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INSTITUTE OF PHYSICS PUBLISHING
EUROPEAN JOURNAL OF PHYSICS
Eur. J. Phys. 24 (2003) 159–168
PII: S0143-0807(03)53589-9
The Young–Laplace equation links
capillarity with geometrical optics
M A Rodrı́guez-Valverde, M A Cabrerizo-Vı́lchez and
R Hidalgo-Álvarez
Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada,
E-18071, Granada, Spain1
E-mail: [email protected]
Received 18 September 2002, in final form 19 November 2002
Published 10 February 2003
Online at stacks.iop.org/EJP/24/159
Abstract
Analogies in physics are unusual coincidences that can be very useful to solve
problems and to clarify some theoretical concepts. Apart from their own
curiosity, analogies are attractive tools because they reduce the abstraction of
some complex phenomena in such a way that these can be understood by means
of other phenomena closer to daily experience. Usually, two analogous systems
share a common aspect, like the movement of particles or transport of matter.
On account of this, the analogy presented is exceptional since the involved
phenomena are a priori disjoined. The most important equation of capillarity,
the Young–Laplace equation, has the same structure as the Gullstrand equation
of geometrical optics, which relates the optic power of a thick lens to its
geometry and the properties of the media.
1. Introduction
Two physical systems are analogous when the respective phenomena are expressed by means
of equations with identical form but different variables. The most well-known analogy in
physics is the patent mechanical equivalence between accelerated motions, i.e. straight-line
motion and rotatory motion. In this case, the mass–moment of inertia and velocity–angular
velocity appear in like equations. Thus, a rotatory motion can be modelled as the superposition
of infinite straight-line motions in different directions even as a straight-line motion can be seen
as a rotatory motion with infinite radius. There are other analogies which involve disjoined
phenomena such as heat conduction and electricity or kinematics. Electric charge, spatial
position and heat are related by means of a similar set of equations. The existence of analogies
usually eases the analysis of the problem. Consequently, the electric resistance of a medium
can visually be described as a damping coefficient, the electric current as a heat flow and
the abstract concept of potential is easily imagined as temperature. In addition, similarities
1
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160
M A Rodrı́guez-Valverde et al
between the forms of differential equations governing mechanical and acoustical phenomena
as well as those of electrical systems enable us to solve the mechanical and acoustical problems
via electrical circuit analysis [1].
Qualitatively, analogies help us to understand why, for example, the surface tension
appears as a force tangential to the liquid surface although the unbalanced net force on a bulk
molecule is directed inward (a vertical force) [2] or why amphiphile (surfactant) molecules
tend to migrate and to locate themselves at free liquid surfaces. Analogous to a mechanical pull
system, a molecule is raised horizontally by surface tension to create a new surface area while
the ‘hydrophobic effect’ associated with surfactants is seen as a microscopic Archimedes’
principle [3]. Those hidden relations and agreements that come from the most distant fields
are, at the same time, the most fruitful and the most difficult to state.
2. The Young–Laplace and Gullstrand equations
An interface is the zone existing between two immiscible substances. The shape of an interface
between two non-solid substances (1 and 2) is described by the generalized Young–Laplace
equation [4]:
1
1
1
P = γ12
+ CH
+
(1)
R1 R2
R1 R2
where γ12 stands for the interfacial tension, P for the difference of pressure along the
interface, C H for the bending stress and R1 and R2 for the principal radii of curvature. Curvature
measures the local bending of a curve or surface. Local surface curvatures are measured by
(1) sectioning the surface with a plane,
(2) best-fitting an osculating circle to the resulting curve at the point of interest, and
(3) reporting its reciprocal radius.
The principal radii of curvature coincide with the maximum and minimum values of the radii
associated with orthogonal osculating circles. The steps mentioned to measure the principal
radii of curvature are briefly shown in figure 1 for the case of a sessile drop.
The shape of a liquid drop on another immiscible liquid is similar to a convergent optic lens
(figure 2). In fact, such a pair of capillary surfaces is frequently labelled as a lens. Moreover, an
optic system is a set of surfaces that separate different refractive indices media. According to
this definition, the above-mentioned drop is considered an optic system because it is composed
of two interfaces (figure 3). This fact is not just a curious coincidence but also points out the
existence of a formal parallel between geometrical optics and capillarity. The first equivalence
comes into sight between Fermat’s principle [5] and the equilibrium state of a liquid–fluid
interface [6], both being extreme conditions2 applied to different quantities: the optic path
and macrocanonical potential. As cited below, there also exists a dual relation in reference to
the correspondence between two interfaces and a lens, that is, the equivalence between two
spherical refractive surfaces and an interface. We will focus on this equivalence due to its
practical use and ease of treatment.
The optical power Pop
of a (thick) lens or a general centred system is defined as the
reciprocal of its corresponding equivalent focal length f properly signed3 (see appendix A.1).
This parameter is linked to the capacity of a lens to focus in a given medium, in such a way
that a lens is said to be more powerful (in absolute value) when its focal point is located closer
(inside or outside) to one of its faces. Although a lens used to be treated as a flat object with no
thickness, real lenses have a given thickness. The optical power of a thick lens (of thickness d)
2
Each of these laws can be formulated in terms of a stationary integral under infinitesimal variations.
The distances to those points placed inside the denser medium (inside the lens), as well as the radii of any convex
faces in the lens, are positive lengths. This is the sign convention employed in this work instead of the usual convention
of analytical geometry (Cartesian sign convention [7, 8]).
3
The Young–Laplace equation links capillarity with geometrical optics
161
Figure 1. Principal planes of curvature (upper drawing) and principal radii of curvature R1 and R2
(lower drawing) of a sessile drop.
Figure 2. Longitudinal section of a liquid lens.
composed of two spherical segments of curvature radii R1 and R2 with a refractive index n 1
and immersed in a medium with a refractive index n 2 is expressed by means of the Gullstrand
equation (see appendix A.2):
2
n2
1
n1
1
1
1
n1
Pop ≡ =
−1
+
−1
d
.
(2)
−
f
n2
R1 R2
n2
n 1 R1 R2
3. Analogy
If equation (2) is compared with equation (1), some analogies are obvious. The interfacial
tension γ12 is equivalent to the quantity n 1 /n 2 − 1, known as refractivity, which also depends
exclusively on the media, i.e. on the material of the lens (phase 1) and on the substance
surrounding it (phase 2). The refractive index is defined as the reciprocal of the relative speed
M A Rodrı́guez-Valverde et al
162
Figure 3. Top view of two water sessile drops on a bitumen film (a) and on a Teflon surface (b).
The images were acquired with the sketched set-up (c) explained in detail elsewhere [9]. The
inner bright rings in pictures (a) and (b) are the images of the annular light source through the
water–air interface. The refracting nature of bitumen causes the appearance of a second inner ring
in picture (a).
of light in the medium. Thus the refractive index is extremely sensitive to any density change
in the medium due to temperature, heterogeneity or anisotropy. As the speed of light is slower
in more dense materials, their refractive indices always exceed one and therefore n 1 is greater
than n 2 . The refractive index is also a function of pressure, increasing as pressure increases,
due to the resulting increase in density, although this dependence is quite weak with liquids
because of its relative incompressibility. The refractive index is an indication of the manner in
which a molecule interacts with light just like interfacial tension is a measure of intermolecular
forces. The curvature radii of each refringent component match the principal curvature radii
of the interface. These radii can be either negative or positive depending on the curvature
direction (figure 4). The analogy between the optical power and the pressure difference along
the interface is not evident although these quantities coincide in sign if the lens and the interface
maintain equal convexity. Therefore, the left term in equations (1) and (2) will be positive
for convex interfaces4 and convergent lenses. Moreover, the minimal interfaces [10] (for
example the catenoid5 ) are those whose pressure difference is zero in the same way as afocal
or telescopic lenses display zero optical power. A more intuitive interpretation of the analogy
between the optical power and the pressure difference along the interface comes from using
the well-known thin lens equation or Gauss equation (see appendix A.1):
1
1
1
=
+
f
x im x ob
(3)
which relates the focal length to the object and image positions, properly signed, with regard
to the two principal planes. The last term of the Gullstrand equation alludes to the bending
stress C H of equation (1), as it refers to the sagging of the lens due to its thickness. To sum
up, the equivalences between the cited quantities are exhibited in table 1.
4
5
Surfaces that bulge outward.
Surface of revolution whose generatrix is a catenary (the chain line).
The Young–Laplace equation links capillarity with geometrical optics
163
Figure 4. Geometrical correspondence between the radii of curvature of a lens and a liquid–fluid
interface. From equation (4) and the sign convention used in this work, there are three types of
> 0): bi-convex (a), plano-convex (b) and concave-convex (c).
convergent lenses (Pop
Table 1. Analogous quantities in capillarity and geometrical optics.
P
Pext
Pop
1
x im
Pint
− x1ob
γ12
n1
n2
−1
−
CH
n1
n2
2
− 1 nn 21 d
For a thin lens, i.e. one for which the condition d R1 + R2 holds, equation (2) can be
simplified as
n1
1
1
Pop −1
+
(4)
n2
R1 R2
which is known as the lens-maker’s equation [11, 12] and is analogous to the classical version
of the Young–Laplace equation [13]:
1
1
P γ12
+
.
(5)
R1 R2
These two equations also coincide in the diverse approaches used in their demonstration. The
lens-maker’s equation can be derived using different proofs [5, 14, 15] based on geometrical
constructions, ray tracing, wave optics, Fermat’s principle, etc. Likewise, the Young–Laplace
equation is obtained by means of force balance,variational method, thermodynamic arguments,
etc, starting from basic cases to the general equation [13, 16, 17].
Finally, the Gullstrand equation is valid in paraxial6 or first-order optics. This
approximation considers only monochromatic light rays that travel close to the optical axis,
where the objects and the apertures are so small that the trigonometric sines and tangents of the
sustained angles can be approximated by their arguments. In this approximation all lenses are
perfect and do not suffer any aberration resulting from the nature of real lenses. Furthermore
this approach avoids the dependence of refractive index on light wavelength7 i.e. with the
type of light used: monochromatic or white (mixture). Likewise, equation (1) is limited to
those interfaces moderately curved where the interfacial tension is curvature-independent and
where liquids and fluids are chemically pure [4].
Geometrical optics also shares several analogies with mechanics, such as the identification
between the equilibrium of an inextensible string subjected to conservative forces and light
6
7
Close to the axis.
Dispersion.
M A Rodrı́guez-Valverde et al
164
Figure 5. Axisymmetric liquid–fluid interfaces with constant mean curvature in terms of different
values of the shape factor (β). The case β < 0 also corresponds to a meniscus confined between
two coaxial cylinders of reduced radii 1 and (β − 1)/β. Equally a drop squeezed from a circular
slit with a radius equal to (β − 1)/β matches the case β > 1. When the slit width (or the supporting
wire thickness) tends to zero then the interface will be a torus.
rays [18], the equivalence of refractive index and nonrelativistic particle velocity [19] and
even an optical–mechanical analogy in general relativity [20]. Likewise thermodynamics and
mechanics are also analogous [21], meeting in interfacial phenomena [13, 17].
4. Application
Examining the noted analogy for thin lenses (equations (4) and (5)), the readers can deduce an
interesting interpretation. Each point on an interface matches an optical configuration with a
focal length equal to γ12 /[(n 1 /n 2 − 1)P]. This argument can be used to design simple lenses
from those capillary menisci that follow the same mathematical law as the image formation
upon paraxial optics.
In the absence of gravity, according to equation (5), the mechanically equilibrium interface
should be uniform in its mean curvature (see appendix A.3). Considering axially symmetrical
interfaces (which are of most practical importance and accessible to the theoretical analysis),
there are only six geometrical surfaces satisfying the above condition [3]. Among them,
the three simplest surfaces (plane, cylinder and sphere) are uniform not only in their mean
curvature, but also separately with respect to each principal curvature. The remaining three
surfaces (forming unduloid, nodoid and catenoid) are uniform only in their mean curvature.
Any axisymmetrical interface, except the sphere8, can only reach the equilibrium state in
contact with solids. On this account, these three capillary menisci are known as liquid bridges
(figure 5).
Any moderately curved interface is determined by an intrinsic parameter (related to the
media) and another purely geometrical one (principal radius of curvature on a reference point).
In this case, the above-mentioned family of interfaces displays a characteristic line similar to
a ‘neck’ or a ‘haunch’, that is, an equator whose radius will be denoted by R0 . Integrating
equation (5) for axial–symmetric interfaces with constant mean curvature (see appendix A.3),
the principal radii of curvature can be written in a reduced form (x̂ ≡ x/R0 ) as follows:
R̂1 =
8
r̂ 2
,
β − 1 + β r̂ 2
R̂2 =
r̂ 2
1 − β + β r̂ 2
A sphere is the only possible variant for surfaces intersecting the axis of revolution.
(6)
The Young–Laplace equation links capillarity with geometrical optics
165
Table 2. Radial range (r̂) of a capillary meniscus according to the value of its shape factor (β).
β
]−∞, 0[
1, β−1
β
r̂
β−1
β
r̂|Ri |→∞
sgn(R1 )
sgn(R2 )
a
−1
±1a
0
[1, ∞[
]0, 21 ]
1, 1−β
β
] 12 , 1]
∓1
1
∓1
1
1−β
β
—
−1
1
1−β
β ,1
1−β
β
]1, ∞[
β−1
β ,1
β−1
β
1
∓1
The upper sign matches the range r̂ < r̂|Ri |→∞ and inversely for the lower one.
where β is the shape factor defined as
β≡
P
R0
2γ12
and determines the type of meniscus as shown in figure 5. Using the analogy described in
section 3, the shape factor can be expressed as
Pop
R0
β = n1
2 n2 − 1
in terms of the lens characteristics and the scale factor R0 .
The choice of a suitable shape factor on the basis of the interface concavity/convexity seems
a simple and excellent criterion to design thin lenses with an imposed geometry. This criterion
requires some equivalences of reference like the cylinder (a plano-convex lens), the sphere (a
bi-convex lens with same-curvature faces), the catenoid (a concave-convex lens but with faces
of the same unsigned curvature), etc. So, if a high curved convergent lens is requested then
the shape factor must be greater than 1, giving two feasible configurations (concave-convex
and bi-convex) according to the value of the radial coordinate r̂ . Similarly, if the lens were
divergent then the shape factor should be negative, choosing between the convex-concave and
bi-concave options. In the region β ∈ ]0, 1] the trends of the radii are switched regarding the
case β > 1, although with smaller curvatures. Owing to this clear asymmetry in the shape
factor, the designer can invert the configuration, taking a negative equatorial radius R0 . Despite
the notation used, the radius of meridian curvature R1 is not related to the radius of the first
face as well as the radius of azimuthal curvature R2 does not match the second face.
Once a value of β is chosen by simple inspection, different reduced radii are available from
equations (6) in accordance with the value of r̂ (table 2). Finally, the real radii are computed
using the respective value of R0 , although for the special case β = 0 this value is arbitrary.
The case of thick lenses and non-moderately curved interfaces is more complicated.
Nevertheless, it is obvious that an afocal thick lens acts as a point of a curved interface where
the difference of pressure is zero. As a result, the following relation is derived:
n2
CH
=
− 1 d.
γ12
n1
Hence, complex phenomena, like the influence of C H on contact angle or the spontaneous
curvature, can become more understandable. There are other effects of second order similar to
the interfacial bending moment that are physically important for systems or phenomena—like
capillary waves, lipid membranes, emulsions and microemulsions.
Aside from the indicated use, this analogy or isomorphism is also interesting by its
noteworthy teaching value, being very useful in understanding some interfacial phenomena
from simple notions in optics. With a preliminary review in geometrical optics, this artifice
can be applied in introductory courses as well as in a graduate classroom.
M A Rodrı́guez-Valverde et al
166
Acknowledgments
Financial support from Ministerio de Ciencia y Tecnologı́a, Plan Nacional de Investigación
Cientı́fica, Desarrollo e Innovación Tecnológica (I+D+I), MAT2001-2843-C02-01 is gratefully
acknowledged. The authors also want to thank the referees for critically reading the manuscript
and making several useful remarks.
Appendix
A.1. Focal length
In Gaussian optics, the light path crossing a refracting spherical surface of radius R (convex by
default), which separates two homogeneous media of refractive indices n 2 and n 1 (n 1 > n 2 ),
is described from Fermat’s principle by [11, 12]:
n1
n2
n1 − n2
+
=
x im x ob
R
(7)
where x ob is the distance of the object and x im stands for the distance of the image from the
pole of the surface.
Applying the definition of focus (back focus F ) to equation (7), the distance between this
point and the surface vertex, known as the focal length, results in
1
n1 − n2
.
=
f
n1 R
(8)
Finally, equation (3) is obtained by substituting equation (8) in (7). As a result of the
reversibility principle of light, a dual focus (front focus F) is similarly defined in the object
space and related to the former as follows:
n2
f = − f .
(9)
n1
A.2. Derivation of the Gullstrand equation
The principal planes in a lens system are two hypothetical planes at which all the refraction
can be considered to happen in such a way that the proper focal lengths are measured with
regard to these planes. For a given set of lenses and separations, these planes are fixed and do
not depend upon the object position. In the case of a thin lens, the principal planes coincide
with the lens itself.
Consider a combination of two systems with a common axis of symmetry (centred system).
From geometrical arguments of the similarity between triangles and the definition of focus,
the focal lengths of the combination are [11]
f1 f2
f f
,
f = 1 2
(10)
t
t
where t is the distance between the foci F1 and F2 . The general paraxial image-forming
properties of a thick lens system are completely determined by the positions of the cardinal
planes. If these are known one can, for example, find the image of any object without making
a detailed study of the passage of the image-forming rays through the system. That is, it is
not necessary to consider the refractions of the rays at the various surfaces. The geometrical
construction for finding the image is the same as the simple method used for a thin lens. Thus,
if a thin lens with a focal length equal to f is placed at the first principal plane, the images
produced are identical to those of the complete optical system except for a lateral shift equal
to the distance between the principal planes. The thin lens is known as the equivalent lens so
the quantities f and f are usually referred to as the equivalent focal lengths. So equation (3)
f =
The Young–Laplace equation links capillarity with geometrical optics
167
can be used to find images formed by thick lenses, although it leaves out the distance between
the principal planes.
Let d be the axial thickness of the lens, i.e. the distance between the poles of the two
surfaces. Then
d = t + f 1 + f2 .
From this equation, the extended Gullstrand equation is easily found using equations (9) and
(10) and considering that the initial and final media are the same:
1
n1 1
1
d
=
.
(11)
−
+
f
n2 f1
f2
f1 f 2
Finally, on substituting into equation (11) for f1 and f 2 given by equation (8) we obtain
equation (2).
A.3. Mean curvature of an axisymmetric meniscus
The radii of principal curvature at a point of a surface of revolution are, as one sees in figure 1,
the radius of curvature of the meridian curve at this point (R1 ), and the radius of curvature of
the curve defined in the portion of the normal to the surface included between the point and the
axis (R2 ). It follows therefore that, if one references the meridian curve of the desired surface
to a pair of rectangular coordinate axes, of which one, the z axis, is the axis of the surface,
that curve will be determined by z = z(r ) in cylindrical coordinates. According to differential
geometry [22] the reciprocal principal radii of curvature are
1
d2 z
1
=
R1
dz 2 dr 2
2
(1 + ( dz
dr ) ) 1 + ( dr )
1
1
dz
1
= R2
r 1 + ( dz )2 dr
dr
and as the local mean curvature holds the following equality [22]:
1
1 1
+
H=
2 R1 R2
then substituting the former radii and rearranging the terms, a differential expression of the
mean curvature can be derived for axially symmetric surfaces:
1
1 d r2
dz
1 d
r
=
.
H=
2r dr
2r dr R2
2 dr
1 + ( dz
dr )
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