Langmuir 1985,1,379-387
Figure 12. Scanning electron micrograph of a eample of microencapsulated particles that was frozen and microtomed. cluded that for the thin-walled layered spherea examined here there is ambiguity associated with quantitative interpretation of the light-scattering data because of the difficultyin determining four parameters (a,6,and the two components of m(*))from the light-scattering data. As in the case of the silicate spheres one can place more confidence in the size and real component of the refractive index than in the shell thickness and the imaginary component of ita refractive index. Figure 12 also shows many particlea with diameters less than 1 in contrast with lightscattering measurements,
379
which generally gave diametera of 3-4 pm. The stability characteristics of a particle in the electrodynamic balance depend on the charge on the particle and the amplitude and frequency of the ac voltage used to trap a particle, and it is likely that the smaller miemspheres were unstable for the ac field used in the experiments. The sensitivity of the particles to the initial transient state of the nitrogen flow when the microvalve was opened indicates that the 3 4 " particles were near an unstable state, for they were readilv lost when the forces on the particle were perturbed due the flow. For silicate ~artielaand liouid dronlets with diameter8 greater than pm there is g& agreement between the aerodynamic diameter and the light-scattering diameter, and modifications to the flow system and jet diameter should permit aerodynamic drag measurements to be made with smaller particles.
to
Acknowledgment. Part of this -cb wan supported by the Defense Nuclear Agency under Contracta DNA 001-82-C-0224 and DNA 001-&L-C-0293. We are grateful to Donald Churchill and Richard Bowman of Appleton Papers Inc. for preparation of the encapsulated ink microspheres.
Electrodynamic Balance Stability Characteristics and Applications to the Study of Aerocolloidal Particlest E.James Davis Department of Chemical Engineering, BF-10, University of Washington, Seattle, Washington 98198 Received December 28,1984. I n Final Form: February 8,1985 The theory of the electrodynamic balance, a device used for the study of single aerosol particles in the size range 0.1-100 lun, is extended to aualyze the particle's &abilitychmacterktica with and without feedba& contmL A new analytical approximation is obtained for the de electric field strength for the bihyperboloidal electrodynamic balance, and the results are applied to the equations of particle motion to examine the marginal stability envelope. On the basis of the stability theory, it is shown that the aerodynamic drag can be determined by measuring the onset of instability. The drag force can ala0 be measured directly by suspending the particle in a jet of air flowing at various velocities, and experimental results are reported for spherical droplets and nonspherical particles. The dc voltage required to maintain the particle position against the opposing forces of aerodynamic drag and gravity was measured to determine the drag force.
Introduction The electrodynamic balance has become an important instrument for the study of the chemical physies of single aemlloidal particles. Although the device dates from the electric mass filter of Paul and Ftaether' in 1955,only in the past 5 years has its potential for aerosol research been recognized. With this device a single charged droplet or particulate is suspended stably by superposition of ac and de electric fields in a cell consisting of two endcap electrodes and a ring electrode located midway between the endcaps. The simplest form, used by Straubel? consists of two flat plates with a ring located at the midplane between the plates. An ac potential applied to the ring focuses the particle, and a dc potential difference applied to the endcaps is used to balance the gravitational force.
In thin way a particle in the size range of about 0.1 to 100 pm can be suspended stably in a laser beam with or without gaa flow through the chamber. Davis3 haa reviewed the development of the electrodynamic balance and its applications to lighhcattering measurements, particle mam and charge measurements, droplet evaporation, and other aerosol phenomena. The symposium for which this paper has been prepared involves more than a dozen applications of the balance, including Raman, infrared, and photophoretic spectroscopy of single droplets or particulates, gas/particle chemical reactions, and droplet solution behavior. New applications of the balance are appearing frequently. It is the purpose of this paper to extend the existing theoretical analyses of the electrodynamic balance and to
'Pressnted at the Symposium "The Chemical Phpica of Aerocolloidal Particles". 188th National Meeting of the American Chemical Society, Philadelphia, PA, Aug 26-31, 1984.
0743-7463/85/2401-0379$01.50/0 0 1985 American Chemical Society
Davis
380 Langmuir, Vol. 1, No. 3, 1985
vectors associated with the ac and dc fields, respectively. El has components El,,and El$, and E2 has components &,r and &,z* For a spherical particle at low Reynolds numbers in the continuum regime the drag force follows Stokes's equation
FD = - 6 r ~ p U and the velocity vector U may be written as
L
bC SOURCE
U = e, dr/dt
I -
Figure 1. Electrodynamic balance with hyperboloidal electrodes and associated circuitry.
examine new applications of the device based on that analysis. Of particular interest here is the use of the balance for aerodynamic drag measurements.
Bihyperboloidal Electrode Configuration Wuerker et ala4introduced the electrodynamic balance with electrodes of hyperbolic geometry shown in Figure 1. This configuration has been used by Davis and Ray,5 Philip et al.? and, in a slightly modified form, Frickel et al.,' and other configurations have been used by Berg et al.8 and Richardson and S ~ a n n .The ~ endcap electrodes shown in Figure 1 are described by the equation 2z2 - r2 = 2zO2
(1)
and the ring electrode satsifies 2z2 - r2 = -2.22
(2)
where 22, is the distance between the endcaps at the center line, which was 26 mm for the balance used in this study. Frickel and his colleagues chose a constant other than zo for the ring electrode, so their device is not geometrically similar to that shown in Figure 1. They also constructed a second balance with spherical electrodes, which are easier to machine. As the electric field and particle motion depend on the specific circuitry, we shall consider the system shown in Figure 1,which has also been used for experimental studies with the dc bias voltage, v b , set to zero. The endcaps are at ground with respect to the ac potential on the ring, and the dc potentials of the upper and lower endcaps are +Vo and -Vo, respectively, when the particle is negatively charged (q < 0). A reversing switch enables the endcap potentials to be reversed when q > 0. Equations of Particle Motion. For a particle of mass m the equations governing particle motion in the radial and axial directions are, respectively,
m d2r/dt2 = F D , r + qE,,, + qE2,r
(3)
+ e, dz/dt
FD = -KopLcU/(l
+ f)
where F,,r and FD,z are the radial and axial components of the drag force, FD, and Eland E2 are the electric field
+
f = Kn[1.234
+ 0.414 exp(-0.876/Kn)]
(8)
and recently Lea and Loyalka15 reviewed the theory of the motion of a sphere in a rarefied gas, provided a new solution based on integral transform techniques, and compared their results with other theoretical analyses and Millikan's results. There is excellent agreement among theoretical and experimental results in the Knudsen regime for spheres, but there are little or no data available for nonspherical particles in the noncontinuum regime. Electric Fields. The electric field vector is given by
E = -AV
(9)
and the electric potential, V, satisfies Laplace's equation in cylindrical coordinates, ( I / r ) d(r dV/dr)/dr
+ d2V/dz2 = O
(10)
The total potential may be written as a superposition of the time-dependent ring potential, VI, and the dc potential associated with the endcaps, V2,that is, by (11)
The potential V1 and the resulting electric field vector El have been presented by several investigators,"' and for the boundary conditions appropriate here El has the components
El,r = -(vb+ V,, cos Ot)r/2z02 (4)Wuerker, R. F.; H. Shelton, H.; Langmuir, R. V. J. Appl. Phys. 1959, 30, 342. ( 5 ) Davis, E. J.; Ray, A. K. J. Colloid Interface Sci. 1980, 75, 566. (6) Philip, M. A.; Gelbard, F.; Arnold, S. J.Colloid Interface Sci. 1983, 91, 507. (7) Frickel, R. H.; Shaffer, R. E.; Stamatoff, J. B. Report No. ARCSL-TR-77041, Chemical Systems Laboratory, Aberdeen Proving Ground, MD, 1978. (8) Berg, T. G. 0.;Trainor, R. J., Jr.; Vaughan, V. J.Atmos. Sci. 1970, 27, 1173. (9) Richardson, C. B.; Spann, J. F. J. Aerosol Sci. 1984, 15, 563.
(7)
where the resistance constant, KO,depends only on the shape and orientation, and KO and the characteristic length, L,, are listed by Dahneke for a number of shapes, including oblate and prolate spheroids, spheroidal disks and needles, and long cylinders. The slip correction factor, (1 f ) , which is a function of the Knudsen number (Kn = X/a, where X is the mean free path of the escaping molecules), was proposed by Knudsen and Weber13 in 1911. Mil1ikanl4 correlated his drag data for oil drops by means of the equation
v = v,+ v, (4)
(6)
where e, and e, are unit vectors in the r and z directions, respectively. Dahnekelo-l2reviewed extensively theoretical results on the aerodynamic drag of nonspherical bodies, and he proposed methods for approximating the slip correction factors for such bodies in the noncontinuum regime. For creeping flow, shape and slip correction factors can be incorporated to give the generalized equation
and m d2z/dt2 = FD,z + qEl,z+ qE2,.- mg
(5)
(12)
and (10) Dahneke, B. E. Aerosol Sci. 1973, 4 , 139. (11)Dahneke, B. E. Aerosol Sci. 1973,4, 147. (12) Dahneke, B. E. Aerosol Sci. 1973, 4 , 163. (13) Knudsen, M.; Weber, S. Ann. Phys. (Leipzig) 1911, 36, 981. (14) Millikan, R. Phys. Reu. 1923, 22, 1. (15) Lea, K. C.; Loyalka, S. K. Phys. Fluids 1982, 25, 1550.
Langmuir, Vol. 1, No. 3, 1985 381
Electrodynamic Balance Stability = (Vi,
+ VacCOS Q t ) z / z o 2
(13)
Note that the radial and axial components of El differ in sign, and the axial component has twice the magnitude of the radial component. The potential Vl has the saddle shape plotted by Philip et al. The determination of the dc potential, V,, is more involved, and two prior attempts have been made to determine it. Philip e t al. solved Laplace's equation numerically for their circuitry and electrode configuration, and for the electric field near the origin they obtained the approximation E2,z
=-~oVo/~o
(14)
where the geometric coefficient, Co, was found to be 0.40 with the upper and lower electrodes maintained a t V0/2 and -V0/2, respectively. For the boundary conditions considered here their results correspond to Co = 0.80. For flat parallel plate endcaps, such as used in the Millikan condenser, Co = 1.0 for our conditions. Frickel et al. did not calculate E, theoretically, so they calibrated their instrument, obtaining a constant they denoted by Cdcequal to 17.7 m-l. This result corresponds to Co = 0.310 for their cell for which zo = 17.5 mm. Because the electric field component E,$ is used to balance the gravitational field and significantly affects the particle motion, it is desirable to obtain a theoretical estimate of electric field vector E2,valid globally rather than just a t the center of the balance. Now V2 satisfies Laplace's equation together with boundary conditions,
+
V2 = Vo a t z = (r2/2)1/2+ 202
Equation 10 transforms to
+
sin q d(sinh f' dV2/df')/df' sinh f' d(sin q dV,/dq)/dq = 0 (23) and the boundary conditions become
V,(E,T/2) = 0
(24)
and
V2(5,?0) = vo The solution of this system of equations is
(25)
VdE,d = V2h) = BVOIn [tan (0/2)1
(26)
where the constant B is defined by B = l / l n [tan (q0/2)] = -1.5186514...
(27)
In prolate spheroidal coordinates V , is a function of q only, provided that we neglect the effect of the ring electrode and the finite dimensions of the endcaps. Thus, the surfaces of constant q are equipotential surfaces. The electric field vector, E,, is obtained by applying eq 9 to yield the components
E2,, = c1(R,Z)Vo/zo
(28)
E2,L= Ca(R,Z)Vo/zo
(29)
and where
(15)
and
V2 = -Vo a t z = -(r2/2)1/2
+ zo2
(16)
V 2 = 0 at z = O
ditions associated with the ring electrode. The potential on the ring surface is undefined here, but we can develop a useful analytical solution of the governing equations by neglecting the edge effect produced by the ring electrode. The geometry of the endcap electrodes suggests suitable transformations to facilitate solution of eq 9, 10,15, and 17, for the endcap surfaces correspond to sheets of constant q in prolate spheroidal coordinates. Prolate spheroidal coordinates are related to cylindrical coordinates by the transformations z = A cosh f COS q (18) (19)
where the geometrical constant A is given by A = 3lI2 20. From eq 18 the midplane corresponds to q = r / 2 , and the surface of the upper electrode, given by eq 1, transforms to 2A2 cosh2 f cos2 qo - A2 sinh2 f sin2 qo = 2A2 cos2 qo (20) which reduces to tan2 qo = 2
(33)
R = r/zo
(34)
z = 2/20
(35)
(17)
For completeness we require additional boundary con-
r = A sinh f sin q
+ 3ll2), + R2]1/2 R2 = [(Z - 31/2)2+ R2]1/2
R1 = [(Z
Because of symmetry about the midplane (z = 0) eq 16 can be replaced by the equation
(21)
Thus, the upper electrode surface corresponds to the constant qo = tan-l 21/2 = 54.7356O (22)
(32)
and We note that the coefficients in the dc field equations depend on both the radial and vertical coordinates. In the limit as r 0 and z 0 the radial component E,,, vanishes, and the z component reduces to
- Ezs
-0.8768Vo/~o
(36)
The constant in eq 36 is approximately 10% larger than the value obtained by Philip et al. from their numerical solution, which included the effect of the ring electrode, so the edge effects are not too large to vitiate further analysis based on these new results. The introduction of holes in the electrodes for observation ports, laser beam tubes, and flow channels will also affect the geometrical constant, so for precise measurements it is necessary to calibrate the electrodynamic balance by techniques discussed below. Solution of the Equations of Motion. To facilitate solution of the equations governing the stability of a particle it is convenient to write the equations in nondimensional form, using dimensionless coordinates R and Z defined by eq 34 and 35 and dimensionless time defined by 7 = Qt/2 (37) The equations of motion transform to
Davis
382 Langmuir, Vol. 1, No. 3, 1985
d2R - + CY-dR dT2
d7
-(
+p
2
vb + cos 27)R = F(R,Z) (38)
Vnc
and
where
a = 2K&#/mQ(l+ f,
(40)
8 = 4Vacq/mn2z02
(41)
F(R,Z) = 4C,(R,Z)V0q/mQ2zo2
(42)
G(R,Z) = 4[qVoC2(R,Z)- mgzo]mQ2z02
(43)
Quations 38 and 39 are coupled equations which govern the stability of the electrodynamic balance. Wuerker et al. and kickel et al. examined the stability characteristics of these equations but only for approximations valid near the origin where F(0,O) = 0 and G(0,O) is given by G(0,O) = 4(qVoCo- mgzo)/mQ2zO2= constant (44) In this case the equations are decoupled. Furthermore, when the dc voltage Vo is adjusted so that the electrostatic force exactly balances the gravitational force, we obtain G(0,O)= 0. Thus,with both F and G equal zero the equations of motion reduce to homogeneous equations. Introducing the transformations R* = R exp(ar/2) and Z* = Z exp(a7/2) (45)
O
-----
Vb/VOC =+0.1
-
a12 and decays with time for -6 < a/2, The marginal stability the equations of motion become inhomogeneous Mathieu state is given by vi = -a/2. The marginal stability state equations, which are is determined by methods outlined by Frickel et al., and we have used the method of continued fractions to solve the eigenvalue problem. The stability is governed by the parameters a,p, and (46) vb/vac, and Figure 2 is a plot of our results for the first three stability envelopes. Table I lists the values of a and and p corresponding to the stability envelopes. The results for the lower two envelopes are identical with the results of Frickel et al. when the differences in the definitions of the parameters are taken into account. The effect of changing the bias voltage, vb,is also indicated in Figure 2, and the effect is to increase or decrease the stability domain, depending on the polarity. The Mathieu equation has been studied e~tensively,'~J~ For a 15q~m-diameterparticle of unit specific gravity and Frickel et al. used a straightforward expansion techsuspended in air a t atmospheric pressure and room temnique in their solution to examine the stability characperature and for an ac frequency of 60 Hz the drag pateristics of the balance. They examined the drag paramrameter is approximately 7.4, and from Figure 2 such an eter region 0 < a < 8 in considerable detail (we note that object would be stable, provided that p < 50, assuming vb their drag parameter is half that defined here), obtaining = 0. As we shall show, experimental data on solid and the first marginal stability envelope as a function of the liquid spheres, which were charged by ejection from an system parameters. Their results are useful for larger electrified capillary tube, correspond to in the range 5-10 particles (a > 10 pm), but for micron and submicron size when the peak-to-peak ac voltage was about 1000 V. Such objects it is desirable to examine a wider range of drag particles are well within the stable region, but it is clear parameters, so we have extended the calculations to exfrom Figure 2 that increasing the ac voltage would lead amine more than the first stability curve computed by to instability when the second stability envelope is crossed. Frickel and his co-workers. To examine particle trajectories and the stability of the In the homogeneous case the solution for 2 has the coupled equations we have solved eq 38 and 39 numeriform" cally, using a Runge-Kutta numerical integration algorm ithm for various initial positions and velocities of the 2 = exp[(iv - a/2)7] C A, exp(i2nm) (48) ne-particle and for various parameters, a, B, and vb/Vnc. Typical calculated results are shown in Figures 3 and 4 for where the coefficients A,, depend on the initial conditions a 6-pm-diameter sphere with a specific gravity of unity in air for V , = 2000 V, v b = 0, a frequency of 60 Hz, a charge (16) McLachlan, N. W. 'Theory and Application of Mathieu of 1.44 X C, and Voselected to balance the gravitaFunctions"; Clarendon Press Oxford, 1947. tional force. For these parameters, which are represent(17) Abramowitz, M.; Stegun, I. A. 'Handbook of Mathematical ative of actual experimental conditions, we calculate a = Functions"; U.S.Government Printing Office: Washington, DC, 1964.
Langmuir, Vol. 1, No. 3, 1985 383
Electrodynamic Balance Stability
Table I. Field Strength and Drag Parameters for the Marginal Stability State for Vb = 0 region I1 region I11 region IV
region I a
B
0.4 0.8 1.2 2.0 2.8 d.6 4.0 4.6 4.8 4.6 4.0 3.2 2.8 2.0 1.6 0.8 0.4 0.2
1.970 2.392 3.014 4.734 7.072 10.17 12.11 16.01 18.30 20.69 20.40 19.02 18.25 16.81 16.21 15.34 15.11 15.05
a
1.2 2.0 2.8 3.6 4.0 4.8 5.2 6.0 6.8 7.6 8.0 8.0 7.6 6.6 6.4 6.0
B
a
B
16.06 17.63 19.93 22.95 24.73 28.85 31.21 36.58 43.04 51.21 56.76 65.0b 64.93 61.56 60.73 59.01
0.4 0.8 1.0 2.0 3.0 4.0 5.0 60.0 7.0 8.0 10.0 11.0 11.56 11.5 11.0 10.0 9.0 8.0
42.70 42.96 43.16 44.81 47.51 51.23 55.98 61.80 68.78 77.08 98.83 114.1 129.7 132.5 133.5 128.6 121.9 115.0
z ye0
a
0
8.0 10.0 11.0 12.0 13.0 14.0
116.6 134.9 146.1 159.0 175.2 192.7
-
!
z
0
v)
2 . DIMENSIONLESS AXIAL POSITION
R x 100,DIMENSIONLESS RADIAL POSITION
Figure 3. Dimensionless axial and radial positions vs. dimensionless time for a = 49.07 and B = 42.69.
2
60
-
‘40
0
t
.
I
H 0.2
I
1
I
I
-
is
I
\\
20
-
0’ 2 x 100, DIMENSIONLESS AXIAL POSITION
Figure 5. Dimensionless axial position for a = 2.00 and 0 = 4.64.
RX100, DIMENSIONLESS RADIAL POSITION
Figure 4. Particle trajectory for the conditions of Figure 3.
49.07 and /3 = 42.69. From Figure 2 it is expected that €or such a large drag parameter the particle will come to a stable final state. Figure 3 shows the dimensionless axial and radial positions as functions of dimensionless time for about 20 cycles after the particle is introduced with zero initial velocity a t a position above the midplane (2 = 0.1) and slightly off the center line (R = 0.002). Both position histories show damped oscillations a t 60 Hz,and the two components are 90° out of phase. The particle trajectory, shown in Figure 4, illustrates the motion produced by the ac field. As the particle moves inward axially it moves outward radially and vice versa until it comes to a stable position a t the center of the balance. For Figures 3 and 4 the calculations were carried out for relatively short times, but convergence is apparent. Additional computations were performed for conditions near the marginal stability curve shown in Figure 2, and
Figures 5 and 6 show axial positions vs. time for conditions just below and just above the marginal stability state. For a = 2.00 and B = 4.64 the oscillation quickly decays and the particle moves to the midplane, but for CY = 2.00 and /3 = 4.86 the amplitude of the oscillation grows until the particle collides with an endcap electrode. For both conditions the radial component of the oscillation is stable. The numerical computations yield a marginal stability state very close to the envelopes shown in Figure 2, indicating that the results obtained by solution of the decoupled equations are satisfactory for prediction of the marginal stability state, but the large amplitude oscillations predicted are different. Stability Enhancement. Figure 2 shows that the stability envelope can be modified by means of a dc bias voltage on the ring electrode, but as the drag parameter decreases the range of ,8 over which the particle is stable becomes smaller, and a particle with a large charge to mass ratio can be difficult to capture and retain. Furthermore, an evaporating or condensing droplet can pass into an unstable condition as the mass changes. Automatic control features can be applied to enhance the stability of the
Davis
384 Langmuir, Vol. 1, No. 3, 1985
I40
where V ois the potential required to balance the gravitational force when the particle is balanced at the null point ( z = O), given by
-
vo = mgzo/Coq
(50)
where Co = -Cz(O,O). Replacing V oin eq 29 by vd, from eq 49 and introducing the result in the vertical equation of motion, we obtain
w l I
9
-[Vb
+ k1Cz(R,Z)zo + V,, COS Q t ] z (51)
Z02
Thus, the effect of adding error rate damping is to increase the effective drag on the particle, and, because the amplification constant k 2 can be varied at will, the drag is readily controlled to maintain stability. Also, inclusion of the term klz has the same effect as adjustment of the dc bias voltage, V,. Error rate damping is essential for operating the balance under vacuum, for the aerodynamic drag is not sufficient to provide stability when the pressure is very low.
7 20
0 -12
-0.4 0 0.4 0.0 2 , DIMENSIONLESS AXIAL POSITION
-0.8
I2
Figure 6. Dimensionless axial position for LY = 2.00 and p = 4.86.
r
Aerodynamic D r a g Measurement Two techniques can be used to determine the aerodynamic drag on a particle. The first is a direct application of the stability analysis discussed above. The marginal stability state shown in Figure 2 relates the field strength parameter p to the drag parameter CY.To apply the marginal stability method the balance is operated in the stable mode using voltage V,, with frequency f a , and voltage Voto balance the gravitational force. The charge to mass ratio is then calculated from V oby using eq 50. Next, the voltage V,, is increased until the marginal stability boundary is reached, at which point the particle will begin to oscillate vertically. The value of P corresponding to this state is then calculated from the observed values of V,, and Q (where Q = Birf,,) by means of the equation
P = 4gVac/Co~oQ~V~ Figure 7. Error rate damping control system currently in use with the electrodynamic balance.
balance. Beams1* introduced “error rate damping” to suspend small magnetic rotors in a magnetic field, and Ashkin and Dziedziclg used the technique to levitate a particle by the photon pressure of a laser beam. Arnold et al.20applied this control strategy to their Millikan balance, and we have installed a similar system in a recently constructed balance. The new control system is shown in Figure 7. A dual photodiode is used to detect light scattered from the suspended object. The two signals from the photodiodes are sent to a log-ratio amplifier. If the particle is located a t the midplane, the two signals balance, and no control signal is generated. When the particle is perturbed from the midplane by an amount z a control signal is generated and used to alter the dc voltage on the lower endcap electrode required to suspend the object. If kl and kz are amplification constants, we may write
Vd, = Vo + K1Z - k,dz/dt ~~~
(49) ~~
~_____
(18)Beams, J. W.Reu. Sci. Instrum. 1950,21,182. (19)Ashkin, A.; Dziedzic, J. M. Appl. Phys. Lett. 1977,30,202. (20)Arnold, S.; Amani, Y.;Orenstein, A. Rev. Sci. Instrum. 1980,5I,
1202.
(52)
It is also possible to reach the marginal stability state by altering the frequency or by a combination of voltage and frequency adjustments. Once p has been calculated from the measured experimental parameters for the observed marginal stability state the value of CY corresponding to that p is obtained from Figure 2. This procedure can also be used to calibrate the device if a sphere of known size and mass is used, e.g., a polystyrene latex microsphere, and we have used this technique to calibrate our electrodynamic balance. Operating in the Stokesian regime, we have CY = 12aap/mQ,so a can be calculated directly. The value of p corresponding to this CY on the marginal stability curve is then used to calculate the geometrical constant Co from the measured values of V,,, Q,and V oin eq 52. An alternate procedure for measuring the aerodynamic drag force has been developed here. By suspending the particle in the balance when a gas flows vertically through the balance the drag force can be measured directly. The dc field is used to balance the combined forces of aerodynamic drag and gravity. Assuming that thermophoretic, photophoretic, and photon pressure forces are negligible, the force balance on a suspended object becomes
cOqvdc/zO = K&,pUO/(l + f , - mg
(53)
where Uois the gas velocity a t the center of the balance chamber. In the absence of a gas flow we recover eq 50,
Electrodynamic Balance Stability
Langmuir, Vol. I , No. 3, 1985 385
Table 11. Experimental Data for DBP Droplets at the Marginal Stability State vo
Vac
a, Pm
a
20.5 18.7 12.0 23.9 21.0 12.1 12.1 9.68 9.11 4.56 3.30 2.20 18.8 18.8 13.0 12.8 11.0 31.9 21.1 19.5
1010
23.1 22.3 21.2 19.1 18.3 14.9 15.4 13.8 13.5 10.9 10.1 8.36 18.4 16.6 14.9 13.9 13.9 26.8 16.2 19.6
0.738 0.790 0.879 1.08 1.17 1.77 1.66 2.07 2.16 3.33 3.89 5.64 1.16 1.43 1.77 2.04 2.04 0.542 3.59 2.72
955 827 1390 1315 1046 1046 1011 1011 933 990 1768 1437 1324 1203 1178 1161 1449 1485 1145
and when the flow rate is adjusted so that the aerodynamic drag force equals the gravitational force no dc voltage is required to suspend the particle. We have used this technique to measure the drag force on solid particles and liquid droplets using the apparatus of Davis and Ray: modified to permit flow through the chamber. Experimental Section The electrodynamicbalance used for drag force measurements was the type shown in Figure 1, and V b = 0 here. The experimental system is described by Davis and Periasamy.21 A 1.5mm-diameter hole was drilled through the bottom electrode to permit a metered gas flow to be introduced into the chamber, and a large hole in the top electrode permitted exit of the gas. The gas was filtered, passed through a coil in a constant temperature bath, metered with a rotameter, and entered the balance chamber as a laminar jet. Liquid droplets were injected into the balance by means of a fine capillary tube connected to a microliter syringe. To electrify it the stainless steel capillary was connected to a capacitor with a dc output of up to 10 kV. A droplet was partly formed at the tip of the capillary, and a dc voltage of 2 to 5 kV was applied to inject the droplet into the balance. The ac field, produced with a variable transformer connected to the 60-Hz laboratory supply, was set at approximately 500 V prior to injection of the droplet, and the dc field was normally set to zero at the start of an experiment. As soon as the charged droplet entered the balance and began oscillating in the ac field, as observed by means of a 70X power microscope attached to the balance, the dc voltage was adjusted to move the droplet to the null point of the balance. A polarized laser beam passed through the center of the balance to permit observation of the object and to perform light-scattering measurements. A window in the ring electrode permitted the scattered light to be detected with a photomultiplier connected to a rotating periscope. The droplet radius was determined via analysis of the light-scattering data by comparison with Lorenz-Mie theory. Solid particles were studied by suspending them in water or some other volatile liquid, and then a droplet of the suspension was injected as discussed above. After capture the particle was levitated for a matter of hours to permit the liquid to evaporate completely. Two types of experiments were performed: (i) measurement of the onset of marginal stability to calibrate the balance and (ii) aerodynamic drag measurements using a gas flow through the chamber. Calibration Experiments. After a particle or droplet was stably suspended the ac voltage was increased until vertical os(21) Davis, E. J.; Periasamy, R. Langgmuir, follcwing paper in this
issue.
II r
I
I
I
I /
-,
/
REGION I V
,
STABLE
SILICATES
o DBP
E+Y ’ f REGION I r
oi1
INSET I
5
10
a , DRAG PARAMETER Figure 8. Comparison of the theoretical marginal stability curve with experimental data for DBP droplets and silicate spheres. cillation of the mass was observed in the microscope. Minor adjustments of the dc voltage were made, if necessary,to maintain stability as the ac voltage was increased. For these experiments the ac frequency was held constant at 60 Hz. Table I1 lists typical experimental data for dibutyl phthalate (DBP) droplets introduced into the balance to study the marginal stability limit. The table gives the dc voltage required for suspension, the ac voltage measured at the onset of instability, the droplet radius, and the calculated value of a. From these values of a and Figure 2 we have determined the geometrical constant Co for the balance. The data of Table I1 together with other data for solid spheres and liquid droplets yield a mean value of Co = 0.471 with a standard deviation of 0.054 and a mean deviation of 9.1% for 32 observations. The scatter in the data can be reduced by more carefully controlled experiments using both frequency variations and voltage variations to determine the point of instability. Preliminary data for a new balance obtained by changing both frequency and voltage indicate that the mean deviation in Co can be reduced significantly. Using the data of Table I1 and the mean value of C, to calculate the experimentally observed values of j3,we obtained the results shown in Figure 8. Data points were obtained at the lower stability boundaries of each of the four regions shown in Figure 8. Most of the data correspond to region I, for most of the droplets used were relatively large. The agreement between the experimental results and theory is good, for the data scatter around the theoretical marginal stability curves. The scatter is due primarily to the fact that the droplet mass was determined from the droplet radius. Because the mass is proportional to the cube of the radius a 3% error in the measurement of the latter accounts for a 9% error in the computed mass. The droplet radii were determined from the light-scattering data by a peak-counting method discussed in a companion paper.21 Aerodynamic Drag Measurements. The dc voltage, Vo, required to levitate the object in the absence of flow through the chamber was recorded once stable suspension was achieved. The gas flow was then set to some low value, and the dc voltage was altered to maintain suspension at the null point of the balance. Next the flow rate was varied in steps until the dc voltage required for suspension passed through zero and changed polarity. The flow rate was increased until it became difficult to maintain stability due to oscillations of the jet or other flow-induced
386 Langmuir, Vol. 1, No. 3, 1985 50
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IO 20 0,NITROGEN FLOW RATE, ml/min
I 30
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Figure 9. Dc potential required for suspension at the midplane of the balance as a function of the nitrogen flow rate for DBP and DOP droplets and spherical solid particles. problems. Figure 9 shows typical results of v d c vs. nitrogen flow rate, Q, for gquid droplets of DBP and DOP and silicate particles. Quantitative measurement of the forces requires calibration of the jet to determine the velocity at the midpoint of the balance chamber, and this in turn requires knowledge of the particle mass and radius. Droplets of known density p p were used, and the radius obtained from light scattering was introduced to calculate the mass from m = 4*a3p,/3. Measurement of the flow rate required for suspension of the particle with no dc field can then be applied to determine the velocity, using the appropriate expression for the drag force in eq 53. For the Reynolds numbers encountered in these experimenta the drag force is given by Stoke’s equation with the Oseen correction FD = 67rapUo(l + 3Re/16)
(54)
where the Reynolds number is defined by &e = 2aU@/p,where p is the gas density. In this case the force balance on the particle is 6mpUo(l + 3aUop/8p) - (4na3p,/3)g = 0
(55)
and the velocity is readily determined from the measured radius and the physical properties of the droplet and the gas. For Re < 0.1 eq 55 leads to the classical result of the terminal velocity of a small sphere
By permitting the droplet to evaporate slowly and repeating the experiment, we were able to calibrated the jet over a suitable range of operating conditions. Calibration data were obtained by using dioctyl phthalate (DOP) and dibutyl phthalate (DBP), which have very low evaporation rates, for a volatile liquid, pinene, and for solid microspheres of polystyrene latex (PSL) and spherical silicate particles. Once the jet has been calibrated, the results can be used to examine the aerodynamic drag for nonspherical particles or for droplets undergoing rapid evaporation or condewtion. The ratio of the drag force to the particle weight is obtained from the dc
Figure 10. Ratio of the drag force to the gravitational force for silicate particles and pinene as a function of flow rate. voltage required to maintain the particle at the null point, for using eq 50 in (53) we obtain FD/mg = l -
vdc/vO
(57)
Figure 10 shows the ratio of the drag force to the gravitational force measured for sphericaland irregular silicate particles (dust) and a volatile pinene droplet as a function of flow rate. At lower flow rates (Q < 15 mL/min for the jet diameter used here) all of the data show that the drag force/weight ratio varies linearly with the flow rate, which is to be expected if the jet velocity is proportional to the flow rate. At higher flow rates the relationship between the drag force and the flow rate is nonlinear as indicated by eq 54.
Discussion The geometrical constant obtained for the electrodynamic balance used in this study, Co = 0.471, is considerably lower than the theoretical value of Co = 0.8768 derived above and is larger than the value obtained by Frickel et al. for their balance (C, = 0.310). Recent results for the picobalance shown in Figure 7, for which zo = 19.0 mm and the outer diameter was 62 mm, yielded Co = 0.92. The existence of holes in the ring electrode and endcap electrodes and windows in the ring electrode contribute to deviations of Co from the theoretical value, and the calibrated balances had different configurations in this respect. Although calibration of the balance is not required for many applications, if both the charge and mass are to be calculated, knowledge of C, is essential. For example, Arnoldz2introduced the method of electron stepping to determine the charge and mass of a particle by using a UV source to alter the charge by one electron. Calculation of the particle mass from the voltages required for suspension before and after charge loss requires knowledge of Co. It should be pointed out that application of Figure 2 and Table I to the calculation of aerodynamic drag parameters is limited to creeping flow conditions because the stability (22) Arnold, S. J. Aerosol Sci. 1979, 10, 49.
Langmuir 1985,1, 387-390
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to maintain the particle in the center of the balance with the electrical field. Flow instabilities and vortex shedding which occur a t higher Reynolds numbers can result in transport of the particle out of the chamber. The flow experiments performed indicate that reasonably large flow rates can be used without losing the particle. This makes it possible to study convective mass transfer, and we have made measurements of mass transfer rates for pinene evaporating into flowing nitrogen. Furthermore, the electrodynamic balance can be used as a chemical reactor for gas-particle chemical reaction studies, for a reactive gas can be passed through the balance chamber.
theory used to determine the marginal stability curve is based on creeping flow. Experimentally, operation of the balance is aided by the effect of higher order terms on the drag. Because of higher order terms (see eq 54) a particle oscillating in the unstable regime experiences a larger drag force than that used in the derivation of the stability theory. The result is that it is frequently possible to operate in the unstable regions of Figure 2 without losing the particle. In fact, we have found it possible to increase the ac voltage beyond that corresponding to region 1of Figure 2 and have the particle undergo a large-amplitude oscillation, reach the next stable region, and become unstable again as the voltage is increased. The flow technique is not limited to low Reynolds numbers, so it is more appropriate to use for the study of aerodynamic drag over a fairly wide range of Reynolda numbers. The only limit to the flow method is the abilit;
Acknowledgment. This work was supported by the Defense Nuclear Agency under Contracts DNA 001-82-C0224 and DNA 001-84-C-0293.
Vesicles from Dimer Acid and Its Derivatives? Jiirgen-Hinrich Fuhrhop,*$ Winfried Kaufmann,t and Fred Schambils Institut f u r Organische Chemie der Freien Universitat Berlin, 0-1000 Berlin 33, West Germany, and Forschungslaboratorium Henkel KGaA,
0 - 4 0 0 0 Diisseldorf 1 , West Germany Received July 6,1984. In Final Form: February 1, 1985 Commercial dimer acid is the only nonexpensive a,w-dicarboxylicacid of appropriate chain length for the formation of vesicle membranes. The acid itself is water insoluble and does not form vesicles on sonication. Addition of cyclohexylamine or amide formation with aniline-2-sulfonic acid, however, converts dimer acid into a vesicle forming bolaamphiphile. Analogous results have been obtained by using esters of commercial dimer alcohol and pyromellitic acid. A stable gel, which produces a vesicular solution on dilution with water, is also described. Dimer acid derivatives constitute candidates for large-scale applications of vesicles.
Introduction Vesicles are sealed, extremely thin (110 nm) membranes of spherical shape, which enclose water volumes of approximately 1-1000 pm3.1v2 Various amphiphiles may aggregate to vesicles with critical concentrations in the range of
