Oct., 1958
EFFECT OF SPREADING SOLVENT ON PROPERTIES OF MONOLAYERS
1291
THE EFFECT OF THE SPREADING SOLVENT ON THE PROPERTIES OF MONOLAYERS BY VICTORK. LA MERAND MAXL. ROBBINS Contribution from the Department of Chemistry, Columbia University, New York 97, New York Received March 5, 1968
Archer and La Mer2have reported that the resistances exhibited by fatty acid monolayers to the evaporation of the aqueous subphase are sensitive to the nature of the spreading solvent and the techniques used in spreading. They attribute these effects to holes produced by solvent molecules occluded in the monolayer. More recently, Ries and Cook3 report that the surface pressure-area (n-a) isotherm for stearic acid shows similar sensitivity to the specific spreading solvent used. They find a variation in curvature in the low pressure region. We investigated the effect upon the r-a isotherm produced by varying the concentration of stearic acid in a benzene spreading solution. The slope and limiting area of the liquid condensed region are dependent upon the concentration of the spreading solution but are independent of the volume of solution delivered t o the surface, the initial area of the surface, or the time allowed for benzene to evaporate. Correlations between our findings and those of Archer and La Mer2 are indicated. The formation of a mixed monolayer of stearic acid and benzene is proposed as a model to explain our observationa.
Introduction Until very recently, the importance of the spreading solvent in determining the measured properties of a monolayer had not been considered. Neither Adam4 nor Harkins6 nor Stenhagena mention the spreading solvent other than to specify that it must be volatile, non-miscible with water and spread readily on water. For example, hexane, petroleum ether and benzene have been considered equally suitable. Since it was supposed that the solvent evaporated completely, no attention was paid to the concentration of the spreading solution. It is pertinent to compare the values for the slope and limiting area of the 7r-a isotherm for stearic acid spread on dilute acid subphase at 25” obtained by two pairs of investigators. Langmuir and Schaefer’ report a slppe, - da/dn, of 0.202 and a limiting area of 25.0 A.2using benzene as the spreading solvent. Nutting and Harkins* repor! a slope of 0.177 and a limiting area of 24.41 A.2 using petroleum ether. Both sets of investigators give no indication of the concentration of the spreading solution. Each claims a precision of the order of 1%. The reported values for the limiting area differ by 3% and those for the slope by 13%. Archer and La Mer2 were the first to recognize the influence of spreading solvent upon the measured properties of monolayers. I n their studies on the retardation of the rate of evaporation of water by fatty acid monolayers, these investigators found that the resistance of the monolayer to evaporation depends markedly upon the concentration of the spreading solution, the specific solvent employed and the techniques used in spreading. For any given concentration, spreading solutions (1) A preliminary report was presented at the Symposium on Monolayers, American Chemical Society National Convention, New York, N. Y. on September 11, 1957. (2) R. J. Archer and V. K. La Mer, THISJOURNAL, 69, 200 (1955). (3) H.D.Cook and H. E. Ries, Jr., ibid., 60, 1533 (1956). (4) N . K. Adam, “The Physics and Chemistry of Surfaces,” 3rd edition, Oxford University Press, London, 1941. (5) W. D. Harkins, “The Physical CbemiBtry of Surface Films,” Reinhold Publ. Corp., New York, N. Y., 1952. (6) E. Stenhagen, “Surface Films,” in “The Determination of Organic Structures by Physical Methods,” by E. A. Brauda and F. C. Nachod, Academic Press, Inc., New York, N. Y., 1955, p. 325 et sep. (7) I. Langmuir and V. T. Schaefer, J . Franklin Inst., 236, 119 (1943). (8) G. C. Nutting and W. D. Harkins, J . A m . Chem. Soc., 61, 1180 (1989).
employing benzene as the solvent yielded consistently lower values for the resistance to evaporation than did petroleum ether solutions. Increasing the concentration of the fatty acid in petroleum ether resulted in an increase in the magnitude of the resistance until a limiting concentration of approximately 0.14 m was reached. The monolayer spread from benzene solution showed a similar, though less marked, increase in resistance with increasing concentration. These variations were attributed t o holes or sites of small resistance produced by solvent molecules occluded in the monolayer. It was shown that if as little as 1% of the monolayer consists of occluded solvent, there could be a twenty-fold decrease in the resistance to evaporation. I n a more recent paper, Ries and Cook3 examined the influence of the spreading solvent on the ?r-a isotherm of stearic acid. They studied the variations caused by hexane, benzene and chloroform. The concentration of the spreading solution was not considered. These investigators found that the curvature of the .Ira isotherm in the region of low surface pressures varied with the specific spreading solvent employed. The curvature was greatest with benzene and smallest with hexane. The present paper treats the effect upon the r-a isotherm of varying the concentration of stearic acid in a benzene spreading solution. Experimental Stearic acid (m.p. 68.8-69.3’) recrystallized three times from petroleum ether was used as the solute and Baker analytical grade benzene as the solvent. The analysis gave the limits of non-volatile impurity in the benzene as 0.0005%. A 50-ml. sample was evaporated to dryness a t room temperature and the residue weighed. Less than 0.2 mg. of residue remained. The spreading solution was delivered from a microburet calibrated with mercury. The volume of solution delivered was approximately 0.1 -1 0.001 ml.; Le., a precision of 0.1%. The stearic acid monolayer was spread on 0.01 m hydrochloric acid contained in a sand-blasted Pyrex tray.* The ground edges of the tray and Pyrex barriers were made hydrophobic by rubbing lightly with paraffin. The interior surface of the tray was kept paraffin-free. Pyrex tubing resting a few mm. below the water surface carried water thermostated at 25“ and assured temperature control of the subphase to within -10.05“. The temperature above the tray remained within 3“ of 25”. The surface pressures were measured with a Wilhelmy wettable plate of depolished platinum suspended from a torsion assembly.* This consisted of a steel wire to which (9) H. 9. Rosano and V. K. La Mer, THISJOURNAL,60, 348 (1956).
1292
VICTOR IC. LA MERAND MAXL. ROBBINS
Vol. 62
where P g hmas
= density of the liquid lifted = acceleration of gravity = height of the lower edge of the plate above the
plane surface a t maximum pull The surface ressure of a monolayer on an aqueous subphase is defmefas
20
T
fi . -
k
a 8 10
5
m
yw
- Ym
(3)
yw = Ym =
surface tension of the clean surface surface tension when a monolayer covers the surface Assuming a zero contact angle, the difference between the pulls exerted on the plate with and without a monolayer in place is given by APmas 2 ~ ( 2-I-t ) -I- pg(lt)(hmax" hmsxm) (4) where hmsxw= height a t max. pull above a clean water surface hmax" = height a t Max. pull when a monolayer covers the surface
8 15 a i$
=
where
c
-
Over the range of surface pressures measured; i.e., beis well under l tween 0 and 25 dyne/cm., (hmaxw h,.,") mm. For a plate with 1 = 2.54 cm. and t = 0.0025 cm., neglect of the second term on the right of equation 4 leads to a maximum error of ap roximately 0.1 d ne/cm. a t a surface pressure of 25 d y n e i m . At lower surhce pressures the error is correspondingly smaller, approaching zero as a limit. This error is well within the experimental precision of approximately 1%. The reading a t maximum pull was reproducible on repeated immersion and withdrawal of the plate provided thaf the Taised $lm of liquid was not ruptured. Since the wettability of the plate, i.e., the zero contact angle, remained unaffected, it may be inferred that no monolayer of stearic acid was deposited on the plate during a series of measurements. In five experiments readings were taken on both compression and decompression to ascertain whether equilibrium conditions had been maintained. Hysteresis in the slopes of the liquid condensed portion of the n-a isotherm, was of the ordef pf l % , in most cases within the limits of experimental precision. Subsequent determinations were carried out on compression only.
-
5
20
22 24 Area/molecule, Asa
26
Fig. 1.-The surface pressure in the liquid condensed region as a function of the molecular area of stearic acid in benzene on 0.01 m HC1 at 25'. Curves b, d and h refer, respectively, to the concentrations of spreading solution (7.72, 5.70 and 2.32) X 10-8 m. The volume of solution spread and initial area are held constant. was firmly fixed a short horizontal arm and a galvanometer muror. The wettable plate was suspended from the arm. The image of a hairline indicator was reflected from the mirror and focussed on a scale. The pull on the torsion arm in the downward direction was registered as a deflection of the hairline on the scale. The torsion wire was calibrated with a series of known weights before each set of measurements. The sensitivity of this surface balance was estimated at better than 0.1 dyne/cm. The entire ap aratus was enclosed in a hood whose walls were coated wit[ Vaseline to eliminate dust settling on the surface. Organic matter W ~ removed B from the platinum plate by flaming. I n use, the plate was raised slowly through the surface and a reading taken when the measured pull on the plate reached a maximum value. This occurred just before rupture of the film of liquid raised by the plate. The method assured a receding contact angle. The pull on the plate in the downward direction exerted by the raised liquid is given by P = 2Y(2 t ) COS e (11'0 where y = surface tension of the liquid 1 = length of the lower edge of the plate t = thickness of the plate e = contact angle between the surface and the plate For a platinum plate and an aqueous subphase, the receding contact angle is generally assumed to be zero. Equation 1 is valid only for the case in which the lower edge of the plate is at the same level as the horizontal portion of the surface. When the plate is used in a maximum pull method, the lower edge of the plate may be several mm. above this level. Equation 1 must, therefore, be corrected for the additional weight of liquid lifted. For muximum pull P,,, = 2Y(z t ) COS e pg(zi)hm.. (2)
+
+
+
(10) A. J. G . Allan, J . CoZZ. Sei.. 13, 273 (1958).
Experimental Results Figure 1 shows the influence of varying the concentration of the spreading solution upon the r-a isotherm of stearic acid. Only three representative curves (b, d, h) of the 9 listed in Table I (a - i) have been drawn to illustrate the effect. Each curve is the average of the number of experiments reported in column 3. Each experiment represents about 20 successive readings of the surface pressure as the area is decreased. In any given experiment these readings fall on a smooth curve which is sensibly linear in the region between 8 to 12 dyne/cm. and 25 dyne/cm. Although the precision for the region below 12 dyne/cm. is relatively poor, there are indications that the curvature below 12 dyne/cm. is greater for less concentrated solutions. The region above 25 dyne/cm. is not reproducible and shows collapse at pressures between 26 and 28 dyne/cm. On distilled water, collapse pressures as high as 43 dyne/cm. were obtained. Table I lists the average values of the reciprocal slopes for the region of sensible linearity (-da/dn) and the limiting areas obtained by extrapolating this linear portion to ?r = 0. These values were obtained by a linear least squares fit to the points above 12 dyne/cm. for each experiment followed by averaging over the experiments performed at
each concentration. Included are the corresponding standard deviations of the mean, i.e., the standard deviation divided by the square root of the number of observations. The standard deviation, (T, has the significance that statistically 75% of all observations will fall within one u of the mean. TABLE I EXPERIMENTAL RESULTSAVERAGEDOVER MENTS WITHIN A SET" Setor curve
Concn.
X 108, m/l.
1293
EFFECT OF SPREADING SOLVENT ON PROPERTIES OF MONOLAYERS
Oct., 1958
No.
expts.
in s e t
Are& (b.): U
THE
0.20
EXPERI-
-da/d r
14.01 2 24.19 f 0.07 0.1616f0.0030 10 23.80 =t .09 ,1626f .0025 7.72 .1672=t .0058 5.80 5 24.40 =k .34 4 24.06 f .OS .17123= .0007 5.70 . 1 8 7 4 f .0014 4.27 5 25.02 f .02 6 23.68 f -09 .1594& .0012 3.92 4 24.44 f .46 .1863& .0111 g 2.93 ,1959f .0008 h 2.32 4 24.91 & .04 4 25.17 .37 .2024f .0035 i 1.67 The slope, -da/dr, refers only to the linear range. The limiting area, a, i s obtained by extrapolating the linear portion of the *-a curve to ?F equal to zero. a b c d e f
Figure 2 shows the dependence of the reciprocal slope -da/dn, upon the concentration of the spreading solution. A similar, though less precise, curve may be drawn showing the dependence of the limiting area on concentration. Note the leveling off of solvent effects with increasing concentration at 8 to 9 X loe3 molar which is about one-eighth the solubility of stearic acid in benzene. This saturation effect was also observed at a slightly higher concentration by Archer and La Mer2 in their studies on evaporation resistance. Figure 3 summarizes data showing that for a given concentration of spreading solutl'bn, -da/dn, the slope of the linear portion of the n-a isotherm, does not change on varying the volume of the solution spread or varying the area before compression. The experimental variables are expressed as initial areas per molecule of stearic acid. The triangles represent experiments in which the initial area was varied between 708.5 and 256.4 em.$ at both constant concentration of spreading solution and constant volume spread. The circles represent data where the initial area and concentration were held constant and the volume of solution spread was varied between 0.0752 and 0.0127 ml. The limiting area, as defined above, is similarly independent of the initial area and the volume of solution spread. The form of the n-a isotherm is also independent of the time allowed for the benzene to evaporate from the freshly spread monolayer. Times ranging between 50 seconds and 1,000 seconds were ,allowed before starting to compress the monolayer. The duration of the experiment was also varied between 800 and 1600 seconds. There resulted no variation in the slope or limiting area outside the limits of experimental precision. I n all experiments where time was varied, the concentration of the spreading solution, the volume of solution spread, and the initial area were held constant. Spreading occurred more rapidly than evapora-
0.19
G
a
\
-3 0.18 I
0.17
0.16
I
I I 5 10 Concn., moles/liter X 103.
Fig. 2.-The
average slope of the linear portion of the
T-a isotherm for stearic acid in benzene on 0.01 m HC1 a t
25' as a function of the concentration of the spreading solution. The volume of solution spread and initial area are held oonstant.
t i
m c . : o.ooe3p n Ih) I
0.19
0.16
'
I
I
I
30
40 50 Initial area/molecule, A . 2
60
Fig. 3.-The individual slopes within the sets b and h of Table I as a function of the initial area per molecule of stearic acid in benzene on 0.01 m HC1 a t 25' a t a concentration of spreading solution equal, respectively, to (7.72 and 2.32) X 10-3 m. In set h, the initial area is varied a t constant volume of solution spread. I n set b, the volume is varied a t constant initial area.
tion of benzene from the spread monolayer. This was determined as follows. With the wettable plate resting at a freshly swept water surface, a drop of stearic acid in benzene solution was delivered t o the surface and the change in surface pressure noted as a function of time. There resulted a very rapid increase of surface pressure t o about 3 dyne/cm. within 1 to 2 seconds followed by a decline reaching a constant value of 0.4 dyne/
1294
VICTORK. LA MER AND MAXL. ROBBINS
em. within 30 seconds. The rise in surface pressure corresponds to an expansion of the film upon spreading while the ensuing fall represents the contraction upon evaporation of benzene and diffusion of benzene into the aqueous subphase. The benzene dissolved in the aqueous subphase makes a negligible contribution to the observed variation in the n-a isotherm. If diffusion is rapid, the volume of benzene solution delivered to the surface determines the concentration of benzene in the subphase. The former and, therefore, the latter, does not influence the n-a isotherm. If diffusion is slow, the layer of subphase adjacent to the surface is saturated with benzene for all concentrations of spreading solution. The subphase would, therefore, make a constant contribution to the observed effect. Each set of experiments referred to in Table I was performed without renewing the subphase. Accordingly, the concentration of benzene dissolved in the subphase should progressively increase with the number of experiments performed. The n-a curve was not influenced by the number of experiments within a given set. This statement is supported by Fig. 3, since no significant trend is exhibited by the individual slopes within the sets b and h of Table I.
Vol. 62
observed dependence of slope and limiting area upon concentration, the experimental objections to this explanation are: 1, spreading takes place before evapokation is completed; 2, the limiting effect in both n-a isotherm and resistance data occurs at high rather than low concentrations; 3, the dependence of resistance upon concentration cannot be explained by the above model. Experiments are in progress to test for the presence of solvent in the monolayer as support for the proposed model. Acknowledgments.-We gratefully acknowledge the award to one of us (M.L.R.) of the Eastman Kodak Fellowship. DISCUSSION V. K. LA ME:R.-our
former associate, Dr. Henri Rosano, has kindly directed our attention to a paper by Mme. Lisbeth Ter Minassian-Sarafa11 which deals in part with an investigation strikingly similar in some respects t o our own reported above. This escaped our attention because the pertinent section 111, p. 88-91, appeared under a different title. Although there are minor differences in the manner in which we report and interpret our results, Mme. VI‘inassianSaraga has reached the same major conclusion that we have, namely, that solvent is retained in a monolayer spread from a benzene solution. Moreover, she concludes that solvent is not retained when spreading a monolayer from petroleum ether. This was the independent and simultaneous conclusion reached by Archer and La Mer2 from their studies Discussion and Conclusions on the retardation of evaporation by monolayers. More recent results obtained with a stearic acid monoOur n-a results correlate well with the resistance layer (to be reported in detail later) also agree with those measurements of Archer and La Mer.2 Rosano of Mme. Ter Minassian-Saraga obtained with a myristic acid and La Mer3 found that the resistance to evapora- monolayer. Both sets of results show that benzene vapor is tion decreases with increasing compressibility. irreversibly adsorbed in the monolayer. Furthermore, she that petroleum ether is reversibly adsorbed in a This fact and the decrease in resistance with de- showed myristic acid monolayer. Likewise, our results obtained by creasing concentration of the spreading solution2 varying the volume of solution spread a t constant concentrspredict that the compressibility, which is essentially tion of the spreading solution are in good agreement with - da/dn, should increase with decreasing concen- Mme. Ter Minassian-Saraga’s although they are re orted in a different but essentially equivalent manner. f h e plots tration. This was observed. total area occupied by the monolayer at constant surface The compressibility, limiting area and curvature pressure vs. volume of solution spread a t constant initial area at low n characterize the extent of expansion of the and concentration whereas our plots are expressed in terms of monolayer. We attribute the increased expansion the area per molecule, a quantity which exposes experimental with decreasing concentration of the spreading errors. Mme. Ter Minassian-Saraga finds for myristic acid spread solution t o the formation of a mixed monolayer from benzene that the total area, S, occupied by the monowith benzene retained in the surface. layer follows the relationship
Interpretations which assume that the solvent = nu’ + b produces differences in molecular orientation with- where S n = number of molecules of myristic acid in the out being retained in the surface are unsatisfactory. monolayer U’ = area per molecule occupied by the fatty acid They involve the assumption that molecular orienin the surface tation is dependent upon the history of the monob = area occupied by benzene in the monolayer layer and, therefore, predict a difference in the ?r-a isotherm on compression as compared with deOur results for a stearic acid monolayer show that the compression. No significant hysteresis was ob- limiting area per molecule, a = S / n (as defined in the body of the paper), follows a relationship which may be expressed served. Likewise, the observed effects cannot be ex- as S / n = constant = u + c plained on the basis that the solvent determines the extent of spreading. It is conceivable, since where u is defined as the molecular area of the “pure” fatty stearic aoid does not spread spontaneously, that acid monolayer and c is a parameter which varies with the spreading is less complete from more concentrated concentration of the spreading solution, Le., the ratio of the of benzene to stearic scid (the amount of benzene solutions. As the concentration of the spreading amount per molecule of stearic acid) delivered t o the surface. Our solution increases, more stearic acid must be results are identical with hers if c = (u’ - u) + b/n. This spread from a given amount of benzene. It is obviously the case since c, (u’ - u), and b are parameters should, therefore, be increasingly difficult to obtain having the Aame physical significance, that is, they describe complete spreading before the evaporation of ben- the area occupied by benzene in the monolayer. Mme. Ter Minassian-Saraga reports a value of (u’ - u) = b = 0 for zene from the surface is completed. Thus, dilute myristic acid spread from petroleum ether which is in acspreading solutions should give more accurate cord with her finding that petroleum ether is reversibly ?r-a isotherms. Although this model explains the adsorbed in the monolayer.
Oct., 1958
BEHAVIOR OF LIQUIDDROPLETS ON SOLID SURFACES
Presumably, Mme. Ter Minassian-Saraga did not suspect that the quantities b and (a’ - c) in her results are functions of the amount of benzene per molecule of stearic acid delivered to the surface, Le., functions of the concentration of the spreading solution. This functional dependence is indicated by the trend of the data of column 4, Table I. If one subtracts the area per molecule obtained for stearic acid spread from the benzene solution of concentration 7.72 X lo-* M (assuming that c = 0 for this case) from the other values reported in Table I, one obtains values of c ranging from 0 to 1.4 A.’J/molecule, increasing as the concentration of the spreading solution is decreased. We believe Mme. Ter Minassian-Saraga’s conclusion that the total area occupied by benzene in the monolayer is independent of the amount of myristic acid in the surface (p. 91, column 1, paragraph 2) t o be erroneous. Our results, in view of a postulated interaction between benzene and the
1295
fatty acid monolayer, can be interpreted such that the total area occupied by benzene in the monolayer depends not only upon the amount of stearic acid in the surface but also upon the relative amounts of benzene and stearic acid delivered to the sarface. Other studies showing that organic solvent vapors are strongly adsorbed on stearic acid monolayers are those by Dean and McBain ( J . Coll. Sci., 2, 383 (1947)); Dean and Fa Si Li ( J . Am. Chem. Soc., 72,3979 (1950)), and Dean and Hayes (ibid., 73, 5583 (1951), and 73, 5584 (1951)). It seems an obvious corollary of their results that there can be retention of certain solvents by the monolayer during the spreading process. Micheli (Phil. Mag., ( 7 ) 3, 895 (1927)) and Jones, Ottewill and Chater (2nd. International Congress of Surface Activity, Acad. Press, vol. 1, p. 188) have found that organic vapors are strongly adsorbed on clean water surfaces.
ON THE BEHAI7IOROF LIQUID DROPLETS AFTER IMPINGING ON SOLID SURFACES’ BY ISAIAHGALLILY AND VICTOR K. LA’MER Contribution from the Department of Chemistry, Columbia University, New York, N . Y . Received March 8 , 1068
The deposition of particles impinging on solid surfaces was investigated for a s stem composed of a two-dimensional jet of glycerol aerosol and Desicote-coated glass microscope slides inclined to it. J h e patterns of the particles deposited in these experiments were found to change with the velocity ol the jet and the radius of the aerosol contrary to the way usually expected. They indicate that a certain fraction of the particles bounces off from the surface on first contact. A qualitative explanation of the phenomena found is offered.
I. Introduction succeeding the impingement of particles is conUntil recently it has been assumed that an aero- trolled by the opposed action of “removing” and sol particle moving at a low Reynolds number and adhesional forces. They hold that the impacting impacting on some solid surface adheres at its first aerosol particles would fail to adhere whenever the point of contact.2 Thus the capacities of collecting ( L removing’’ force (or energy) exceeds that due to devices such as cylinders, slides and fibrous filters, adhesion. However, these forces are evaluated difto remove aerosols were calculated and checked ferently in the two studies. While Jordan visualizes the adhesional force as experimentally without paying attention to the “sticking probability’’ of the impinging particle~.~-7 operating be$ween an impinging solid sphere and a On the other hand, in sampling sprays by impaction plane that have only one point of contact regardless the probability of noli-adherence usually has been of the velocity at impact, Gillespie and Rideal taken into account and in practice the collecting allow for the compression of the particle and the surfaces are coated with suitable materials to affect subsequent generation of elastic strains withln it. adhesion. The problem of the “sticking prob- Whereas the “removing” action is attributed by ability” of aerosol particles impinging on solid sur- Jordan to the kinetic energy of the impacting parfaces was treated lately in a series of studies which ticles, Gillespie and Rideal consider the viscous drag questioned the complete-adherence assumption of the medium flowing past the surface as the prineven for the sub-micron and the low Reynolds cipal operating agent. Thus, according to Gillespie and Rideal, the number ranges of particle sizes and velocities.*-l‘ A. Previous Work.-According to JordanlO and impinging particle would not adhere when Gillespie and Rideal,9 the sequence of events (1) Thia work was supported in part by Contract AT(30-1)2017 between Columbia University and the U. S. Atomic Energy Commission and in part by the Government of Israel. Isaiah Gallily: P. 0. Box 7057, Hakiryah, Tel-Aviv, Israel. (2) (a) I. Langmuir, O.S.R.D. 865, Sept. 4, 1942, reprinted by De partment of Commerce; (b) C. Y. Chen, Chem. Reus., 56,595 (1955). (3) C. N. Davis, Inst. Mech. Engrs. (London) PTOC.( b ) , 1B, 185 (1952). (4) F. Albrecht, Physik. 2..82, 48 (1931). (5) S. IC. Friedlander, A.I.Ch.E., J . , 3, 43 (1957). (6) V. K. La Mer, el aE., Final Report NYO 512, Contract AT(30-1)651, Columbia, University, N. Y., (7) E. A. Ramskill and W. L. Anderson, J . Colloid Sci., 6 , 416 (1951). ( 8 ) T. Gillespie, ibid., 10, 266 (1955). (9) T. Gillespie and E. Ridesl, ibid., 10, 281 (1955). (10) D. W. Jordan, Brit. J . A p p . Phys., 8 , S194 (1954). (11) I. Gallily, J . Colloid Sei., 12, 161 (1957).
1951.
where a is the radius of the particle, R, is the radius of its common circle of contact with the collecting surface, h is its separation (at contact) from the plane, BA is the surface energy of adhesion/cm.21p is the viscosity of the dispersing medium, Va is its velocity parallel to the surface and at a distance a from it, and y is a dimensionless constant smaller than 1. With respect to the interplay between drag and adhesion one should mention also Rumpf’s paper12 where the forces acting on deposited particles imbedded within a turbulent boundary-layer flow are (12) H. Rumpf, Chem. Ing. Tech., 6 , 317 (1953).
