A
The Astrophysical Journal, 614:781–795, 2004 October 20 Copyright is not claimed for this article. Printed in U.S.A.
LABORATORY EXPERIMENTS ON ROTATION AND ALIGNMENT OF THE ANALOGS OF INTERSTELLAR DUST GRAINS BY RADIATION M. M. Abbas,1 P. D. Craven,1 J. F. Spann,1 D. Tankosic,2 A. LeClair,2 D. L. Gallagher,1 E. A. West,1 J. C. Weingartner,3 W. K. Witherow,1 and A. G. G. M. Tielens4 Received 2004 March 12; accepted 2004 May 26
ABSTRACT The processes and mechanisms involved in the rotation and alignment of interstellar dust grains have been of great interest in astrophysics ever since the surprising discovery of the polarization of starlight more than half a century ago. Numerous theories, detailed mathematical models, and numerical studies of grain rotation and alignment with respect to the Galactic magnetic field have been presented in the literature. In particular, the subject of grain rotation and alignment by radiative torques has been shown to be of particular interest in recent years. However, despite many investigations, a satisfactory theoretical understanding of the processes involved in grain rotation and alignment has not been achieved. As there appear to be no experimental data available on this subject, we have carried out some unique experiments to illuminate the processes involved in the rotation of dust grains in the interstellar medium. In this paper we present the results of some preliminary laboratory experiments on the rotation of individual micron /submicron-sized, nonspherical dust grains levitated in an electrodynamic balance evacuated to pressures of 103 to 105 torr. The particles are illuminated by laser light at 5320 8, and the grain rotation rates are obtained by analyzing the low-frequency (0–100 kHz) signal of the scattered light detected by a photodiode detector. The rotation rates are compared with simple theoretical models to retrieve some basic rotational parameters. The results are examined in light of the current theories of alignment. Subject headings: astrochemistry — dust, extinction — methods: laboratory Online material: color figures
1. INTRODUCTION
mechanisms involved in interstellar grain rotation and alignment is of great interest, because if the grains are aligned by processes involving magnetic fields, the observation of the polarization of starlight would provide a valuable and independent means of determining the morphology of the Galactic magnetic fields. In addition to the issues of polarization of starlight and the Galactic magnetic fields, the rotation of interstellar dust is also of great interest, in view of the possibility of detecting the radio waves emitted by fast-spinning, charged dust grains ( Erickson 1957; Hoyle & Wickramasinghe 1970). On the basis of some recent studies, it has been proposed that the anomalous 10–100 GHz component of the diffuse Galactic background emission that correlates with the 100 m thermal emission from dust is produced by electric dipole emission from very small rotating interstellar dust grains ( Draine & Lazarian 1998a, 1998b). In addition, rotational fragmentation of sufficiently fastspinning, elongated dust grains in the interplanetary and interstellar medium, driven by torques induced by radiation pressure, has been proposed and has attracted some interest in the interplanetary community (e.g., Radzievskii 1954; Paddack 1969; Paddack & Rhee 1975; Misconi 1993). This process may provide a mechanism for the breakup of large interplanetary dust particles as they spiral toward the Sun because of the Poynting-Robertson effect, eventually becoming small enough to be blown out of the solar system by radiation pressure. As a rule, interstellar dust grains are believed to be nonspherical and are subject to rotation and alignment by a variety of processes. Numerous processes for rotation and alignment have been proposed and detailed theoretical models developed. A brief outline of the nature of the basic processes is given here for the convenience of interpreting the experimental results
Ever since the surprising discovery of the polarization of starlight more than half a century ago ( Hall 1949; Hall & Mikesell 1949; Hiltner 1949a, 1949b), a great deal of attention has focused on the physical and dynamical properties of interstellar dust grains as the primary source of the observed phenomenon. From the very earliest, it was recognized that the selective extinction properties of dust grains determined by their size, configuration, and orientation must be responsible for the polarization. The first explanations of the observed polarizations based on alignment of rotating, elongated interstellar dust grains were provided by Spitzer & Schatzman (1949) and Spitzer & Tukey (1949). The relationship between the existence of aligned dust grains and the polarization of starlight was inferred on the basis of a correlation of the degree of polarization with interstellar extinction. Central to the requirements of the interstellar dust grains as a source for the observed polarization are the existence of a suitable source or mechanism for the rotation of nonspherical dust grains in the interstellar medium to sufficiently high speeds and some mechanism or process for the alignment of the major axes of the grains along the Galactic magnetic field or with respect to some preferred direction. Understanding the processes and 1 NASA Marshall Space Flight Center, 320 Sparkman Drive, Huntsville, AL 35812;
[email protected],
[email protected], james.f.spann@nasa. gov,
[email protected],
[email protected], william.k.witherow@ nasa.gov. 2 University of Alabama in Huntsville, Huntsville, AL 35899; dtankosic1@ cs.com,
[email protected]. 3 George Mason University, Fairfax, VA 22030;
[email protected]. 4 Kapteyn Astronomical Institute, Groningen, Netherlands; tielens@astro .rug.nl.
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discussed in this paper and for evaluating the feasibility of carrying out any future experiments. Several comprehensive and excellent review articles have appeared in the literature, with the details discussed in the original references (e.g., Weingartner & Draine 2003; Lazarian 2003). 1.1. Rotation Mechanisms Numerous processes have been discussed in the literature that are capable of exerting torques on interstellar grains (assumed to be elongated, nonspherical, or with uneven surfaces), leading them to high rotational speeds. These processes may be classified into two categories: Collisional processes.—These include (1) thermal rotation of grains induced by random collisions with interstellar gas atoms, as originally considered by Davis & Greenstein (1951), leading to a thermal equilibrium rotational rate !T ¼ ð2kT = IÞ1=2 ; (2) streaming rotation produced by collisions with the bulk motion of a tenuous gas, as proposed by Gold (1952); and (3) suprathermal rotations induced by collision with the ambient hydrogen gas. The formation and ejection of H2 from nonspherical and irregular surface features of the grains, as considered by Purcell (1979), has been proposed as a mechanism for suprathermal rotations at rates many orders of magnitude higher than the thermal rotation rates. Of the collisional processes, the H2 formation mechanism is considered to be the most effective process. Radiation pressure–induced torques.—Torques due to radiation pressure on grains by radiation from the Sun or the interstellar radiation field have been considered an effective mechanism for spinning dust grains to high speeds (Paddack 1969, 1973; Paddack & Rhee 1975; Harwit 1970, 1998; Misconi 1976, 1993; Dohnanyi 1978). However, more recently, a detailed theory of suprathermal spin-up and alignment of interstellar dust grains by radiative torques induced by anisotropic starlight has been presented in a series of papers by Draine & Weingartner (1996, 1997) and Weingartner & Draine (2003). These detailed studies indicate some surprising results, with radiative torques as the most effective mechanism for the rotation and alignment of interstellar grains in regions where sufficient anisotropic radiation intensity exists. 1.2. Aliggnment Mechanisms Detailed reviews of the proposed alignment mechanisms have been given in some recent articles (e.g., Mathis 1986; Lazarian 2003; Weingartner & Draine 2003). The alignment of rotating grains may be considered along two separate lines: (1) alignment of the principal axis of the greatest moment of inertia along the angular momentum axis and (2) alignment of the angular momentum axis in space. The basic alignment schemes involve some type of internal dissipation of the rotational kinetic energy of the grain, or alternatively, a direct alignment process with no explicit involvement of dissipation. The first mechanism involving internal dissipation that has attracted a great deal of attention was proposed by Davis & Greenstein (1951), who suggested that alignment results from paramagnetic dissipation in the grains. This mechanism would require a magnetic field in the medium and the grain material to be paramagnetic. A paramagnetic grain rotating in a magnetic field is continuously magnetized and remagnetized along the field direction, with a lag that results in internal dissipation of rotational energy. The Davis-Greenstein (D-G) mechanism tends to drive a grain to
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rotate along its principal axis of maximum moment of inertia, which then approaches alignment along the interstellar magnetic field. Interstellar grains are likely to be charged by UV photoemission or electron impact, positively or negatively, respectively. It was shown by Martin (1971) that the angular momentum of a charged grain would lead it to precess about the magnetic field on a timescale less than the relaxation time of the D-G mechanism. However, Dolginov & Mytrophanov (1976) proposed that a much faster mechanism arises from the Barnett effect ( Landau & Lifshitz 1960, p. 144), which refers to the process of a paramagnetic or ferromagnetic grain rotating in a field-free medium spontaneously developing a magnetic moment along the axis of its rotation. In the Barnett effect, a rotating body decreases its kinetic energy while keeping its total angular momentum constant, but with some of the angular momentum taken up by unpaired spins of electrons or nuclei in the grain ( Lazarian & Draine 1999, 2000). Purcell (1979) introduced the concept of ‘‘Barnett dissipation,’’ in which a grain in nonsteady rotation is driven to rotate about its greatest moment of inertia. Another internal dissipation mechanism is viscoelasticity, associated with the periodic alternating mechanical stresses by centrifugal forces ( Purcell 1979). This process, however, is estimated to be generally less significant than the Barnett dissipation. In addition to the alignment mechanisms based on magnetic moments of the grains, the possibility of direct alignment of grains has been considered. This suggestion is based on the observation that every photon carries an intrinsic angular momentum f ¼ h=2 (where h is Planck’s constant) that makes the incident light an effective carrier of angular momentum ( Harwit 1970). In recent detailed studies of the rotation of interstellar grains by radiative torques, numerical calculations often indicate direct rapid alignment of the principal angular momentum axis along the Galactic magnetic field without the need for D-G alignment ( Draine & Weingartner 1996, 1997; Weingartner & Draine 2003). The exact mechanism involved in this process remains unclear. To our knowledge, no successful experimental investigations of the rotation of dust grains have been carried out. In the Dusty Plasma Laboratory at NASA Marshall Space Flight Center, we have developed an experimental facility for investigating the physical and optical properties of cosmic dust grains by levitating micron /submicron-sized laboratory analogs in an electrodynamic balance in simulated astrophysical environments. Some recent experiments on micron-sized particles of astrophysical interest have included UV photoelectric emission and radiation pressure measurements (e.g., Abbas et al. 2002a, 2002b, 2003). In this paper, we present the first direct measurements of the rotation of a wide range of micron / submicron-sized levitated dust grains illuminated by laser light. The measurements were made on nonspherical, irregularly shaped SiC grains levitated in an electrodynamic balance, with radiative torques exerted by laser light over a range of radiation intensities. The rotation rates determined from the observed spectral features arising from time variation in the light reflected by the rotating grains are compared with the model values, and the rotational parameters are retrieved from the measurements. The basic equations of motion of a grain in an electrodynamic balance are given in x 2, and the experimental technique is discussed in x 3. The results of our laboratory experiments on micron-sized SiC grains are given in x 4, with the comparison with model calculations discussed in x 5.
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2. BASIC EQUATIONS OF AN ELECTRODYNAMIC BALANCE FOR ROTATIONAL EXPERIMENTS Rotational measurements on dust grains have been made with an experimental facility based on an electrodynamic balance that has been effectively employed for photoelectric emission and radiation pressure measurements on individual micron-sized dust grains (Spann et. al. 2001; Abbas et al. 2002a, 2002b, 2003). The rotation measurements presented here constitute an extension of the radiation pressure measurements that employ 5320 8 laser radiation pressure to rotate irregularly shaped dust grains levitated in an electrodynamic balance. In the experimental setup, an individual, well-characterized grain is levitated in the trap. Radiation torque by a focused, visible laser light beam sets the particle spinning, with the motion counteracted by drag exerted by the ambient gas. The rotation rate of the particle is monitored through low-frequency (1000) and the gas drag on the dust grain is in the free molecular regime, as opposed to the viscous regime, for which the slip correction factor was defined in equation (8). For the free molecular regime applicable to the low pressures employed in the experiments discussed here, and for the interstellar medium, we use the following expression derived for the drag torque acting on a stationary sphere in a slowly
b¼
The quantity QT ¼ QPR rma in equation (17) has been introduced to represent the equivalent ‘‘torque efficiency.’’ The radiation pressure efficiency is determined by using Mie scattering theory calculations, and the effective moment arm representing the asymmetry factor of the dust grain is retrieved from the laboratory experimental data, as shown in the following equations. The solution to equation (15) is ! R ðt Þ ¼
b ð1 eat Þ: a
ð18Þ
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In the limiting case with vanishing gaseous drag, the constant a ! 0, and equation (18) reduces to !R ðt Þ ¼ bt:
ð19Þ
Equation (18) is similar in form to equation (4) of Paddack & Rhee (1975), who assumed that the drag on the particle was exerted by the ambient magnetic field. In equilibrium, the steady-state angular velocity in rotations per second is given by !ss R ¼
b 43:6Ik QT ¼ : a Dm Ptorr CFT
ð20Þ
The growth constant a can be retrieved from the measurement of !R ðt Þ obtained experimentally with a curve-fitting retrieval procedure employing equation (18), simultaneously retrieving values of the constants a and b and the steady-state rotation rate !ss R in equilibrium. The effective particle diameter is determined by the ‘‘spring-point’’ measurements, as discussed in the Appendix. The effective torque correction factor may be calculated by solving equation (16) for CFT as CFT ¼
5:52 ; 104 aDm : Ptorr
ð21Þ
Since Ik and !ss R are directly measured quantities, and since QPR may be calculated by using Mie scattering theory for the known particle size and characteristics and is a relatively slowly varying function of the size parameter, the effective torque efficiency is determined by solving equation (20) for QT : QT ¼
2:3 ; 102 !ss R Dm Ptorr CFT : Ik
ð22Þ
The effective asymmetry factor, or the effective moment arm ratio, rma of the grain may then be calculated from the retrieved effective torque efficiency by using the definitions rma ¼
lm 1 QT ¼ : Dm Dm QPR
ð23Þ
In addition to the curve-fitting retrieval procedure discussed above, the constant a may also be determined from measurements between any two times t1 and t2 with a significant change in the rotation rate, by using the time rate of change of !R ðt Þ from equation (18), leading to a¼
ln½!˙ R ðt2 Þ=!˙ R ðt1 Þ ; t
ð24Þ
where !˙ R represents the derivative with respect to time. Equations (16)–(24) are used in x 3 to retrieve the rotational data from the measured rotation rates as a function of time. 3. EXPERIMENTAL SETUP AND PROCEDURE FOR ROTATIONAL MEASUREMENTS The experimental setup and the procedure for rotational measurements of dust grains employed are straightforward extensions of those used for direct measurements of radiation pressure on individual dust grains and reported in a previous publication (Abbas et al. 2003).
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3.1. Experimental Setup The experimental setup for rotational measurements with an electrodynamic balance, shown in the schematic ( Fig. 2), is equipped with the following main components: 1. Electrodynamic balance.—Top and bottom electrodes and a ring electrode, of hemispherical configuration, enclosed in a chamber with appropriate viewing ports. 2. Power supplies.—Computer-controlled electrical power supplies for the electrodes, to provide AC voltage (VAC 100 2000 V, at fAC 10 1000 Hz) and DC voltage (VDC 0:01 100 V ), and a high DC voltage (Vh 0 1000 V ) power supply for inductive charging of the dust particles with a pressure-impulse particle injector. 3. Particle injector.—A device to inject an inductively charged particle ( positive or negative) of known composition and density into the balance through a suitable port at the top. A device utilizing pressure impulse on a liquid, with a mixture of the desired particles exiting through a high electric field and an orifice, has been described in a previous publication (Spann et al. 2001). 4. Particle imaging system.—A He-Ne laser and an optical system to project a magnified image of the levitated particle on a monitor. 5. Vacuum system.—A vacuum pump system with leak valves and pressure gauges, capable of controlled evacuation of the system to pressures of 105 to 106 torr. 6. Radiation power laser.—A Spectra-Physics Nd:YVO4 5320 8 laser with variable power and an optical system to focus the radiation to a beam width of 175 m ( FWHM ) at the trap center and the particle. 7. Photodiode detector.—A silicon photodiode with an optical system to focus the dust grain image illuminated with the laser light at 5320 8 on a sensitive 1 ; 1 mm photodiode. The low-frequency signal (0–100 kHz) from the detector, containing spectral information about the scattered light, is digitized with an A /D converter, recorded, and analyzed by a spectrum analyzer as a function of time. 8. Fast Fourier transform (FFT ) spectrum analyzer.— Spectrum Laboratory software, version 2.0, and a HewlettPackard Spectrum Analyzer, to analyze and interpret the observed frequency spectrum of the scattered light measured by the photodiode signal as a function of time. 3.2. Frequency Spectrum of an Illuminated RotatinggGrain A Fourier transform spectral analysis of the low-frequency signal (0–100 kHz) generated by scattered light from the particle and focused on a photodetector, as discussed in x 3.1, provides information about the rotational-vibrational motions of the particle. The particles are known to be nonspherical and of uneven surfaces. The observed spectral features include strong fundamental components, as well as higher harmonics of (1) the 60 Hz component of the local power supply through the photodiode, (2) the driving frequency fAC of the AC voltage of the ring electrode, generally of 100–500 Hz, (3) the vertical ( fz) and radial ( fAC) frequency vibrations, or oscillations, of the particles, generally of 10–50 Hz, and (4) spectral features due to the rotation of a grain arising from the uneven nature of the grain surface, with the fundamental spectral frequency representing the rotation rate per second of the grain. For constant electrical parameters, the low-frequency components in features 1–3 remain constant with time and may be
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Fig. 2.—Schematic of the experimental setup employed for rotational measurements on an electrodynamic balance.
easily filtered out. The spectral feature arising from rotation, however, represents the rotational dynamics of the particle. For a given radiation intensity on the grain, the frequency increases with time to reach an equilibrium value representing a steady-state rate of rotation, limited by the drag of the ambient gas determined by the pressure. This identification of the low-frequency spectral feature with the rotation rate of the grain is fundamental to the measurements presented here. Ultimate confirmation of our interpreted correspondence between the spectral measurements and rotation rates will be sought in the future using images captured by an ultrafast camera. 3.3. Experimental Procedure The procedure involved in the rotational measurements is similar to that for the radiation pressure measurements presented in a previous publication (Abbas et al. 2003), with the exception that the illuminating radiation scattered by nonspherical dust grains is focused on a photoconductive detector and recorded. Briefly, positively or negatively charged dust particles of known composition, such as SiC, are selected from bins of various size ranges and levitated in the trap. The voltages and the frequency are appropriately adjusted within the domain of the stability conditions to retain only one
particle stably trapped at the balance center. A controlled evacuation procedure is started while the field parameters are adjusted to keep the particle stably trapped. A few springpoint measurements, which constitute a determination of the critical field parameters at marginal instability of the levitated particle, are performed at pressures of 1–10 torr. The ‘‘effective’’ diameter of the particle is then calculated by using the calculated viscous drag and the stability conditions, through the equations outlined in the Appendix and discussed in detail in previous publications (e.g., Spann et al. 2001). For large particles (k5 m), an approximate estimate or an upper limit on the diameter is also obtained from the monitor images using scale factors developed from particles of known size. After the size determinations, the pressure is allowed to reach the lower values of 103 to 105 torr, and rotation measurements are carried out at various pressure levels and a series of radiation intensities incident on the levitated particles. 4. EXPERIMENTAL RESULTS The rotational measurements presented in this paper were made on nonspherical dust grains of SiC of radii in the range 0.17–8.2 m obtained from South Bay Technology, Inc., of Temple City, California. The lower limit on the particle size in our current experiments is set by the sensitivity of detection of
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Fig. 3.—Sequence of magnified images of a 2.7 m radius SiC particle illuminated with 5320 8 laser light at 10 W cm2. The images are selected at 8 s intervals, starting with predominantly oscillatory motions, gradually progressing into a stable rotational motion, as indicated by the last image, after 40 s.
the scattered light from small particles and may be improved in future experiments. The SiC particles were chosen for the convenience of availability over a wide range of sizes, although it is known that interstellar grains are likely to be amorphous silicates or amorphous carbon. This difference, however, does not change the significance of the results, in that the experiments are for examination of the basic physics of rotation and alignment, and the results may be appropriately scaled to interstellar parameters. The particles are levitated at pressures of 103 to 105 torr and illuminated by laser light at 5320 8, and the spectral characteristics in the time variation of the scattered light intensity are recorded as a function of time. The rotation rates of the illuminated particles are represented by the peak amplitude frequency of the measured spectrum. The levitated particles are stabilized to minimum oscillations at the desired pressure in the trap by appropriately adjusting the electrical parameters, as discussed in x 3. Illumination by laser light triggers rotation of the dust grains only at sufficiently high intensity. With insufficient radiation, the particle motions remain oscillatory, and the observed spectrum is composed only of random frequencies above the constant low-frequency components discussed in x 3.2. In the following discussion, we present some laboratory data relating to experiments on the rotation of dust grains in the laboratory. The time in seconds shown in the plots refers to the start of data recording for each particle. Figure 3 represents selected magnified images on a monitor of an SiC particle (497) of 2.2 m radius levitated at a pressure of 3:3 ; 103 torr, with radiation intensity increased from 8 to 10 W cm2. The first image in Figure 3 represents the oscillatory particle motion (at frequencies of 20–300 Hz) at the
time the radiation is turned on, changing to progressively more stable motion with time, as indicated by the subsequent images. The image in the steady, stable condition represents a predominantly rotational motion, with the intensity at 10 W cm2 after 40 s. The exact mechanism involved in the damping out of the oscillatory motions remains to be investigated. The three plots shown in Figure 4 represent the rotation of SiC particles when the radiation intensity is marginal for stable rotation of the particle. Figure 4a shows a 2.2 m radius particle at a pressure of 2:3 ; 103 torr, illuminated with radiation at 6 W cm2 that was gradually increased to 10 W cm2, with pressure at 3:4 ; 103 torr. At lower intensities the oscillatory motions are dominant, and the particle spontaneously locks into a stable rotational mode after some 15 minutes only when the radiation is increased to 10 W cm2 at a slightly higher pressure. Before locking into a rotational mode, the observed spectrum ( Fig. 4a) exhibits weak periodic features representing rotational motion, with random motion dominating. A similar example is shown in Figure 4b, where a 0.45 m radius SiC particle at a pressure of 2:1 ; 103 torr with 8 W cm2 radiation initially exhibits a strong oscillatory motion, then begins to rotate, and later falls back into oscillatory motion after a couple of minutes. For marginal intensities, the grain may first trigger into a rotational mode and later fall back into a mode dominated by oscillatory/random motions in a random, periodic manner. This phenomenon is shown in Figure 4c, where a 4.1 m radius particle subjected to a 10 W cm2 radiation at 3:9 ; 103 torr periodically falls in and out of a simple rotational mode. As indicated by equations (16)–(20), the rotation rate !R is proportional to the radiation intensity Ik and the torque
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Fig. 4a
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Fig. 4b
Fig. 4c
g. 4aFig. 4bFig. Fig. 4c 4.—Rotational plots of three SiC particles initially in oscillatory/random motions and illuminated with marginal radiation intensity. (a) A 2.2 m particle with radiation intensity gradually increasing from 6 to 10 W cm2, at which it triggers into a rotational mode. (b) A 0.45 m particle, exposed to 8 W cm2 radiation, triggers into oscillatory motion after some 70 s and falls back to oscillatory/random motions after a couple of minutes. (c). A 4.1 m particle with 10 W cm2 incident radiation triggers in and out of rotational mode periodically. [See the electronic edition of the Journal for a color version of this figure.]
efficiency QT and inversely proportional to the particle size and drag represented by the atmospheric pressure with the drag correction factor, CFT. A series of experiments was carried out to evaluate the dependence of rotation rates on these factors. A set of selected plots is presented here to evaluate and validate the dependence on the various parameters. Figure 5 shows a ‘‘roller-coaster’’ rotational motion of an SiC particle of 3.9 m radius levitated in the trap at a pressure of 3:3 ; 104 torr. The particle is initially triggered into rotation by radiation with an intensity of 10 W cm2 at t ¼ 4200 s. At t ¼ 4250 s, the intensity is increased to 12 W cm2, and the observed rotation rate reaches 4800 s1 at t ¼ 4600 s, at which time the intensity is increased to 16 W cm2. The rotation rate increases to 6200 s1 at t 4900 s, when the pressure is increased to P 5:9 ; 104 torr. The rotation rate now decreases to 5000 s1 at t 5025 s, at which time the pressure is decreased to P 3:0 ; 104 , and the rotation rate increases again to 6000 s1. When the pressure is continuously allowed to increase to 2 ; 103 , the rotation rate falls back to 4000 s1. This plot shows the initial response of the grain’s rotation rate to the incident radiation intensity, followed by a series of responses to several variations in the atmospheric drag, in qualitative accordance with equations (16)–(20).
Fig. 5.—Variation in the rotation rate of a 3.9 m SiC particle as a function of radiation intensity and pressure. [See the electronic edition of the Journal for a color version of this figure.]
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Fig. 6.—Rotation rate of a 0.17 m particle as a function of radiation intensity. The line represents a linear fit to the observed data and permits determination of the torque correction factor CFT as a measure of the atmospheric drag. [See the electronic edition of the Journal for a color version of this figure.]
Figure 6 shows the functional relationship between the particle rotation rates and the radiation intensity incident on an SiC particle of 0.17 m radius levitated at an average pressure of 3:5 ; 105 torr. With the radiation intensity varying from 4 to 21.5 W cm2, the measured rotation rate is seen to vary from 4600 to 21,500 s1. The dashed line indicates a linear fit to the data, in accordance with equation (20), with the slope of the linear fit providing a measure of the torque efficiency QT, in conjunction with the torque correction factor CFT , as QT @ !ss R ¼ 2:3 ; 102 Dm Ptorr ¼ 2:7 ; 104 : CFT @Ik
ð25Þ
If a torque correction factor of 1 is assumed, an efficiency of ð2 3Þ ; 104 for this particle is inferred. However, in x 5 we examine the consistency of this value with the retrieved data for the torque correction factors and efficiencies derived from these experiments. Another illustration of the dependence of rotation rate on the radiation intensity, as well as pressure, is shown in Figure 7, where at t ¼ 3200 s, an SiC particle of 0.98 m radius trapped at a pressure of 0:7 ; 103 torr is rotating at 9200 s1 with a radiation intensity of 12 W cm2. The pressure rises to 1:1 ; 103 torr at the end of the experiment. The radiation is successively decreased to 4 W cm2 at t 3588 s and then successively increased to 16 W cm2 at t 4160 s, with the rotation rates falling and rising accordingly. At each incremental increase in intensity, the rotation rate is seen to rise exponentially to an equilibrium steady-state value, in accordance with equation (18), with the rise rate dependent on the constant a. The steady-state values on the right-hand side of the plot are seen to be lower than those in the falling portion on the left-hand side, where the pressure is significantly lower. Modeling calculations for the retrieval of rotational parameters based on some portions of the data plotted in this figure is discussed x 5. As discussed in x 2, the alignment of a rotating grain is to be considered in two aspects. The first is the alignment of the grain’s axis of greatest moment of inertia along the axis of rotation. This is the mode in which the rotating particle with a constant moment of inertia has the minimum kinetic energy.
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Fig. 7.—Rotation rate of a 0.98 m particle subjected to radiation intensities decreasing from an initial value of 12 to 4 W cm2 and then increasing to 16 W cm2. [See the electronic edition of the Journal for a color version of this figure.]
The second is the alignment of the spin axis along some preferred direction in space. If the grain begins to rotate along some initial, arbitrary axis, the presumed alignment mechanism will change the axis of rotation to be along the axis of greatest moment of inertia. Many cases of this apparent change in the axis of rotation, on a timescale of the order of 20–50 s, were observed in the laboratory. Figure 8 shows a typical example of an SiC particle with radius am ¼ 2:7 m, at a pressure of 1:1 ; 103 torr with 12 W cm2 incident radiation, spontaneously rising to a rotation rate of 4500 s1, falling back to 3600 s1 in about 100 s, and then picking up speed to an equilibrium spin rate of greater than 6200 s1, with an adjusted orientation. Similar examples of this phenomenon that involve a significant and observable change from the initial orientation of the spin axis of the grain were seen in any many cases for particles of various sizes exposed to radiation intensities of 12–16 W cm2. A possible explanation of the phenomenon exhibited in Figure 8 is that the particles begin to rotate along an axis other than the axis of greatest moment of inertia and rise to rotation rates high enough to reorient the axis of rotation to the principal axis. This change reduces the rotation rate to a minimum value consistent with the change in the angular momentum,
Fig. 8.—Typical rotation rate of a 2.7 m SiC grain, indicating an apparent change in the orientation of the axis of rotation during the buildup, with the rate falling to a minimum value before rising to the equilibrium value.
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Fig. 9.—Measured rotation rates (dotted lines), from low values to the steady-state values reached at equilibrium, for a selected set of particles over the 0.98– 8.25 m size range. The rotational profiles modeled in accordance with eqs. (15)–(18) (red lines) indicate a good fit between the observed and the calculated data. The experimental data and the retrieved parameters for all particles included in the modeling analysis are shown in Table 1. The particle and experiment numbers, particle radii, pressures, and radiation intensities are (a) SiC particle 521, experiment 1d, 0.45 m, P ¼ 2:3 ; 103 torr, 6 W cm2; (b) SiC particle 527, experiment 2a, 0.98 m, P ¼ 5:0 ; 104 torr, 8 W cm2; (c). SiC particle 492, experiment 1, 1.27 m, P ¼ 4:2 ; 103 torr, 6 W cm2; (d) SiC particle 487, experiment 3, 2.6 m, P ¼ 5:9 ; 103 torr, 8 W cm2; (e) SiC particle 510, experiment 5, 3.3 m, P ¼ 1:2 ; 103 torr, 8 W cm2; ( f ) SiC particle 528, experiment 2, 8.25 m, P ¼ 4:9 ; 104 torr, 4 W cm2.
assuming that the change in the kinetic energy of the particle is negligible during the transition period. A set of selected plots indicating the rotation growth rates for SiC particles of radii in the range 0.45–8.25 m illuminated with various radiation intensities is shown in Figure 9.
The measured rotation rates are shown with the dotted lines, with the red lines representing model calculations to be discussed in the following section. These plots do not show any observable change in the spin axis of the particles relative to their initial orientations and may be easily modeled using
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LABORATORY EXPERIMENTS ON GRAIN ROTATIONS TABLE 1 Summary of Measured Rotational Data and Retrieved Parameters
Radius ( m)
Particle
0.45........
521
0.98........
527
1.27........ 2.1.......... 2.6.......... 3.35........
492 501 487 510
3.9.......... 4.05........
499 509a
5.7..........
526
8.25........
528
Experiment
P (torr)
1c 1d 3d 2a 2b 3a 3b 3c 1 2 3 5 7b 2 4a 6a 1 2 6 2
2.3 ; 10 2.3 ; 103 2.3 ; 103 5.0 ; 104 5.0 ; 104 1.1 ; 103 1.1 ; 103 1.1 ; 103 4.2 ; 103 1.2 ; 102 5.9 ; 103 1.2 ; 103 1.4 ; 103 8.0 ; 103 2.1 ; 103 2.1 ; 103 1.3 ; 103 1.3 ; 103 5.1 ; 104 4.9 ; 104
Kn 3
2.4 ; 10 2.4 ; 104 2.4 ; 104 5.1 ; 104 5.1 ; 104 2.3 ; 104 2.3 ; 104 2.3 ; 104 4.7 ; 103 1.1 ; 103 1.6 ; 103 6.2 ; 103 5.3 ; 103 8.0 ; 102 2.9 ; 103 2.9 ; 103 3.5 ; 103 3.5 ; 103 5.9 ; 103 6.1 ; 103 4
Ik (W cm2)
a
6 6 6 8 10 6 8 10 6 6 8 8 12 7 4 6 16 21.5 27 4
2.0 ; 10 4.1 ; 101 3.7 ; 101 7.6 ; 102 1.7 ; 101 1.7 ; 101 1.8 ; 101 1.8 ; 101 2.1 ; 101 9.8 ; 101 1.3 ; 101 4.0 ; 102 4.0 ; 102 1.3 ; 101 1.3 ; 101 1.6 ; 101 5.9 ; 102 8.4 ; 102 6.5 ; 102 2.6 ; 102
equations (16)–(24) and some basic rotational parameters retrieved from the measured data. The results of the model calculations and the retrieval of rotational parameters are discussed in the following section. 5. MODEL CALCULATIONS AND RETRIEVAL OF ROTATIONAL PARAMETERS 5.1. Model Calculations The rotation rates determined from the observed spectra of the illuminated grains are compared with the calculated model values by using equations (16)–(24) to retrieve some significant rotational parameters. The radiation intensity Ik incident on the particle is calculated from measurements of the laser power at the source and the laser beam width at the trap center (Abbas et al. 2003). The radiation pressure efficiency QPR expresses the efficiency of momentum transfer from the radiation to the particle and may be written in a simple form as QPR ¼ Qext hcos iQsca , where hcos i represents the average cosine of the scattering angle. We used Mie scattering theory (e.g., van de Hulst 1957; Wiscombe 1979; Bohren & Huffman 1983) and employed the program of Wiscombe (1979) for numerical calculations, assuming values of the real and imaginary parts of refractive index of 2.55 and 0.05, respectively. The effective spherical particle radii obtained by carrying out spring-point measurements, discussed in x 2 and the Appendix, were used in the efficiency calculations. For the 5320 8 wavelength and particle radii in the 0.17–8.2 m range, the size parameter x ¼ 2r=k varies from 2 to 100, while the radiation pressure efficiency QPR is calculated to be a slowly varying function of the size parameter, with values varying from 1.0 to 2.7. The rotational data obtained from experiments on 10 different particles of 0.45–8.25 m radii illuminated by various radiation intensities at various pressures were analyzed for retrieval of rotational parameters a, b, !ss, CFT, QT , and rma by solving equation (18) with an iterative least-squares technique. Figure 9 shows typical examples comparing the
b 1
QPR
!ss
2.8 ; 10 4.5 ; 103 5.6 ; 103 6.6 ; 102 1.6 ; 103 1.1 ; 103 1.2 ; 103 1.3 ; 103 7.9 ; 102 1.6 ; 103 5.2 ; 102 2.2 ; 102 2.3 ; 102 4.6 ; 102 1.1 ; 103 1.7 ; 103 2.3 ; 102 4.8 ; 102 4.6 ; 102 7.0 ; 102 3
1.43 ; 10 1.09 ; 104 1.52 ; 104 8.63 ; 103 9.38 ; 103 6.27 ; 103 6.97 ; 103 7.62 ; 103 3.80 ; 103 1.67 ; 103 4.13 ; 103 5.39 ; 103 5.61 ; 103 3.69 ; 103 8.53 ; 103 1.06 ; 104 3.83 ; 103 5.73 ; 103 7.01 ; 103 2.72 ; 103 4
1.44 1.44 1.44 1.23 1.23 1.23 1.23 1.23 1.18 1.12 1.10 1.08 1.08 1.06 1.06 1.06 1.04 1.04 1.02 1.02
CFT
QT
rma
0.133 0.279 0.249 0.515 1.142 0.522 0.537 0.541 0.219 0.619 0.193 0.386 0.332 0.210 0.852 1.049 0.934 1.320 3.625 1.492
1.5 ; 10 2.4 ; 102 3.0 ; 102 1.3 ; 102 2.4 ; 102 2.7 ; 102 2.3 ; 102 2.0 ; 102 3.3 ; 102 1.9 ; 101 7.0 ; 102 4.8 ; 102 3.3 ; 102 1.6 ; 101 7.2 ; 101 7.3 ; 101 7.3 ; 102 1.2 ; 101 1.8 ; 101 1.9 ; 101 2
1.0 ; 102 1.7 ; 102 2.1 ; 102 1.0 ; 102 2.0 ; 102 2.2 ; 102 1.9 ; 102 1.7 ; 102 2.8 ; 102 1.7 ; 101 6.4 ; 102 4.5 ; 102 3.1 ; 102 1.5 ; 101 6.8 ; 101 6.9 ; 101 7.0 ; 102 1.1 ; 101 1.8 ; 101 1.8 ; 101
observed rotation rates (dotted lines) for SiC particles of radii 0.45–8.25 m under various conditions with the rates calculated by using equation (18) with the retrieved parameters. A nearly perfect fit between the observed and the calculated values (red lines) for most of the particles appears to validate the functional form of the model used, as well as the values of the retrieved rotational parameters. The results of rotational experiments conducted on 10 particles are tabulated in Table 1 with the parameters a, b, !ss, QPR, CFT, QT, and rma, along with the radiation intensity and pressure. Also listed in the table is the Knudsen number, defined as ratio of the mean free path to the particle diameter: Kn ¼ lmfp =D. The three retrieved quantities of particular interest listed in Table 1 are (1) CFT, representing the experimentally determined correction to the atmospheric drag with respect to the free molecular formulation employed in the calculations, (2) QT, representing the radiative torque efficiency of the grain, equivalently expressed as QT ¼ QPR rma , and (3) rma , which can be interpreted as the effective asymmetry factor, or the effective moment arm ratio, of the grain and is a measure of the nonsphericity and irregular nature of the grains. Plots of the retrieved quantities a, CFT, QT, and rma listed in Table 1 for the 10 particles are shown in Figure 10, with the solid lines representing linear fits to the measured data. The large scatter in the retrieved data in all the plots is mainly due to the highly nonspherical and irregular configuration of the dust grains and does not represent noise in the measured or the retrieved data. Figure 10a shows a plot of the growth time constant a versus the particle radius. The linear fit to the data, as expected, indicates smaller values for larger particles over the range in radii from 3.3 to 8.25 m. The larger particles in Figures 9e and 9f, for example, show slower rates of growth than the smaller particles in Figures 9a and 9b. A plot of the torque correction factor CFT versus the Knudsen number is shown in Figure 10b. The linear leastsquares fit to the measured data exhibits a near-constant value of 0.5–0.6 over a wide range of Knudsen numbers,
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Fig. 10a
Fig. 10c
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Fig. 10b
Fig. 10d
g. 10aFig. 10bFig. Fig.10cFig. 10.—Retrieved 10d values for all particles listed in Table 1 with radii in the 0.45–8.25 m range. The line represents a linear fit to the retrieved data, with the scatter representing large variations in the configuration of randomly selected particles. (a) Growth constant a vs. particle radius, indicating a downward trend in a for large particles. (b) Torque correction factor CFT vs. Knudsen number, with the linear trend indicating a near-constant value. (c) Torque efficiency QT vs. Knudsen number, with the linear fit indicating a downward trend for small particles ( large Knudsen numbers). (d ) Effective moment arm ratio, or asymmetry factor rma (eq. [23]) vs. particle radius, with the linear fit indicting an upward trend with particle radius, as the smaller particles tend to be more spherical. [See the electronic edition of the Journal for a color version of this figure.]
indicating the general validity of using the formalism of equation (12) for torque in the free molecular regime. A plot of the torque efficiency QT, defined in equation (22), is shown in Figure 10c, indicating decreasing values varying from 0.2 to 102 over a large range of Knudsen numbers. The decrease in torque efficiency at higher Knudsen numbers (and smaller particle sizes) is attributed to smaller particles’ being less asymmetric and less effective in inducing radiative torques. The retrieved data for the torque correction factors and efficiencies shown in Figures 10b and 10c may be employed to examine the torque efficiency in conjunction with the correction factor independently inferred in equation (25) from the data plotted in Figure 6 as QT =CFT ¼ 2:7 ; 104 . The Knudsen number for the 0.17 m radius particle at a pressure of 3:5 ; 105 torr and a temperature of 300 K is calculated to be 4 ; 105 . Although this high value falls outside the range of data plotted in Figure 10c, extrapolated values of QT in the range of ð1 4Þ ; 103 may be estimated, considering the large scatter in the data due to the variability of the particle configurations. The torque correction factors plotted in Figure 10b show a considerable variability, from 0.1 to 4. The extrapolated efficiency for the particle in Figure 6 would thus require a torque correction factor of 4 in order to be consistent with
the data plotted in Figure 10c. This is not an unreasonable value, considering the smallest size of the particle in the data set and the variability of the range of CFT plotted in Figure 10b. The low torque efficiencies for small particles are also indicated by the data in Figure 10d, with the linear fit to the plot of the effective asymmetry factor rma ¼ lm =Dm increasing from 2 ; 102 to 2 ; 101 for increasing particle radii. These experimental values for the asymmetry factor for the laboratory-prepared, nonspherical SiC dust grains are found to be orders of magnitude larger than the value of 5 ; 104 for interplanetary dust particles ( IDPs) estimated by Paddack (1969), based on experiments using a hydrodynamic analogy. In the absence of any other experimental data, this value has generally been assumed in discussions of the rotation of IDPs ( Paddack & Rhee 1975; Ratcliff et al. 1980; Misconi 1993). The data presented in this paper, however, appear to be the first experimental data based on spinning, micron-sized dust grains exposed to radiation. 5.2. Evvaluation of Aliggnment Mechanisms in the Laboratory The experimental data for rotations of dust grains presented in this paper exhibit micron-sized dust grains in initial oscillatory/random motions that change to stable rotational motions
No. 2, 2004
when the grains are illuminated with laser light. The time periods involved in this change vary from a near-spontaneous transition to several minutes, depending on the particle size, radiation intensity, atmospheric drag, and initial level of oscillatory motions. The rotation rates increase exponentially from the initial rate to a steady-state value, implying equilibrium between the radiative torque and the atmospheric drag. The rise to the equilibrium rate is sometimes a continuous, exponential buildup, but it sometimes exhibits a dip to a local minimum in the rotation rate followed by an exponential rise to the equilibrium value. The latter may indicate a change in the rotational configuration, e.g., a transition to rotation about the principal axis of greatest moment of inertia. It is of interest to examine the measurements presented here in view of the alignment mechanisms discussed in the literature, as briefly outlined in x 1. One of the most interesting mechanisms is the Barnett effect, which may involve both electron and nuclear paramagnetic dissipations and predicts alignment of the greatest moment of inertia aˆ with the angular momentum vector J ( Dolginov & Mytrophanov 1976; Purcell 1979). A modified form of the timescale for this alignment has been given by Weingartner & Draine (2003): Bar 1 ; 105 yr
; !2 4
am 2 1 m 1 1 7 ; 104 þ ; 1 þ 3 ; 108 !24 1 þ !24
T 300 K
3 g cm3
ð26Þ
where !4 is the angular velocity in units of 104 rotations s1. This expression, involving both electron and nuclear dissipations, is similar to that given by Purcell (1979), who considered electron dissipation only; the modification and the nuclear dissipation included in this process are incorporated in the last term. However, for the laboratory measurements on an SiC particle with am 1 10 m, 3 g cm3, and T 300 K, the Barnett relaxation time predicted by equation (26) is found to be many years and is not likely to be observed in the laboratory. The magnetic moment mBar of a spinning grain due to the Barnett effect interacting with the external magnetic field produces a torque &B ¼ mBar < B, causing the angular momentum vector J to precess about B with angular frequency
B. For interstellar grains, the precession timescale is more rapid than the alignment timescale, and the time-averaged component of the magnetic moment mBar perpendicular to J vanishes, while the parallel component remains constant. The effective orientation of the grains for correlation with the starlight must thus be considered to be in the direction of the magnetic field. The precession angular frequency B is given by Weingartner & Draine (2003) as
B 25 yr
1
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LABORATORY EXPERIMENTS ON GRAIN ROTATIONS
0:1 m am
2
B 3 g cm3 0 ; 1 3:3 ; 104 5 G ð27Þ
where is a geometric factor for the grain ( Draine & Weingartner 1996) and 0 is the magnetic susceptibility of the material. With electron paramagnetism dominating nuclear paramagnetism, 0 1:7 ; 104 to 5 ; 106 , particle radii of 0.2–1 m, and B 0:6 G, the precession timescale is 2– 40 minutes and may be observable in the laboratory. The effect
of the precession of a rotating particle on the rotation frequency spectrum should appear as a modulation of the rotation frequency at the precession frequency. Such a low-frequency modulation was often observed in the laboratory experiments. However, the amplitude of the modulation depends on a number of parameters and may be too small to be observed in many cases. No quantitative analysis of this phenomenon has been carried out yet. Inelastic dissipation, as shown by Lazarian & Efroimsky (1999), is somewhat less important than Barnett dissipation and may not be observed in the laboratory experiments under consideration. In addition, the timescale for grain alignment by the D-G mechanism, as given by Weingartner & Draine (2003), is DG 1:5 ; 106 yr
1 3 g cm3
2
am 0:1 m
Td 15 K
5 G B
2 :
ð28Þ For particles of 0.1–1.0 m radii, 1:5, 3 g cm3, and B 0:6 G, the D-G timescale for alignment is in terms of days to months and may not be observed for grain alignment along the Earth’s magnetic field. However, with the introduction of an external magnetic field of the order of 20– 100 G at the trap center, the D-G alignment timescale predicted by equation (28) is reduced to the order of a few minutes and could be observed in the laboratory. An effect analogous to the Barnett effect arises from the magnetic moment ¼ ZeJ =2mc of a spinning charge Ze on a grain of mass m with angular momentum J. The precession frequency resulting from the torque due to coupling with the external magnetic field is
B 20 yr
1
3 g cm3
1
Z 1000
am 1 m
3
B : 1G ð29Þ
For the range of parameters considered here, the contribution of the spinning charge to the magnetic torque and the precession of the grain predicted by equation (29) is negligible, compared with the Barnett effect. 6. SUMMARY AND CONCLUSIONS In this paper, we report the first laboratory measurements of rotations induced by radiative torques on the analogs of micronsized interstellar dust grains. The measurements were made on irregularly shaped particles of 0.17–9 m effective radii levitated in an electrodynamic balance and illuminated with laser light at a wavelength of 5320 8, at pressures of 103 to 105 torr. The rotation rates were determined by analyzing the low-frequency spectrum of the light scattered by the fastspinning particles with an FFT spectrum analyzer. The following are the principal features of the results: 1. The rotation rates of irregularly shaped SiC particles of 0.17–9 m effective radii at ambient pressures of 103 to 105 torr illuminated with radiation intensities of 4–30 W cm2 were observed in the range of 1000–22,000 rotations s1. 2. The grain rotation rates were observed to follow the expected functional relationships, being directly proportional to the incident radiation intensities and inversely proportional to the drag represented by the ambient pressure.
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This is so for the charged grains, even with several hundred thousand electron charges (negative or positive) on the grains. However, the D-G mechanism could be observed on laboratory timescales by introducing an external magnetic field of 20– 100 G at the trap center. The D-G timescale for alignment predicted by equation (28) for high magnetic fields is reduced to the order of a few minutes or seconds.
3. Average values of the effective torque efficiency (QT), the torque correction factor (CFT) to the atmospheric drag torque D, and the effective asymmetry factor representing the irregular and nonspherical configuration of the particles were retrieved as a function of the Knudsen number from rotational measurements on SiC particles of 0.2–8.2 m radii. 4. The timescale for the Barnett effect for alignment of the dust grains along the Earth’s magnetic field (0.6 G) is calculated to be many years and may not be observed in the laboratory. However, the precession timescale, arising as a result of the Barnett torque’s representing coupling between the Barnett magnetic moment of the spinning grains and an external magnetic field, is much shorter. This timescale for the Earth’s magnetic field and for particles of 0.2–1 m radii is 2–40 minutes and was observed in several cases as a lowfrequency modulation of the rotational spectrum. The amplitude of the modulation precession frequency, however, is found to be too weak to be clearly detected in many cases. 5. The alignment timescales due to the D-G mechanism’s requiring paramagnetic dissipation, as well as those of the mechanism based on the magnetic moment due to charges on spinning grains, are too long to be observed in the laboratory.
This work was partly supported by the Science Directorate at NASA Marshall Space Flight Center, and we are grateful to Ann Whitaker, Ron Koczor, Frank Six, and John Davis, for their encouragement and support. We thank J. Nuth for a very comprehensive review of the manuscript and many useful suggestions. We are grateful to Hugh Comfort for his support during the course of this work, to D. Hathaway and R. Phillips for their assistance in preparing the videos and images for the experiments, to A. Ewing for several helpful suggestions during the course of this work, and to V. Coffee, W. Fountain, N. Martinez, C. Ryland, and J. Redmon for their assistance in the laboratory setup and upkeep.
APPENDIX Assuming that the electrostatic and gravitational forces in equation (6) are balanced, with the dimensionless variables ¼ t=2, R ¼ r=z0 , and Z ¼ z=z0 , together with the transformations Rˆ ¼ Re d and Zˆ ¼ ze d , the two equations of motion take the form of homogeneous Mathieu equations: d 2 Rˆ þ ða 2cr cos 2 ÞRˆ ¼ 0 d 2
ðA1Þ
d 2 Zˆ þ ða 2cz cos 2 ÞZˆ ¼ 0; d 2
ðA2Þ
and
with a ¼ d2 ¼ cr ¼
18 ; d 2
cz 4VAC q g VAC ¼ F ¼ ¼ ; 2 V 2 z0 2 mz0 2 C0 z0 fAC DC
ðA3Þ ðA4Þ
and q gz0 ¼ : m VDC C0
ðA5Þ
Since the AC and DC potentials and the frequency fAC ¼ =2 are known, the charge-to-mass ratio q/m in equation (A5) is a directly measurable quantity. If the effective particle size is determined by some independent means, such as the spring-point method (e.g., Spann et al. 2001), or by scattering measurements, both the charge and mass, and therefore the effective particle diameter, may be determined separately. The quantities d and F in equations (A3) and (A4) are referred to as the drag and field strength parameters, respectively, and determine the stable solutions of the radial and axial Mathieu equations ([A1] and [A2]). A general solution is of the form Y ¼ exp½ði d Þ
þ1 X
An expði2n Þ;
ðA6Þ
n¼1
where the coefficient An depends on the initial conditions and ¼ r þ i i is a complex number. Here and An are functions of the parameters ai and ci in equations (A3) and (A4). For i > d , the solution grows without limit and the particle motion is unbounded and unstable, whereas if i < d , the value of Y is bounded and the particle motion is confined and stable. The motion stability boundary is therefore given by i ¼ d. The stability of the homogeneous Mathieu equations, and thus the stability of the particle motion, is governed by the field strength and drag parameters F and d as functions of . The electrical parameters that
No. 2, 2004
LABORATORY EXPERIMENTS ON GRAIN ROTATIONS
795
control a stable confinement of the particle in the trap are thus VDC, VAC, and fAC. The plots of F and d that represent the stability conditions have been given in the literature (e.g., Davis 1985) and are utilized in this work. An analysis of the equations of motion of the particle indicates the physical description as that of a harmonic oscillator damped by the viscous drag of the ambient gas ( Wuerker et al. 1959; Richardson & Spann 1984). In the absence of any viscous drag with d ¼ 0, and if the particle’s motion is stable, the resultant motion will consist of a slow vibration with a small ripple due to the superimposed AC drive, which has the average force toward the center of the balance and the consequent vertical and radial frequencies of motion ( Wuerker et al. 1959). The vertical oscillation frequency is given by fz ¼ k
1 VAC ; fAC VDC
ðA7Þ
where the constant k depends on the balance’s geometric configuration and has to be determined experimentally. In the balance employed here, the vertical oscillation frequency in equation (A7) is determined to be fz 34:3
1 VAC : fAC VDC
ðA8Þ
With the electrical parameters employed during the course of the rotational measurements presented here, fz is generally in the range of 10–150 Hz.
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