The present invention relates to fluidic mixing and, in particular, to a method of multi-axis non-contact mixing of magnetic particle suspensions.
In the last few years it has been shown that a wide variety of triaxial magnetic fields can produce strong fluid vorticity. See J. E. Martin, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 79, 011503 (2009); J. E. Martin and K. J. Solis, Soft Matter 10, 3993 (2014); K. J. Solis and J. E. Martin, Soft Matter 10, 6139 (2014); J. E. Martin and K. J. Solis, Soft Matter 11, 241 (2015); and U.S. application Ser. No. 12/893,104, each of which is incorporated herein by reference. These fields are comprised of three mutually orthogonal field components, of which either two or three are alternating, and whose various frequency ratios are rational numbers. These dynamic fields generally lack circulation, in that a magnetically soft ferromagnetic rod subjected to one of these fields does not undergo a net rotation during a field cycle. Yet these fields do induce deterministic vorticity, which might seem counterintuitive. For this deterministic vorticity to occur it must be reversible. This reversibility is possible if the trajectory of the field and its physically equivalent converse, considered jointly, is reversible. This field parity occurs because the symmetry of this union of fields is shared by vorticity, which is reversible.
An analysis of the symmetry of these fields enables the prediction of the vorticity axis, which is determined solely by the relative frequencies of the triaxial field components. For these fields changing the relative phases of the components enables control of the magnitude and sign of the vorticity—and in some cases changing the sign of the dc field also reverses flow—but not the axis around which vorticity occurs. Thus, when the frequency of one of the field components is detuned slightly to cause a slow phase modulation, the vorticity will periodically reverse, but it remains fixed around a single axis. Such flows produce a simple form of periodic stirring, as occurs in a washing machine.
The present invention goes well beyond this simple form of stirring and is based on transitions in the symmetry of the triaxial field.
According to the present invention, a method for non-contact mixing a suspension of magnetic particles comprises providing a fluidic suspension of magnetic particles; applying a triaxial magnetic field to the fluidic suspension, the triaxial magnetic field comprising three mutually orthogonal magnetic field components, at least two of which are ac magnetic field components wherein the frequency ratios of the at least two ac magnetic field components are rational numbers, thereby establishing vorticity in the fluidic suspension having an initial vorticity axis parallel to one of the mutually orthogonal magnetic field components; and progressively transitioning the symmetry of the triaxial magnetic field to a different symmetry, thereby causing the vorticity axis to reorient from the initial vorticity axis to a vorticity axis parallel to a different mutually orthogonal magnetic field component. For example, the volume fraction of magnetic particles can be greater than 0 vol. % and less than 64 vol. %. The magnetic particles can be spherical, acicular, platelet or irregular in form. The magnetic particles can be suspended in a Newtonian or non-Newtonian fluid or suspension that enables vorticity to occur at the operating field strength of the triaxial magnet. For example, the strength of each of the magnetic field components can be greater than 5 Oe. For example, the frequencies of the at least two ac field components can be between 5 and 10000 Hz. The ac frequency can be tuned along at least one of the ac magnetic field components. The relative phase of at least one of the ac magnetic field components can be adjusted.
It has recently been shown by the inventors that two types of triaxial electric or magnetic fields can drive vorticity in dielectric or magnetic particle suspensions, respectively. The first type—symmetry-breaking rational fields—consists of three mutually orthogonal fields, two alternating and one dc, and the second type—rational triads—consists of three mutually orthogonal alternating fields. In each case it can be shown through experiment and theory that the fluid vorticity vector is parallel to one of the three field components. For any given set of field frequencies this axis is invariant, but the sign and magnitude of the vorticity (at constant field strength) can be controlled by the phase angles of the alternating components and, at least for some symmetry-breaking rational fields, the direction of the dc field. In short, the locus of possible vorticity vectors is a one-dimensional set that is symmetric about zero and is along a field direction.
According to an embodiment of the present invention, continuous, three-dimensional control of the vorticity vector is possible by progressively transitioning the field symmetry by applying a dc bias along one of the principal axes. Such biased rational triads are a combination of symmetry-breaking rational fields and rational triads. A surprising aspect of these transitions is that the locus of possible vorticity vectors for any given field bias is extremely complex, encompassing all three spatial dimensions. As a result, the evolution of a vorticity vector as the dc bias is increased is complex, with large components occurring along unexpected directions. More remarkable are the elaborate vorticity vector orbits that occur when one or more of the field frequencies are detuned. These orbits provide the basis for highly effective mixing strategies wherein the vorticity axis periodically explores a range of orientations and magnitudes.
More specifically, applying a dc field parallel to a carefully chosen alternating component of an ac/ac/ac rational triad field can create a field-symmetry transition. By exploiting this transition, theory and experiment show that the vorticity vector can be oriented in a wide range of directions that comprise all three spatial dimensions. The direction of the vorticity vector can be controlled by the relative phases of the field components and the magnitude of the dc field. Detuning one or more field components to create phase modulation causes the vorticity vector to trace out complex orbits of a wide variety, creating very robust multiaxial stirring. This multiaxial, non-contact stirring is attractive for applications where the fluid volume has complex boundaries, or is congested. Multiaxial stirring can be an effective way to deal with the dead zones that can occur when stirring around a single axis and can eliminate the accumulation of particulates that frequently occurs in such mixing.
The detailed description will refer to the following drawings, wherein like elements are referred to by like numbers.
Mixing with triaxial magnetic fields has some unique and attractive characteristics. See J. E. Martin, Phys. Rev. E79, 011503 (2009); and J. E. Martin et al., Phys. Rev. E 80, 016312 (2009). Only a small volume fraction of magnetic particles is needed (˜1-2 vol. %); only modest, uniform fields (˜150 Oe) are required; the mixing torque is independent of field frequency and fluid viscosity (within limits); and the mixing torque is independent of particle size, making this technique suitable for use in a variety of systems ranging in size from the micro to industrial scale. Furthermore, the torque density is uniform throughout the fluid, creating a ‘vortex fluid’ capable of peculiar dynamics. Finally, unlike traditional magnetic stir bars, which can experience instabilities that result in fibrillation or stagnation, there are no such instabilities associated with this technique, making it a simple, robust means of creating non-contact mixing.
This approach to mixing can eliminate or reduce the fluid stagnation that can occur in conventional stirring, in which the stirring axis is stationary. Fluid stagnation is a problem in simple geometries, such as near the corners of a cylindrical volume, and is even worse in complex or obstructed volumes, such as those that occur in engineered microfluidic systems. See C. Gualtieri, “Numerical simulation of flow and tracer transport in a disinfection contact tank,” Third Biennial Meeting: International Congress on Environmental Modeling and Software (iEMSs), 2006; and S. Suresh and S. Sundaramoorthy, in Green Chemical Engineering: An introduction to catalysis, kinetics, and chemical processes, CRC Press, USA 2014. Moreover, in a single-axis, rotary mixing scheme, the fluid flow profile is typically non-uniform and assumes the form of an irrotational vortex, wherein the fluid velocity is inversely proportional to the radial distance from the mixing axis. See S. Kay, in An introduction to fluid mechanics and heat transfer, 2nd Ed., The Syndics of the Cambridge University Press, New York, USA, 1963.
The method of inducing flow in bulk liquids complements advances in liquid surface mixing using magnetic particles driven by an alternating magnetic field. See G. Kokot et al., Soft Matter 9, 6767 (2013); A. Snezhko, J. Phys.: Cond. Mat. 23, 153101 (2011); M. Belkin et al., Phys. Rev. Lett. 99, 158301 (2007); and M. Belkin et al., Phys. Rev. E 82, 015301 (2010). In the surface mixing method, the field organizes the particles into complex aggregations, such as “snakes,” and the induced motion of these aggregations creates significant near-surface vorticity. These surface mixing techniques share an important similarity with the bulk mixing techniques: viscosity as a means of control. In the surface flow experiments increasing the viscosity causes a transition from the formation of “snakes” to the formation of “asters,” which have less vigorous flow. See P. L. Piet et al., Phys. Rev. Lett. 110, 198001 (2013). When the liquid viscosity in the suspensions is increased, there is a transition from inducing vorticity to creating static particle aggregations.
Two methods have previously been discovered by the inventors of inducing fluid vorticity in magnetic particle suspensions. In the first method, two orthogonal ac components whose frequency ratio is a simple rational number are applied to the suspension. Vorticity is induced when an orthogonal dc field is applied, because this field creates the parity needed for deterministic vorticity. A theory of these symmetry-breaking fields has been developed that predicts the direction and sign of vorticity as functions of the frequencies and phase. See J. E. Martin and K. J. Solis, Soft Matter 10, 3993 (2014). The second method is based on rational triad fields, comprised of three orthogonal ac fields whose relative frequencies are rational numbers (e.g., 1:2:3). These fields also have the parity and symmetry required to induce deterministic vorticity and a symmetry theory has been developed that allows computation of the direction and sign of vorticity as functions of the frequencies and phases. See J. E. Martin and K. J. Solis, Soft Matter 11, 241 (2015).
According to an embodiment of the present invention, by progressively biasing one particular ac component of a rational triad to dc, competing symmetries can be generated that lead to a continuous reorientation of the vorticity vector, providing full three-dimensional control of fluid vorticity. Therefore, symmetry transitions between certain classes of alternating triaxial magnetic fields are used to produce time-dependent, non-contact, multi-axial stirring in fluids containing small volume fractions of magnetic particles. In this approach to mixing, the vorticity axis continuously changes its direction and magnitude, executing elaborate, periodic orbits through all three spatial dimensions. These orbits can be varied over a wide range by phase-modulating one or more field components, and a wide variety of orbits can be created by controlling the phase offset between the field components. This method provides an entirely new approach to efficient mixing and heat transfer in complex geometries.
The symmetry-transition method of the present invention is based on the observation that both ac/ac/dc (symmetry-breaking) and ac/ac/ac (rational triad) fields can generate fluid vorticity. The axis around which this vorticity occurs is the critical factor enabling field-symmetry-driven vorticity transitions.
For the symmetry-breaking ac/ac/dc fields the vorticity axis is determined by the reduced ratio l:m of the two ac frequencies. Because l and m are relatively prime then at least one of these numbers is odd. A consideration of the symmetry of the field trajectory and its equivalent converse jointly shows that if only one of these numbers is odd the vorticity is parallel to the odd field component and reversing the dc field reverses the vorticity. If both of these numbers are odd the vorticity is parallel to the dc field component. For odd:odd fields reversing the dc field direction does not reverse the flow, which suggests that for these fields the dc component can be replaced by an ac field and vorticity can still occur. In all cases, the sign and magnitude of the vorticity can be controlled by the phase angle between the two ac components. See J. E. Martin and K. J. Solis, Soft Matter 10, 3993 (2014).
For the fully alternating rational triads (ac/ac/ac), the direction of vorticity is controlled by the three relative field frequencies l:m:n, where l, m, and n are integers having no common factors. There are four classes of such fields: I) even:odd:odd; II) even:even:odd where even:even can be factored to even:odd; III) even:even:odd where even:even can be factored to odd:odd and; IV) odd:odd:odd. See J. E. Martin and K. J. Solis, Soft Matter 11, 241 (2015). By analyzing the symmetries of the 3-d Lissajous trajectories of the field and its converse jointly it is possible to show that the direction of vorticity is parallel to the field component that has unique numerical parity. The fourth class (odd:odd:odd) has no component with a unique numerical parity and so does not possess the symmetry required to predict a vorticity axis. However, off-axis vorticity exists in that case.
Consider now the possibility of creating a continuous symmetry transition by gradually transitioning one of the three ac field components of a rational triad into a dc field, while keeping the root-mean-square (rms) field amplitude constant. To be definite, let l, m, and n lie along the x, y, and z components, respectively. If it desired to transition the z component of the field to dc the relevant expression is
where f is a characteristic frequency determined by the operator. Note that all three field components have equal rms values and the ac-to-dc transition is effected by increasing c from 0 to 1 or from 0 to −1. The z axis ac and dc fields have equal rms amplitudes when c=1/√{square root over (2)}. The effect of this ac-dc transition on field symmetry depends on both the class of rational triad as well as the component that is transitioned.
Consider the odd:even:odd field 1:2:3 for one particular set of phases.
Transitioning either of the odd field components to dc is much more interesting. The x and z axis are antisymmetric under a 180° rotation. If the x component is fully transitioned to dc a change of field symmetry occurs that causes the vorticity vector to reorient from the y to the z axis. The progression of this symmetry change can be seen in
The same considerations hold when the z component is continuously transitioned from ac to dc, only in this case the final vorticity axis is along x. The vorticity vector can thus be expected to orient anywhere in the x-y plane, but a strong contribution to the vorticity occurs around the z axis, which is surprising.
To summarize, for even, odd, odd fields applying a dc field along one odd ac component causes the vorticity to rotate from the even component direction to the other odd component. But applying a dc bias along the even component does not cause a change in the vorticity direction.
The simplest field of this class is 1:2:4. For these fields the vorticity is around the odd axis (e.g., relative field frequency l=1), which is x in this case. As a result, if the y or z component of the field is transitioned to dc the direction of the vorticity axis will not change. The sign of the vorticity might change, however, because in this case it is dependent on the sign of the dc field. Thus a symmetry-driven transition that gives rise to flow reversal can be effected by a proper selection of the dc field sign.
If the x component transitions to dc, the vorticity axis will reorient from the x to the y axis, with the vorticity sign again dependent on the dc field sign. In this case it is expected that the vorticity vector can be continuously oriented in the x-y plane, but the torque density functional described below predicts a surprising component along the z axis during this transition. Therefore, applying a dc field along the odd component (i.e., the x axis in this case) will cause the vorticity to reorient from the odd field axis to the odd axis that arises from factoring even:even (i.e., the y axis is the odd axis that arises from factoring the remaining 2:4 fields to 1:2).
In summary, for odd, even, even fields applying a dc field along either even component will not change the orientation of the vorticity axis, but might cause it to reverse. Applying a dc field along the odd component will cause the vorticity to reorient from the odd field axis to the odd axis that arises from factoring even:even.
For this class of fields, such as 1:2:6, the vorticity is around the odd field axis, which in this case is again along x. If one ac component of such an odd:even:even field is fully transitioned to dc there are three possible outcomes: dc:odd:odd (i.e., the remaining 2:6 fields factor to 1:3), odd:dc:even, or odd:even:dc. In each case the symmetry rules show that the vorticity remains around the x axis (underlined). Therefore no change in the orientation of the vorticity axis is expected, though its sign and magnitude might change during the transition. In other words, such fields produce robust vorticity that is not strongly affected by stray dc fields. Note that only if the even field is transitioned does the final vorticity sign depend on the sign of the dc field.
The final case of odd:odd:odd (e.g., 1:3:5) fields is interesting, because any field component that is transitioned to dc becomes the vorticity axis. This suggests that applying a dominant dc field in any direction along any field component will induce vorticity around that component, enabling fine control of the vorticity direction.
A measure of the torque density produced in a magnetic particle suspension subjected to a triaxial field has previously been proposed that is based on both theory and experiment. See J. E. Martin and K. J. Solis, Soft Matter (2015). This functional was found to conform to all of the predictions of the symmetry theories but can also be applied to those cases where the trajectories of the triaxial fields do not possess the symmetry of vorticity, such as the field-symmetry-driven vorticity transitions of the present invention. This functional also makes useful quantitative predictions for the amplitude of the torque density as a function of field frequencies and phases. The functional is given by
where the dependence on the phase angles is indicated. Here J{φ}(s) is the instantaneous torque density, h(s)=H0−1H0(s) is the reduced field, and s=ft is the reduced time in terms of the characteristic field frequency in Eq. 1. The experimentally measured, time-average torque density is related to this functional by T{φ}=const×φpμ0H02J{φ}, where μ0 is the vacuum permeability and φp is the particle volume fraction.
Before giving the predictions of the torque density functional it is informative to consider what one might reasonably expect to occur. Returning to the example case of a 1+dc:2:3 field, for zero dc field the vorticity is parallel to the y axis (along the “2” field component) and for the full dc case, dc:2:3, it is parallel to the z axis, both in accordance with symmetry theory. For intermediate values of c a simple ‘rule of mixing’ consistent with a field-squared effect is
J
{φ}(c)=(1−c2)|J{φ}(0)|ŷ+c2|J{φ}(1)|{circumflex over (z)}. (3)
This expression confines the vorticity vector to the y-z plane, which seems reasonable, but how does this expression compare to the predictions of Eq. 2? It is clear that inserting Eq. 1 into Eq. 2 does not result in an expression in which the ac and dc terms are separable, but it is not clear how important this is.
To obtain an appreciation for the complexity of this symmetry transition, in
One aspect of the nature of the vorticity transition from the rational triad 1:2:3 to the symmetry-breaking rational field dc:2:3 is illustrated in
The full range of three-dimensional control of the torque density is given in
It is interesting to determine how the torque density produced at any given pair of phase angles evolves as the dc bias is progressively increased.
In
However, the behavior predicted by the torque functional is much richer than that predicted by the simple mixing law. When the functional in Eq. 2 is used to predict the torque density the result is dramatically different. In
Phase modulating components of the applied 1+dc:2:3 field produces a rich variety of vorticity orbits that are both interesting and potentially useful for a number of applications. These orbits have been numerically investigated for the dc bias c=0.5. In
The fourth transect is a bit of a disappointment, as the torque density barely changes, but the other transects produce striking results. The first and second transects produce orbits with a net torque around the z axis (averaged over one orbital cycle) but with zero net torques around the other axes. For these orbits the mixing is persistent. The orbit for the third transect is interesting in that it produces zero net torque around any of the principal axes, which would enable complex mixing in freestanding droplets without incurring any net migration of the droplet. This mixing strategy would be ideal for the development of parallel bioassays of container-less droplet arrays, perhaps comprised of millions of droplets. The fourth transect produces a non-zero net torque around the x axis alone.
The phenomenology of these vorticity orbits is much richer than indicated.
The same phase shifts were used to generate a set of orbits for a set of transects parallel to the second transect, generating the set of widely varying vorticity orbits in
Finally, vorticity orbits for a few cases were investigated where the field frequencies along the x and z axes are detuned by unequal amounts, specifically by 2:1. These orbits, shown in
The magnetic particle suspension consisted of molybdenum-Permalloy platelets ˜50 μm across by 0.4 μm thick dispersed into isopropyl alcohol at a low volume fraction.
For the 1:2:3 rational triad field, the fundamental frequency was 36 Hz (f in Eq. 1) and all three field components were 150 Oe (rms). The spatially uniform triaxial ac magnetic fields were produced by three orthogonally-nested Helmholtz coils. Two of these were operated in series resonance with computer-controlled fractal capacitor banks. See J. E. Martin, Rev. Sci. Instrum. 84, 094704 (2013). The third coil was driven directly in voltage mode by an operational power supply/amplifier. The phase shift of this coil at its operational frequency of 36 Hz was measured as +68° with a precision LCR meter. To compensate for this phase shift, this phase was added to the signal that drives the amplifier.
The signals for the three field components were produced by phase-locked via two function generators, allowing for stable and accurate control of the phase angle of each field component. Note that if these signals are simply produced from separate signal generators there will be a very slow phase modulation between the components due to the finite difference in the oscillator frequency of each function generator. And simply running two separate signal generators off the same oscillator does not control their phase relation. All of the measurements are strongly dependent on phase.
To quantify the magnitude of the vorticity, the torque density of the suspension was computed from measured angular displacements on a custom-built torsion balance. In this case the suspension (1.5 vol %) was contained in a small vial (1.8 mL) attached at the end of the torsion balance and suspended into the central cavity of the Helmholtz coils via a 96.0 cm-long, 0.75 mm-diameter nylon fiber with a torsion constant of ˜13 mN·m rad−1.
All of the above predictions depend on one key point: the appearance of torque along the x axis when a dc field is applied parallel to this axis. Recall that this torque does not exist for c=0 or 1, but is only expected for intermediate values, i.e. during the symmetry transition. In fact, upon the application of the dc bias this torque does appear, and is strong.
The set of measured vorticity vectors is plotted in
The progression of the measured torque density as the dc field is increased from c=0 to 1 is shown in
The vorticity orbits can be obtained by detuning one or more field components. To be clear about the experimental parameters the field can be written
where f1=f, f2=2f, and f3=3f. The parameters Δf1 and Δf3 have been included to indicate detuning of the first and third field components. The principal vorticity orbits for the 1+dc:2:3 field, in
The complexity of these orbits can be appreciated by one single phase modulation example, wherein the x component of the torque was monitored for the frequencies 36.1, 72, and 108.2 Hz and recorded the torque density as a function of time. The time dependence of this single component of the vorticity orbit is plotted in
The present invention has been described as a method of multi-axis non-contact mixing of magnetic particle suspensions. It will be understood that the above description is merely illustrative of the applications of the principles of the present invention, the scope of which is to be determined by the claims viewed in light of the specification. Other variants and modifications of the invention will be apparent to those of skill in the art.
This invention was made with Government support under contract no. DE-AC04-94AL85000 awarded by the U. S. Department of Energy to Sandia Corporation. The Government has certain rights in the invention.