Dynamics of a 4 He quantum crystal in the superfluid liquid
aa r X i v : . [ c ond - m a t . o t h e r] N ov Dynamics of the fall of a quantum He crystal in superfluid liquid
V. L. Tsymbalenko ∗ NRC Kurchatov Institute, 123182 sq.Kurchatov 1Institute for Physical Problems RAS, 119334 Kosigin st.2Moscow, Russia
The motion of helium crystals has been experimentally studied when the crystals fall in thesuperfluid liquid owing to gravity at temperatures above the roughening transitions where the wholecrystal surface is in the atomically rough state. The rate of crystal fall at T = 1 .
25K is higher thanat T = 1 . T = 1 .
54K does not change the pressure significantly. The high surface mobilityat T = 1 .
25K results in decreasing the pressure in the container in the course of the fall of a crystal.The pressure drop exceeds the difference in the hydrostatic pressure for the initial and final positionsof the crystal. After the stop the pressure in the container relaxes to the difference mentioned above.This fact demonstrates an additional growth of the crystal in the flow of a superfluid liquid.
PACS numbers: 67.80. -s, 68.08. -p
INTRODUCTION
The fast kinetics of the atomically rough surfaces of He crystals [1] displays a remarkable interplay betweenthe fluid flow and the superfluid-solid interface. This factis mentioned by Nozieres, Uwaha [2], and M.Kagan [3],who theoretically studied the tangential instability of aflat liquid-solid interface. It is found that the instabil-ity of the cylindrical crystal shape is associated with thefast interface kinetics. The fluid flow in the direction ofthe cylindrical axis of a He crystal leads to developingthe instability on the smaller scales [4]. As it concernsthe faceted interfacial segments with low mobility, thefluid flow has no observable effect. However, there existconditions when the crystal facets acquire high mobilitycomparable with that of the atomically rough surface.Provided that a large overpressure is applied to the crys-talline facet, the growth rate of the facet increases dras-tically by 2 - 3 orders of magnitude in jump-like man-ner [5]. At the initial stage the He crystal grows athigh growth rates and giant accelerations of the interface.This situation is theoretically analyzed within the frame-work of the Rayleigh-Taylor and Richtmyer-Meshkov in-stabilities [6].The experimental test of theoretical conclusions iscomplicated by the requirement to prepare the fluid flowwith controlled parameters while the He crystal is inthe container. It has been observed that the fluid jetcreated with the heat flux [7], the oscillating loop [8] orthe motion of charges [4] distorts the crystalline surface.Observations of the crystal shapes at the stage of fastgrowth demonstrate the instability due to accelerationof liquid-solid interface [6]. All these observations havebeen performed with immobile crystals under the fluidflow induced by the methods mentioned above.Since the direct measurement of the velocity of fluid flow is difficult, the alternative method is to set a Hecrystal into motion in the immobile fluid. The veloc-ity of the crystal can be directly measured using videorecording of the crystal motion. The force exerted to thecrystal is determined by the method of setting the crystalinto motion. In a series of experiments a superconduct-ing loop is used for forcing the crystal to oscillate. Thecrystal is either pierced by the crossbar or is fixed on thecrossbar [8]. The oscillation frequency of such system isdetermined by the crystal mass and force arising from thefluid flow. The maximum velocity of the crystal, whichone can reach, is as large as 3 cm/s. In the first case, afurther enhancement of the oscillation amplitude resultsin the fast remelting of the crystal before its detachingfrom the crossbar. In the second case the crystal escapesfrom the crossbar.Another method is the study of free fall of a He crys-tal in the field of gravity. Here the motive force is theweight of the crystal, compensated partially by the forceof buoyancy. A direct observation of the crystal shapetransformation during the fall of the crystal has beenperformed by the Japanese group at 0.3K using a high-speed camera [9]. At these temperatures the shape ofthe He crystal is governed by the facets with low growthrates. As can be seen from this study, the effect of facetinterface kinetics upon the motion of the He crystal isinsignificant.In experiments which continue the study of burst-likegrowth effects within the 0.1 – 0.2K range we have no-ticed a curious feature [10]. As the crystal stops to grow,the mobility of crystalline facets remains very high [11].The crystal starts to remelt in the hydrostatic pressuregradient and then detaches from its nucleation site atthe wall in the upper part of the container. The fall ofthe crystal is accompanied by a decreasing pressure inthe container. It is unexpected and intriguing that thepressure changes non-monotonically. At first, the pres-sure drops below the difference in hydrostatic pressuresbetween the crystal nucleation site and the container bot-tom. Then the pressure relaxes to the pressure differenceindicated. Such behavior of the pressure implies two es-sential facts. Firstly, the pressure decrease in the courseof fall is possible only provided that the crystal is growingand its volume increases. In other words, as the crystalstops to grow in the burst-like growth regime, the crys-tal facets keep their high mobility. Secondly, the extrapressure drop and the following pressure relaxation proveunambiguously the increase of the crystal volume duringthe fall of the crystal not only due to hydrostatic pressurebut also due to an additional factor.In Ref. [10] the influence of fluid flow is involved as afactor explaining such an effect. The simplified model issuggested for the fall of an isotropic sphere with a mo-bile interfacial boundary which is encircled by the fluidflow. The fluid flow is supposed to be laminar as in alltheoretical works known so far [2–4, 6]. The model hasexplained the pressure behavior and given the estimatefor the magnitude of the averaged kinetic growth coeffi-cient K = 0 .
22 s/m, V = K ∆ µ . Here V is the interfacegrowth rate and ∆ µ is the difference in chemical poten-tials. However, the time behavior of pressure p ( t ) mea-sured experimentally in the course of the fall differs no-ticeably from the prediction of the model. The reasonsmay be as follows: growth anisotropy of crystal facets,specific trajectory of the fall and orientation of the crys-tal, specific features of fluid flow at such velocities ofencircling and so on.Many of these difficulties can be avoided by observ-ing the fall of helium crystals at 1.2 - 1.6K. Within thistemperature range all the liquid-solid He interface is inthe atomically rough state. The kinetic growth coeffi-cient is isotropic. The crystals have approximately spher-ical shape [12]. The numerical magnitude of the kineticgrowth coefficient for such temperatures lies within therange K ∼ . − . He crystals.
EXPERIMENTAL METHODS AND RESULTS
The crystals are grown in optical container similar tothose used in experiments previously [13, 14]. The innerdiameter of the container is 20 mm and the volume is10 cm . A capacitive pressure sensor is fixed on the up-per flange of the container. The pressure is measured at2 ms time intervals. The crystal nucleates at a tungstentip located at 10.5 mm distance from the bottom of thecontainer. A high voltage is applied to the tip. After FIG. 1: The crystal fall at two temperatures. The upper seriesis made at T = 1.54K, and the lower series at T = 1.25K. Theintervals between the frames from left to right: 60ms, 40msand 40ms. The light pulse duration is 5 µs . nucleation and growth, the crystal has 1-2 mm diameter.The hydrostatic pressure gradient leads to a remeltingof the crystal, while its center shifts down and it subse-quently separates from the tip [12]. The fall of the crystalis recorded by a television system with a ccd-matrix. Thecrystal is illuminated by a light pulse of an infrared LEDlasting 5 µs and synchronized with the frame. An exam-ple of shooting the fall of a crystal at two temperaturesis shown in Fig.1. The relaxation time of the pressurein the container to the pressure of the external system is ∼ ∼
160 ms atalmost constant mass of helium in the container. Notethat when the crystal is separated from the tip, a protru-sion remains on its surface, as was previously observedin Ref. [12]. Then, the protrusion flattens and at lowertemperature this occurs faster due to the higher kineticgrowth coefficient. The observation for the protrusionreveals that the crystal rotates in the course of its fall.The experiments were carried out at two temperatures.At T=1.54K, the crystal has the bcc structure, and atT=1.25K the crystal has the hcp structure. The kineticgrowth coefficient of an atomically rough surface in thistemperature range is determined by the energy dissipa-tion processes in the liquid. These processes are insen-sitive to the crystal structure [15]. The surface growthrate in both cases is linearly dependent on supersatura-tion and isotropic for small deviations from equilibrium.The kinetic growth coefficients measured from surface re-laxation [16] and from pressure relaxation [15] are con-sistent and equal to K = 0 . ± . m/s at 1.54K and K = 0 . ± . m/s at 1.25K.Fig.2 shows the results of measurements on the posi-tion of the crystal center H as a function of the fall time. H , c m t , sec FIG. 2: The position of crystal center versus the square ofthe fall time. The ordinate corresponds to the distance fromthe center of the crystals to the bottom of the container. Thesquares refer to the crystals grown at 1.25K and circles at1.54K V , c m / s e c a cc , c m / s e c t, sec FIG. 3: Above: The time behavior of the fall rate of crystalsshowing the velocity V ( t ) as a function of time. Below: thechange in the acceleration of fall over time. The squares referto the crystals grown at 1.25K and circles at 1.54K The solidcurves refer to 1.54K and the dashed lines to 1.25K. It can be seen that the temperature growth deceleratesduring the fall. This can also be seen in the video ofFig.1, where the crystals starting from the same posi-tion are at different heights after 140 ms. In Fig.3, toppanel, we see the time behavior of the fall rate V ( t ) ofcrystals, obtained by differentiating the smoothed data H ( t ). Crystals fall with acceleration. Lowering the tem-perature increases the fall rate of crystals. The wave onthe dependence V ( t ) at T = 1.54K is probably causedby the rotation of the crystal and the non-sphericity ofits shape. The Fig. 3, bottom panel, shows the change -0.4 -0.2 0.0 0.2 0.4-0.050.000.05 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.35-0.30-0.25-0.20-0.15-0.10-0.050.000.05 p , m b a r t, sec FIG. 4: Upper curve: the pressure change in the containerduring the fall of the crystal at 1.54K. The pressure differencebetween the beginning and the end of the fall is 0.026mbar.This value lies within the measurement noise magnitude of0.03mbar. Bottom curve: ∆ p ( t ) during the fall of the crystalat 1.25K. The arrows show the time of crystal fall. The dashedline corresponds to the difference in the hydrostatic pressurebetween the initial and final positions of the crystals. in acceleration in the process of fall, obtained by numer-ical differentiation of the upper graph. Acceleration isdetermined by the points included within the 20 ms timeinterval. To avoid dispersion in the velocity graph atT = 1.52K, the time interval is enlarged to ±
30 ms forthe point at 130 ms. The low accuracy in determiningthe acceleration allows us to draw qualitative conclusionsonly. The acceleration of the crystal motion decreaseswith time at both temperatures. The magnitudes of ac-celeration are within the range of 30 − cm/s . It ispossible that the acceleration at ∼ . K becomes con-stant after ∼ ms .Averaged over all measurements the results for thepressure variations are shown in Fig.4. To reduce noise,the records of ∆ p ( t ) in each series are combined by thetime of separating the crystal from the tip and summa-rized. One can see that the pressure variation differsdrastically for two temperatures. At high temperature,the change in pressure during the fall, if it occurred, doesnot exceed the measurement noise of ∼ . p = 0 .
16 mbar. The same kineticgrowth coefficients for the crystals that grow at differenttemperatures in the completely different states lead to aqualitatively identical change in pressure upon falling.The observations on the fall of the crystals at 1.16Kshow that the crystals have their flat facets facing down,resulting in floating during the motion, see Fig.5. Thelower facet is oriented perpendicular to the direction of
FIG. 5: The crystal fall illuminated with stroboscopic light:pulse duration 5 µ s, pulse interval 20 ms, 1.16K fall. The moment of forces acting on the body from theside by the liquid flow turns the crystal in this way [17].Below 0.6 K, the crystal facets have such low growth ratethat the nucleated crystals remain hanging on the tip fortens of minutes, as was previously observed [12]. DISCUSSION OF THE RESULTS
The crystal falls at temperatures where the normalcomponent density of superfluid helium is significant.Since the rates of fall are small and the fluid can beconsidered as incompressible, the equations of hydrody-namics are separated into the equations for the superfluidand normal components [18]. The equation for the nor-mal component reads ∂~v n ∂t + ( ~v n ∇ ) ~v n = − ρ n ∇ p n + η n ρ n △ ~v n (1)Here ~v n , p n , η n and ρ n are the velocity, pressure, viscosityand density of the normal component, respectively. Theequation can be transformed to the dimensionless repre-sentation similar to the classical Navier-Stokes equation: t → t RU , r → rR, v n → v n U, p n → ρ n U p n ∂v n ∂t + ( v n ∇ ) v n = −∇ p n + 1 Re n △ ~v n , (2) Re n = U R ρ n η n = U Rν n . Here U is the velocity of the body, R is its radius and Re n is the Reynolds parameter for the normal compo-nent flow. The normal component density is 0 . ρ at1.54K, 0 . ρ at 1.25K and 0 . ρ at 1.16K [19]. The vis-cosity of the normal component is 20 µP at 1.54K and23 µP at 1.25K [20]. For the crystal radius of ∼ mm and falling rates of 1-10 cm/sec, the Reynolds parame-ter varies within Re n = 400 – 4000 at 1.54K and Re n =100 – 1000 at 1.25K. There exist a lot of studies on themotion of spheres in helium at saturated vapor pressure.Without attempting to provide a complete description ofthese results, we use only a few works for illustrative pur-poses. The resistance to the motion of a ball in normalhelium is proportional to velocity squared and estimatedby the relation [21–23] F n = Cx n ρ n U πR . (3)The parameter Cx n is 0.2-0.5 in the indicated range ofthe Reynolds numbers.The superfluid component flow is described by theideal fluid equation △ ϕ = 0 , ~v s = ∇ ϕ. (4)The experimental studies of the motion of spherical bod-ies in superfluid helium show that resistance to motionis observed starting at velocities of ∼ . v crit = ( h/m R ) ln ( R/A ) [25], where h is the Plankconstant, m is the helium atom mass and A is the in-teratomic distance. The resistance to the motion of thesphere is approximately described by a relation similarto expression (3) [24, 26, 27] F s = Cx s ρ s U πR . (5)The parameter Cx s in the indicated range of theReynolds numbers is 0.3-0.5. A number of authors usethe Reynolds parameter Re s = Rv s /ν s to evaluate thenature of superfluid flow and turbulence. The value ν s = h/m ≃ − cm /s is taken as the kinematicviscosity. For the rates of crystal fall in these experi-ments, this corresponds to the values Re s = 100 − v surf = v n , that is, themass flow through the phase boundary is provided onlyby the superfluid component. This statement is consis-tent with the experimental fact that near λ -transition,when ρ s →
0, the kinetic growth coefficient also tends tozero [15]. Therefore, the motion of a sphere in the normalcomponent (2) occurs as the motion of an impenetrableball, whose size and shape, generally speaking, changeduring the motion. The superfluid flow around a spherewith the mass flow through its surface is considered inRef. [10] under the assumption that the superfluid flowis laminar. In this case, the crystal grows mainly in thedirection perpendicular to the flow of the liquid. Thischange in the shape leads to increasing the added massand, as a result, to decreasing the acceleration of move-ment. In our experiments, as is seen in Fig.1, the crystalsrotate during the fall. This leads to more uniform sur-face growth and small change in the shape of the crystal.It follows that the added mass of the superfluid liquidalso changes insignificantly, and does not lead to reduc-ing the acceleration during a fall within the frameworkof the model in Ref. [10].In our experiments, the flow around a crystal differssignificantly from that considered in Ref. [10]. The rota-tion of the crystal leads to the appearance of a Magnusforce perpendicular to the direction of motion. The for-mation of vortices in the normal and superfluid compo-nents creates a complex picture of the pressure distribu-tion on the crystal surface. For these reasons, the flow ofliquid around the crystal is very complicated. The prob-lem of crystal growth taking into account these factorshas not yet been solved.For the values of the Reynolds parameter in these ex-periments (see above), the crystal falls down with theformation of a vortex pattern in the normal componentand the creation of quantum vortices in the superfluidcomponent. Combining formulas (3) and (5), we obtainan expression for the total force F acting on the crystal F = ( Cx n κ n + Cx s κ s ) 12 ρU πR , κ n + κ s = 1 . (6)Here the coefficients κ s and κ n are the fractions of the su-perfluid and normal components, respectively. The equa-tion of motion of the crystal reads ρ ′ α π R ˙ U = − ( Cx n κ n + Cx s κ s ) 12 ρU πR + ∆ ρ π R g, (7)where ρ ′ is the density of the crystal, ∆ ρ is the differencein density between the solid and liquid phases, and α takes the added mass of the liquid into account. For asphere, the parameter α is equal to α = 1 + ρ ρ ′ ≃ . U ( t ) = U tanh (cid:18) tτ (cid:19) , < Cx > = Cx n κ n + Cx s κ s , (8) U = (cid:18)
83 ∆ ρρ gR< Cx > (cid:19) ≃ . r gR< Cx > ,τ = α (cid:18)
83 ( ρ ′ ) ρ ∆ ρ < Cx > Rg (cid:19) ≃ . s Rg < Cx > .
From the form of equation (7) it follows that if the con-dition Cx n > Cx s is fulfilled, then the lower rate ofcrystal fall at higher temperatures is explained by in-creasing the normal component fraction. The values ofthe Cx n,s coefficients at a liquid pressure 25 bar are un-known. Assuming these values to be of the order of Cx n,s at the saturated vapor pressure for solid balls [24], weobtain an estimate for the parameters U and τ . For < Cx > = 0 . U ≈ τ ≈
120 ms. For < Cx > = 0 . U ≈
17 cm/s and τ ≈
260 ms. As can beseen from Fig. 2, the second pair of parameters is betterconsistent with the experimental data. These estimatesshould be considered as qualitative since the essential fea-tures of the process, for example, such as the rotation ofthe crystal and the possible dependence of the Reynoldsparameter on velocity, are not taken into account whenderiving equation (7).The decrease in pressure during the fall of the crys-tal is clear evidence for the growth of its volume. Thechange in crystal volume is due to two factors. First,due to growth in the denser liquid layers at the bot-tom of the container, where the pressure exceeds thatfor the liquid in the initial position of the crystal by∆ p = ρg ∆ h = 0 . p rel ≃ . mbar . The total increasein the crystal volume with an initial size of ∼ mm willbe ∼ p ( t ) at two tempera-tures shows, see Fig. 4, that a decrease in the fraction ofthe superfluid component leads to a practically constantvolume of the crystal during the fall. This results fromthe small pressure variation in the container. As is notedabove, this conclusion agrees with the observation thatthe crystal grows from the superfluid component. CONCLUSION
The study of falling the crystals in superfluid heliumhas shown a significant effect of the normal componenton the resistance to motion. This is clearly seen in thedependencies V ( t ) in Fig.3. For the fluid of the 41%normal component fraction, the velocity of motion is no-ticeably lower than that in a fluid with the 14% normalcomponent concentration. The influence of the interfacegrowth kinetics is clearly shown by the pressure mea-surements during the fall. The low kinetic coefficient ofinterface growth at 1.54K does not result in a noticeablecrystal growth either due to gradient of hydrostatic pres-sure or pressure of the liquid flow. The increase in theinterface mobility at 1.25K demonstrates these both ef-fects. When falling, the pressure decreases due to crystalgrowth under influence of both effects, and then increasesafter stopping the crystal at the bottom of the containerdue to melting the excess crystal volume as a result ofthe liquid flow effect. The experimental results have con-firmed the hypothesis that the nonmonotonic behaviorpressure observed for the fall of crystals nucleated in theburst-like growth mode at ∼ . ∼ .
6s af-ter growing a crystal in the burst-like growth mode [10]exceeds the growth rate of crystal facets in the normalstate by ∼
20 times [29].The author is grateful to V. V. Dmitriev for the pos-sibility of performing these experiments at Kapitza In-stitute for Physical Problems RAS. The author is alsograteful to V. V. Zavyalov for supporting this work,S.N.Burmistrov for helpful comments and V. S. Kruglovfor interest to the work. ∗ [email protected][1] A. F. Andreev and A. Ya. Parshin, Sov. Phys. JETP ,763 (1978).[2] P. Nozieres and M. Uwaha, J.de Physique , , 263 (1986)[3] M. Yu. Kagan, JETP , 288 (1986).[4] L. A. Maksimov and V. L. Tsymbalenko, JETP , 455(2002).[5] V. L. Tsymbalenko, Physics-Uspekhi , , 1059 (2015)[6] S. N. Burmistrov, L. B. Dubovskii and V. L. Tsym-balenko,, Phys.Rev.E , , 051606 (2009)[7] A. V. Babkin, D. B. Kopeliovich and A. Ya. Parshin, Sov. Phys. JETP , 1322 (1985).[8] V. L. Tsymbalenko, J. Low Temp. Phys. , 21 (2013)[9] R. Nomura, Y. Okuda, A. Tachiki and T. Yoshida,
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