Dynamics of fine particles due to quantized vortices on the surface of superfluid 4 He
NNoname manuscript No. (will be inserted by the editor)
Dynamics of fine particles due to quantized vorticeson the surface of superfluid He Sosuke Inui · Makoto Tsubota · PeterMoroshkin · Paul Leiderer · KimitoshiKono.
Received: date / Accepted: date
Abstract
We have conducted calculations of the coupled dynamics of quan-tized vortices and fine metallic particles trapped at a free surface of superfluid He. The computational result so far indicates that a particle-vortex complexmay produce quasi-periodic motions along the surface and that the motionscan be enhanced if the metallic particles are heated and induce local radialflows. Our results qualitatively reproduce recent experimental observations oftrapped particle motion.
Keywords quantized vortices · free surface of He A new dynamical aspect of superfluid He that involves both bulk quantizedvortices and the free surface of the system may have been revealed by theexperiment utilizing a number of electrically charged fine metallic particles
S. InuiOsaka City University, Osaka, JapanTel.: +81-66-605-2530E-mail: [email protected]. TsubotaOsaka City University, Osaka, JapanP. MoroshkinOkinawa Institute of Science and Technology, Okinawa, JapanP. LeidererUniversity of Konstanz, Konstanz, GermanyK. KonoNational Chiao Tung University, Taiwan a r X i v : . [ c ond - m a t . o t h e r] S e p Short form of author list trapped just below the surface of superfluid He by Moroshkin et al . Theyproduce the particles by a laser ablation of a metallic target submerged insuperfluid He [1, 2], and trap them just below the free surface by using acombination of surface tension and the applied static electric field. The parti-cles are illuminated with the expanded beam of a cw laser, and their motionsare tracked. Most of the particles rest at their equilibrium positions; however,some of them are observed to exhibit anomalous quasi-periodic motions thatare largely classified into two types; (1) circular motions with specific frequen-cies and radii, and (2) quasi-linear oscillations with sharp turning points. [3]The injection of fine particles into superfluid He is a method that has beenappreciated for decades for various purposes and has achieved a big successin various cases [4–6], but first observations of particles at a free surface havebeen reported only recently [7]. We discuss that the motion of type (1) can becaused by a relatively simple vortex-particle configuration: a vertical vortexfilament whose edges are terminated at the bottom of the vessel and at theparticle just below the surface of He. In a number of experiments with He + ions (or “snow balls”) and negative ions (or “electron bubbles”) at the surfaceof superfluid He ever conducted [8,9], the particles of type (1) and (2) were notobserved either, yet these experimental set-ups and the one used by Moroshkin et al . have similarities. However, there is a decisive difference. That is that theilluminated metallic particles may act as local “heaters” in the experiment byMoroshkin et al ., and it can be estimated that a particle can be substantiallyheated by the intense laser beam and create a local radial counter flow with avelocity at the particle surface of order of 30 cm/s.
The anomalous motions of type (1) and (2) are presumably due to the inter-actions between quantized vortices and particles. Thus, the coupled dynamicsof quantized vortices and particles has to be considered. That calculation iscarried out based on the vortex filament model [10–12]. In this model, vor-tices are treated as very thin filaments, and they are discretized into finiteamount of segments. Each segment is represented by a point sss ( ξ ), where ξ is the arc length parameterization along the vortices. The distance betweenthe neighbouring vortex points determines the vortex resolution ∆ξ . In orderto take account for the particle-vortex interaction, we introduce the equationof motion for a trapped particle as described in Ref. [13]. By making threeassumptions, we are able to let a vortex point represent the trapped particle.The assumptions are: (i) the particle radius is much smaller than the vortexresolution, (ii) a vortex reconnects with the center of the particle, and (iii) theparticle and the vortex segment trapping it, in average, travel with the samevelocity.The initial vortex-particle configuration considered in this set of calcu-lations is shown in Fig. 1(a). A straight vertical vortex filament trapping aparticle on its top end sits still at the center of the system, and we excite the itle Suppressed Due to Excessive Length 3 (a) Initial configuration Mirror Vortex
Helium II Surface
Vortex Filament (b) Zoomed in on the particle
Fig. 1: (a) Illustration of the initial configuration of the system. The top of thebox corresponds to the He surface, and a particle of radius 5 µ m (a red ball)is trapped at the top of a straight vertical vortex filament (a blue line). Weexcite the vertical filament by letting a small vortex ring (a blue ring) collidewith it. (b) The trapped particle at the surface after the excitation. Only thetip of the vortex bends and starts to oscillate. −15 −10 −5 0 5 10 15 x [μm] −10−50510 μ [ μ m ] (a) Top view Time [s] −0.02−0.010.000.010.02 P a r t i c l e t r a j e c t o r y [ mm ] in x-directionin y-direction (b) In x , y -directions Fig. 2: A typical particle trajectory on the surface. At the beginning it showsa large scale motion, but it decays quickly as a small scale circular motion ofperiod T traj and radius R traj appears at around 0 . He surface. Inorder to reproduce the phenomena found in the experiment, we give suitablevalues to the parameters; the system is a cubic box of length 2 mm, and aparticle of radius a few µ m is trapped at the top of the box filled with He.At the top and the bottom surfaces we assume that the superflow is subjectto the solid boundary condition. The vortex resolution ∆ξ is set to be 10 µ m,which is larger than the particle radius, and the temperature is set to be 1 . Short form of author list
Density of particle ρ p [kg/m ] O s c ill a t i o n p e r i o d T t r a j [ s ] (a) x-directionT traj =0.0001ρ p ^0.6925+0.0009 Density of particle ρ p [kg/m ] O s c ill a t i o n r a d i u s R t r a j [ μ m ] (b) x-directionR traj =0.2808ρ p ^0.1603−0.1713 Fig. 3: (a) Period and (b) radius of the circular motion as a function of theparticle density ρ p . Both T traj and R traj values are obtained by averagingover some time after the spiral motion decays sufficiently, and error bars areobtained by taking the standard deviations. Since there are no noticeabledifferences in the time-averaged motion in x and y -directions, each plot isrepresented by the motion in x -direction. Here, the particle radius a p is fixedto be 5 µ m. Both plots are fitted with a function of form y = αx β + γ as shownin the legends and represented by the black dashed curves. ρ p = 3594 . corresponds to the density of Ba at room temperature. A typical motion of a particle trapped by a vertical vortex is shown in Fig. 2.The particle is initially at rest at the origin which is an equilibrium position.Once it is excited, the particle is kicked out of the origin and starts to spiralout. However, the spiral motion decays as a small circular motion appears.While the detail of spiral motion depends on the excitation, the small circularmotion seems to be intrinsic for the particle-vortex complex. We have examinedthe small circular motion by changing (i) the density and (ii) the volume ofthe particle.It turns out that the period T traj and trajectory radius R traj of the intrinsiccircular motion strongly depend on its density ρ p as shown in Fig. 3. Althoughthe accurate asymptotic behaviours as ρ p approaches zero are not determined,we need to note that the fitted plot of Fig. 3(b) starts to drop quickly when ρ p (cid:46)
200 kg/m . This indicates that it is difficult to detect the type (1) circularmotions in the experiments with the light particles such as solid hydrogen( ∼ ) . Not only the density but the volume of the particle also determinesthe motion. The plots of the period and the radius of the circular motion asfunctions of the particle radius a p are shown in Fig. 4.Although the Maryland group succeeded in visualizing a vortex filamentby letting solid hydrogen particles be trapped onto it in series [4], meaningthey have directly observed the particles trapped onto the vortices, the type(1) motions were not reported. It seems that that is largely because solidhydrogen was used for the tracer particles. In the experiment by Moroshkin et al ., on the other hand, heavier metals such as Ba (3594 . at room itle Suppressed Due to Excessive Length 5 Particle radius a p [μm] O s c ill a t i o n p e r i o d μ t r a j [ s ] (a) x-directionμ traj =0.0011ρ p ^2.2198+0.0008 Particle radius a p [μm] O s c ill a t i o n r a d i u s μ t r a j [ μ m ] (b) x-directionμ traj =0.1386ρ p +0.1730 Fig. 4: (a) Period T traj and (b) radius R traj of the circular motion as a functionof the particle radius a p . The T traj and R traj values are obtained by time-averaging in the same way as in Fig.3, and the plots are represented by themotions in x -direction. The particle density is set to be that of barium. Theplots are fitted with the functions shown in the legends.temperature) are used, which makes it possible to observe the new types ofmotions.Intuitively, the origin of the type (1) motion with this configuration canbe qualitatively understood as follows: We consider a straight vertical vortexfilament trapping a particle at the surface. The vortex creates a concentricvelocity field vvv s,BS that drops inversely proportional to the distance from thecenter. We assume that the small displacements of the particle and the tipof the vortex trapping the particle (see Fig. 1(b)) do not perturb the velocityprofile significantly. When the particle is out of the origin, it shows the circularmotion about a new equilibrium location which is determined by balancing thetwo forces acting on the vortex segment; the Magnus force FFF M and the tension FFF T . The Magnus force is given by FFF M ∝ sss (cid:48) × [˙ sss − vvv s,BS ] , (1)where sss is the filament, and the prime and the dot symbols represent thederivatives with respect to arc length and time, respectively. This force actsoutward and is dominant when vvv s,BS is large. On the other hand, the tension,given by FFF t ∝ sss (cid:48)(cid:48) , (2)only depends on the local curvature (1 / | sss (cid:48)(cid:48) | ) of the filament. As the particlegets farther the local curvature of the vortex (and its mirror vortex) becomessmaller, and the tension force that pulls the particle backward becomes domi-nant. Although we can qualitatively explain the origin of the type (1) motion,we need to note that the simulated radius of the trajectory is smaller by a fac-tor of more than 10 than that found in the experiment, while the periods arein the same order. This discrepancy may be explained by an effective radiusmentioned in Sec.4. Short form of author list
Normal flow Super flow (a) (b)
Fig. 5: (a) Illustration of the local radial counter flow around a heated particle.Normal flow is radiated away from the particle, while superflow is toward theparticle. The figure (b) shows how a particle of radius 5 µ m attracts a vortexthat is initially separated by a distance 50 µ m and reconnects with it in acubic system of length 200 µ m. Filament positions are shown with a time stepof 4 × − s, and the whole process occurs within 0 .
04 s.
If a particle is heated, it should establish a local radial counter flow as shown inFig.5(a), and the relative velocity of normal and superfluid components shouldbe given by v = QρST , (3)where Q is the heat flux, ρ is the density, S is the entropy of the normal compo-nent, and T is the temperature of the system. In the experiment by Moroshkin et al ., the largest possible heat flux at the particle surface illuminated by thelaser is estimated as several W/cm . The counterflow velocity predicted byEq. (3) for Q = 1 W/cm is 37 cm/s, which is significantly larger than thecritical vortex tangle velocity ( ∼ ∼ . a p , where a p is the particle radius.That could result in two effects; one is in the increase of effective particleradius by carrying both normal and superfluid with it, and the other is inattracting free quantized vortices toward it. The latter effect is confirmedcomputationally by introducing a particle that sets up a steady local counterflow, neglecting the motion of the particle for the simplicity.Figure 5( b ) shows the case where the particle of radius 5 µ m emits a heatflow of 1 W/cm and depicts how a vertical vortex filament initially placed50 µ m away from the particle is moving toward it. This attraction seems toenhance the possibility that a particle is trapped and exhibits the type (1)motion. The origin of the type (2) motions, quasi-linear oscillations, is stillunder investigation, but the effects of particles creating local counter flowsseem to be a clue to understand the phenomena. itle Suppressed Due to Excessive Length 7 In summary, we have conducted calculations of the coupled dynamics of aquantized vortex and a fine particle to investigate the vortex-particle inter-actions at the surface of superfluid He based on the vortex filament model.By considering a simple initial configuration, we can reproduce a motion thatqualitatively explains the origin of the type (1) motion. This scenario also sug-gests that the type (1) motion due to a particle whose mass density is less thanthat of solid helium would be difficult to observe. It is also confirmed that aheated particle is likely to attract free vortices through the local counter flow,which enhances the chance for a particle to be trapped by vortices and toexhibit the anomalous motions.The analysis so far has been made mainly without treating the particle aslocal heaters. In the future work we would like to investigate more closely theeffects of the local counter flows and discuss how the motion of type (2) comesabout.
References
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