experimental cell [15]. In the second type of measurements this sudden heating is detected when the interface passes a sidewall thermometry probe [12,16]. In both cases the temperature of the super#uid is detected well away from the interface using a sidewall probe in the He-II phase, which is isothermal for Q)10 lW cm\ [15,16]. In the range 0.5)Q)5 lW cm\ these two di!erent types of measurement agree relatively well. This surprising result suggests that the breakdown in counter#ow heat transport as the cell transitions from its He-II phase to its He-I/He-II interface phase must occur with the interface forming a few correlation lengths away from the heated endplate [42,43]. This suggests that a hysteresis in the super#uid transition temperature under a heat #ux should exist with a magnitude of about the temperature rise associated with the singular Kapitza resistance [44}48]. Two groups have searched for such a hysteresis using high-resolution thermometry, but none has been observed [49,50]. Recently Chatto et al. [24] have shown that the measurements of the depression of the super#uid transition temperature by a heat #ux [15] near the heated solid endplate of the cell may be alternatively described as a consequence of the singular Kapitza resistance alone, without invoking the breakdown of bulk counter#ow. But why does the observed depression of the super#uid temperature by a heat #ux have the same value both at, as well as away from, the solid endplate? These preliminary results suggest that the super#uid order parameter may have the same boundary conditions at the He-I/He-II interface as it does at a solid endplate. Alternatively, the depression of the super#uid transition temperature may be insensitive to the order parameter's boundary conditions, however this seems counter-intuitive. These measurements and calculations suggest that the relationship between the singular Kapitza resistance and the failure of super#uid counter#ow is complicated, and not well understood. The future resolution of these questions may provide an understanding of the dynamic
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R.V. Duncan et al. / Physica B 280 (2000) 45}49
boundary conditions on the super#uid order parameter, which has been understood to be a very interesting and complex issue for some time. In addition to the thermal conductivity measurements discussed above, a new class of heat transport measurements have been made recently with the heat #ux Q along the direction of gravity, resulting in self-organized heat transport [51]. In the self-organized critical state the thermal gradient across the sample equals the pressureinduced gradient in ¹ [13], independent of the heat #ux, H which was varied from 20 nW cm\ to 6.5 lW cm\. No direct indication of the predicted [52] onset of the nonlinear region was observed in these measurements [51], however such features may become observable when the measurements are repeated with better thermal stability. Opportunities for future work in this self-organized critical state include a search for an upper heat #ux beyond which the state cannot form, and a study of second sound noise and propagation which may provide an indication of the microscopic mechanism that permits this state to form [9}11]. It is interesting to note that the upper heat #ux used in these self-organized critical measurements is just below the heat #ux at which bulk He-II dissipation has been observed [15,16]. At this upper heat #ux, the self-organized critical helium had a remarkable e!ective thermal conductivity in excess of 5 W cm\ K\. Will the bulk He-II dissipation make this self-organized state fail, or will it persist to some maximum counter#ow velocity, such as the Landau velocity? Measurements of this self-organized state at higher Q are clearly required. In addition to the intrinsic change in the "nite-Q thermal conductivity discussed above, gravity has been predicted [11] to create qualitative di!erences in the depression of the super#uid transition temperature ¹ (Q) by a heat #ux in the limit that QP0. HD predict [2,3] that both ¹ (Q) and ¹ (Q) will equal ¹ in the QP0 H limit in the absence of gravity. Recently this di!erence between ¹ (Q) and ¹ (Q) in the QP0 limit has been explained quantitatively by Haussmann [11], who predicts that this di!erence is purely a gravitational e!ect. Haussmann argues that in gravity these temperatures ¹ (Q) and ¹ (Q), which are predicted identical in the QP0 limit in microgravity [2,3], will di!er under Earth's gravitational acceleration. On Earth, ¹ (Q) and ¹ (Q) will correspond to the temperature on opposite sides of the interface. The thickness of this interface is approximately m , the gravity-induced width of a correla tion length along Q in the low-Q limit. Hence the appropriate reference temperature, ¹ , on each side of the H interface will di!er by the observed variation of ¹ with H height under gravity, which is a"1.27 lK/cm under saturated vapor pressure [13]. Hence the extrapolation of t (Q)"[¹ (Q)!¹ ]/¹ and t (Q)"[¹ (Q)!¹ ]/ H H H ¹ to zero Q should be t (0)!t (0)"am /¹ "6.4; H H 10\. This means that the zero-Q extrapolation of ¹ (Q) should fall 14 nK above the zero-Q extrapolation of
¹ (Q), assuming m "110 lm. As displayed in the "gure, this prediction agrees well with the experimental measurements. Haussmann's analysis [11] suggests that ¹ (0)"¹ (0)"¹ when the measurements H shown in the "gure are repeated in the microgravity environment.
4. Conclusions Measurements of nonlinear heat transport within the microgravity laboratory are required in order to test our most comprehensive theories of nonlinear dynamics near continuous phase transitions [2,3,6}9]. Gravity has been predicted to a!ect the nonlinear dynamics in genuinely new ways that can be tested only in the microgravity laboratory [9}11]. Microgravity measurements are planned that will test these theories. The microgravity data will also be compared to extensive measurements on Earth to determine the intrinsic e!ects of gravity on this model phase transition.
Acknowledgements We thank Dr. Rudolf Haussmann, Dr. Peter Weichman, and Dr. Jonathan Miller for numerous helpful discussions, and for preprints of their publications. We also thank Mr. Andrew Chatto, Ms. Alexa Harter, and Prof. David Goodstein for many interesting discussions. This work has been supported by the Microgravity Science Program O$ce of NASA under Contract 960494, and conducted in collaboration with the Jet Propulsion Laboratory.
References [1] R.V. Duncan et al., Dynamic and gravitational e!ects on the correlation length: experimental methods, the Second International Chernogolovka Workshop on Low Temperature Physics in the Microgravity Environment (CWS-99), Chernogolvka, Russia, 1999, to be presented. [2] R. Haussmann, V. Dohm, Phys. Rev. Lett. 67 (1991) 3404. [3] R. Haussmann, V. Dohm, Z. Phys. B 87 (1992) 229. [4] B.I. Halperin, P.C. Hohenberg, E.D. Siggia, Phys. Rev. Lett. 32 (1974) 1289. [5] B.I. Halperin, P.C. Hohenberg, E.D. Siggia, Phys. Rev. B 13 (1976) 1299. [6] A. Onuki, J. Low Temp. Phys. 50 (1983) 433. [7] A. Onuki, J. Low Temp. Phys. 55 (1984) 309. [8] A. Onuki, J. Low Temp. Phys. 104 (1996) 133. [9] P. Weichman, A. Prasad, R. Mukhopadhyay, J. Miller, Phys. Rev. Lett. 80 (1997) 22. [10] P. Weichman, J. Miller, preprint. [11] R. Haussmann, J. Low Temp. Phys. 114 (1999) 1.
R.V. Duncan et al. / Physica B 280 (2000) 45}49 [12] P. Day, W. Moeur, S. McCready, D. Sergatskov, R. Duncan, Phys. Rev. Lett. 81 (1998) 2474. [13] G. Ahlers, Phys. Rev. 171 (1968) 275. [14] F.-C. Liu, R. Duncan, U. Israelsson, T. Chui, D. Hensinger, A. Nash, M. Adriaans, W. Moeur, Czech. J. Phys. (Suppl. S1) 46 (1996) 87. [15] R.V. Duncan, G. Ahlers, V. Steinberg, Phys. Rev. Lett. 60 (1988) 1522. [16] H. Baddar, G. Ahlers, K. Kuehn, H. Fu, Proceedings of the 1999 NASA/JPL Microgravity Fundamental Physics Workshop, Washington, DC, April 29}May 1, 1999. [17] C.J. Gorter, J.H. Mellink, Physica 15 (1949) 285. [18] J. Lipa, B.C. Leslie, T.C. Walstrom, Physica B 107 (1981) 331. [19] B.J. Klemme, M.J. Adriaans, P.K. Day, D.A. Sergatskov, T.L. Aselage, R.V. Duncan, J. Low Temp. Phys. 116 (1999) 133. [20] H. Fu, H. Baddar, K. Kuehn, M. Larson, N. Mulders, A. Schegolev, G. Ahlers, J. Low Temp. Phys. 111 (1998) 49. [21] P.B. Welander, I. Hahn, Bull. Am. Phys. Soc. 44 (1999) 1050. [22] F.-C. Liu, G. Ahlers, Phys. Rev. Lett. 76 (1996) 1300. [23] D. Murphy, H. Meyer, Phys. Rev. B 57 (1998) 536. [24] A. Chatto, A. Harter, D. Goodstein, unpublished. [25] A.W. Harter, R.A.M. Lee, T.C.P. Chui, D.L. Goodstein, Physica B, these Proceedings (LT-22). [26] R. Haussmann, V. Dohm, J. Low Temp. Phys. 89 (1992) 429. [27] R. Haussmann, V. Dohm, Phys. Rev. B 46 (1992) 6361. [28] T.C.P. Chui, D.L. Goodstein, A.W. Harter, R. Mukhopadhyay, Phys. Rev. Lett. 77 (1996) 1793. [29] R.V. Duncan, P.K. Day, S.S. McCready, W.A. Moeur, F.-C. Liu, D.A. Sergatskov, J. Low Temp. Phys. 113 (1998) 861.
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[30] J.A. Lipa, D.R. Swanson, J.A. Nissen, T.C.P. Chui, U.E. Israelsson, Phys. Rev. Lett. 76 (1996) 944. [31] J.A. Lipa, D.R. Swanson, J.A. Nissen, P.R. Williamson, K. Geng, D.A. Stricker, T.C.P. Chui, U.E. Israelsson, M. Larson, J. Low Temp. Phys. 113, in press. [32] R.A. Ferrell, N. Menyhard, H. Schmidt, F. Schwabl, P. Szepfalusy, Phys. Rev. Lett. 18 (1967) 891. [33] V. Dohm, Phys. Rev. B 44 (1991) 2697. [34] V. Dohm, R. Folk, Phys. Rev. Lett. 46 (1981) 349. [35] G. Ahlers, Phys. Rev. Lett. 21 (1968) 1159. [36] J. Kerrisk, W.E. Keller, Phys. Rev. 177 (1969) 341. [37] W.Y. Tam, G. Ahlers, Phys. Rev. B 32 (1985) 5932. [38] W.Y. Tam, G. Ahlers, Phys. Rev. B 33 (1986) 183. [39] M. Dingus, F. Zhong, H. Meyer, J. Low Temp. Phys. 65 (1986) 185. [40] A. Singsaas, G. Ahlers, Phys. Rev. B 30 (1984) 5103. [41] G. Ahlers, J. Low Temp. Phys. 115 (1999) 143. [42] R.V. Duncan, Bull. Am. Phys. Soc. 44 (1999) 192. [43] R.V. Duncan, Proceedings of the 1998 NASA/JPL Microgravity Fundamental Physics Workshop, Oxnard, CA, June 1998. [44] R.V. Duncan, G. Ahlers, V. Steinberg, Phys. Rev. Lett. 58 (1987) 377. [45] D. Murphy, H. Meyer, J. Low Temp. Phys. 105 (1996) 185. [46] J.A. Lipa, Q. Li, Czech. J. Phys. (Suppl. S1) 46 (1996) 185. [47] Q. Li, Ph.D. Thesis, Stanford University, Department of Physics, 1990. [48] H. Baddar, H. Fu, F.-C. Liu, G. Ahlers, Czech. J. Phys. (Suppl. S1) 46 (1996) 191. [49] F.-C. Liu, G. Ahlers, Physica A 194B}196B (1994) 597. [50] P. Day, W. Moeur, S. McCready, D. Sergatskov, R. Duncan, unpublished. [51] W.A. Moeur, P.K. Day, F.-C. Liu, S.T.P. Boyd, M.J. Adriaans, R.V. Duncan, Phys. Rev. Lett. 78 (1997) 2421. [52] G. Ahlers, F.-C. Liu, J. Low Temp. Phys. 105 (1996) 255.
Onset of super#uidity far from equilibrium: dynamical e!ects on the correlation length R.V. Duncan *, D.A. Sergatskov , S.T.P. Boyd , T.D. McCarson , A. Babkin , P.K. Day, D. Elliott Department of Physics and Astronomy, University of New Mexico, 800 Yale Blvd. NE, Albuquerque NM 87131-1156, USA Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
Abstract Nonlinear heat conduction has recently been measured near the super#uid transition in pure He at very low heat #ux Q. Since both dynamic e!ects and gravity limit the divergence of the super#uid correlation length near the transition at low-Q, these measurements must be repeated in the microgravity environment in order to observe the dynamic e!ects in isolation. Comparison of the microgravity data to similar data obtained on Earth will provide experimental insight into the e!ect of gravity on this nonlinear conduction region at low heat #ux where theoretical predictions are lacking. While some measurement advantages exist in the microgravity laboratory, it is the study of the direct e!ect of gravity on the nonlinear conduction measurements that motivate the microgravity need. 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Critical phenomena; Heat transport; Microgravity; Nonequilibrium super#uidity
1. Introduction We report on recent studies of dynamic e!ects near the super#uid transition in pure He, which is a continuous phase transition in the static limit. The physics of these dynamic e!ects is the focus of this report, and the experimental design of a prototype instrument for space #ight is discussed in a companion paper [1]. Such e!ects are studied experimentally in this work by placing a heat #ux through the He either along, or in the opposite direction to, the gravitational acceleration g. A quantitative theory for nonlinear heat transport, based on renormalization group techniques, has been developed by Haussmann and Dohm (HD) (Fig. 1) [2,3] for all heat #ux Q and for temperatures ¹'¹ , where ¹ is the static super#uid H H transition temperature under its vapor pressure. This analysis by Haussmann and Dohm, which is based on &Model F' of Halperin et al. [4,5] does not consider
* Corresponding author. E-mail address: [email protected] (R.V. Duncan)
gravitational e!ects. Calculations of gravitational e!ects on the nonlinear heat conduction have been made using mean-"eld theories [6}10], however these e!orts lack a fundamental treatment of the e!ect of #uctuations. These #uctuation e!ects become large when Q becomes very small. It is in this same limit, however, that the e!ects of gravity become most important [2,3,9,10]. Recently, a renormalization theory of nonlinear heat transport under gravity has been developed [11]. While this theory provides predictions over the full nonlinear heat transport region, its range of applicability is limited to Q'70 nW cm\, which is at the upper end of our measurement range. This predicted [2,3] nonlinear region is displayed in Fig. 1, along with recent experimental data [12]. In Fig. 1 t (Q) is de"ned as [¹ (Q)!¹ ]/¹ , where ¹ (Q) H H is de"ned by HD as that temperature where nonlinear e!ects make a 5% correction to the predicted heat transport within linear response region. By convention, t (Q) H and ¹ (Q) are now referred to as t (Q) and ¹ (Q), to H avoid confusion with the true critical temperature ¹ , H which is only de"ned in the static (Q"0) limit.
0921-4526/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 4 4 5 - 3
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R.V. Duncan et al. / Physica B 280 (2000) 45}49
2. Recent results
been observed by Liu and Ahlers [22] and Murphy and Meyer [23], however, the interpretation of the nature of this nonlinear region is not yet determined. These measurements, and a recent Caltech study [24] at lower Q, suggest that the depression of the super#uid transition temperature and the Kapitza resistance singularity may be closely related, as discussed later in this paper. For the intermediate range (70 nW cm\(Q(10 lW cm\) the isothermal nature of the helium just below ¹ (Q) permits the heat capacity of the helium under helium counter#ow to be measured. This pioneering measurement has recently been made by Harter et al., and their initial results are reported here at LT-22 [25]. Such an enhancement of the heat capacity just below ¹ (Q) had been predicted on the basis of numerous theories [26}28]. In the lowest heat #ux region Q(70 nW cm\, where gravitational e!ects dominate in nonlinear heat transport, experiments have only been conducted recently [12]. Experiments have measured these nonlinear e!ects down to 10 nW cm\, and deviations from theory [2,3] at these low values of Q are thought to arise from gravitational in#uences [11,29]. Future measurements, currently in de"nition for the International Space Station, will permit this nonlinear region to be studied in the absence of gravitational bias. Such measurements will also elucidate the fundamental way in which gravitational acceleration in#uences nonlinear thermal conductivity through its e!ect on the correlation length near the super#uid transition, as described below. The intrinsic e!ect of gravity on the nonlinear dynamics near the super#uid transition arises from a combination of gravitational e!ects and dynamical e!ects on the correlation length in the direction of heat transport [2,9}11]. The near-divergence of this correlation length, made possible by the chemical and physical purity of the helium sample, results in the sharp heat capacity singularity that gives the lambda point its name. This singularity, and its rounding by "nite size e!ects, have been extensively studied on orbit by the lambda point experiment and the con"ned helium experiment in the static (Q"0) limit [30,31]. In the dynamics, it is the #uctuations of the super#uid order parameter in normal #uid He that create the divergence of the normal #uid thermal conductivity, which has been studied over the last three decades [32}39]. It is the modi"cation of these #uctuations by the heat #ux that gives rise to the recently observed failure of Fourier's law near the super#uid transition [12]. Furthermore, it is the convolution of these dynamical e!ects with the e!ect of gravity on the correlation length [12], discussed in more detail below, that make a comprehensive experimental study of these
New phenomena have been discovered recently in all three ranges of heat #ux discussed above. In the highest range of heat #ux (Q'10 lW cm\) a new, dissipative phase of heat transport near the super#uid transition has
Critical dynamics in Microgravity (DYNAMX) is part of the &M1' Mission Set, in de"nition and competition for #ight on the International Space Station in year 2004.
Fig. 1. Predictions for the extent of the nonlinear region from Haussmann and Dohm (HD) [2,3], and experimental data from Day et al. [12]. Here t (Q) is the reduced temperature where H counter#ow heat transport fails catastrophically, and t (Q) is the reduced temperature where nonlinear heat transport makes only a 5% correction from linear heat transport, both consistent with HD.
Three distinct ranges of Q are of interest in studies of dynamic e!ects near the super#uid transition. First, for Q(70 nW cm\, the hydrostatic variation in ¹ across H the helium column [13] pushes the system away from criticality faster than does the temperature pro"le that results from Q [2,3]. It is desirable to conduct experiments in this range of Q aboard an orbiting laboratory to remove these gradients in the super#uid transition temperature, making the nonlinear region near the super#uid transition much wider and easier to study [14]. As discussed below, gravity intrinsically a!ects the divergence of the correlation length close to criticality, which should result in dramatic di!erences in the nonlinear conductivity observed on Earth and on orbit. Just the opposite situation exists for Q'70 nW cm\, where the temperature gradient in the middle of the nonlinear region pushes the system further from criticality than does the gravitationally induced gradient in ¹ . Measurements H made by Duncan, Ahlers, and Steinberg (DAS) [15], and recent measurements made by Baddar et al. [16] indicate that, above about 10 lW cm\, the bulk super#uid develops a measurable thermal gradient before counter#ow heat transport fails catastrophically at ¹ (Q). Finally, in the region Q'10 lW cm\, a Gorter}Mellink-like [17] bulk thermal gradient becomes detectable with subnanokelvin resolved thermometry [18}21], making the sharp transition to the thermally resistive state more di$cult to study in this high Q region [16].
R.V. Duncan et al. / Physica B 280 (2000) 45}49
47
lower Q the data show a clear, systematic departure from the HD prediction due to the gravitational e!ects discussed above, which have not been considered in the renormalized theory at these low values of Q [2,3]. A clear understanding of the gravitational e!ects on the nonlinear region will not be possible until the data reported in Ref. [12] are repeated on earth-orbit.
3. Depression of the super6uid transition temperature by a heat 6ux
Fig. 2. Recent measurements [12] of ¹ (Q), as de"ned by Day et al. [12], and ¹ (Q), as de"ned by Haussmann and Dohm [2,3]. The fact that these two temperatures do not extrapolate to the same value in the QP0 limit has recently been explained by Haussmann [11].
predicted nonlinear e!ects impossible within the Earthbased laboratory [29]. In zero gravity, as the super#uid transition is approached from the static (Q"0) normal #uid phase by lowering the He temperature at constant pressure, the correlation length diverges as m(t)"m t\J, where M m "3.4;10\ cm, l"0.672, and t"(¹!¹ )/¹ M H H [40]. Under gravity, however, the pressure variation across the correlation length in the direction of gravity creates a gradient in the super#uid transition temperature that limits the divergence of the correlation length along gravity to m +110 lm, corresponding to a min imum closest approach to the transition of t+3;10\ in the bulk, when "nite-size e!ects are not a concern [12,41]. The Q-induced limit to the correlation length divergence at ¹ in the absence of gravity is predicted to H equal m when Q"60 nW cm\ [2,3]. This implies that these convolved gravitational and dynamical e!ects must be studied at the lowest values of Q that are tractable experimentally today [12]. Fig. 2 of Ref. [12] displays the thermal resistivity of He very close to the super#uid transition. These data, which were taken on Earth, show better agreement with the Q"0 theoretical predictions of HD as Q is increased. This is expected, since gravitational e!ects are much less important at higher Q. At
As a rough estimate we do not distinguish between the correlation length above and below the super#uid transition.
The depression of the super#uid transition temperature ¹ (Q) by a heat #ux Q has been measured by many experimentalists [12,15,16]. These measurements have been made using two di!erent procedures. In one type of measurements, the breakdown of counter#ow is detected by observing rapid heating at the heated endplate of the experimental cell [15]. In the second type of measurements this sudden heating is detected when the interface passes a sidewall thermometry probe [12,16]. In both cases the temperature of the super#uid is detected well away from the interface using a sidewall probe in the He-II phase, which is isothermal for Q)10 lW cm\ [15,16]. In the range 0.5)Q)5 lW cm\ these two di!erent types of measurement agree relatively well. This surprising result suggests that the breakdown in counter#ow heat transport as the cell transitions from its He-II phase to its He-I/He-II interface phase must occur with the interface forming a few correlation lengths away from the heated endplate [42,43]. This suggests that a hysteresis in the super#uid transition temperature under a heat #ux should exist with a magnitude of about the temperature rise associated with the singular Kapitza resistance [44}48]. Two groups have searched for such a hysteresis using high-resolution thermometry, but none has been observed [49,50]. Recently Chatto et al. [24] have shown that the measurements of the depression of the super#uid transition temperature by a heat #ux [15] near the heated solid endplate of the cell may be alternatively described as a consequence of the singular Kapitza resistance alone, without invoking the breakdown of bulk counter#ow. But why does the observed depression of the super#uid temperature by a heat #ux have the same value both at, as well as away from, the solid endplate? These preliminary results suggest that the super#uid order parameter may have the same boundary conditions at the He-I/He-II interface as it does at a solid endplate. Alternatively, the depression of the super#uid transition temperature may be insensitive to the order parameter's boundary conditions, however this seems counter-intuitive. These measurements and calculations suggest that the relationship between the singular Kapitza resistance and the failure of super#uid counter#ow is complicated, and not well understood. The future resolution of these questions may provide an understanding of the dynamic
48
R.V. Duncan et al. / Physica B 280 (2000) 45}49
boundary conditions on the super#uid order parameter, which has been understood to be a very interesting and complex issue for some time. In addition to the thermal conductivity measurements discussed above, a new class of heat transport measurements have been made recently with the heat #ux Q along the direction of gravity, resulting in self-organized heat transport [51]. In the self-organized critical state the thermal gradient across the sample equals the pressureinduced gradient in ¹ [13], independent of the heat #ux, H which was varied from 20 nW cm\ to 6.5 lW cm\. No direct indication of the predicted [52] onset of the nonlinear region was observed in these measurements [51], however such features may become observable when the measurements are repeated with better thermal stability. Opportunities for future work in this self-organized critical state include a search for an upper heat #ux beyond which the state cannot form, and a study of second sound noise and propagation which may provide an indication of the microscopic mechanism that permits this state to form [9}11]. It is interesting to note that the upper heat #ux used in these self-organized critical measurements is just below the heat #ux at which bulk He-II dissipation has been observed [15,16]. At this upper heat #ux, the self-organized critical helium had a remarkable e!ective thermal conductivity in excess of 5 W cm\ K\. Will the bulk He-II dissipation make this self-organized state fail, or will it persist to some maximum counter#ow velocity, such as the Landau velocity? Measurements of this self-organized state at higher Q are clearly required. In addition to the intrinsic change in the "nite-Q thermal conductivity discussed above, gravity has been predicted [11] to create qualitative di!erences in the depression of the super#uid transition temperature ¹ (Q) by a heat #ux in the limit that QP0. HD predict [2,3] that both ¹ (Q) and ¹ (Q) will equal ¹ in the QP0 H limit in the absence of gravity. Recently this di!erence between ¹ (Q) and ¹ (Q) in the QP0 limit has been explained quantitatively by Haussmann [11], who predicts that this di!erence is purely a gravitational e!ect. Haussmann argues that in gravity these temperatures ¹ (Q) and ¹ (Q), which are predicted identical in the QP0 limit in microgravity [2,3], will di!er under Earth's gravitational acceleration. On Earth, ¹ (Q) and ¹ (Q) will correspond to the temperature on opposite sides of the interface. The thickness of this interface is approximately m , the gravity-induced width of a correla tion length along Q in the low-Q limit. Hence the appropriate reference temperature, ¹ , on each side of the H interface will di!er by the observed variation of ¹ with H height under gravity, which is a"1.27 lK/cm under saturated vapor pressure [13]. Hence the extrapolation of t (Q)"[¹ (Q)!¹ ]/¹ and t (Q)"[¹ (Q)!¹ ]/ H H H ¹ to zero Q should be t (0)!t (0)"am /¹ "6.4; H H 10\. This means that the zero-Q extrapolation of ¹ (Q) should fall 14 nK above the zero-Q extrapolation of
¹ (Q), assuming m "110 lm. As displayed in the "gure, this prediction agrees well with the experimental measurements. Haussmann's analysis [11] suggests that ¹ (0)"¹ (0)"¹ when the measurements H shown in the "gure are repeated in the microgravity environment.
4. Conclusions Measurements of nonlinear heat transport within the microgravity laboratory are required in order to test our most comprehensive theories of nonlinear dynamics near continuous phase transitions [2,3,6}9]. Gravity has been predicted to a!ect the nonlinear dynamics in genuinely new ways that can be tested only in the microgravity laboratory [9}11]. Microgravity measurements are planned that will test these theories. The microgravity data will also be compared to extensive measurements on Earth to determine the intrinsic e!ects of gravity on this model phase transition.
Acknowledgements We thank Dr. Rudolf Haussmann, Dr. Peter Weichman, and Dr. Jonathan Miller for numerous helpful discussions, and for preprints of their publications. We also thank Mr. Andrew Chatto, Ms. Alexa Harter, and Prof. David Goodstein for many interesting discussions. This work has been supported by the Microgravity Science Program O$ce of NASA under Contract 960494, and conducted in collaboration with the Jet Propulsion Laboratory.
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