Particles for tracing turbulent liquid helium

Abstract

We address the problem of making quantitative measurements of local flow velocities in turbulent liquid helium, using tracer particles. We survey and evaluate presently available particles and previous work to establish the need to develop new particles for the purpose. We present the first practical solution for visualizing fluid motions using a suspension of solid hydrogen particles with diameters of about 2 μm. The hydrogen particles can be used to study flows with Taylor-microscale Reynolds numbers between 85 and 775. The particles can be used equally well with the PIV, LDV, or particle-tracking techniques.

Appendix

We have observed the aggregation of several types of particles in liquid helium, and we argue here that aggregation may be unavoidable in this fluid. The forces that draw particles together are the same as the ones that binds the molecules of the particle together to form a solid, known as the van der Waals forces. This force overwhelms Brownian motion for particles that come close to each other, as we see if we consider the attractive potential, U _D, between two spheres with center to center separation r and with radius a,

$$ U_{{\text{D}}} = \frac{{ - A}} {6}{\left[ {\frac{{2a^{2} }} {{r^{2} - 4a^{2} }} + \frac{{2a^{2} }} {{r^{2} }} + \ln {\left( {\frac{{r^{2} - 4a^{2} }} {{r^{2} }}} \right)}} \right]}, $$

(19)

which is a function of Hamaker’s constant, A (Vold and Vold 1983). Thermal energy is measured by E _T = k _B T ≈ 6 × 10⁻²³ J, where k _B is Boltzmann’s constant, and T is the temperature, and the value is given for T = 4.2 K. The value of r for which U _D = E _T is approximately r _c = 4a, using Hamaker’s constant for polystyrene, A = 8 × 10⁻²⁰ J, although the Hamaker constant for most materials is of the same order. Note that an exception is liquid helium, and since Hamaker’s constant for liquid helium is much smaller (Paalanen and Iye 1985), we do not need to account for it as the intervening fluid (Hiemenz and Rajagopalan 1997). Within 2 diameters, the van der Waals potential rises sharply, and it is therefore unlikely that thermal fluctuations will separate particles under its influence.

Most methods to prevent particles from coalescing under the influence of the van der Waals potential are inapplicable in liquid helium. The first, steric stabilization, can be accomplished by manufacturing particles with a surface coating or by adding a surfactant. This solution depends on the flexibility of polymer molecules (Vold and Vold 1983), and in liquid helium the mechanism fails because its temperature is below the glass transition temperature of all polymers. The second method uses electrostatic repulsion. To exploit this force, one can manufacture particles with polar groups on the particle surface, or cause the particles to accumulate free ions. The dissociation of an ion that allows the formation of a polar group requires a polar solvent, such as water. Since helium is nonpolar, this solution is inapplicable. However, adding an electrostatic charge to particles was implemented successfully in liquid nitrogen (Huber and Wirth 2003), and the method is potentially useful for dispersing particles in liquid helium.

Since the van der Waals potential given in (19) diverges to infinity as the separation between particles becomes smaller, the theory predicts that particles are inseparable once they come into contact. Without a method to prevent particles from coalescing, one must know the mechanical properties of the particles, or perform experiments to measure the strength of the bond between particles under the relevant conditions, to understand how to break particle aggregates apart. We are not aware of such work having been done.

It is known that the shear inherent to turbulence both causes particle collisions (Saffman and Turner 1956) and breaks apart particle aggregates (Sonntag and Russel 1986). Based on the above discussions, we assume that turbulence cannot break aggregates at the intensity we are able to generate it. If particles are inseparable, we estimate the rate of aggregate growth as follows. Saffman and Turner (1956) give a solution for the particle number density as a function of time under the assumption that the collision frequency is independent of particle size:

$$ n = {n_{0} } \mathord{\left/ {\vphantom {{n_{0} } {{\left( {1 + \alpha n_{0} t} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \alpha n_{0} t} \right)}}. $$

(20)

Here, n is the total number of particles per unit volume, with initial value n ₀, and α is the collision frequency, where α = 1.30a ³ νRe ^3/2/L ². The time for the particle number to halve is

$$ \tau _{{1/2}} \approx {\pi L^{2} } \mathord{\left/ {\vphantom {{\pi L^{2} } {{\left( {\Phi \nu Re^{{3/2}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\Phi \nu Re^{{3/2}} } \right)}}, $$

(21)

where Φ is the volume fraction of particles relative to the fluid, and is equal to 4πa ³ n ₀/3.

For typical conditions in turbulent liquid helium, the time constant given by (21) is about 5 min. This indicates that if particles of the desired initial size were dispersed, there exists a useful period of time during which they are fluid tracers. However, τ _1/2 is inversely proportional to the particle volume fraction. If the particles must be injected into the fluid as a concentrated solution, and this process is necessarily turbulent, then at the time of injection and near the injector the volume fraction of particles is larger than in the fully dispersed solution. For particles at an initial volume fraction of 0.1 injected at 10 cm/s through a 3-mm injector, τ _1/2 is reduced to about 0.01 s. This leaves little time for the particles to leave the injector before they irreversibly clump into larger particles.