Diffusional phoresis of aerosol particles

DIFFUSIONAL

PHORESIS

B. V. D e r yagin,

OF AI~'ROSOL

S. P . B a k a n o v ,

PARTICLES

S. S. D u k h i n ,

and G. A. Batova Institute of Physical Chemistry. Academy of Sciences of the USSR

The action of surface forces on aerosol particles in a diffusional field is essentiaUy different in the two limiting cases characterized by the conditions

r~Z,

(1)

where r Ls the radius of the particle, assumed to be spherical, and ~, is the mean free path of the surrounding gas molecules. In the first case, the pr=ence of the aerosol particle causes a negligibly small perturbation in the velocity distribution of the surrounding gas molecules which are about to collide with it. This makes it possible zo calculate the resultant force on the aerosol particle from the Chapman and Enskog molecular velocity distribution for the diffu;ing gaseous mixture [1]. This force gives to the particle a stationary velocity relative to coordinates fixed in the walls of the containing vessel which is determined from the expression

.

~"

= -

3imoMzMzkT

"

2,

".

I"

Izrad,,= X {.q, A.~:p---;

=o ~

/H=

[ l--Vn-~l~

[l ~--;,,7-~

,

;nz~ -- t B z -t- .'~zsB_ z M,/. / (3) I + n:l " -;- 2 (n:l + l[nJD'_l~ ~" J

Here o zz is one-half the sum of the molecular diameters; p z is the density of the second component; B~, B[, B: z, A are numerical coefficients; m n = ml/m=: mzz = ms/ms; m0 = mz + ms; Ms = ml/me: nzz = uz/nz; k is the Boltzmann constant; T is the absolute temperature; ms. mz are the masses of the molecules: and el, nz are the numbers of molecules per cubic centimetez, It is seen from Eq. (3) that the tint member in curved brackets reduces to zeso when the components of the mixrare have the same molecuhaz masses (molecular weights) and the effect is then due m the difference in the molecular d iameters. If diffusion results through evaporation from a liquid surface (droplets or a large volume of liquid, for example), the gaseous component can be considered to be at rest with respect to the latter and the rate of movement of the diffusing particles then expressed through the equation ; IT should be noted that the first term in the curved brackets of Eq. (3) is related to the coefficient of diffusion of the mixture. In fact. in the zemth approximation

uz =

--

D~s grad nz

n (nhm..) '/'

pn=

(m"=-- t) ( nltm~ .~- t) '

where n is the total number of gaseous molecules pea cubic centimeter of mixture. P is the density of the mixture. and Dzz is the coefficient of interdiffesion of the components.

277

278

B.V. Derjaguin u'z = -

~ { moMtA,b kT

~'/, grad na

I

B°Mz

J 1_~ ~ i

I -i-n2,rn~

+

2 (n2~ + ]/n~d)

I

(4)

Here the v o l a t i l i t y or l a c k of v o l a t i l i t y of the particles t h e m s e l v e s has no effect on either the resultant force or the p a r t i c l e velocity, since evaporation takes p l a c e s y m m e t r i c a l l y and therefore does not give rise to a resultant impulse. The situation is essentially more involved in a stream of diffusing particles which satisfy relation (2). It is then necessary to draw on the e m p i r i c a l equations of c o n v e c t i o n a l diffusion and diffusional hydrodynamiCs in order to d e t e r m i n e the t a n g e n t i a l and n o r m a l forces a c t i n g on the p a r t i c l e surface [2]. Account must be taken here of the boundary conditions, a t the p a r t i c l e surface and a t infinity, for both the diffusion and the velocities in the gaseous mixture. The k i n e t i c theory of gases must be drawn on only to establish the c o e f f i c i e n t of proportionality k between the slip v e l o c i t y and the diffusioaal flow, or concentration gradient, t a n g e n t i a l to the particle surface. Here lies the principal d i f f i c u l t y of the problem, yet to be solved. We w i l l consider two l i m i t i n g eases: 1) the particles and the diffusing vapors are of the same m a t e r i a l ( ' v o l a t i l e " particles), and, 2) the particles are i n c a p a b l e of even p a r t i a l l y absorbing the vapors ( ' n o n v o l a t i l e " particles). In the first ease, the concentration and pressure of vapor must be constant over the e n t i r e p a r t i c l e surface and equal, r e s p e c t i v e l y , to the concentration and pressure of the saturated vapor. This is the boundary condition for the diffusion p r o b l e m . It follows that the t a n g e n t i a l component of the diffusional flow must be equal to zero and the slip v e l o c i t y of the gaseous m i x t u r e at the drop surface w i l l therefore be e q u a l to zero as well. This s i m p l i f i e s the problem considera b l y s i n c e it e l i m i n a t e s the necessity of d e t e r m i n i n g the c o e f f i c i e n t referred to above. In the second case, the diffusional flow cannot have a n o r m a l c o m p o n e n t and the normal c o m p o n e n t of the c o n c e n t r a t i o n gradient must therefore be e q u a l to zero. This is the boundary condition for the equations of c o n v e c t i o n a l diffusion. S. S. Dukhin and B. V. Deryagin [2, 3] have shown that a " n o n v o l a t i l e " p a r t i c l e w i l l be at rest with respect to the c e n t e r of mass of the surrounding gaseous m i x t u r e when the effect of diffusional phoretic slip is neglected. This means that such particles must m o v e with the "Stefan current" whose vector v e l o c i t y is given by

is

polo q- pll P0 + 9~

(5) '

where p I and P2 are the densities of the two components of the gaseous mixture, and I0 and l I are the m e a n l i n e a r v e l o c i t i e s of the molecules. In the case of diffusienal phoretic slip, the "intrinsic" diffnsinnal phoretic rate of transfer I D is to be added to the total p a r t i c l e v e l o c i t y

I = Is + I o , here

Io ~ ( I t - - Is) = k g r a d Pi and grad P 1 is the density gradient in the diffusing vapors. The experiments of P. S. Prokhrov and L. F. Leenov on silvered glass Sl~aexes in a diffusion current of water vapor with I0 = 0 have shown that the components I s and I D are of approximately the s a m e order of magnitude. Lack of knowledge of the coefficient k makes it impossible to carry out a complete theoretical calculation of the diffusional phoresis of large "nonvolatile" particles. An exact and complete theory can be set up for the diffusional phoresis of large " v o l a t i l e " particles. Here the disappearance of the combined boundary condition involving the rate of slip and the tangential diffusinnal flow simplifies the work so that a first approximation to the concentration field can be obtained independently of a solution of the hydrodynamlcal problem, n e g l e c t i n g nonlinear terms in" the equations of convectional diffusion and hydrod y n a m i c s . On this basis, one obtains a solution in which the gaseous component is everywhere at rest with respect to the drop, with the vapor diffusion current passing through it [2-5]. For this case, the concentration (partial density) field of the vapor p ' is found by taking into account the fact that the vapor concentration is fixed on the drop surface and varies linearly in the direction of the assumed unidi-

Selected Works - 2

279

m e n s i o n a l flow at infinity grad

p' =

const.

o n c e the fields of concentration anddiffusion have been d e t e r m i n e d in this way, the velociJ.y field can be obtained from Eq. (5) and the assumption h = 0; from this the tensor for e a c h direction is found by use of the hydrodynamieal equations for viscous liquids. A first approximation to the resultant force a c t i n g on the drop can be had by summing over the drop surface F=

4 a x ~ R

p ~ p" -p°

(6)

Here p " is the Partial density of the gas, P ] is the density of the saturated vapors at the drop surface, p ~ is the vapor density a t distances which are large in comparison with the drop radius, and

D

(6')

when D is the diffusion coefficient, ~ is the conventional viscosity of the mixture, ~ is the bulk viscosity, and p " a n d / J " are, respectively, the m o l e c u l a r masses of gas and vapor, x > 0 for air and water vapor. Using Stokes' law for the rate of migration of the p a r t i c l e s in combination with (6) and (6') one obtains D

p.] gradp'(Pl--Po)

(6")

Thus, the direction of diffnsional phoresis depends not o n l y on the direction of the diffnsional flow but also on the sign of ( p ~ - p ~), being d e t e r m i n e d by whether the drop evaporates or grows through condensation of the surrounding vapors. Equation (6) is analogous m the expression for the e l e c t r o s t a t i c force which acts on a charged body where the rate of mass exchange is replaced by the charge and the diffnsional flow by the field strength [4]. The following expression can be obtained for the diffusional force F which acts on the drop as a resuR of the diffusional flow arising from another drop (cited in [4]) :

F ' = 4nx

R1R= h

(7)

Here 6 p l = p i - P L and 6p~ = p~ - p~ (the indices 1 and 2 referred to the first and second drops, respectively) and the other symbols have the s a m e s i g n i f i c a n c e as before. The a n a l o g y with Coulomb's law is striking. These equations take into a c c o u n t the fact that the t e m p e r a t u r e of a drop involved in mass transfer differs by a d e f i n i t e amount from the t e m p e r a t u r e of the ~Jrrormdings ( p s y c h m m e t r i c temperature). Q u a l i t a t i v e confirmation of this theory of diffnsional phoresis can be found in the observations of Facy on the repulsion of dust P a r t i c l e s from evaporating drops. The t h e o r e t i c a l trealrnent of F a c t is in error, however, since it m a k e s no a l l o w a n c e for differences in behavior of large and s m a l l , and v o l a t i l e and n o n v o l a t i l e , particles. The direct measurements of P. S. Prokhnrov and L. F. Leonov have confLrmed the fact that the force of diffusional repulsion of l a r g e nonvolatile particles by a water drop is inversely proportional to the square of the distance and d i r e c t l y proportional to the evaporation rate. The p r i n c i p a l difficulty here lay in the e l i m i n a t i o n of t h e r m a l conv e c t i o n b y w a r m i n g the evaporating drop to the temperature of the surroundings. Much significance attaches to the diffusional phoresis of the particles which enter the a l v e o l i of the lungs w i t h the inhaled air. The evolution of CO 2 and the absorption of O 2 set up two diffusional countercurrents near the lung surfaces. Although the volume of evolved CO 2 is less than the volume of absorbed O 2 by 10-15°/o, the greater m o l e c u l a r w e i g h t of the f'u-st gas assures that the Stefan current w i l l be d i r e c ~ d away from the a l v e o l i walls, thereby protecting the l a t t e r from c o n t a m i n a t i o n by those foreign aerosol particles for which r >> X. The coefficient k would h a v e to be e v a l u a t e d in order to d e t e r m i n e whether the diffusional p h o r e t i c s l i p would h a v e an essential effect here in one direction or the other.

280

B.V. Derjaguin

It is c l e a r that particles for which r << X will always be repelled from the w a l k of the a l v e o l i ; on the other hand, most such particles do not p e n e t r a t e to the a l v e n l i because intensive Brownian m o v e m e n t leads to p r e c i p i t a tion in the upper part of the bronchial passages. The diffusional phoresis of aerosol particles can be s i g n i f i c a n t in condensation filters where it would f a c i l i t a t e p a r t i c l e condensation on moist surfaces [4]. The fact that particles of the same composition as the diffusing vapors do not behave in the diffusion current in the s a m e way as particles of different composition can be used for carrying out q u a l i t a t i v e analysis and separation of c h e m i c a l l y nonhomogeneous aerosols. The same m e c h a n i s m which leads to the surface forces responsible for diffnsional phoresis of aerosol particles c a n also g i v e rise to the opposite effect, n a m e l y , the flow of a gaseous mixture through a porous m e m b r a n e when the concentrations on the two sides of the m e m b r a n e are different (or, in general, with two different gases). This e f f e c t can be treated in a very e l e m e n t a r y fashion and is w e l l known in the l i m i t i n g case of Knudsen flow in the pores under the condition

a , ~ ~.,

(8)

(a is the maximur~n pore diameter). In such Knudsen diffusion, the lighter component diffuses more rapidly than the heavier and there is an inc r e a s e in pressure.on the side where the content of t h e h e a v i e r c o m p o n e n t was higher i n i t i a l l y (when the pressures were equal). This effect disappears when the m o l e c u l a r weights are identical. The theory becomes quite c o m p l e x when condition (8) is no longer fulfilled. This s o - c a l l e d c a p i l l a r y osmosis has been detected and studied e x p e r i m e n t a l l y , by B. V. Deryagin and G A. Batova [6]. In the i n t e r m e d i a t e case of a m e m b r a n e in which the condition for " p s e u d o - m o l e c u l a r flow" [7] • (~ - - 6) so <"~ ~" ~

2b so

(9)

is f u l f i l l e d (5 is the r e l a t i v e pore v o l u m e , a quanti~'y a p p r o x i m a t e l y equal to unity; and SO is the s p e c i f i c surface of the m e m b r a n e expressed in c m 2 / c m s v o l u m e of m e m b r a n e ) a n e x a c t theory can be obtained d i r e c t l y from the theory of diffusional phoresis of " s m a l l " aerosol particles [1, 7]. C a l c u l a t i o n of the effect for larger pores requires a knowl e d g e of the coefficient of diffusional phoretic slip k which at the present t i m e can be obtained o n l y from e x p e r i m e n t . When condition (8) is no longer fulfilled, e x p e r i m e n t shows that a c a p i l l a r y osmotic effect c a n arise from differences in the radii of the two components even if the m o l e c u l a r masses are the same [7]. An e x a m p l e is found in n i t r o g e n - e t h y l e n e vapors. C a p i l l a r y osmosis must be t a k e n into account in the theory and c a l c u l a t i o n s of diffusional hygrometers. Because of failure to a l l o w for the effect of diffusions1 phoretin slip, it was formerly impossible to understand certain aspecl'~ of the behavior of hygrometers in the face of h u m i d i t y changes. LITERATURE CF~'ED

1. 2. 3. 4. 5. 6. 7.

B.V. B.V. B.V. S.S. S.S. B.V. B.V.

Detyagin and S. P. Bakanov, Dokl. AN SSSR 11'7, 6 (1957). Deryagin and S. S. Dukhin, Dokl. AN SSSR 10_~6, 5 (1956). Deryagin and S. S. Dukhin, Dokl. AN SSSR 111, 3 (1956). Dukhin and 13. V. Deryagin, Dokl. AN SSSR 11....~2, 3 (1957). Dukhin and B. V. Deryagi.n, Dokl. AN SSSR 11.._.~5, 1 (1957). Deryagin and G. A. Batova. Dokl. AN SSSR 12.._~8, 2 (1959). Deryagin and S. P. Bakanov. Dokl. AN SSSR 115, 2 (1957); Zhtu. Tekh. Fiz. 27, 9 (1957).