Page:EB1911 - Volume 05.djvu/277

so that, if the preceding argument be correct, no such thing as mixture of two liquids could ever take place.

There are two apparent exceptions to Marangoni’s rule which call for a word of explanation. According to the rule, water, which has the lower surface-tension, should spread upon the surface of mercury; whereas the universal experience of the laboratory is that drops of water standing upon mercury retain their compact form without the least tendency to spread. To Quincke belongs the credit of dissipating the apparent exception. He found that mercury specially prepared behaves quite differently from ordinary mercury, and that a drop of water deposited thereon spreads over the entire surface. The ordinary behaviour is evidently the result of a film of grease, which adheres with great obstinacy.

The process described by Quincke is somewhat elaborate; but there is little difficulty in repeating the experiment if the mistake be avoided of using a free surface already contaminated, as almost inevitably happens when the mercury is poured from an ordinary bottle. The mercury should be drawn from underneath, for which purpose an arrangement similar to a chemical wash bottle is suitable, and it may be poured into watch-glasses, previously dipped into strong sulphuric acid, rinsed in distilled water, and dried over a Bunsen flame. When the glasses are cool, they may be charged with mercury, of which the first part is rejected. Operating in this way there is no difficulty in obtaining surfaces upon which a drop of water spreads, although from causes that cannot always be traced, a certain proportion of failures is met with. As might be expected, the grease which produces these effects is largely volatile. In many cases a very moderate preliminary warming of the watch-glasses makes all the difference in the behaviour of the drop.

The behaviour of a drop of carbon bisulphide placed upon clean water is also, at first sight, an exception to Marangoni’s rule. So far from spreading over the surface, as according to its lower surface-tension it ought to do, it remains suspended in the form of a lens. Any dust that may be lying upon the surface is not driven away to the edge of the drop, as would happen in the case of oil. A simple modification of the experiment suffices, however, to clear up the difficulty. If after the deposition of the drop, a little lycopodium be scattered over the surface, it is seen that a circular space surrounding the drop, of about the size of a shilling, remains bare, and this, however often the dusting be repeated, so long as any of the carbon bisulphide remains. The interpretation can hardly be doubtful. The carbon bisulphide is really spreading all the while, but on account of its volatility is unable to reach any considerable distance. Immediately surrounding the drop there is a film moving outwards at a high speed, and this carries away almost instantaneously any dust that may fall upon it. The phenomenon above described requires that the water-surface be clean. If a very little grease be present, there is no outward flow and dust remains undisturbed in the immediate neighbourhood of the drop.]

Fig. 6.

On the Rise of a Liquid in a Tube.—Let a tube (fig. 6) whose internal radius is $r$, made of a solid substance $c$, be dipped into a liquid $a$. Let us suppose that the angle of contact for this liquid with the solid $c$ is an acute angle. This implies that the tension of the free surface of the solid $c$ is greater than that of the surface of contact of the solid with the liquid $a$. Now consider the tension of the free surface of the liquid $a$. All round its edge there is a tension ${\mbox{T}}$ acting at an angle $a$ with the vertical. The circumference of the edge is $2\pi r$, so that the resultant of this tension is a force $2\pi r{\mbox{T}}\cos \alpha$ acting vertically upwards on the liquid. Hence the liquid will rise in the tube till the weight of the vertical column between the free surface and the level of the liquid in the vessel balances the resultant of the surface-tension. The upper surface of this column is not level, so that the height of the column cannot be directly measured, but let us assume that $h$ is the mean height of the column, that is to say, the height of a column of equal weight, but with a flat top. Then if $r$ is the radius of the tube at the top of the column, the volume of the suspended column is $\pi r^{2}h$, and its weight is $\pi \rho gr^{2}h$, when $\rho$ is its density and $g$ the intensity of gravity. Equating this force with the resultant of the tension

$\pi \rho gr^{2}h=2\pi r{\mbox{T}}\cos \alpha ,$

or

$h=2{\mbox{T}}\cos \alpha /\rho gr.$

Hence the mean height to which the fluid rises is inversely as the radius of the tube. For water in a clean glass tube the angle of contact is zero, and

$h=2{\mbox{T}}/\rho gr$

For mercury in a glass tube the angle of contact is 128° 52′, the cosine of which is negative. Hence when a glass tube is dipped into a vessel of mercury, the mercury within the tube stands at a lower level than outside it.

Rise of a Liquid between Two Plates.—When two parallel plates are placed vertically in a liquid the liquid rises between them. If we now suppose fig. 6 to represent a vertical section perpendicular to the plates, we may calculate the rise of the liquid. Let $l$ be the breadth of the plates measured perpendicularly to the plane of the paper, then the length of the line which bounds the wet and the dry parts of the plates inside is $l$ for each surface, and on this the tension ${\mbox{T}}$ acts at an angle $\alpha$ to the vertical. Hence the resultant of the surface-tension is $2l{\mbox{T}}\cos \alpha$. If the distance between the inner surfaces of the plates is a, and if the mean height of the film of fluid which rises between them is $h$, the weight of fluid raised is $\rho ghla$. Equating the forces—

$\rho ghla=2/{\mbox{T}}\cos \alpha ,$

whence

$h=2{\mbox{T}}\cos \alpha /\rho ga$

This expression is the same as that for the rise of a liquid in a tube, except that instead of $r$, the radius of the tube, we have a the distance of the plates.

Form of the Capillary Surface.—The form of the surface of a liquid acted on by gravity is easily determined if we assume that near the part considered the line of contact of the surface of the liquid with that of the solid bounding it is straight and horizontal, as it is when the solids which constrain the liquid are bounded by surfaces formed by horizontal and parallel generating lines. This will be the case, for instance, near a flat plate dipped into the liquid. If we suppose these generating lines to be normal to the plane of the paper, then all sections of the solids parallel to this plane will be equal and similar to each other, and the section of the surface of the liquid will be of the same form for all such sections.

Fig. 7.

Let us consider the portion of the liquid between two parallel sections distant one unit of length. Let ${\mbox{P}}_{1}$, ${\mbox{P}}_{2}$ (fig. 7) be two points of the surface; $\theta _{1}$, $\theta _{2}$ the inclination of the surface to the horizon at ${\mbox{P}}_{1}$ and ${\mbox{P}}_{2}$; $y_{1}$, $y_{2}$ the heights of ${\mbox{P}}_{1}$ and ${\mbox{P}}_{2}$ above the level of the liquid at a distance from all solid bodies. The pressure at any point of the liquid which is above this level is negative unless another fluid as, for instance, the air, presses on the upper surface, but it is only the difference of pressures with which we have to do, because two equal pressures on opposite sides of the surface produce no effect.

We may, therefore, write for the pressure at a height $y$

$p=-\rho gy$