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Stability of multi-electron bubbles in liquid helium

The stability of multi-electron bubbles in liquid helium is investigated theoretically. We find that multi-electron bubbles are unstable against fission whenever the pressure is positive. It is shown that for moving bubbles the Bernoulli effect can result in a range of pressures over which the bubbles are stable.


I. INTRODUCTION
Multi-electron bubbles in liquid helium were first observed by Volodin et al. 1 In their experiment a layer of electrons was held in place just above the free surface of a bath of liquid helium by an electric field. The field was produced by a positive voltage applied to an electrode immersed in the liquid. The electrons remained outside the helium because for an electron to enter liquid helium it has to overcome a potential barrier of height approximately 1 eV. 2 When the field reached a critical value, the surface of the liquid became unstable and a large number of electrons entered into the liquid through the formation of bubbles. Each of these bubbles typically contained 10 7 ∼ 10 8 electrons. The multi-electron bubbles are of interest because they could possibly provide a way to study a number of properties of an electron gas on a curved surface. 3 As a first approximation, one can consider that the radius of a spherical multi-electron bubble (MEB) is such as to minimize the sum of the energy associated with the Coulomb repulsion of the electrons and the surface energy of the bubble. This gives an equilibrium radius of where Z is the number of electrons, σ is the surface tension of helium (0.36 erg cm −2 at 1.3 K), 4 ǫ is the dielectric constant (1.0573 at low temperature), and the applied pressure has for the moment been taken to be zero. Thus, for example, for N = 10 7 the radius is 106 µm.
So far, there have been a very limited number of experimental studies of these bubbles. 1,5,6,7 In this paper we first consider the stability of an MEB that is at rest in the liquid (section II). We find that, at least when the simplest model of the energy of the electron system is used, the bubble is unstable against fission whenever the applied pressure is positive. In section III we investigate how the stability of a bubble is changed when it is moving through the liquid. We have been able to determine the region in the pressure-velocity plane where the bubble is stable.

II. STABILITY OF BUBBLES AT REST
Since MEB's were first observed, there have been several theoretical investigations of the stability of these objects.
The first discussion was given by Shikin 8 and further analysis has been given by Salomaa and Williams, 9,10,11 and Tempere, Silvera and coworkers. 3,12,13,14 In the simplest model, the electrons are taken to be distributed over the inner surface of the bubble in a way such that the electric field is everywhere exactly normal to the surface. This ensures that the charge distribution is in equilibrium. The electrons are treated classically and so are localized at the surface in a layer of zero thickness (see below). Thus the total energy of the bubble is taken to be Here E S = σS is the surface energy with S the surface area and σ the surface tension, E V = P V the volume energy with P the applied pressure and V the bubble volume, and E C is the Coulomb energy given by Since the electrons can move freely around the surface, the field E inside the bubble must be zero and so the integral in Eq. (3) can be restricted to the region outside the bubble. If the bubble is spherical, the bubble radius that gives the minimum value of the energy is the solution of the equation For zero applied pressure, this gives the total energy of an MEB as Since the energy is proportional to Z 4/3 the energy is always reduced if the bubble breaks into two. Hence, in the discussion of stability given here we are not considering whether the energy of the bubble can be lowered if it breaks into pieces, but are trying to determine whether there is an energy barrier that prevents the bubble from breaking.
To consider whether the spherical shape is stable, write where η l,−m = η * lm . It is straightforward to show that to second order in the parameters η lm , the three contributions to the energy can be written as Hence the total energy is where the spring coefficients α l are given by and for l ≥ 1. From this one can see that the bubble is stable against spherically symmetric perturbations provided that 3σR 2 0 + 2P R 3 0 > 0. This leads to the condition P > P c where For l = 1, the spring coefficient α 1 is zero; this is to be expected since a perturbation of the form η 1m Y 1m (θ, φ) corresponds to a simple translation of the bubble in some direction. For l = 2 the spring constant α 2 is zero if the pressure is zero, and so this analysis of the effect of small perturbations to the initial spherical shape does not determine the stability of the bubble. The higher l spring constants are all positive at zero pressure but each becomes negative if the pressure is increased to a sufficiently positive value. It was noted by Tempere et al. 12 that if the pressure is negative (but not negative with respect to P c ), all of the spring constants will be positive 15 and so the bubble must be stable. The stability of the bubble at zero pressure is of especial importance since in the experiments that have been performed so far there has been no applied pressure apart from the very small hydrostatic pressure due to the distance the bubble is below the free surface. At zero pressure α 2 is zero, and so we need to go beyond the lowest order in perturbation theory in order to investigate the stability of an MEB at zero pressure. One approach would be to calculate the terms in the energy that are fourth order in the η lm parameters. Instead we have performed numerical calculations of the total energy as a function of bubble shape.
To do this, we describe the shape of the bubble using Eq. (6) but now do not restrict the parameters η lm to being small. When the bubble shape changes, the electrons will redistribute themselves over the surface so as to minimize the energy and to make the electric field inside the bubble zero. For each choice of shape we use the finite element method 16 to compute the surface charge distribution and the Coulomb energy. The simulation uses 1280 triangle patches. We start with a spherical shape and vary the parameters η lm to see if a state of lower energy can be reached without passing over a barrier. We have done this using a maximum value of 5 for l in Eq. (6). This process was then repeated for a series of different pressures. We also performed similar calculations with a maximum value of l of 15 but taking only m = 0. Both procedures gave the same results for the stability.
The result of this investigation is that for all positive pressures there is no barrier to fission, whereas for negative pressures there is a barrier. This result holds for all values of Z. To illustrate the path to fission, we describe results obtained for a simplified calculation in which only l = 0 and l = 2 contributions are retained. Thus we write Within this simplified model, fission occurs when a 2 = a 0 and the bubble develops a hole along the z-axis, i.e., takes on a donut shape, when a 2 = −a 0 /3. In Fig. 1(a), we show examples of contour plots of the energy in the a 0a 2 plane. The pressure is −0.03 mbar and Z = 10 6 . There is a stable minimum with a 2 equal to zero, i.e., the bubble is spherical. When the pressure is zero ( Fig. 1(b)), there is still a point in the plane at which the energy of the bubble is stationary with respect to both a 0 and a 2 (at a 0 = 23.8 µm and a 2 = 0), but it is now possible to reach the fission line from this point without passing over any energy barrier. Note that along this path there is, of course, an increase in the value of a 2 but also a substantial decrease in a 0 . Once the pressure becomes positive (see, for example, Fig. 1(c)), there is no point in the a 0 -a 2 plane where the energy is stationary.
These results can be compared with the earlier calculations by Tempere et al. 14 who also investigated the stability against fission. They used an ingenious method in which the bubble was described by 6 parameters chosen so that the shape of a bubble undergoing fission could consist of two spheroids connected by a hyperboloidal neck. The choice of parameters was such that the bubble could vary from consisting of a single sphere, to an ellipsoid, and then all the way to separated spheres. They minimized the total energy of the bubble by adjusting these parameters subject only to the constraint that the total length L of the bubble had to have a given value. They then investigated how the total energy varied with L starting from a value of L equal to 2R 0 . If the energy decreased monotonically as L increased from 2R 0 to a large value, this indicated that the MEB was unstable against fission. If the energy first increases before decreasing, this indicates that the bubble is stable. To simplify the calculation, Tempere et al. made the approximation that the charge density was uniform over the surface of the bubble. They concluded that at zero pressure even though there is a mode of deformation (the l = 2 mode) which can grow without increasing the energy of the bubble, there should be an energy barrier which prevents fission, 17 whereas we find no barrier.
This difference in the results arises from the treatment of the charge distribution on the bubble. If the bubble is assumed to have surface charge density that remains uniform when the shape changes, it is straightforward to show that the Coulomb energy for small changes from the equilibrium spherical shape is In this case the spring constant α ′ l for the l-th mode (considering only l ≥ 1) becomes Comparing this with the spring constant α l when the charge redistributes (Eq. (12)) gives Thus for all modes, except l = 0 and l = 1, making the approximation of a uniform surface charge gives an increase in stiffness and makes it harder for the bubble to undergo fission. The increase in stiffness is to be expected since a redistribution of surface charge can only lower the total energy. In Fig. 2, we show energy contour lines in the a 0 -a 2 plane for an MEB with 10 6 electrons at zero pressure calculated by taking a uniform surface charge.
One can see that within this approximation the spherical bubble is stable.
There are several physical effects that are not included in the simplified model used so far. It is possible that allowance for these effects would change the stability of an MEB at zero pressure. A more detailed consideration of the Coulomb energy (the total electron energy, to be more precise) for a spherical bubble has been given by Salomaa and Williams 9,10 using the density functional formalism of Hohenberg and Kohn. 18 This makes possible the inclusion of the kinetic, exchange and correlation energies, but how these extra contributions affect the spring constants is not clear and is difficult to calculate. Salomaa and Williams show that these extra contributions to the energy make a very small contribution to the energy when Z is large. For example, for Z = 10 8 the extra terms make a contribution that is roughly 4000 times smaller than the form Z 2 e 2 /2ǫR 0 for the energy used in the simple model. The calculation could also be improved, for example, by using a density functional theory to treat the surface of the liquid helium, and by allowing for the penetration of the electron wave function into the liquid. All of these effects appear to be very small corrections to the total energy and hence are unlikely to change the spring constants by a large amount. However, it is important to note that even a small correction could lead to a positive value for α 2 which would in turn lead to a finite (but small) energy barrier against fission. As an example, consider corrections that arise as a result of using a density-functional scheme to describe the helium. For a bubble with radius large compared to the thickness of the liquid-vapor interface the first correction to the energy can be represented by considering the surface tension σ to contain a correction that is proportional to the total curvature κ of the surface. Based on a simple density functional scheme used previously, 19 it is straightforward to show that the correction to the surface tension is ∆σ = σ ′ κ, where σ ′ = 0.9 × 10 −8 erg cm −1 , (18) and the sign of the correction is such that the surface tension is increased for a concave surface of the liquid. Inclusion of this term changes the total energy by an amount ∆E which for a bubble at zero pressure is given by It is straightforward to show that the spring constant for an l = 2 deformation at zero pressure now becomes Because α 2 is now positive at zero pressure there will be a barrier against fission, but clearly for large Z (e.g. Z ∼ 10 8 ), this barrier will be very small.

III. STABILITY OF MOVING BUBBLES
The above results indicate that one way to stabilize an MEB is to produce it in liquid that is under a small negative pressure. We now consider an alternate way to maintain a stable bubble. A bubble moving through a liquid will be affected by the local pressure change associated with the liquid moving around it. For a spherical bubble moving at velocity v through an incompressible inviscid fluid with density ρ, the Bernoulli effect results in a pressure variation over the surface of the bubble which is given by 20 P (θ) = P 0 + 1 8 ρv 2 (9 cos 2 θ − 5) For a bubble in liquid that is at zero pressure far removed from the bubble (P 0 = 0), this changes the shape of the bubble in two ways. The term proportional to Y 00 by itself would provide a negative pressure around the surface of the bubble and since bubbles are stable at negative pressure, this contribution serves to stabilize the bubble. The second term gives a positive pressure at the poles of the bubble and a negative pressure around the waist. This pressure distribution will distort a spherical bubble so as to make the parameter η 20 in Eq. (6), or a 2 in Eq. (14), to be negative. This tends to stabilize the bubble since, as can be seen from Fig. 1, for fission to occur a 2 has to become positive.
We have performed computer simulations in order to find the shape of moving bubbles and the range of velocity and pressure for which they are stable. We start with a guess at the bubble shape and then calculate the charge distribution on the surface. This then gives the pressure ∆P el (θ) exerted on the surface by the electrons. We then find the flow in the liquid. To do this we expand the velocity potential as where B l are some coefficients and the sum includes terms from l = 1 to l = 20. The coefficients are determined so as to give a velocity distribution in the liquid such that in the frame of reference of the moving bubble, the liquid velocity at the bubble surface in the direction normal to bubble surface is as close to zero as possible. This gives a pressure at the bubble surface of where P 0 is the pressure in the bulk liquid far removed from the bubble and ∆P B (θ) is the Bernoulli pressure.
The net inward force acting on unit area of the bubble surface is then where κ is the total curvature of the surface and ∆P el (θ) is the outward pressure exerted by the electrons. Each part of the bubble surface is then moved inward a distance proportional to this force, and the process repeated until the equilibrium shape is found. The calculation used a maximum value of l of 15. The calculation as just described is based on the assumption that the bubble shape and velocity field have axial symmetry around the direction in which the bubble is moving. In order to test this assumption, we also performed a calculation in which axial symmetry was not assumed. This calculation used values of l up to 5 and |m| ≤ 5. This calculation showed that the axially symmetric shape was stable. For an MEB with Z = 10 6 shapes for three velocities are shown in Fig. 3. We are able to perform the numerical calculation until the bubble becomes concave at the poles. This is shown by the dashed line in Fig. 5. In Fig. 4, the distance R pole from the bubble center to the pole and the radius R waist of the waist are shown as a function of the velocity. In Fig. 5 we show a plot of the region in the pressure-velocity plane in which the bubble is stable. This region is bounded by two lines. For small velocities there is a critical positive pressure at which the bubble undergoes fission. At negative pressures the bubble becomes unstable against expansion. For zero velocity this expansion is isotropic.
Note that the change in the shape of the bubble even for a small velocity is surprisingly large. This comes about simply because the Bernoulli pressure contains a finite term varying with angle as Y 20 (θ, φ) but the spring constant α 2 for this pressure component is zero. Thus, for an MEB the changes in R pole and R waist are linearly proportional to the bubble velocity whereas for a gas bubble in a liquid the spring constant α 2 is finite and so the changes in dimensions are proportional to the square of the velocity. where f is a dimensionless function, A = (σ 4 ǫ/e 2 ) 1/3 and B = ρ(e 2 /σ 4 ǫ) 1/3 . Thus, for zero velocity the critical negative pressure at which a bubble becomes unstable is proportional to Z −2/3 , and at zero applied pressure the critical velocity at which the bubble becomes concave at the poles occurs is proportional to Z −1/3 . We note that in this paper we have treated the liquid as inviscid although, of course, helium above the lambda point has a finite viscosity and below the lambda point the liquid still has a normal fluid component. At sufficiently low temperatures the density of the normal fluid becomes very small and, in addition, the mean free path of the excitations making up the normal fluid becomes comparable to the radius of an MEB. Under these conditions, it appears that the only effect of the normal fluid is to determine the mobility of an MEB and there should be no effect on the shape change or the stability. For a bubble with Z = 10 6 the mean free path of the thermal excitations becomes equal to the radius at around 0.6 K and at this temperature the normal fluid density is less than the total density by a factor of 4 × 10 −5 . But as far as we are aware, there have been no experiments with MEB's at low temperature.
At high temperatures where the helium is in the normal state, the situation is not so clear. It is known that when the Reynolds number is large (but not so large that the flow becomes turbulent) the viscosity results in a thin boundary layer on the surface of the bubble and the pressure at the bubble surface is close to the value that would result from potential flow. 21 This general idea would suggest that the inviscid approximation should give reliable results for the stability of MEB's over a wide range of Reynolds number. To determine this range one could calculate the effect of viscosity using the method developed by Li and Yan 22 and applied by them to calculate the shape and drag on gas bubbles moving through a liquid. We have not attempted to do this. We note that, Albrecht and Leiderer 7 in their experiments at 3.5 K found that the mobility of the MEB's was between one and two orders of magnitude larger than expected on the basis of ordinary hydrodynamics. The reason for this is unknown.
In the experiments of Volodin et al. 1 and Khaikin 5 which were performed at 1.3 K, velocities of the order of 10 4 cm s −1 were reported. At these velocities the normal fluid component would be in the turbulent regime and the bubble is moving so fast that it should lose energy through the production of quantized vortex rings.

IV. CONCLUSION
We have examined the stability of multi-electron bubbles in liquid helium and found that stationary bubbles at positive pressures are unstable. We show that because of the Bernoulli effect moving bubbles can be stable even at small positive pressures.