Page:EB1911 - Volume 05.djvu/273

form $\chi -\chi _{0}$, that is to say, the difference between the energy in two different states. The only cases, however, in which we have experimental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas. It is impossible to make direct measurements of the properties of particles of the substance within the insensible distance ε of the bounding surface.

When a liquid is in thermal and dynamical equilibrium with its vapour, then if $\rho '$ and $\chi '$ are the values of $\rho$ and $\chi$ for the vapour, and $\rho _{0}$ and $\chi _{0}$ those for the liquid,

$\chi '-\chi _{0}={\mbox{JL}}-\rho (1/\rho '-1/\rho _{0}),$

(21)

where ${\mbox{J}}$ is the dynamical equivalent of heat, ${\mbox{L}}$ is the latent heat of unit of mass of the vapour, and $\rho$ is the pressure. At points in the liquid very near its surface it is probable that $\chi$ is greater than $\chi _{0}$, and at points in the gas very near the surface of the liquid it is probable that $\chi$ is less than $\chi '$, but this has not as yet been ascertained experimentally. We shall therefore endeavour to apply to this subject the methods used in Thermodynamics, and where these fail us we shall have recourse to the hypotheses of molecular physics.

We have next to determine the value of $\chi$ in terms of the action between one particle and another. Let us suppose that the force between two particles $m$ and $m'$ at the distance $f$ is

${\mbox{F}}=mm'(\phi (f)+{\mbox{C}}f^{-2}),$

(22)

being reckoned positive when the force is attractive. The actual force between the particles arises in part from their mutual gravitation, which is inversely as the square of the distance. This force is expressed by $m\,m'\,Cf^{-2}$. It is easy to show that a force subject to this law would not account for capillary action. We shall, therefore, in what follows, consider only that part of the force which depends on $\phi (f)$, where $\phi (f)$ is a function of $f$ which is insensible for all sensible values of $f$, but which becomes sensible and even enormously great when $f$ is exceedingly small.

If we next introduce a new function of $f$ and write

$\int _{f}^{\infty }\phi (f)df=\Pi (f),$

(23)

then $m\,m'\,\Pi (f)$ will represent—(1) The work done by the attractive force on the particle $m$, while it is brought from an infinite distance from $m'$ to the distance $f$ from $m'$; or (2) The attraction of a particle $m$ on a narrow straight rod resolved in the direction of the length of the rod, one extremity of the rod being at a distance $f$ from $m$, and the other at an infinite distance, the mass of unit of length of the rod being $m'$. The function $\Pi (f)$ is also insensible for sensible values of $f$, but for insensible values of $f$ it may become sensible and even very great.

If we next write

$\int _{f}^{\infty }\phi (f)df=\Pi (f),$

(24)

then $2\pi m\sigma \psi (z)$ will represent—(1) The work done by the attractive force while a particle $m$ is brought from an infinite distance to a distance $z$ from an infinitely thin stratum of the substance whose mass per unit of area is $\sigma$; (2) The attraction of a particle $m$ placed at a distance $z$ from the plane surface of an infinite solid whose density is $\sigma$.

Fig. 2

Let us examine the case in which the particle $m$ is placed at a distance $z$ from a curved stratum of the substance, whose principal radii of curvature are ${\mbox{R}}_{1}$ and ${\mbox{R}}_{2}$. Let ${\mbox{P}}$ (fig. 2) be the particle and ${\mbox{PB}}$ a normal to the surface. Let the plane of the paper be a normal section of the surface of the stratum at the point $B$, making an angle $\omega$ with the section whose radius of curvature is ${\mbox{R}}_{1}$. Then if ${\mbox{O}}$ is the centre of curvature in the plane of the paper, and ${\mbox{BO}}=u$,

${\frac {1}{u}}={\frac {\cos ^{2}\omega }{{\mbox{R}}_{1}}}+{\frac {\sin ^{2}\omega }{{\mbox{R}}_{2}}}$

(25)

Let $POQ=\theta ,\quad {\mbox{PO}}=r,\quad {\mbox{PQ}}=f,\quad {\mbox{BP}}=z$,

$f^{2}=u^{2}+r^{2}-2ur\cos \theta .$

(26)

The element of the stratum at Q may be expressed by

$\sigma u^{2}\sin \,\theta \,d\theta \,d\omega$

or expressing $d\theta$ in terms of $df$ by (26),

$\sigma ur^{-1}f\,df\,d\omega .$

Multiplying this by $m$ and by $\pi (f)$, we obtain for the work done by the attraction of this element when $m$ is brought from an infinite distance to $P_{1}$,

$m\sigma ur^{-1}f\Pi (f)dfd\omega .$

Integrating with respect to $f$ from $f=z$ to $f=a$, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the radii of curvature, we obtain for the work

$\int m\sigma ur^{-1}(\psi (z)-\psi (a))d\omega ,$

and since $\psi (a)$ is an insensible quantity we may omit it. We may also write

$ur^{-1}=1+zu^{-1}+\,\&{\mbox{c.}},$

since $z$ is very small compared with $u$, and expressing $u$ in terms of $\omega$ by (25), we find

$\int _{0}^{2\pi }m\sigma \psi (z)\left\{1+z\left({\frac {\cos ^{2}\omega }{{\mbox{R}}_{1}}}+{\frac {\sin ^{2}\omega }{{\mbox{R}}_{2}}}\right)\right\}d\omega =2\pi m\sigma \psi (z)\left\{1+{\frac {1}{2}}z\left({\frac {1}{{\mbox{R}}_{1}}}+{\frac {1}{{\mbox{R}}_{2}}}\right)\right\}.$

This then expresses the work done by the attractive forces when a particle $m$ is brought from an infinite distance to the point ${\mbox{P}}$ at a distance $z$ from a stratum whose surface-density is $\sigma$, and whose principal radii of curvature are ${\mbox{R}}_{1}$ and ${\mbox{R}}_{2}$.

To find the work done when $m$ is brought to the point ${\mbox{P}}$ in the neighbourhood of a solid body, the density of which is a function of the depth $\nu$ below the surface, we have only to write instead of $\sigma \rho dz$, and to integrate

$2\pi m\int _{z}^{\infty }\rho \varphi (z)\,dz+\pi m\left({\frac {1}{{\mbox{R}}_{1}}}+{\frac {1}{{\mbox{R}}_{2}}}\right)\int _{z}^{\infty }\rho z\varphi (z)\,dz,$

where, in general, we must suppose $\rho$ a function of $z$. This expression, when integrated, gives (1) the work done on a particle $m$ while it is brought from an infinite distance to the point ${\mbox{P}}$, or (2) the attraction on a long slender column normal to the surface and terminating at ${\mbox{P}}$, the mass of unit of length of the column being $m$. In the form of the theory given by Laplace, the density of the liquid was supposed to be uniform. Hence if we write

${\mbox{K}}=2\pi \int _{0}^{\infty }\psi (z)\,dz,\qquad {\mbox{H}}=2\pi \int _{0}^{\infty }z\psi (z)\,dz,$

the pressure of a column of the fluid itself terminating at the surface will be

$\rho ^{2}\left\{{\mbox{K}}+{\tfrac {1}{2}}{\mbox{H}}\left(1/{\mbox{R}}_{1}+1/{\mbox{R}}_{2}\right)\right\},$

and the work done by the attractive forces when a particle $m$ is brought to the surface of the fluid from an infinite distance will be

$m\rho \left\{{\mbox{K}}+{\tfrac {1}{2}}{\mbox{H}}\left(1/{\mbox{R}}_{1}+1/{\mbox{R}}_{2}\right)\right\}.$

If we write

$\int _{z}^{\infty }\psi (z)\,dz=\theta (z)$

then $2\pi m\rho \theta (z)$ will express the work done by the attractive forces, while a particle $m$ is brought from an infinite distance to a distance $z$ from the plane surface of a mass of the substance of density $\rho$ and infinitely thick. The function $\theta (z)$ is insensible for all sensible values of $z$. For insensible values it may become sensible, but it must remain finite even when $z=0$, in which case $\theta (0)={\mbox{K}}$.

If $\chi '$ is the potential energy of unit of mass of the substance in vapour, then at a distance $z$ from the plane surface of the liquid

$\chi =\chi '-2\pi \rho \theta (z)$

At the surface

$\chi =\chi '-2\pi \rho \theta (0)$

At a distance $z$ within the surface

$\chi =\chi '-4\pi \rho \theta (0)+2\pi \rho \theta (z).$

If the liquid forms a stratum of thickness c, then

$\chi =\chi '-4\pi \rho \theta (0)+2\pi \rho \theta (z)+2\pi \rho \theta (z-c).$

The surface-density of this stratum is $\sigma =c\rho$. The energy per unit of area is

$e=\int _{0}^{c}\chi \rho dz=c\rho (\chi '-4\pi \rho \theta (0))+2\pi \rho ^{2}\int _{0}^{c}\theta (z)\,dz+2\pi \rho ^{2}\int _{0}^{c}\theta (c-z)\,dz.$

Since the two sides of the stratum are similar the last two terms are equal, and

$e=c\rho (\chi '-4\pi \rho \theta (0))+4\pi \rho ^{2}\int _{0}^{c}\theta (z)\,dz.$

Differentiating with respect to $c$, we find

${\tfrac {d\sigma }{dc}}=\rho ,\,{\tfrac {de}{dc}}=\rho (\chi '-4\pi \rho \theta (0))+4\pi \rho ^{2}\theta (c).$

Hence the surface-tension

${\mbox{T}}=e-\sigma {\tfrac {de}{d\sigma }}=4\pi \rho ^{2}\left(\int _{0}^{c}\theta (z)\,dz-c\theta (c)\right).$

Integrating the first term within brackets by parts, it becomes

$c\theta (c)-0\theta (0)-\int _{0}^{c}z{\tfrac {d\theta }{dz}}dz.$

Remembering that $\theta (0)$ is a finite quantity, and that ${\tfrac {d\theta }{dz}}=-\psi (z)$, we find

${\mbox{T}}=4\pi \rho ^{2}\int _{0}^{c}z\psi (z)\,dz.$

(27)

When $c$ is greater than $\epsilon$ this is equivalent to ${\mbox{2H}}$ in the equation of Laplace. Hence the tension is the same for all films thicker than $\epsilon$, the range of the molecular forces. For thinner films

${\frac {d{\mbox{T}}}{dc}}=4\pi \rho ^{2}c\psi (c).$

Hence if $\psi (c)$ is positive, the tension and the thickness will increase together. Now $2\pi m\rho \psi (c)$ represents the attraction between a particle $m$ and the plane surface of an infinite mass of the liquid, when the distance of the particle outside the surface is $c$. Now, the force between the particle and the liquid is certainly, on the whole, attractive; but if between any two small values of $c$ it should be repulsive, then for films whose thickness lies between these values the tension will increase as the thickness diminishes, but for all other cases the tension will diminish as the thickness diminishes.

We have given several examples in which the density is assumed to be uniform, because Poisson has asserted that capillary