On the stochastic nature of the motion of nonspherical aerosol particles

On the Stochastic Nature of the Motion of Nonspherical Aerosol Particles V. The Orientation-Averaged Diffusion Tensor for a Simple Shear Flow, and Related Experiments ALFRED D. EISNER 1 AND ISAIAH G A L L I L Y 2 Department of Atmospheric Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel Received October 17, 1983; accepted March 16, 1984 The particle-ensemble, orientationally averaged translational diffusion tensor of nonspherical aerosol particles in the molecular regime was calculated for a typical though not limiting case of Brownian cylindrical fibers immersed in a simple shear flow. For usually encountered conditions, the value of the averaged tensor of very small particles was found to be insensitive to the strength of the gradient flow, indicating the preponderance of the randomizing over the orienting physical factor which influences particle orientation. Experiments designed to test the calculational method by the effect of the flow on light scattered from (cylindrical) particles proved the correctness of the method. I. INTRODUCTION

In two previous studies (1, 2) we calculated for the molecular regime the rotational diffusion tensor rD of cylindrical and discoidal aerosol particles and their gradient rotational velocity co within a simple shear flow. It was found that the values of the mid-diameter rotational diffusion coefficient rD± for typical diffusing cylindrical particles were about one and one half orders of magnitude greater than those calculated from classical continuum theory while the values of o~ were essentially the same as those derived according to Jeffery for (rotational) ellipsoids of an axial ratio identical to that of our particles (3). Thus, from the latter near equality, it became possible to use a solution for the orientation distribution function F of spheroidal particles which is based on the equations of Jeffery, and deduce the particle-ensemble, orientationaUy averaged translational diffusion tensor defined by (1, 4)3: Present address: High Temperature Chemical Reaction Engineering Laboratory, Yale University, Box 2159 Yale Station, New Haven, CT 06520. 2 To whom all correspondence should be directed. 3 See Appendix for nomenclature.

= -cl f~ crtDd~

[1.1]

where c = f , ~d~b and ~ is the number concentration in an orientation-location hyperspace. It is this ensemble-average which is important for treating Brownian diffusion of nonspherical particles. II. AIM OF STUDY

We set out here to calculate the ensemblediffusion tensor ~ in a simple shear flow of cylindrical aerosol particles which are much smaller than the mean free-path of the gas molecules and which show a non-negligible Brownian motion. Also, we intended to test our theoretical results by suitable experiments. The particles we considered simulate the important asbestos fibers, long chain-aggregates, and some viruses; however, as pointed out beforehand (1, 2), our method of dealing with the problem of this study is by no means limited to the above particle shape. The flow, which is encountered quite frequently, can approximate many real fluid dynamic situations (5).

356 0021-9797/84 $3.00 Copyright© 1984by AcademicPress,Inc. All rightsof reproductionin any formreserved.

Journalof Colloidand InterfaceScience.Vol. 101.No. 2, October 1984

NONSPHERICAL AEROSOL PARTICLES III. THE ORIENTATIONALLY AVERAGED DIFFUSION TENSOR The solution for the distribution function we chose was that of Peterlin (6): f--~--£~ 1 + p

(l-~cos2q~+~--p-~eSin2q~ 3 )

X 1 +3sin20,,o/v ""''r~e2"I + p2 E_

+ ~

+

9

(35

COS 4 0 - -

<(60) 1-~

3 (3 COS2 0 -- 1)

16

sider particles' orientation as a state that is simultaneously influenced by the randomizing action of the rotational Brownian motion and the orienting effect of flow gradients, the relative importance of both of which is given in the rotational Peelet number rPe, rpe = ]o~]/ rD±. For a quiet medium, rpe = 0, one obtains that F ~ F" as t ---o oo (8), and for a very strong velocity gradient or a very small rotational diffusion coefficient rD±, when rpe --" ~ , one gets the (Peterlin) asymptotic solution

(6):

30 cos 0 + 3)

c o s 4 ~ b - r ~ esin4~b

357

>

F~=41r

3 1 - p ~ c o s 2 q ~ s i n 20

+p213(15cos40-30cos20+ 15 sin 4 0 1 × 16(1 + l ~ f f P e 2 i J

7)

+ i 6 cos 44~ sin 4 0 - p3 cos 2q~320 sin2 0

1

}

× 1 + (36/rpe 2) + p 3 [ . . . ]

[3.1]

which is written as a pertubation to the random distribution F " = 1/47r with terms including increasing powers ofp. This expression holds for the steady state established almost instantaneously for very small aerosol particles (7); it can be conveniently put into the form: F = F" + q

[3.2]

where q denotes the contribution of the gradient flow to the value of F. From a physical point of view, one can con-

35 × (105 cos 4 0 - 470 cos2 0) + ~ cos 64~ Xsin60+20el+...}.

[3.3]

Here, to calculate the components of the orientationally averaged diffusion tensor 2) in shear flow, we used the method employed by us for a still environment (1) since the concentration-dependent term of the diffusion equation is identical in the two cases. The essential part of the method is the evaluation of the orientation-average of some trigonometric functions f(0, qS)which can be formally written for our axisymmetric particles as (1):

o2~ fo~f(O, ch)F sin OdOd4~
fo~"fo~FsinOdOd49 f(O, c~)F" sin OdOd4~+

f?fo _A1 +A2

B1 + B2

F " sin

OdOd¢ +

f(O, ¢b)qsin OdOdcb

f?f0 q

sin

OdOd4~ [3.4]

Journal of Colloid and Interface Science, Vol. 101, No. 2, October 1984

358

EISNER A N D GALLILY

where A1, A2, B1, B2 correspond to the similarly situated terms in the middle portion of [3.4]. In the typical case of diffusing cylindrical particles of tobacco mosaic virus whose diameter, 2p, is 0.015 #m and whose length, L, is 0.1 #m, we obtain from [3.4] and [3.1], neglecting terms proportional to pE, that 1


[3.5]

Consequently, by the previously presented calculational technique (1), we get for the considered typical particles at normal atmospheric conditions that ~911 = tD± + (tDii - tD± )

[3.9]

[3.10]

The above conclusions are rather striking; however, one should be cautioned to note that, though here the flow field does not essentially affect the value of 1) which has to be introduced into the equation of translational diffusion, the velocity of the fluid is still of a great influence through the convective terms of that equation. IV. E X P E R I M E N T A L

1 + (36/rpe2)[ etc.

[3.61

which, if we put for brevity sake: A~)ii = d ~ ) i i - *~)ii(r = 0 ) 1

and

I

[3.7]

0.05AtD A~933 = 1 -t-(36/rpe2) "

[3.8]

A t D = tDii -- t D ± '

can be formulated as A~011 =

-0.17AtD 1 + (36/rPe 2) '

A ~ 2 2 --

0.22AtD 1 + (36/rpeE) '

and

Equations [3.8] immediately show that, unless rpe becomes much greater than unity, which is very rarely the case, the effect of the shear flow on the (meaningful) components of the ensemble-diffusion tensor of our typical small TMV particles is negligible. This hints that, in this situation, the ranJournal of Colloid and Interface Science,

0.27AID,

A~gEE/A;DEE(F = 0) -----0.27.

1 0.22
×

A~)2E ~

which can be written also as the relative contribution of the gradient flow:

0.17 1 + (36/rpeE) '

1 0.05 (cos 2 O) = ~ + 1 + (36/rpeE) "

domizing rotational Brownian motion overwhelms the field-orientation of the particles. For comparison, we calculated A~aEfor very large Peclet numbers where the orientation function is given by [3.3]. In the latter case we obtain that, for our particles:

Vol. 101, No. 2, October 1984

In searching for an appropriate experiment to confirm the theoretically predicted insensitivity of the diffusion tensor ~9 to a shear flow, we could either adopt a direct approach, where the values of the tensor components are directly determined, or an indirect approach in which the calculational method used to reach the above result is examined. The first choice involves experiments of convective diffusion where variations in the values of :D are expected to fall within the experimental noise; consequently, we had to return to the second choice. Due to the closed structure of our calculations, it seemed that an experimental support of a fundamental intermediate finding is sufficient to establish confidence in the validity of the other conclusions. We decided to test the calculations by checking the value of the particles' diffusion coefficient since it is in this coefficient where the enormous difference between our, and the classical theoretical results, lies (1, 2). Furthermore, because of the envisaged difference between those two results, we could even be satisfied with an order of

NONSPHERICAL

AEROSOL

magnitude-type of an answer; instead of a direct measurement of the coefficient such as that performed by (the difficult) relaxation techniques (9), we were able to employ an indirect method in which the correct value of tD± is decided upon through the role of the Peclet number in an orientation-sensitive effect. It should be mentioned at this place that an independent indication of the validity of our calculations was supplied by the experiments of Adams and Lloyd (10) where the torque operating on a (macro) discoidal pendulum undergoing damped torsional vibrations in a rarefied gaseous atmosphere was found to have essentially the same value as that derived from our formula (1). 1. M e t h o d G e n e r a l The experiments themselves consisted of a determination of light scattered from an aerosol suspension subjected to various rates of shear under free molecular and normal atmospheric conditions. The effect of gradient field-orientation on light scattering from liquid suspensions of tobacco mosaic virus particles has been measured already (11) and shown to be quite great as explained also by theory (12); so, we anticipated obtaining a similar magnitude of the phenomenon. In essence, the rationale of our experiments was based on the dependence of the orientational distribution of particles within a shear flow on the characteristic Peclet number, rpe. Thus, one could choose here operational conditions for which, if the true value of rpe is that deduced from our calculated rD±, no preferred orientation can be expected while if its value coincides with that formed from the classical expression for the diffusion coefficient, a preferred alignment with the flow lines can be assumed. Confidence in our calculational method could be established just by the absence or existence of a flow-orienting effect. Specific. The shear flow carrying the aerosol particles was produced in a gap between two

359

PARTICLES

concentric cylinders by rotating the inner one with a constant velocity o~c. The (incompressible) Couette flow in such a gap is given by the (known) solution:

u(R) =

_

2

R R ~ wc

[4.1]

which was experimentally tested and confirmed a long time ago. Likewise, the stability of the flow was already theoretically analyzed and experimentally checked (13). Here, taking the latter factor into consideration, we adjusted the working conditions so that, for our employed gap-width, the rotational velocity ~Oc of the inner cylinder was about two orders of magnitude lower than the critical value for the emergence of instability; the Reynolds number of the flow was very small too. The cylindrical particles used in the experiments were chosen to be much larger than the tobacco mosaic particles taken as a typical example in the calculations. If the latter particles would have been employed, then one would have to create very high flow gradients so as to get Peclet numbers of the order of unity; likewise, the scattered signal-to-noise ratio would have been too small at achievable number concentrations, all of which seemed to us prohibitive. In order to still have the free-molecular flow conditions of the theoretical analysis, we were obliged to produce a suitable vacuum in the intercylinder gap. 2. A p p a r a t u s

The experimental apparatus is schematically shown in Fig. 1. It was composed of the following five operational units: (1) A set of two coaxial vertical cylinders (A9) in the gap between which our aerosol particles were subjected to the action of the flow gradient; (2) an electromechanical device (A4, As) which rotated the inner cylinder at a measured velocity; (3) an aerosol generator (A3); (4) a light source (A~, A2), and (scattered) signal receiving (A6, A7) and recording (A8) instrumentation; (5) a vacuum pump (A10) and pressure gauge Journal of Colloid and Interface Science, Vol. 10 I, No. 2, October 1984

360

EISNER AND GALLILY

A9

FIG. 1. Blockdiagramof the apparatus(general).

The laser beam was absorbed after traversing the gap by a trap (A12 , Fig. 2) owing to which, and to a suitable black paint on the surface of the inner cylinder, the optical noise in our experimental system was reduced to the photomultiplier dark current level. The aerosol source used consisted mainly of a glass-made vessel which contained a preprocessed mass of the aerosol material and which was shaken by a vibrator whose amplitude and frequency were controlled by a frequency generator. Preceding the glass vessel there was a train of three capillaries placed to facilitate a slow introduction of a stream of dried and filtered air which carried the particles loosened from the vibrating material through the exit into the intercylinder space.

3. Aerosol Particles, Experimental (A11), all mounted on a heavy table to elimConsiderations and Parameters inate mechanical vibrations. In more detail (Fig. 2), the first unit was The aerosol particles were glass fibers of comprised of a glass outer cylinder (Al0) of (mostly) a cylindrical shape. The size distri14 cm i.d. and 60 cm length, and a brass inner bution of these particles was determined before cylinder (All) of 9 cm o.d.; the second unit and after each experiment, and subsequently consisted of a controlled and stabilized-speed to sampling by a thermal precipitator (Casella, 12 V dc electric motor (A3) which drove the England), with the aid of a scanning electron inner cylinder through a flexible coupling (A4) microscope. Instead of our usual photogramand a pulley-belt assembly (A~) by the agency metric technique of size determination (15), of two pairs of strong horseshoe magnets (A6, we employed now a more convenient proAT); the third unit was essentially the Spumy cedure in which the microscope stub collecting generator (14); the fourth unit contained a cw 15 mW He-Ne laser (Spectra Physics), an orientationally adjustable plane-mirror (A7) which deflected the light beam to pass through A1 __ A4 an optical window into the intercylinder space, a photomultiplier (AT, Fig. 1) (EMI 9558 B) A7 As with its power supply (A6, Fig. 1) (Fluke 412 As B), and a suitable electric output recorder (A8, A9 Fig. 1) (Sunbeam 320); the fifth unit included A~o. [ a vacuum pump and a Pirani gauge (LKB, type 3294 B). The rotational velocity of the inner brass cylinder was determined by a photoelectric tachometer (A1, Fig. 1) in which the slotted disc (A2, Fig. 2) revolving with the cylinder formed an integral part. A12 A~a The aerosol particles were introduced into FIG. 2. Block diagram of the apparatus (detailed). the cylinders' gap through the opening (A3).

AA3 I

Journal of Colloid and Interface Science, Vo¿. 10h No. 2, October 1984

NONSPHERICAL AEROSOL PARTICLES the deposited particles was smeared by a layer of kerosene, a relatively involatile liquid. When this layer was later evaporating, the glass fibers were forced by surface tension effects to lie flat on the stub's surface, thus obviating the need of photographing every particle from at least two directions of sight. The biparametric size distribution of the cylinders is presented in Fig. 3 from which one can note the typical diameter and length values, viz. 0.18 and 4.2 #m, respectively. Based on these particles sizes, we could decide that a total air pressure within the intercylinder gap of 5-7 m m Hg would secure a Knudsen number Kn (= i/p) great enough to provide the required experimental conditions of a molecular regime. If we would have calculated the mid-diameter diffusion coefficient rD. of the fibers according to classical theory, then we would be able to expect a noticeable influence of the flow gradient on the scattered light at the innercylinder rotational velocities of the order of magnitude of 1 sec -~, as shown in the study of Heller et al. for liquid, continuous fluid suspensions (1 1); but, on the other hand, if we would use our theoretical formula for rD±, we could not assume any gradient effect at that velocity. Thus, to be on the safe side, we extended the range of the rotational speeds ~0c in our experiments up to o0c = 1.5 sec -L.

30

~ z

20

10

I==1

.....

Ill

5

A

_ _ B

361

As the glass particles were not neutrally bouyant, we had to estimate the interference of their gravitational settling to their pure gradient-field motion. For our fibers of the above typical sizes (and aspect ratios), the calculated velocity of fall was about 0.01 times the tangential velocity of the medium within the cylinders' gap at the worst experimental case; so the gravitational interference could be justifiably ignored. The curvature of the real flow lines in the gap was neglected too, due to the small dimensions of the sensed volume in which the actual light scattering determination was performed.

4. Experimental Procedure The experimental procedure consisted first of processing the glass fiber mass from which the aerosol was produced by grinding it in a coffee machine before its placement in the container vessel of the generator. In the next stage, the intercylinder gap was overevacuated to a total pressure of about 0.15 m m Hg and the vacuum pump disconnected. The vibrator of the aerosol source was then operated, and dried and filtered air was slowly introduced through the container to achieve aerosolization. During the passage of the particles into the gap, the inner cylinder was started to rotate at a high speed so as to suppress any local excessive deposition of the in-going fibers on the surface of the former. The determination stage in the experiments was carried out after switching on the H e - N e laser and the photomultiplier power supply by measuring the scattered light signal at various rotational velocities ¢oc.

5. Results

10

15

FIG. 3. Sizedistribution of the cylindricalparticles. (A) Diameters, 20 × 105 cm; (B) lengths, L × 104 cm.

The experimental results are brought out in Fig. 4 where the relative change of the scattered light signal in a gradient field,/, from the signal in still air, I0, is plotted against the rotational velocity wc once for molecular flow conditions (branch A) and second time for an Journal of Colloid and Interface Science, Vol. 101, No. 2, October 1984

362

EISNER AND GALLILY

1.0

0.5 I--4 o

0.0

I I

0.5

[

I A

I

I

1.0

1.5

~(sec-,) FIG. 4. Relative increase of the intensity of scattered light from cylindrical aerosol particles at various rotational velocities. (A) Discontinuous medium; total gas pressure, 6.5 mm Hg. (B) Continuous medium; total gas pressure, 695 mm Hg. T = 297°K. 2 d / I o = ( I - Io)/Io.

atmospheric, continuous regime situation (branch B). Out of this figure it is seen that, in the molecular flow case, there is no perceivable effect of the velocity gradient on the scattered light whereas in the continuous fluid regime there is a pronounced change of the signal with we. This means that the orientation distribution of our particles is not affected by the flow, as expected from our calculations. The true rotational Peclet numbers in the experiments are bounded so from above by the value of 0.016. In passing, it should be noted that the atmospheric pressure determinations not only served as a blank test to our hypothesis but also indicated that disturbances due to phoretie factors could not have played a role in the overall found results. V. DISCUSSION AND FINAL CONCLUSIONS

The results of our experiments with a real aerosol system seem to deafly prove that, under ratified gas conditions (which conform to a case of very small particles of Kn ~> 1), the calculational method of evaluating the rotational mid-diameter diffusion coefficient of our particles is essentially correct. Thus, by Journal of Colloid and Interface Science, Vol. 101, No. 2, October 1984

establishing confidence in a major finding of our method, the other theoretical conclusions get support. The main outcome of this study is that, according to calculations checked by experiments, the particle-ensemble, orientationally averaged diffusion tensor E) of very small, Brownian cylindrical particles is essentially independent of usually encountered flow gradients; the value of this tensor does coincide with that for a quiet medium for all practical purposes. It should be stressed here again that the flow velocity itself still influences the diffusion process through the convective terms of the equation describing the phenomenon. The calculational method presented in the two previous studies (1, 2) and this reported one can be extended to many other systems of diffusing nonspherical aerosol particles in the molecular regime. APPENDIX: NOMENCLATURE A I , A2, B I , BE

tD

rD

E)

F

I

Translational diffusion tensor of particles; tDNN,tensor component parallel to particle's axis of rotational symmetry; tDz, component perpendicular to axis Rotational diffusion tensor of particles; rDz, rotational diffusion coefficient around a mid-diameter of the particle Particle-ensemble, orientationally averaged translational diffusion tensor; ~ ) i i , (diagonalized) tensor component related to (external) axis i Orientation distribution density; F ' , random orientation density; F ~ , asymptotic density as rpe -~ oo Scattered light intensity signal; Io signal at a still medium; AI/Io = (I - Xo)/L Knudsen number defined by Kn -

Kn

C o n s t a n t s d e f i n e d in t e x t

=

i/o

NONSPHERICAL AEROSOL PARTICLES

7 L P rpe

R

R I , R2 u(R)

M e a n free-path o f the gas molecules Length o f cylindrical particle Particle size parameter defined by p = [(L/2) 2 - p2]/[(L/2)2 + p2l Rotational Peclet n u m b e r o f the particles, defined in text contribution o f the gradient flow to the orientation distribution density Radial distance from axis o f the twocylinder set to a point within the intercylinder gap Radius o f inner and outer cylinders, respectively (Tangential) velocity o f m e d i u m within the intercylinder gap

Greek Letters

F p o~ Wc 4, 0

Flow gradient Radius o f cylindrical particle Rotational velocity o f the particle Rotational velocity of the inner cylinder Euler angles

ACKNOWLEDGMENTS This study has been supported by a grant from the Branch for Basic Research, Science Division, the Israeli

363

Academy of Sciences and Humanities, whose kind help is hereby gratefully acknowledged. REFERENCES 1. Eisner, A. D., and Gallily, 1., J. Colloid Interface Sci. 81, 214 (1981). 2. Eisner, A. D., and Gallily, I., J. Colloid Interface Sci. 88, 185 (1982). 3. Jeffery, A. D., Proc. Roy. Soc. A 102, 167 (1923). 4. Brenner, H., and Condiff, D. M., J. Colloid Interface Sci. 41, 228 (1972). 5. GaUily,I., and Eisner, A. D., J. Colloid Interface Sci. 68, 320 (1979). 6. Petedin, A., Z. Phys. 111,232 (1938). 7. Brenner, H., and Condiff, D. M., J. Colloid Interface Sci. 47, 199 (1974). 8. Vadas, E. B., Cox, R. G., Goldsmith, H. L., and Mason, S. G., J. Colloid lnterface Sci. 57, 308 (1976). 9. Stoylov, S. P., Colloid Szech. Chem. Commun. 31, 2866 (1966); Stoylov, S. P., and Sokerov, S., J. Colloid Interface Sci. 27, 542 (1968). 10. Adams, R. D., and Lloyd, D. H., J. Phys. E. 8, 475 (1950). 11. Heller, W., Wada, E., and Papazian, L. A., J. Polym. Sci. 47, 481 (1961). 12. Okano, K., and Wada, E., J. Chem. Phys. 34, 405 (1961). 13. Taylor, G. I., "The Scientific Papers of Taylor" (G. B. Batchelor, Ed.), Vol. IV. Mechanics of Fluids, Miscellaneous Papers, p. 34 Cambridge Univ. Press, London/New York (1971). 14. Spumy, K. R., Boose, C., and Hochrainer, D., StaubReinhalt Luft 35, 440 (1975). 15. Gallily, I., J. Colloid Interface Sci. 37, 403 (1971).

Journal of Colloid and lnte(ace Science, Vol. 10 I, No. 2. October 1984