The coagulation of smokes and the theory of smoluchowskirspa.royalsocietypublishing.org/content/royprsa/116/775/540.full.pdf · of the apparatus and the form of the cell we have finally

540

The Coagulation o f Smokes and the Theory o f Smoluchowski. By G. Nonhebel, J. Colvin, H. S. Patterson and R. Whytlaw-Gray.

(Communicated by R. Whiddington, F.R.S.—Received July 14, 1927.)

In a previous communication* it has been shown that smokes are unstable disperse systems which spontaneously coagulate from the moment of formation. In this respect they differ from hydrosols, which are stable normally and coagulate only on the addition of an electrolyte. For these latter systems there is evidence to show that the so-called “ rapid coagulation ” begins when the complete discharge of the double layer has been effected, and that the particles are brought together mainly by Brownian motion. With smokes, too, chance collision brought about by molecular bombardment is probably the chief factor in aggregation, so that a close analogy between the rapid coagulation of hydrosols and the spontaneous coagulation of smokes would appear to exist. Further, if it is admitted that every chance collision between the particles in the two types of system results in a union, then the equations developed by Smoluchowskif for the rate of coagulation of colloids might be expected to be equally valid for smokes after making allowance for the difference in properties of the two dispersion media. Reliable experimental data on sol coagulation are not easily obtained, for the process is completed in a relatively short period, but the measurements of Zsigmondy, J Westgren,§ and others|| show a satisfactory agreement with theory. In a smoke, conditions are more favourable, for coagulation can be followed at much greater dilutions, so that the whole process is slowed down, and observations may be extended over several hours.

On account of the interest presented by the above analogy, and the possibility of testing it quantitatively, we have measured the velocity of coagulation of various smokes by a method which we believe to be free from any serious experimental errors.

Experimental.In the previous work the particles were counted with the ultramiscroscope,

using a modification of the slit instrument of Zsigmondy. Most of the number* W hytlaw-Gray and Speakman, ‘ Roy. Soc. Proc.,’ A, vol. 102, p. 600 (1923).t ‘ Z. f. Phys. Chem.,’ vol. 92, p. 129 (1918).t ‘ Z. f. Phys. Chem.,’ vol. 92, p. 600 (1918).§ ‘ Z. f. Phys. Chem.,’ vol. 92, p. 750 (1918).|| K ruy t and Arkel, ‘ Rev. Trav. Chim. Pays-Bas,’ vol. 39, p. 656 (1920); Tuorila,

* Kolloid Z.,’ vol. 38, p. 3 (1926).

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time curves contained in the previous paper are not in conformity with the equation of Smoluchowski, the rate of coagulation observed being smaller than that predicted by the equation, and the deviation increasing progressively with time. To explain this it was suggested that a diminution of the proportion of effective collisions might be brought about by the progressive adsorption of gas around the particles. Alternatively, the breaking up of the large aggregates formed as the cloud aged would yield an apparent rate of coagulation slower than that demanded by the equation. Support for both these hypotheses could be found in the literature.* Finally, the divergence between experimental and theoretical results might be attributed to errors which are inherent in the method of counting, and the first step evidently was to examine this possibility. Some attention must now be given to a discussion of the experimental arrangements used in the earlier work.

Originally the depth of the volume in which particles were counted was taken as that of the depth of the illuminating ribbon of light. This depth was determined by the Zsigmondy method of rotating the slit through an angle of 90°. It was assumed that only particles lying within the ribbon of light were counted, all others outside this region being invisible, and it is upon this assumption that the validity of the method is entirely dependent. For the observation of coarse clouds a weak source of illumination was used in conjunction with a microscope of low numerical aperture. Under these conditions the results were in satisfactory agreement with the equation of Smoluchowski. The investigation of fine clouds, however, necessitated powerful illumination and a lens system of high aperture, in which case the results showed a progressive discrepancy from theory as the cloud aged. In view of this divergence when using lenses of high numerical aperture, it seemed probable that scattered or reflected light either from the illuminated particles themselves, or from the cell, was rendering visible particles outside the defined volume. The correctness of this view was finally proved by a direct measurement of the depth of the zone in which particles could be counted at different periods during the coagulation of the smoke. This depth should, of course, be constant and agree with the thickness of the beam of light measured by the usual Zsigmondy procedure. To test this the viewing microscope was replaced by an instrument with a fine, graduated focussing adjustment, and a number of measurements were made of the limiting distance between particles which were just visible on the near and far sides of the illuminating ribbon of light. The same lenses were used as in the counts

* Bancroft, ‘ Applied Colloid Chemistry,’ p. 147 (1921); Gibbs, 4 Clouds and Smokes,’ p. 93 (1924).

Coagulation o f Smokes and Theory 541



542*

G. Nonhebel and others.

of fine clouds, namely, a 16-mm. Watson’s Holos on the microscope and a 25-mm. Holos on the projection lens, but a pointolite lamp was substituted for the arc in order to avoid fluctuations of the light intensity during the observations. Although no special accuracy was claimed for the measurements, it was clear that with fine particles in the early stages of aggregation, such as tobacco smoke, the depth of the beam did not differ markedly from that estimated in the usual way ; but that as coagulation proceeded and the particles grew larger and larger, the apparent depth of the beam increased progressively and particles well outside the true zone of illumination became visible. I t appeared that this effect increased rapidly when the particles had grown on the average beyond some definite size. When the microscope was focussed on the centre of the illuminated zone, the same effect was shown by the appearance at a certain stage of coagulation of a large number of minute points and ill-defined discs of low luminosity. These corresponded to particles outside the beam on its near and far sides. I t is evident that the counts were vitiated by this error, for a practice was made of counting every particle, including those on the limit of visibility, and it was not possible to discriminate between faintly illuminated particles outside the beam and minute particles within it. The explanation of the first results, which agreed with Smoluchowski’s equation, is that the illuminating beam was of such low intensity that particles outside the beam did not become visible at any stage during coagulation. I t must therefore be concluded that a progressive error is inherent in the Zsigmondy ultramicroscope when it is used for particles as large as those found in smokes, and that this error may be very considerable when high-intensity illumination and lenses of high numerical aperture are used. Similar criticism has been made by Westgren* when counting the particles in gold sols, and he claimed that the errors could be reduced by using a broad slit and cutting down the intensity of illumination.

To eliminate this error, experiments were undertaken by Mr. W. Sever to test the possibility of using a cell in which the depth of the observed volume was limited by the walls of the cell itself, as in the cardioid ultramicroscope used for sols. On account of the lively Brownian movement of the particles, the concentration of a smoke contained in a shallow cell rapidly diminishes, so that provision had to be made for sucking a stream of smoke from a reservoir through the cell. In a series of experiments with a cell of this type, Mr. Sever established the possibility of counting by this means. The wedge-shaped form of the cell was entirely his design; but in the first cell made it was difficult to ensure that the depth of the observed volume remained constant. The arrangement

* c Arkiv. Matem. Astron. F y s ./ vol. 11, No. 8 (1918).



of the apparatus and the form of the cell we have finally adopted is shown in the diagram.

The cell consists essentially of two glass blocks AA fitting together on optically ground surfaces to form an airtight join, and hollowed out in the manner shown in fig. 1, which shows a horizontal section through the cell. Glass plates BB


F ig . 1.

I

END ELEVATION

SI DE E L E V A T I O N

fitting on to the ground surfaces serve to close the cell at each end of the hollo wed- out portion. The smoke enters by a 2-mm. bore tube C through the end plate opposite to the illuminating objective I), and passes into a wedge-shaped chamber 1 cm. long, the apex of which merges into a narrow slit 0 • 1 mm. wide between two optically plane surfaces E, 2 mm. broad. With this arrangement it is possible to take the cell to pieces for cleaning, and to reassemble it without altering the depth of the illuminated volume. The smoke is viewed in this part of the cell by an optical system, the depth of focus of which is greater than the distance between the two plane surfaces E. It then passes into a similar wedge-shaped chamber, 0-5 cm. long, and out by a tube F at right angles to the illuminating band of light. The angle at the apex of the wedge is approximately 20°. Fig. 1 (side elevation) shows a vertical section through the cell at the essential surfaces E, and indicates the line of junction of the two glass blocks. The cell is held tightly together by spring bands in a brass case, as shown in fig. 1 (end elevation), which gives a view of the cell from the position of the observation microscope F. The cell and case were constructed to our design by Messrs. Hilger.

The illumination system consisted of a 5-ampere Zeiss automatic arc, with a water-cooled slit placed within 1 cm. of the carbons. A small condensing lens



544 G. Nonhebel and others.

was placed immediately behind the slit, and the image of the anode was focussed on to the back of the illumination objective D, a 25-mm. Holos of N.A. 0*3. The light from this objective fell on to a slit in the brass case and so into the cell, being focussed into the space between the essential surfaces B, and completely filling it from side to side. Any light which struck the sides of the wedge was reflected internally in the cell, and was unable to pass into the viewing objective.

The observation microscope consisted of a 25-mm. Holos objective, stopp'ed down to approximately N.A. 0-07, in combination with a series of X 10 eyepieces carrying square diaphragms of various sizes. The depth of focus of this combination was somewhat greater than the depth of the cell, and consequently both walls of the cell were held in focus, and there was no parallax. This is important; earlier counts with an objective of N.A. 0-3 gave irregular results with the larger diaphragms, which were traced to the fact that both walls of the cell were not completely in focus at once. That the system finally used suffered from no parallax was proved by substituting a micrometer scale for the cell, and moving it relatively to the microscope over a range of 0-1 mm., when the size of the diaphragm remained unchanged.

In the old method of counting the particles, a continuous slow stream of smoke was sucked through the cell, and the number of particles in each field was counted as it passed across the field of view. I t will readily be understood that the correct estimation of the number of fields which contained no particles was difficult. By placing taps L and M (see fig. 1) on each side of the cell and rotating them in phase, however, it was possible to suck fresh quantities of smoke into the cell and count the number of particles in the fields obtained when the taps were closed. As it was essential that no particles should be lost in their passage from the smoke chamber to the cell, the tap L and tubing were made of 1 cm. internal bore. The tube projected 15 cm. into the tank, but the tap was almost flush with its side, and was then drawn off sharply to a jet of 2-mm. diameter, which was connected with the cell by a short length of rubber tubing. A T-piece carrying a small tap N served to flush out tap L and the leading tube with smoke before a fresh charge was admitted to the cell. The cell exit F was connected to the second rotating tap M, which was of fine-bore capillary, so that it opened and shut sharply. From this, connection was made to the aspirator through a fine needle valve, by means of which the flow of smoke through the cell when the taps were open could be delicately adjusted. I t was found essential that all the connections between the taps L and M and the cell should be airtight and rigid, otherwise the particles in



the field of view were not steady during the time the taps remained closed. The taps were rotated mechanically at the rate of 30 revolutions per minute by means of a geared-down electric motor, and were carefully synchronised.

In our first experiments it was found that particles which struck and adhered to the surfaces E appeared as bright spots of light which obscured the field. But by coating the cell walls with a thin film of quinoline these particles were rendered invisible. The quinoline was applied by sweeping down the surfaces with a swab of cotton-wool wrapped round the end of a thin glass rod and moistened with a mixture of dust-free redistilled quinoline and acetone. A fresh coating of quinoline had to be applied only after every six counts. Several other substances, such as immersion oil, were tried in place of quinoline, but none of them gave such a thin and stable film upon the glass.

The method of making a count was as follows : Taps L and N were opened and the tubes flushed out with smoke. N was then closed, M opened, and a • slow stream of smoke sucked through the cell until all the air had been displaced. The needle valve was then closed down so that the particles appeared as streaks as they passed across the field of view. The motor was switched on, and the number of particles in each field obtained when the taps were closed was counted. Except during the early stages of a run, when the Brownian motion was considerable, the particles appeared as steady points of light and were easy

. to count. The average of 70 fields was taken for each point, and the time taken over each such count was approximately 1 | minutes. Within limits the speed of the motor was immaterial. The number of particles per c.c. of smoke at the mean time of counting could then be calculated from the depth of the cell, and the area viewed by the eyepiece. During the course of a run about 10 litres of smoke were sucked out of the chamber and replaced by dust-free air, which entered at a point remote from the intake tube for the cell.

The smokes were generated in an air-tight metal cistern of 0 • 87 cubic metre capacity, with two detachable glass windows. The internal arrangements were similar to those previously described,* and the air inside was kept dry and free from carbon dioxide by trays containing fused calcium chloride and soda lime.

Numerous clouds of various concentrations of ammonium chloride were investigated. The clouds were dispersed by heating a weighed amount of resublimed ammonium chloride in a porcelain boat; but in an effort to discover

v some method of quantitatively reproducing a cloud, various types of heater were tried. These were :—


* Whytlaw-Gray and Speakman, loc. cit.




1. A silica plate heated electrically by bare nichrome wire raised to red heat. 15 mgms. of ammonium chloride could be dispersed in one minute.

2. Two mica plates, placed horizontally one above the other at a distance of 2 cms., the boat being inserted between the two plates, which were wound with nichrome wire. In this case evaporation was extremely rapid, and 15 mgms. were dispersed in less than 30 seconds.

3. A small silica U-tube containing the heating coil inside and sealed so that no electrons could escape from the heated metal into the atmosphere around the boat. The smoke should therefore be formed in an atmosphere comparatively free from ions. The boat was of thin aluminium, resting across the arms of the U-tube. Sublimation of the chloride was slow, and two minutes were required for complete dispersal of 15 mgms.

The fan in the chamber was kept running for three minutes after the first appearance of smoke, in order to ensure complete homogeneity.

Results.

If the particulate volume, a (i.e., the reciprocal of the number of particles per cubic centimetre), or the volume containing a single particle, was plotted against the age of the cloud in minutes, t, a straight line was in all cases obtained. This is one of the predictions of Smoluchowski’s theory of coagulation and will be discussed later. The coagulation was usually followed for about 120 mins. Straight lines have been obtained up to 250 mins., but counting became less accurate- after two hours because of irregularities in the uniformity of the cloud, due to dilution and to settling out of the particles. In order to eliminate a personal error in counting, the counts were made by two observers taking alternate readings. The following table gives the results obtained during a typical ru n :—



Coagulation o f Smokes and Theory o f Smolnchoivski. 547

Table I.—Run, LXYI.Weight of ammonium chloride = 30 mgms. Heater, type II.

Cloud dispersed in 25 secs. Fanned for 3 minutes.

Time in minutes. Average number per diaphragm.

Volume of smoke viewed, cu. cm.

No. per cubic centimetre X 10~6.

Particulate vol., cu. cm.

(7 X 107.

6 0 2-67 1*576 X lO"6 1-70 5-ff901 1 0 1-87 — 1-19 8-401 6 0 1-79 — 1 1 4 8-819-5 1-37 — 0-87 11-524 0 1-31 — 0-83 12-03 2 0 2-81 4*034 x 10-6 0-694 14-436-5 2-40 — 0-595 16-842-5 2-49 — 0-617 16-248-0 1-97 — 0-488 20-560-5 1-56 — 0-388 25-867 0 1-49 — 0-369 27-172-0 1-44 — 0-357 28-079-0 1-43 — 0-355 28-283 0 2-20 7*373 X KM 0-299 33-589-5 2-20 — 0-299 33-5

103-5 1-91 _ 0-259 38-698-0 1-76 — 0-239 41-9

A graph of a against time is shown in fig. 2.

40 60A G E OF CLOUD IN M IN U T E S

F ig . 2.

Owing to the intense Brownian motion of the particles in a freshly dispersed cloud, the accuracy of the counting was always less during the early stages of a run. The constants of the equation

g == o'o + (1 )

were, therefore, always calculated—by the method of least squares—from the points obtained with the larger diaphragms.




g is the reciprocal of the number of particles at time t, <70 at the time of dispersal, and K is the coagulation constant of the cloud.

A few runs were also made with smokes of antipyrin and arsenious oxide, both prepared by sublimation, and with smokes of cadmium oxide prepared by striking an arc between two electrodes of the metal. In all cases straight- line graphs of a against time were obtained. The experimental results are summarised in Table II, which gives the constants o0, n0, and K of equation (1). The quantity S is the initial number of particles per c.c. per mgm. of smoke dispersed. I t will be seen that it is not constant, but that the values are more nearly constant if we compare the figures for each weight concentration. In general also the numbers obtained with the closed heater were smaller than those obtained when the smoke was formed in a highly ionised atmosphere such as that which would exist in the immediate vicinity of the red-hot elements of heaters 1 and 2.

Table II.

Run.Wt. dispersed. Type

heater. n0 X 10~8. a X 107 cu. cm.

K c.c./min. X 10s. IQ-® S. s.

mg.

Ammonium Chloride.63 4-2 2 1-47 6-79 4-73 3-5064 4-2 2 1-08 9-23 4-89 2-5769 4-2 2 0-89 11-27 4-71 2-1171 4-2 2 0-96 10-37 5-20 2-2944 8-4 3 0-61 16-46 6-00 0-72456 8-0 2 1-30 7-71 4-66 1-6357 8-2 2 0-96 10-44 4-50 1-1758 8-0 2 0-91 10-95 4-57 1-1443 14-6 3 0-89 11-22 5-60 0-61045 15-0 3 1-27 7-88 4-93 0-84646 15-6 3 1-24 8-09 4-03 0-79447 15-2 2 1-49 6-71 4-74 0-95051 1 5 0 2 1-24 8-04 4-34 0-82652 15-0 2 1-19 8-40 4-03 0-84053 14-8 2 1-05 9-54 3-15 0-70832 28-6 1 2-48 4-04 3-88 0-86333 30-6 1 3-04 3-29 4-68 0-99035 32-2 1 2-56 3-91 4-50 0-79238 31-6 3 1-33 7-53 3-24 0-42039 28-4 3 3-22 3-11 4-77 1-1365 30-0 2 2-40 4-17 3-29 0-79966 30-0 2 3-80 2-63 3-61 1-2767 30-0 2 1 1 4 8-77 3-01 0-38068 60 0 2 17-2 0-58 3-60 2-8770 60-0 2 4 1 2-44 3-67 0-690

Arsenious Oxides19 || 78-6 |1 1 1 1-92 | 5-20 | 4*40 1 0*244 |

Antipyrin.54 Ij 40-0 1 2 1 4-60 1 2-17 1 3-86 1 1-15 |55 I 40-0 I1 2 1 2-02 1 4-91 I 3-01 I 5-20 |

2-542-772 - 763- 00 4 1 2 2-842 532 - 943- 93 3-31 2-733 12 2-922- 71 2 1 73- 34 3 0 7 3-052- 383- 08 2-22 2-29 2-241- 952- 54

3 10

2-572-23




Table II—(continued).Cadmium Oxide.

The weight concentrations were unknown, but were of the order of 50 mg. Consequently only values of a0 and K can be calculated from the experimental results.

Run. 59 60 61 62 28 29 —

na X 10-6 .... 1-36 3 44 6-54 1-44 1-15 1-04 _CT0 X 107 .... 7-35 2-90 1-53 6-94 6-69 9-60 —K X 10® .... 5 1 0 5-30 5-50 3-96 5-20 4-60

Accuracy of Results.One of the largest sources of error is due to averaging over so few particles ;

in general the numbers counted in 70 fields were between 100 and 200. If a larger number of fields were viewed, inaccuracies might arise due to settling of particles in the narrow leading tube of the cell, and if a larger diaphragm were used errors of observation would be magnified because of the difficulty of counting the particles in the large groups which occasionally appear. Errors in the calculation of the volume of smoke viewed may amount to 5 per cent., and actual errors in counting may be of the same magnitude. We do not think that our individual points have a greater accuracy than 10 per cent., but the value of the coagulation constant K has probably a higher accuracy. The error in the constant o0, the reciprocal of the initial number of particles, may be as great as 20 per cent., partly because of the error in determining the zero time itself, and partly because of the smaller accuracy of the counts during the first few minutes when a very small diaphragm was used, and the particles were in rapid Brownian motion. As already mentioned, these counts were neglected in calculating the equation to the lines, and consequently the errors due to extrapolation over a longer time-interval become magnified.

These errors concern only the particles which actually reached the field of view. There was certainly a slight loss of particles during the passage of the smoke from the chamber to this point, but, as the leading tube was wide, no serious error was introduced in this way. The numbers showed no tendency to fall off during a count of 70 fields, except when the taps were rotated very slowly, and the amount of smoke passed through the cell was very small. Usually about 10 c.c. were sucked through the cell during a count of 70 fields. A certain proportion of particles were lost by striking the walls of the cell at the narrow portion, but this was estimated to be only about 2 per cent. There was also the possibility of a slight leakage of air into the cell at the junction of

VOL. CXVI.—A. 2 O




the end plate B with the prisms AA, but this again would not lead to serious error. Taking the whole of the possible errors into account, it is believed that the actual numbers counted are not likely to be more than 10 per cent, too low, and this is offset slightly by the error of observation which lies on the side of counting too many particles. Nevertheless, this latter error was small because analyses of the numbers counted in 100 fields shows them to lie well on distribution curves.

It is to be noted that the initial numbers we have obtained are distinctly smaller than those given in the paper by Why flaw-Gray and Speakman* for clouds of similar weight concentration, and it is obvious that the particulate number-time curves are of a different shape. We have been privileged to see some figures of Mr. H. L. Green (which will be published shortly), obtained by another method of counting, involving the use of the Aitken effect, which are in satisfactory agreement with our own. The concordance between these two independent methods indicates that there can be no serious error in our numbers.

4

The Theory of Smoluchowski.

On the assumption that the particles in a colloidal solution unite when they approach within a definite distance of each other not much greater than twice their radius, Smoluchowski has shown that coagulation can be formulated in terms of diffusion. If all the particles are initially of the same size, the total number present at any time can be expressed by a simple equation similar to the bimolecular equation of chemical kinetics. If n is the total number of particles present at time t and nQ the initial number, then

n1 + nQ ’

where K should be a constant if certain simplifying assumptions are made. This equation has been experimentally verified for colloidal solutions by many workers. I

If we assume that the mechanism of coagulation in smokes is similar, the same equation should be valid and a linear relationship should be obtained when the particulate volumes are plotted against the time, for equation (2), when rearranged, becomes the same as equation (1). The theory of Smoluchowski predicts, therefore, the correct form of the coagulation curve for smokes.

* Loc. cit.f Zsigmondy, Westgren, K ruyt, Tuorila, loc. cit.




The constant K is given by the formulaK = 4tcDS, (3)

where D is the diffusion constant of Einstein and S is the radius of the sphere of action of the particles initially. Einstein’s equation for D is

D = RT . B/N, (4)where N is the Avogadro number and B is the mobility. For colloidal solutions we may assume that the mobility is given, to a first approximation, by Stokes’s equation, so that

D RT 1 N 6tct]-r ’ (5)

where ris the radius of the particles and 7] the viscosity of the medium. If we write S = sr,equation (3) becomes

If now the sphere of influence is always the same for a given smoke, i.e., if s is a constant, K should always have the same value. Actually, as inspection of Table II will show, K is not a constant, but varies from cloud to cloud. The same is true also of hydrosols.

At first sight it would seem possible that this may be due to the fact that in smokes the size of the particles is comparable with the mean free path of the molecules of the medium, so that the mobility cannot be deduced from Stokes’s law. If we substitute the Stokes-Cunningham expression for the mobility, namely,

B = (1 -j- AX/r)/6wv]r,wrhere X is the mean free path and A is a constant, we obtain

K = I • g d + A P)

In this case it is evident that K is no longer a constant, but becomes smaller as the average size of the particles increases. Using Millikan’s value for A, AX = 9 X 1(T6, and putting R = 8-3 X 107, N = 6 -1 X 1023, T = 293, 7] = 1 • 82 X 10~4, and reckoning time in minutes, we find that

K = 8-76 x 10~9 (1 + 9 X 10~6 . r_1) s cms.3/min. (8)Values of s, the ratio of the radius of the sphere of action to the initial radius of the particles calculated from this equation, are given in Table II, the values of r being calculated from the weight of the particles and the density* of the

* The density of antipjTin was found to be 1 • 18 a t 18° C.



552 Coagulation o f Smokes and Theory o f Smoluchowski.

dispersed substance in tbe solid state. I t will be seen that the values of s are far from constant, but, nevertheless, lie in groups for each weight concentration. The most consistent results were obtained when the cloud was dispersed from the open heater, type II, probably because dispersal was accomplished more rapidly. That the Stokes-Cunningham expression gives a slightly better representation of the curves is shown by the fact that the rate of coagulation is generally greater th e . smaller the initial size of particles. On the other hand, the curves for a do not show the curvature predicted by the equation, but this maybe due to the error introduced by the settling out of the particles, which tends to make the experimentally found values of a too large. The curvature to be expected is small in all cases.

A possible explanation of the lack of constancy of the values of s and of their high value, approximately 3 instead of 2, is that the particles are charged electrically, both positively and negatively; but until more reliable data on this branch of the subject are available, it is not possible to test this hypothesis. I t is also possible that the weight concentrations themselves might be in error, due to incomplete dispersal of the heated substance into a fine particulate state ; but calculation shows that the values are not greatly altered even if only half of the material weighed out was present as smoke. Thus, for run 47, where the weight concentration was 15 mgm., and the value of 3 • 12, the latter is only lowered to 2-83 on the assumption that the actual weight of solid in the smoke was 7*5 mgm. Again, in calculating the radius of the particles, the densities of the substances have been taken as those of the material in bulk; but if we were to use a lower density, such as has been found by us* for cadmium oxide particles, the values of s obtained would be even greater. Assuming a normal density of cadmium oxide, and a weight concentration of 50 mgm, the average value for s for the clouds given in Table II is approximately 2 • 8.

Finally, it must be remembered that Smoluchowski derived his equation on the assumption that the particles are spherical and coagulate to spherical aggregates. Actually, it has been shown* that such is not the case for particles of cadmium oxide, which form chains of smaller particles. The particles of ammonium chloride appear in the ultramicroscope to be loose but apparently spherical aggregates, whilst those of antipyrin, by their brightness, appear to be liquid. If this is so, spherical particles would be formed on coagulation. We hope at some later date to make a further study of this substance and of substances which are known to yield liquid particles, such as heavy paraffin.

* Patterson and Whytlaw-Gray, ‘ Roy. Soc. Proc.,’ A, vol. 113, p. 302 (1926).



Uniaxial Optically Active . 553

In conclusion, we must express our thanks to Professor Conrady for valuable advice on the optical and illuminating system we have employed.

Summary.1. The slit ultramicroscope has been criticised as an instrument for deter

mining the number of particles in a smoke.2. A special cell has been designed for counting smoke particles and has been

shown to give reliable results.3. The coagulation of clouds of ammonium chloride, antipyrin and cadmium

oxide has been studied and has been shown to obey the same general law, which is probably valid for all smokes composed of non-volatile particles.

4. The experimental data agree as closely as those obtained for sols with the theory of Smoluchowski.

Investigations of the Molecular Arrangement of Uniaxial OpticallyActive Crystals.

By W. G. Burgers (Ramsay Memorial Fellow).

(Communicated by Sir William Bragg, F.R.S.—Received July 15, 1927.)

The present paper deals with the X-ray investigation of certain optically active uniaxial crystals. The investigation was undertaken for several reasons. In the first place it was hoped to obtain evidence on the disputed question as to whether their optical activity must be ascribed to pseudo-symmetrical intergrowths of biaxial lamellae (their rotatory power thus being produced in a similar way as that exhibited by definite spiral piles of thin mica plates) or whether special arrangements of the atoms within the unit cells proved to be present in the structures of such crystals. I t was therefore necessary to discuss in how far X-rays are able to reveal the presence and the true nature of a lamellar structure. The cases of both relatively large lamellae and of lamellae which are too small to give individual X-ray reflections must be discussed. The latter case has been dealt with especially in connection with recent statements of G. Friedel* with regard to the value of X-ray analysis of structures of this type. The crystals discussed in this paper have been investigated with particular reference to this difficulty. The conclusions arrived at are given in

* ‘ L e ^ n s de Cristallographie,’ Paris (1926); ‘ C. R .,’ vol. 182, p. 741 (1926).



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The coagulation of smokes and the theory of smoluchowskirspa.royalsocietypublishing.org/content/royprsa/116/775/540.full.pdf · of the apparatus and the form of the cell we have finally