[1] The descending series for J0(z) appears to have been first given
by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy.
Irish Trans._, 1840.
[2] Airy, loc. cit. "Thus the magnitude of the central spot is
diminished, and the brightness of the rings increased, by covering
the central parts of the object-glass."
[3] _"Man kann daraus schliessen, was moglicher Weise durch
Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B,
dessen Durchmesser = ([lambda]) ist, und der aus zwei Theilen
besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt
werden. Dieses zeigt uns eine Grenze des Sehvermogens durch
Mikroskope"_ (_Gilbert's Ann._ 74, 337). Lord Rayleigh has recorded
that he was himself convinced by Fraunhofer's reasoning at a date
antecedent to the writings of Helmholtz and Abbe.
[4] The last sentence is repeated from the writer's article "Wave
Theory" in the 9th edition of this work, but A. A. Michelson's
ingenious echelon grating constitutes a realization in an unexpected
manner of what was thought to be impracticable.--[R.]
[5] Compare also F. F. Lippich, _Pogg. Ann._ cxxxix. p. 465, 1870;
Rayleigh, _Nature_ (October 2, 1873).
[6] The power of a grating to construct light of nearly definite
wave-length is well illustrated by Young's comparison with the
production of a musical note by reflection of a sudden sound from a
row of palings. The objection raised by Herschel (_Light_, S 703) to
this comparison depends on a misconception.
[7] It must not be supposed that errors of this order of magnitude
are unobjectionable in all cases. The position of the middle of the
bright band representative of a mathematical line can be fixed with a
spider-line micrometer within a small fraction of the width of the
band, just as the accuracy of astronomical observations far
transcends the separating power of the instrument.
[8] "In the same way we may conclude that in flat gratings any
departure from a straight line has the effect of causing the dust in
the slit and the spectrum to have different foci--a fact sometimes
observed." (Rowland, "On Concave Gratings for Optical Purposes,"
_Phil. Mag._, September 1883).
[9] On account of inequalities in the atmosphere giving a variable
refraction, the light from a star would be irregularly distributed
over a screen. The experiment is easily made on a laboratory scale,
with a small source of light, the rays from which, in their course
towards a rather distant screen, are disturbed by the neighbourhood
of a heated body. At a moment when the eye, or object-glass of a
telescope, occupies a dark position, the star vanishes. A fraction of
a second later the aperture occupies a bright place, and the star
reappears. According to this view the chromatic effects depend
entirely upon atmospheric dispersion.
[10] In experiment a line of light is sometimes substituted for a
point in order to increase the illumination. The various parts of the
line are here _independent_ sources, and should be treated
accordingly. To assume a cylindrical form of primary wave would be
justifiable only when there is synchronism among the secondary waves
issuing from the various centres.
[11] H. Necker (_Phil. Mag._, November 1832); Fox Talbot (_Phil.
Mag._, June 1833). "When the sun is about to emerge ... every branch
and leaf is lighted up with a silvery lustre of indescribable
beauty.... The birds, as Mr Necker very truly describes, appear like
flying brilliant sparks." Talbot ascribes the appearance to
diffraction; and he recommends the use of a telescope.
DIFFUSION (from the Lat. _diffundere; dis-_, asunder, and _fundere_, to
pour out), in general, a spreading out, scattering or circulation; in
physics the term is applied to a special phenomenon, treated below.
1. _General Description._--When two different substances are placed in
contact with each other they sometimes remain separate, but in many
cases a gradual mixing takes place. In the case where both the
substances are gases the process of mixing continues until the result is
a uniform mixture. In other cases the proportions in which two different
substances can mix lie between certain fixed limits, but the mixture is
distinguished from a chemical compound by the fact that between these
limits the composition of the mixture is capable of continuous
variation, while in chemical compounds, the proportions of the different
constituents can only have a discrete series of numerical values, each
different ratio representing a different compound. If we take, for
example, air and water in the presence of each other, air will become
dissolved in the water, and water will evaporate into the air, and the
proportions of either constituent absorbed by the other will vary
continuously. But a limit will come when the air will absorb no more
water, and the water will absorb no more air, and throughout the change
a definite surface of separation will exist between the liquid and the
gaseous parts. When no surface of separation ever exists between two
substances they must necessarily be capable of mixing in all
proportions. If they are not capable of mixing in all proportions a
discontinuous change must occur somewhere between the regions where the
substances are still unmixed, thus giving rise to a surface of
separation.
The phenomena of mixing thus involves the following processes:--(1) A
motion of the substances relative to one another throughout a definite
_region_ of space in which mixing is taking place. This relative motion
is called "diffusion." (2) The passage of portions of the mixing
substances across the _surface_ of separation when such a surface
exists. These surface actions are described under various terms such as
solution, evaporation, condensation and so forth. For example, when a
soluble salt is placed in a liquid, the process which occurs at the
surface of the salt is called "solution," but the salt which enters the
liquid by solution is transported from the surface into the interior of
the liquid by "diffusion."
Diffusion may take place in solids, that is, in regions occupied by
matter which continues to exhibit the properties of the solid state.
Thus if two liquids which can mix are separated by a membrane or
partition, the mixing may take place through the membrane. If a solution
of salt is separated from pure water by a sheet of parchment, part of
the salt will pass through the parchment into the water. If water and
glycerin are separated in this way most of the water will pass into the
glycerin and a little glycerin will pass through in the opposite
direction, a property frequently used by microscopists for the purpose
of gradually transferring minute algae from water into glycerin. A still
more interesting series of examples is afforded by the passage of gases
through partitions of metal, notably the passage of hydrogen through
platinum and palladium at high temperatures. When the process is
considered with reference to a membrane or partition taken as a whole,
the passage of a substance from one side to the other is commonly known
as "osmosis" or "transpiration" (see SOLUTION), but what occurs in the
material of the membrane itself is correctly described as diffusion.
Simple cases of diffusion are easily observed qualitatively. If a
solution of a coloured salt is carefully introduced by a funnel into the
bottom of a jar containing water, the two portions will at first be
fairly well defined, but if the mixture can exist in all proportions,
the surface of separation will gradually disappear; and the rise of the
colour into the upper part and its gradual weakening in the lower part,
may be watched for days, weeks or even longer intervals. The diffusion
of a strong aniline colouring matter into the interior of gelatine is
easily observed, and is commonly seen in copying apparatus. Diffusion of
gases may be shown to exist by taking glass jars containing vapours of
hydrochloric acid and ammonia, and placing them in communication with
the heavier gas downmost. The precipitation of ammonium chloride shows
that diffusion exists, though the chemical action prevents this example
from forming a typical case of diffusion. Again, when a film of Canada
balsam is enclosed between glass plates, the disappearance during a few
weeks of small air bubbles enclosed in the balsam can be watched under
the microscope.
In fluid media, whether liquids or gases, the process of mixing is
greatly accelerated by stirring or agitating the fluids, and liquids
which might take years to mix if left to themselves can thus be mixed in
a few seconds. It is necessary to carefully distinguish the effects of
agitation from those of diffusion proper. By shaking up two liquids
which do not mix we split them up into a large number of different
portions, and so greatly increase the area of the surface of separation,
besides decreasing the thicknesses of the various portions. But even
when we produce the appearance of a uniform turbid mixture, the small
portions remain quite distinct. If however the fluids can really mix,
the final process must in every case depend on diffusion, and all we do
by shaking is to increase the sectional area, and decrease the thickness
of the diffusing portions, thus rendering the completion of the
operation more rapid. If a gas is shaken up in a liquid the process of
absorption of the bubbles is also accelerated by capillary action, as
occurs in an ordinary sparklet bottle. To state the matter precisely,
however finely two fluids have been subdivided by agitation, the
molecular constitution of the different portions remains unchanged. The
ultimate process by which the individual molecules of two different
substances become mixed, producing finally a homogeneous mixture, is in
every case diffusion. In other words, diffusion is that relative motion
of the molecules of two different substances by which the proportions of
the molecules in any region containing a finite number of molecules are
changed.
In order, therefore, to make accurate observations of diffusion in
fluids it is necessary to guard against any cause which may set up
currents; and in some cases this is exceedingly difficult. Thus, if
gas is absorbed at the upper surface of a liquid, and if the gaseous
solution is heavier than the pure liquid, currents may be set up, and
a steady state of diffusion may cease to exist. This has been tested
experimentally by C. G. von Hufner and W. E. Adney. The same thing may
happen when a gas is evolved into a liquid at the surface of a solid
even if no bubbles are formed; thus if pieces of aluminium are placed
in caustic soda, the currents set up by the evolution of hydrogen are
sufficient to set the aluminium pieces in motion, and it is probable
that the motions of the Diatomaceae are similarly caused by the
evolution of oxygen. In some pairs of substances diffusion may take
place more rapidly than in others. Of course the progress of events in
any experiment necessarily depends on various causes, such as the size
of the containing vessels, but it is easy to see that when experiments
with different substances are carried out under similar conditions,
however these "similar conditions" be defined, the rates of diffusion
must be capable of numerical comparison, and the results must be
expressible in terms of at least one physical quantity, which for any
two substances can be called their coefficient of diffusion. How to
select this quantity we shall see later.
2 _Quantitative Methods of observing Diffusion._--The simplest plan of
determining the progress of diffusion between two liquids would be to
draw off and examine portions from different strata at some stage in the
process; the disturbance produced would, however, interfere with the
subsequent process of diffusion, and the observations could not be
continued. By placing in the liquid column hollow glass beads of
different average densities, and observing at what height they remain
suspended, it is possible to trace the variations of density of the
liquid column at different depths, and different times. In this method,
which was originally introduced by Lord Kelvin, difficulties were caused
by the adherence of small air bubbles to the beads.
In general, optical methods are the most capable of giving exact
results, and the following may be distinguished, (a) _By refraction in a
horizontal plane._ If the containing vessel is in the form of a prism,
the deviation of a horizontal ray of light in passing through the prism
determines the index of refraction, and consequently the density of the
stratum through which the ray passes, (b) _By refraction in a vertical
plane._ Owing to the density varying with the depth, a horizontal ray
entering the liquid also undergoes a small vertical deviation, being
bent downwards towards the layers of greater density. The observation of
this vertical deviation determines not the actual density, but its rate
of variation with the depth, i.e. the "density gradient" at any point,
(c) _By the saccharimeter._ In the cases of solutions of sugar, which
cause rotation of the plane of polarized light, the density of the sugar
at any depth may be determined by observing the corresponding angle of
rotation, this was done originally by W. Voigt.
3. _Elementary Definitions of Coefficient of Diffusion._--The simplest
case of diffusion is that of a substance, say a gas, diffusing in the
interior of a homogeneous solid medium, which remains at rest, when no
external forces act on the system. We may regard it as the result of
experience that: (1) if the density of the diffusing substance is
everywhere the same no diffusion takes place, and (2) if the density of
the diffusing substance is different at different points, diffusion will
take place from places of greater to those of lesser density, and will
not cease until the density is everywhere the same. It follows that the
rate of flow of the diffusing substance at any point in any direction
must depend on the density gradient at that point in that direction,
i.e. on the rate at which the density of the diffusing substance
decreases as we move in that direction. We may define the _coefficient
of diffusion_ as the ratio of the total mass per unit area which flows
across any small section, to the rate of decrease of the density per
unit distance in a direction perpendicular to that section.
In the case of steady diffusion parallel to the axis of x, if [rho] be
the density of the diffusing substance, and q the mass which flows
across a unit of area in a plane perpendicular to the axis of x, then
the density gradient is -d[rho]/dx and the ratio of q to this is
called the "coefficient of diffusion." By what has been said this
ratio remains finite, however small the actual gradient and flow may
be., and it is natural to assume, at any rate as a first
approximation, that it is constant as far as the quantities in
question are concerned. Thus if the coefficient of diffusion be
denoted by K we have q= -K(d[rho]/dx).
Further, the rate at which the quantity of substance is increasing in
an element between the distances x and x+dx is equal to the difference
of the rates of flow in and out of the two faces, whence as in
hydrodynamics, we have d[rho]/dt =-dq/dx.
It follows that the equation of diffusion in this case assumes the
form
d[rho] d / d[rho] \
------ = -- ( K ------ ),
dt dx \ dx /
which is identical with the equations representing conduction of heat,
flow of electricity and other physical phenomena. For motion in three
dimensions we have in like manner
d[rho] d / d[rho]\ d / d[rho]\ d / d[rho]\
------ = -- ( K ------ ) + -- ( K ------ ) + -- ( K ------ );
dt dx \ dx / dy \ dy / dz \ dz /
and the corresponding equations in electricity and heat for
anisotropic substances would be available to account for any parallel
phenomena, which may arise, or might be conceived, to exist in
connexion with diffusion through a crystalline solid.
In the case of a very dilute solution, the coefficient of diffusion of
the dissolved substance can be defined in the same way as when the
diffusion takes place in a solid, because the effects of diffusion will
not have any perceptible influence on the solvent, and the latter may
therefore be regarded as remaining practically at rest. But in most
cases of diffusion between two fluids, both of the fluids are in motion,
and hence there is far greater difficulty in determining the motion, and
even in defining the coefficient of diffusion. It is important to notice
in the first instance, that it is only the relative motion of the two
substances which constitutes diffusion. Thus when a current of air is
blowing, under ordinary circumstances the changes which take place are
purely mechanical, and do not depend on the separate diffusions of the
oxygen and nitrogen of which the air is mainly composed. It is only when
two gases are flowing with unequal velocity, that is, when they have a
relative motion, that these changes of relative distribution, which are
called diffusion, take place. The best way out of the difficulty is to
investigate the separate motions of the two fluids, taking account of
the mechanical actions exerted on them, and supposing that the mutual
action of the fluids causes either fluid to resist the relative motion
of the other.
4. _The Coefficient of Resistance._--Let us call the two diffusing
fluids A and B. If B were absent, the motion of the fluid A would be
determined entirely by the variations of pressure of the fluid A, and by
the external forces, such as that due to gravity acting on A. Similarly
if A were absent, the motion of B would be determined entirely by the
variations of pressure due to the fluid B, and by the external forces
acting on B. When both fluids are mixed together, each fluid tends to
resist the relative motion of the other, and by the law of equality of
action and reaction, the resistance which A experiences from B is
everywhere equal and opposite to the resistance which B experiences from
A. If the amount of this resistance per unit volume be divided by the
relative velocity of the two fluids, and also by the product of their
densities, the quotient is called the "coefficient of resistance." If
then [rho]1, [rho]2 are the densities cf the two fluids, u1, u2 their
velocities, C the coefficient of resistance, then the portion of the
fluid A contained in a small element of volume v will experience from
the fluid B a resistance C[rho]1[rho]2v(u1- u2), and the fluid B
contained in the same volume element will experience from the fluid A an
equal and opposite resistance, C[rho]1[rho]2v(u2 - u1).
This definition implies the following laws of resistance to diffusion,
which must be regarded as based on experience, and not as self-evident
truths: (1) each fluid tends to assume, so far as diffusion is
concerned, the same equuibrium distribution that it would assume if its
motion were unresisted by the presence of the other fluid. (Of course,
the mutual attraction of gravitation of the two fluids might affect the
final distribution, but this is practically negligible. Leaving such
actions as this out of account the following statement is correct.) In
a state of equilibrium, the density of each fluid at any point thus
depends only on the partial pressure of that fluid alone, and is the
same as if the other fluids were absent. It does not depend on the
partial pressures of the other fluids. If this were not the case, the
resistance to diffusion would be analogous to friction, and would
contain terms which were independent of the relative velocity u2 - u1.
(2) For slow motions the resistance to diffusion is (approximately at
any rate) proportional to the relative velocity. (3) The coefficient of
resistance C is not necessarily always constant; it may, for example,
and, in general, does, depend on the temperature.
If we form the equations of hydrodynamics for the different fluids
occurring in any mixture, taking account of diffusion, but neglecting
viscosity, and using suffixes 1, 2 to denote the separate fluids,
these assume the form given by James Clerk Maxwell ("Diffusion," in
_Ency. Brit._, 9th ed.):--
Du1 dp1
[rho] --- + --- - X1[rho]1 + C12[rho]1[rho]2(u1 - u2) + &c. = 0,
Dt dx
where
Du1 du1 du1 du1 du1
--- = --- + u1 --- + v1 --- + w1 ---,
Dt dt dx dy dz
and these equations imply that when diffusion and other motions cease,
the fluids satisfy the separate conditions of equilibrium dp1/dx -
X1[rho]1 = 0. The assumption made in the following account is that
terms such as Du1/Dt may be neglected in the cases considered.
A further property based on experience is that the motions set up in a
mixture by diffusion are very slow compared with those set up by
mechanical actions, such as differences of pressure. Thus, if two gases
at equal temperature and pressure be allowed to mix by diffusion, the
heavier gas being below the lighter, the process will take a long time;
on the other hand, if two gases, or parts of the same gas, at different
pressures be connected, equalization of pressure will take place almost
immediately. It follows from this property that the forces required to
overcome the "inertia" of the fluids in the motions due to diffusion are
quite imperceptible. At any stage of the process, therefore, any one of
the diffusing fluids may be regarded as in equilibrium under the action
of its own partial pressure, the external forces to which it is
subjected and the resistance to diffusion of the other fluids.
5. _Slow Diffusion of two Gases. Relation between the Coefficients of
Resistance and of Diffusion._--We now suppose the diffusing substances
to be two gases which obey Boyle's law, and that diffusion takes place
in a closed cylinder or tube of unit sectional area at constant
temperature, the surfaces of equal density being perpendicular to the
axis of the cylinder, so that the direction of diffusion is along the
length of the cylinder, and we suppose no external forces, such as
gravity, to act on the system.
The densities of the gases are denoted by [rho]1, [rho]2, their
velocities of diffusion by u1, u2, and if their partial pressures are
p1, p2, we have by Boyle's law p1 = k1[rho]1, p2 = k2[rho]2, where
k1, k2 are constants for the two gases, the temperature being constant.
The axis of the cylinder is taken as the axis of x.
From the considerations of the preceding section, the effects of
inertia of the diffusing gases may be neglected, and at any instant of
the process either of the gases is to be treated as kept in
equilibrium by its partial pressure and the resistance to diffusion
produced by the other gas. Calling this resistance per unit volume R,
and putting R = C[rho]1[rho]2(u1 - u2), where C is the coefficient of
resistance, the equations of equilibrium give
dp1 dp2
--- + C[rho]1[rho]2(u1 - u2)= 0, and --- + C[rho]1[rho]2(u2 - u1)= 0 (1).
dx dx