Download Mechanism of Tear Film Rupture and Formation of Dry Spots on

Mechanism of Tear Film Rupture and Formation of Dry Spots on Cornea
A. SHARMA AND E. RUCKENSTEIN l
Department of Chemical Engineering, State University of New York, Buffalo, New York 14260
Received June 8, 1984; accepted October 4, 1984
The rupture of the tear film covering the cornea and the formation of dry spots is of importance
in various pathological states associated with a dry eye. The purpose of this paper is to identify its
mechanism and to compute the time required for such a rupture. A two-step, double-film model is
proposed which accounts for the real structure of the tear film as composed of a mucous layer and
an aqueous layer. The mechanism identifies the key step in the process of rupture to be the instability
and eventual rupture of the mucous layer of about 200 to 500 A thickness, which covers the
epithelium, due to the dispersion interactions. This in turn exposes the aqueous layer of the tear film
to the hydrophobic cornea, resulting in the observed end result of a spontaneous tear-film rupture or
"beading-up." The proposed mechanism is shown to be consistent with the range of observed breakup
times and other clinically observed characteristics of the tear-film rupture. The implications of this
mechanism in the normal wearing and tolerance of contact lenses are also indicated. © 1985 Academic
Press, Inc.
INTRODUCTION
than 10 s is usually considered abnormal and
may have mild to serious consequences, such
as: an instantaneous sensation of local irritation, a persistent feeling of ocular irritation,
or a foreign body sensation. In severe cases,
a breakup time of less than the interblink
period may result in an irreversible dewetting
of cornea, epithelial damage, and corneal
ulceration. An abnormally rapid tear breakup
also plays an important role in determining
the contact lens tolerance. It has been shown
to be responsible for the adhesion of the lens
to cornea (and consequent epithelial damage)
and excessive, rapid deposit formation on
the anterior lens surface (12, 13).
The measurement of BUT thus plays an
important role in the characterization and
differentiation of a normal cornea from a
pathological one. In view of this, it is not
surprising that the clinical measurements of
BUT are widespread in assessing the severity
of conditions associated with a dry eye. It is,
however, surprising that in spite of its significance, none of the models proposed for the
instability of the tear film address the issue
of the time of rupture. This information may
The stability of the lacrimal liquid film
covering the cornea (or the contact lens) has
attracted much attention (1-11) because of
the important role it plays in the optimal
conditions of vision; both from the optical
and physiological points of view. The normal
tear film forms a continuous smooth cover
over the corneal surface. The structural integrity of this continuous film is maintained
by involuntary periodic blinking, with a normal interblink interval of about 5 to 10 s. If,
however, the eyelids are held open for a
longer time period, the stability of the tear
film is threatened, resulting in the eventual
formation of randomly distributed holes in
the lacrimal liquid.
The average time elapsed between a blink
and the first hole to appear is defined as the
tear breakup time (BUT). This is found to
be in the range of 20 to 50 s for a normal
adult eye (2). For a pathological eye, however,
BUT is found to be considerably less than
the normal BUT. A breakup time of less
To whom correspondence should be addressed.
12
0021-9797/85 $3.00
Copyright © 1985 by Academic Press, Inc.
All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
TEAR
FILM
be used as an important criterion for model
discrimination. The following two mechanisms have been suggested as possible causes
for the rupture of the tear film.
MARANGONI
INSTABILITY
It has been suggested that the lipids secreted
from the tarsal glands decrease the overall
wetting ability of the aqueous layer which
rests on a mucus-coated corneal epithelium
(12-15). The tear film consists of a mucous
layer covering the epithelium, an aqueous
layer and a lipid-mucin bilayer located at
the air-tear interface (a brief exposure to the
structure of the tear film is provided later).
In a series of papers, Holly and his associates
(1-3, 12-17) have advanced a mechanism of
film rupture based on the assumption that
the lipids present at the tear-air interface
migrate rather rapidly to the mucous-aqueous
interface and eventually overwhelm the protective capacity of the mucous layer, thus
creating areas of high hydrophobicity. The
surface-tension driven motion (Marangoni
effect) was thought to be responsible for such
a process (14). This may happen if the convective motion along the lipid-air interface
(from the regions of low surface tension to
those with high surface tension) is in the
same direction as the convective flow induced
by the interfacial perturbations. However,
any uneven distribution of interfacial lipids,
due to interfacial perturbations, has a tendency to equilibriate. This is so, because the
surface-tension gradient driven Marangoni
motion is in the direction opposite to the
convective flow caused by interfacial perturbations. This stabilizing effect of surface active
agents on the stability of a falling film (18)
and a thin film subjected to dispersion forces
(19) has been reported. It should however be
noted that even in the absence of Marangoni
convection, the diffusion of lipids in the
aqueous layer is present. Based on the magnitudes of the thickness of aqueous layer (ho
6-9/~m) and the diffusion coefficient, D
10-5 cm2/s., the penetration time (hg/D),
13
RUPTURE
for the lipids is easily shown to be of the
order of 10 -2 s. Thus the aqueous layer
becomes saturated with lipids in less than a
second. The solubility of lipids, which are
made of waxy and cholesteryl esters (2), in
the aqueous medium is rather low. In addition, since the lipids are much less surface
active than the mucous material covering the
epithelium, their adsorption on the mucousaqueous interface is thermodynamically unfavorable, viz., it increases the mucousaqueous interfacial tension. As shown later,
an increased mucous-aqueous interfacial
tension makes the mucous layer more resistant to the interfacial deformations and thus
contributes to its stability. In addition, it
may be noted that the presence of lipids is
not necessary for the tear film breakup which
is observed even in the event of a complete
destruction of meibomian gland openings (2).
Based on an upper estimate for the mucinconcentration gradient between the epithelium and the air-tear interface, Lin and
Brenner (5) show that a mucin-concentration
driven Marangoni motion may exist in the
aqueous tear film. The presence of surfaceactive agents (in this case, interfacial lipids),
however, makes it less likely.
RUPTURE
DUE
TO DISPERSION
FORCES
Lin and Brenner (6) carded out a linear
stability analysis of a nondraining, micrometer-sized tear film, under the influence of
retarded van der Waals dispersion forces.
The thickness of neutrally stable tear film
was computed for various values of the retarded Hamaker constant, by taking the
wavelength of the perturbation to be the
same as the linear dimension of the eye ( ~ 1
cm). A tear film with a thickness less than
this critical thickness eventually ruptures because of the dominance of the dispersion
forces over the viscous and surface tension
resistances of the fluid to any interfacial
deformation. Additional information about
the time of rupture is, however, needed for
the purpose of model differentiation.
Journal of Colloid and Interface Science. Vol. 106. No. 1. July 1985
14
SHARMA
AND RUCKENSTEIN
The main purpose of this work is thus to air-tear interface is a lipid-mucin layer, which
formulate a mechanism of the tear-film rup- is about 1000 to 2000 A thick (2). Lipids
ture which is consistent with the observed present in this layer retard the evaporation
breakup times and other characteristics of of the aqueous phase so that only about 7%
the rupture process. For this purpose, we of the aqueous layer evaporates in 1 min
establish a unified linear instability criterion under the normal circumstances (13). An
which incorporates the van der Waals inter- increased rate of evaporation has an adverse
actions and the hydrodynamics of the tear effect both on the thinning of the tear film
flow. An estimate for the time of rupture is and the dehydration of cornea. In the presobtained from this linear stability analysis. ence of a normal lipid layer, the drying of
An application of this formalism to the entire the epithelial surface cannot be caused by
micrometer-sized tear film (considered as a the evaporation, since it would take about
homogeneous film) shows that such a rupture 10 to 20 min. Underneath this lipid layer is
cannot occur within the observed breakup an aqueous electrolyte middle phase which
times. Therefore a different mechanism of is about 6-9 ~tm thick immediately after
the tear-film rupture is required to explain blinking. The thickness then decreases in an
the observed BUT. The fact that the tear almost linear manner because of evaporation
film is, in reality, composed of a hydrophobic and osmotic transfer of tears across the corepithelium, a mucous layer, and an aqueous nea, and in approximately 20 to 50 s (BUT),
layer, naturally suggests a mechanism which its thickness is reduced to about 4 #m. At
should incorporate this heterogeneity. The this point, the film has been observed to
cause for the instability of the entire tear film rupture almost instantaneously (4). Sandis sought in the instability and the rupture wiched between the aqueous phase and the
of the thin mucous layer, which covers the corneal epithelium is a mucoid layer, which,
epithelium, under the influence of van der in a normal eye, is about 200 to 500 A thick
Waals dispersion forces. This mechanism is (2). Most of the mucous material covering
shown to be consistent with the observed the superficial epithelium originates in the
breakup times and other characteristics of conjunctival goblet cells and is distributed
the rupture process. In addition, it aids in a over the preocular surface by the shear created
rational understanding of a diversity of clin- by the lid motion during blinking. The reical observations about the pathological con- newal rate of this layer is very small and
ditions of a mucus-deficient eye and the only a small fraction of it is removed during
tolerance of contact lens. A clear understand- each blink. In addition to serving such vital
ing of the proposed mechanism and model functions as the maintenance of corneal and
differentiation thus necessitates a brief expo- conjunctival surfaces in the proper state of
sure to the structure and functioning of the hydration and lubrication, the mucous layer
tear film, as well as to the observed charac- provides a hydrophilic base for an even
teristics of the rupture process. This is fol- spreading of the aqueous tear film (15).
lowed by linear stability analysis and deriThe corneal and conjunctival surfaces are
vation of expressions for BUT. The impli- highly hydrophobic and indeed, they are
cations of the suggested mechanism are then incapable of sustaining a continuous, aqueous
tear film without the presence of the mucous
discussed.
layer coating the epithelium (2, 3, 15). There
is even some experimental evidence for a
THE STRUCTURE AND THE RUPTURE
highly hydrophobic lipid monolayer sandCHARACTERISTICS OF A TEAR FILM
wiched between the epithelium and the muThe tear film which covers conjunctiva cous layer (14). The presence of the mucous
and cornea consists of at least three distinct layer is thus necessary to effectively mask the
fluid layers. The outermost layer making the hydrophobic character of the epithelial surface
Journal of Colloidand Interface Science, Vol. 106, No. 1, July 1985
TEAR FILM RUPTURE
and to impart a stable, hydrophilic base to
the tear film. Experiments of Dolhman et al.
(8) also demonstrate that a reduction in the
glycoproteins contents (a mucous-layer constituent) is associated with a decrease in the
tear-film breakup time and was detected in
some pathological states of the dry eye. The
various components of a typical, normal tear
film are depicted in Fig. 1.
The production rate of tear is approximately 1.2 #1/min, which is drained continuously between each blink. Calculations based
on the cross-sectional area of the tear film
( ~ 2 × 10-3 cm 2) and the production rate
give an average tear velocity of about 10 -2
cm/s for such a flow (6).
Perhaps, it is also important to note that
a meniscus extends along the entire margin
o f both the upper and lower eyelids. Similar
menisci are present surrounding the bubbles
and debris found in the tear film (11) and
contact lenses. The locally thin areas, the socalled "black lines," appear adjacent to these
thick menisci. Such locally thin (couple of
micrometers thick) areas have been thought
to be instrumental in accelerating the process
of film rupture (6).
With this brief exposure to the tear film,
we now proceed to derive a stability criterion
for a tear film.
THE INSTABILITY CRITERION AND THE
TIME OF RUPTURE OF A THIN FILM
From the geometry and the dimensions of
eye as shown in Fig. 2, it is readily shown
that the angle/30 is less than 1 10 °. Thus the
deviation of the tear film from a vertical film
(/30 = 7r/2) is only slight and the small
curvature of the vertical plane may be neglected. Thus we essentially consider the stability of the tear film on a two-dimensional
cylindrical surface as shown in Fig. 3. The
AIR
LIPID LAYER
15
ANTERIOR
CORNEAL ~
EPITHELIUM ~
/ ho
--I
:
TEAR FILM
!
/
FIG. 2. Side view of a human eye.
relevant equations describing the hydrodynamics are the Navier-Stokes equations
which include the van der Waals dispersion
forces acting on the tear film as a body force.
This body force becomes important when
the film is thinner than the range of interaction of the dispersion forces between the
epithelium and the film resting on it. The
effect of van der Waals dispersion forces is
incorporated by using a Hamaker-type approximation, which has been used by Scheludko (20), Ruckenstein and Jain (19), and
G u m e r m a n and Homsey (21), with many
others for various problems of thin-film stability. For thin films (h0< 500 A), the van
der Waals dispersion forces acting on a unit
volume located at the interface are derived
from the potential
49' = ,~/67rh'3,
[ 1]
where A is an effective Hamaker constant
and h' is the thickness of the film. For films
of higher thicknesses, the retardation effect
has to be included. Consequently, while investigating the stability of the micrometer-
~
ho
EAR FILM
z
i000-2000 A
AQUEOUS LAYER
4-8/,trn
v~a:,'.c:c:zi.:c,~:.:~:c~;:~':~;~• MUCOUS LAYER,200-500
r
EPITHELIUM
FIG. 1. Structure of the precorneal tear film.
FIG. 3. Definition sketch.
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
16
SHARMA
AND
RUCKENSTEIN
sized homogeneous tear film, the following
retarded potential may be used:
dp' = B/h '4,
Selecting the initial film thickness, h0, and
the free interface velocity of a film flowing
down a flat vertical wall, v~0 = pgh2/2#, as
the basic units of length and velocity, the
time and pressure may be scaled by ho/v~o
and pv~, respectively. In this way, one obtains the following nondimensional form of
the Navier-Stokes equations in the cylindrical
coordinates
[2]
where B = B~ - B I 2 and B o is the retarded
Hamaker constant for the interactions between molecules of type i and j (1 refers to
the tear film and 2 to the epithelium).
OUr _1_ OUr 1)0 OVr ,
0-7
+ r-ff +
_
031)r 1)2
a-7- r
op o, ±(o vr
Or
1
1921)r 021)r
~ Or + R e \ o r 2 + ~ - ~
1
20Vo
01.) r
+-~z 2 + r Or
r 2 ao
vr~
~5),
[31
Vo 01)o + OVo + "1)rVo
OVo + Vr OVo + -0-7
-fir
r-ffff
Vz-~z
r
= -
1 8p
~704~+
r O0
r O0
1 (a2Vo
R-e l~Or 2
021)0 10Vo 20Vr
+ ~15 02vo
-~+~
+'+
r Or r 2 O0
VO~
)
-~
[4]
and
O1)z + 1)rOVz + Vo OV~ + Vz OVz
o-7
-fir
r
2
Re
Op
Oz
Od~
1 [02Vz 1 02Vz 02Vz 10v~
nOz+ Re\~rZ + ~ +
Oz----2+r Or]"
[5]
The continuity equation is
Ovr vr 1 OVo Ov~
O--r- + --r + - r - ~ + ~ = 0.
at
[61
r :
(ao/ho) : a,
13r = VO = Vz = 0.
[7]
The following three boundary conditions are
to be satisfied at the tear-air interface, viz.,
at r = a + h(z, O, t): two conditions of zero
shearing stresses,
In Eqs. [3] to [6], vr, Vo, v=, and p are the
three nondimensional velocity components
and the nondimensional pressure, respectively. The coordinate system is shown in
O
(Oh~21
Fig. 3. The Reynolds number, Re, is defined pr._.zrOho_._+pr°[
r
1---\aO] I
as pv~oho/~. Although the interaction potential
4~ is in general a function of the spatial
Pr~ Oh Oh poz Oh
Oh
coordinates, it is only the potential at the
r O00z
r O0 poz-~z = 0 [8]
film interface that is required within the
framework of the approximation employed and
h e r e (Appendix A). At the interface, the
Oh Pro Oh Oh
parameter • is given by B'/h~pv2zo and q~ = 1/ Prr OZ
r Oz O0
h c. The quantity 4~ is related to the potential
4J = B'/h '~ by the relation 4~ = h~c#/B'.
Clearly, B' = B and c = 4 apply to the
+ prz 1 kOzl l
r O0
retarded case, and B' = .~/67r and c = 3 to
Oh
the nonretarded case. Here h is the nondi- P~z --~ = O, [91
mensional thickness of the film.
On the anterior epithelium, the following and the relation between the pressure jump
and the surface tension
adhesion conditions are satisfied:
-~21
[_(oh 21
Journalof ColloidandInterfaceScience,Vol. 106, No. 1, July 1985
TEAR FILM RUPTURE
IPrr_.} Poo(Oh]2
[Oh~ 2
r2 \-~]
2p~oOh+2po~OhOh
r O0
r O00z
+Pzz~oz,]
\-~](Oh]
2
__ S
~po
~75
1[1
r
+ -
+
(Ohm21
\-~zl J
1 02h [1 + (Oh] l
-
-r 2 -002
-
02h [
Oz 2
Here Po is the static nondimensional pressure
of the nonviscous, semi-infinite medium (air),
S is a nondimensional parameter defined as
S = a/pho v 2 , and a is the interfacial tension.
Finally, the description of the interface is
made complete by adding the kinematic condition
Oh
Ot
Vo Oh
Oh
Vr + r O0 + Vz Oz
O,
at
r=a+h.
[11]
For a Newtonian fluid, the various components of the nondimensional pressure tensor
are related to the velocity gradients by the
following constitutive relations:
P,r=-P
2 Ov~
+ Re Or'
poo = - p
+ ~
-gr +
'
20v~
Pz~ = - P + R e Oz '
Pro = ~
- ~ + Or
'
1 (Ovo l OVz]
l (Ovr OVz]
Prz = ~
17
2 Oh Oh
\-~z] [ + r 2 O00z O00z
1 Oh 2
1 (Oh~ 2
(0h]21-3/2
[lO]
where the subscript b denotes the base case
velocity and pressure distributions. Now in
order to investigate the instability of the basic
solution [13], which is caused by the dispersion forces and flow, we make use of a
perturbation approach similar to that of
Shlang and Sivashinsky (22). This approach
has the advantage of yielding a nonlinear
equation of evolution for the interface, from
which the information about the linear stability of the interface may also be extracted.
In what follows, we make use of the fact that
S = ~/phov~ is a large parameter for thin
liquid films (S ~ 108 for the tear film) and
thus defining S = l/e 2, the estimate e ~ 1
holds. This may be referred to as the "largesurface tension approximation" combined
with the "thin-film approximation." At this
stage, the problem involves two time scales
which are related to the deformation of the
free surface at a given cross section and to
the vertical wave motion, respectively. This
complication may be circumvented by introducing a moving coordinate system and thus
eliminating the time scale associated with the
vertical wave motion
z' = z + 7t,
[12]
[14]
\ OZ + Or] "
The domain of the solution is bounded by
the layer
a <~ r <~ a + h(z, O, t).
where 2/is a nondimensional velocity, which
is determined in what follows. The following
space-time stretching transformations are now
introduced
The basic, steady, laminar flow solution to
the set of Eqs. [3] to [12] is given by
x = r - a,
'/-)rb
O0b =
0,
1 2
~)zb = y(r
-- a 2)
--(1 +a)21n(r/a),
h= 1
[13a]
and
Pb = PO + S/(1 + a),
[ 13b]
Z1 ~
6Z',
O~ =
ar,
y = aO,
T = E2t. [15]
The film thickness, velocities, and the pressure
may thus be expanded around their base
values as the following power series perturbation expansions
JournalofColloidandInterfaceScience,Vol.
106, No. 1, July 1985
18
SHARMA AND RUCKENSTEIN
h =- 1 + ~
Enhn(y, Z1, T),
[16a]
n=l
T),
[ 16b]
Vo -- ~ e"Vo,(X, y, zl, T),
[16c]
l)r -~" ~
~nVrn(X, y~ Z I ,
n=l
n-I
1)z = Vzb(X , e) + E
enl)zn(X, Y, Z1, T ) ,
[16d]
n=l
and
P -- Pb(~) + Z ~"P,-
The dispersion potential, q~, is, however, not
expanded in power series, since we wish to
retain the dispersion nonlinearities in the
subsequent derivation. The derivation pursued here retains both the convective and
the dispersion force contributions to the instability of the film and allows one to study
the synergism between the two. Transforming
to the coordinate system [14]-[15] and
equating the like powers of e, one obtains a
hierarchy of perturbed equations, which are
solved successively for the deviation in the
film thickness, hi. Details of these calculations
are given in Appendix A. The nonlinear
equation of evolution for the deviation in
the film thickness, hi, is obtained,
2 Oh1
3a Oz t
X V2hl +
4hi Ohi + 8Re 02hi + R e
Oz'
15 OZ t2 3a 2
S 1/2
Ren
V4hl - ~qZ V2q~ = 0,
[17]
where
v 2=
hi ~ exp(flt + ik'z' + il'O),
[ 16e]
n=l
Ohl
Ot
available presently for various parameters
and the breakup times to be predicted, an
estimate of the breakup times obtained by
linearizing Eq. [ 17] is deemed satisfactory for
model discrimination.
The linear stability of the interface is determined by examining the response of the
interface to a Fourier component of an arbitrary traveling wave perturbation, viz.,
.
This may be solved numerically to obtain an
accurate information about the kinetics of
rupture, for a given initial interfacial disturbance. We, however, do not undertake this
calculation here, since such an endeavor demands that an equally accurate information
be available for various physical parameters
(most crucially, the Hamaker constant) and
the amplitude of interracial perturbations. As
only an order of magnitude information is
Journal of Colloidand InterfaceScience, Vol. 106, No. 1, July 1985
[18]
where k' and l' are the wavenumbers of the
disturbances in the z' and 0 coordinates.
Note that k' is dimensionless. Substituting
[18] in Eq. [17] and discarding the nonlinear
terms in Eq. [ 17], one arrives at the dispersion
relation in terms of the original variables,
_- 2 /
3\
k
ao]
1
+
[g8 hov2p
+~
- a{(kho) 2 + (Iho)2}
× {(kh0)2 + (lh0)a},
[19]
where l = (/'/ao), k = (k'/h0), fl is the rate of
instability parameter, and k and l are the
wavenumbers of perturbations in the dimensional coordinates corresponding to z' and 0
coordinates, respectively. Any arbitrary perturbation imposed on the surface may be
Fourier-decomposed into various wave-like
components (Eq. [18]) with different wavenumbers. We seek to determine the wavenumber for which the growth rate, /3, is a
maximum. The waves with these wavenumbers are selectively amplified at a faster rate,
thus causing the eventual rupture of the film.
The wavenumber of this fastest growing perturbation is thus obtained by maximizing the
real part of fl with respect to (k 2 + /2), viz.,
from
dfl/d(k 2 + l 2) = 0.
[20]
Denoting the wavenumbers of the most dangerous perturbations by km and Im and noting
that for the parameters of tear film (Table I),
(~)
pho
~
8 hoGp,
~-
TEAR FILM RUPTURE
19
TABLE I
The expression, V~o = ogh2/2#, has been used
in the derivation of Eq. [23].
As is expected, our results without the van
Property
Average value
Ref.
der Waals interactions (B' = 0) recover the
results of Benjamin (23), who considered the
Maximum thickness of the
stability of the flow of a viscous liquid film
tear film
7.8 #m
(2, 4, 5)
Minimum thickness of the
down a vertical plate. For a nonretarded
tear film
4.0/~m
(4)
potential, viz., C = 3 and B' = A/67r and in
Mucous layer thickness
200-500 A
(2)
the absence of flow, it reduces to the result
Surface tension for the
air-tear interface (a)
40 dyn/cm
(5)
derived by Ruckenstein and Jain (19) for the
Density (o) of the tear film
1.0 g/cm 3
(5)
rupture of thin films. It may be pointed out
Viscosity (t~) of the tear
that the interfacial instability of the film due
film
0.035 g/cm-s
(5)
(t~a) for the mucous layer
1.0 dyn g/cmZ-s
to the flow (which is reflected in the first
S (dimensionless)
5.1 × 108
term
of the denominator of Eq. [23]) does
Re(dimensionless)
3.1 × 10-4
not lead to the film rupture, but results in a
stable self-fluctuating wave or the formation
of ripples on the film surface (22). The van
one obtains
der Waals dispersion forces on the other
hand are capable of rupturing a thin film.
cB'
2
[21]
This happens because, although the surface
free energy increases with the increasing surAll of the two-dimensional wave disturbances face area associated with the deformation of
which satisfy Eq. [21] are now selectively the interface, the total free energy of the
amplified at the fastest rate. A characteristic system decreases due to van der Waals diswavelength for these perturbations may be persion interactions.
Williams and Davis (24) obtained the time
defined by letting lm ~ 0 (one-dimensional
disturbance) in which case
of rupture of a nonflowing, thin film with a
~km = 27r/km.
[22] nonretarded potential (C = 3, B' = .~/67r),
by solving a nonlinear equation which they
The above definition is chosen only for con- derived by a long-wavelength approximation.
venience, since only an order of magnitude Their results indicate that the rupture times
for the wavelength is desired. An alternate computed after retaining the nonlinearities
choice such as km ~ lm may also be used to are no more than a factor of two lower than
define the characteristic wavelength. We that derived from a linear theory, even for
compute this characteristic wavelength to amplitudes of perturbations as large as oneshow that the wavelength of the fastest grow- third of the film thickness. It is in view of
ing perturbation is smaller than the linear this, and of the fact that only an order of
magnitude is available for the Hamaker condimension of the eye.
Substituting the wavenumbers of the fastest stant, that Eq. [23] can be used to evaluate
growing perturbations (Eq. [21]), in the real the breakup time, ~-m.
part of 13, the maximum growth coefficient,
/3m, may be obtained. Finally, an estimate R U P T U R E O F A H O M O G E N E O U S T E A R F I L M
for the time of rupture is obtained from rm
/3m~, which leads to the expression for the
Equation [23] is now applied to the entire
time of rupture:
tear film considered as a homogeneous film
of several micrometers in thickness. The
shaded region in Fig. 4 depicts the feasible
~-m = 12#aho
1'1"2
+ h~--~. ] .
[23]
ranges for the retarded Hamaker constant
Physical Properties of the Tear Film
c.,l
Journal of Colloid and Interface Science, Vol. 106, No. I, July 1985
20
SHARMA AND RUCKENSTEIN
a model of rupture predicts a continuous
process of thinning prior to the rupture and
thus cannot explain the observed instantaneous rupture or "beading-up" of tear film
at a thickness of about 4 ~zm (4). Finally, it
fails to recognize the key role played by the
mucous layer in determining the stability
characteristics of the tear film. This last point
is amply supported by various experiments
(8, 29) and clinical tests (12-17). Thus, although the instability of the air-tear interface
due to the dispersion forces and flow exists,
it is not sufficient to cause the observed rapid
rupture.
10 -12
5x.lO-~3
r m =15 sec
B
10-ts
(ergs- cm)
"rm = 14-0 sec
5x10 -14
10 -14
5x10 -15
10 -t5
5x10 -
1
2
5
4,
5
6
7
8
ho (/~m)
FIG. 4. Feasible region for the tear-film rupture in the
H a m a k e r constant-tear-film thickness parameter space.
(B) and the tear-film thickness which are
consistent with the observed breakup times.
Although the retarded Hamaker constant has
not been measured for the epithelium-tear
fluid or the contact lens-tear fluid systems,
various experiments for other substances indicate an upper bound of 10-19 erg-cm for B
(25-28). Thus even assuming local micrometer-sized inhomogenities in the tear film
and a rather large time for the tear breakup
(-~ 140 s), the estimated Hamaker constant
by this mechanism is about three orders of
magnitude higher than that expected. For a
more realistic tear-film thickness of 4 ~m
and a breakup time of 50 s, the rupture of
the tear film requires a Hamaker constant of
about 3 × 10 -13 erg-cm, which is about six
orders of magnitude higher than that reported.
In other words, even assuming a retarded
Hamaker constant of 10 -18 erg-cm and a
micrometer-sized local inhomogenity in the
tear film, Eq. [23] predicts the time of rupture
of such a film to be about 3 years!
In view of these considerations, it may be
concluded that the retarded van der Waals
interactions are not strong enough to break
the micrometer-sized tear films within the
observed times of rupture. In addition, such
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
THE R U P T U R E OF U N D E R L Y I N G M U C O U S
LAYER: A N E W M E C H A N I S M OF
TEAR-FILM INSTABILITY
It is well known that the presence of the
mucous layer on the epithelium is absolutely
necessary for the wetting of cornea (2, 1416, 30). These studies indicate that the epithelial mucus is capable of adsorbing onto
the low-energy surfaces, converting them into
hydrophilic surfaces. Even Teflon, which
cannot be rendered hydrophilic by any commercially available surfactant, becomes water
wettable when exposed to a layer of mucin.
The experiments of Doughman et al. (17)
and Padday (31) demonstrate the rupture of
aqueous films of several hundred micrometers
thick on hydrophobic surfaces such as Teflon,
paraffin, and polyethylene. The hydrophobicity and water-wetting characteristics of a clean
(without mucus) anterior epithelium are
found to be similar (32).
The tear film is seen to rupture spontaneously only after it thins from its initial
thickness of 7 to about 4/zm. This is much
smaller than the critical thickness of water
over hydrophobic surfaces at which rupture
occurs (17, 31) and much too large for the
retarded dispersion forces to cause a rupture
within observed BUT. A plausible mechanism
which emerges from this discussion is that
the initially mucus coated cornea converts
its wettability characteristics within the observed breakup times, thus effectuating a
spontaneous rupture of the aqueous layer.
21
TEAR FILM RUPTURE
The rupture of the thin mucous layer, which
coats the hydrophobic epithelium and provides a hydrophilic base for the aqueous
tears, due to dispersion interactions indeed
leads to such a hydrophilic-hydrophobic
transition. The issue whether the time of
rupture of the mucous layer is consistent
with the observed range of BUT is addressed
next.
The thickness of the aqueous film is at
least two orders of magnitude larger than the
mucous layer, and hence a slow deformation
of the air-aqueous interface may be neglected
while investigating the stability of the mucous-aqueous interface. The superficial epithelial cell walls, which are coated by the
mucous layer, have a complex structure of
protrusions known as microvilli (2). The
characteristic dimensions of these protrusions
are, however, about an order of magnitude
larger than the thickness of the mucous layer
(2) and thus the mucous-epithelial interface
may be considered to be planar while investigating the rupture of the mucous layer. In
other words, we consider the instability of a
thin (200-500 A) mucous layer sandwiched
between epithelium and a semi-infinite viscous, aqueous phase (Fig. 1). Since the film
is thin, the following potential is used for the
dispersion forces:
49' = A/67rh '3.
[24]
The effective Hamaker constant for this geometry is given by
[25]
where A 0 is the Hamaker constant for the
interaction between the molecules of type i
and j (1 refers to the aqueous phase, 2 to the
mucous layer, and 3 to the anterior epithelium). The aqueous layer is viscous and thus
the boundary conditions [8]-[10], which are
established for a nonviscous (air) bounding
medium, do not hold rigorously. However,
the ratio of the viscosities of the aqueous
layer and the mucous layer is much smaller
than unity. In this case, the effects of the
viscosity of the bounding medium on the
stability of the mucous layer may be shown
to be insignificant (less than 1%) (Appendix
= A I 2 -]- A 2 2 - A13 - A 2 3 ,
B). Thus, the entire formalism developed
earlier holds and we now investigate the
amplification o f fluctuations or the initial
nonhomogeneities arising at the mucousaqueous interface. In addition, v~0 ~ 0 since
the mucous layer is very thin and viscous.
The time of rupture for such a layer is
derived from Eq. [23] by making use of the
potential [24]. Thus the substitutions C = 3,
B' = A/6~-, and g = 0 (no-flow) in Eq. [23]
give
rm = 48~rZ~#h~/A2.
[26]
This result has been derived earlier by Ruckenstein and Jain (19) by making a lubrication
approximation for a thin film.
The discrepancy between the time of rupture as determined from the linear and nonlinear equations depends on the magnitude
of the perturbations (24). Since no a priori
information is available about the amplitude
of perturbations at the mucous-aqueous interface, a factor of 2 may again be used to
convert the times of rupture as computed by
Eq. [26] to more realistic times of rupture
computed from the solution of a nonlinear
equation (24). The feasible range of Hamaker's constant for the rupture of the mucous
film to occur may thus be obtained by rewriting Eq. [26] as
A m . . . . in = ( 2 4 a # T r 2 h ~ / r m i . . . . . )1/2.
[27]
The breakup time estimates "rmin = 20 S and
7ma x =
50 S are used to find the range of the
Hamaker constant which is consistent with
the observed BUT. These are shown as the
shaded region in Fig. 5. The wavelength of
the most dangerous perturbation is
Xm = ( 8 7 r 3 f f / d ) l / 2 h 2 ,
[28]
which, for the parameters values falling in
the shaded region of Fig. 5, is easily verified
to be several orders of magnitude smaller
than the dimensions of the eye.
The Hamaker constant thus obtained for
a mucous-film thickness of 200 to 500 A is
in the range of 10 -14 to 4 × 1 0 -13 erg. This
magnitude of the Hamaker constant is realistic (19, 20, 27) and thus the dispersion
forces acting on the mucous layer are indeed
Journal of Colloid and Interface Science, Vol. 106, No, 1, July 1985
22
SHARMA AND RUCKENSTEIN
10"12
8x10 "15
6x10 -13
4x10 -13
A 2x10-t3
Vm=2Osec
(ergs)
13
10-t4
8xlO"
6xtO "14
4xlO -t4
2x10 -14
rrn = 5 0 sec
i 0 -i4
8xfO -le
6x10 -l~
normal eye. At this point, the inherent hydrophobicity of the exposed epithelium is
responsible for an instantaneous (fast compared to the previous step) rupture of the
aqueous tear film.
Due to its sequential nature and the fact
that the model incorporates the structure of
the tear film as made up of two distinct films,
the proposed mechanism may be referred to
as a "two-step, double-film" mechanism of
tear-film rupture.
DISCUSSION
AND
IMPLICATIONS
4x10 "15
In a normal eye, tear-film rupture does
not occur because the interblink time is small
compared to the breakup time and the blinkl
O
1
5
~
_
_
J
100
200
500
400
500
ing is instrumental in restoring the structural
o
ho(A)
integrity of the mucous layer. This rupture
and
the consequent formation of dry spots
FIG. 5. Feasible region for the mucous-filmrupture in
the Hamaker constant-tear-film thickness parameter on cornea, however, are crucial for a mucusspace.
deficient pathological eye, for which the rupture would occur rather frequently within
the interblink period. For instance, assuming
capable of rupturing it within a time period
an effective Hamaker constant of 10 -14 erg,
consistent with the observed breakup times.
an abnormally thin ( ~ 100 ~ ) mucous layer
With this understanding, the basic steps of
would break within 4-5 s, thus exposing the
the proposed mechanism may now be sumhydrophobic epithelium to the aqueous layer.
marized:
This may result in the extensive dewetting
(i) The blinking movement exerts shear and desiccation of cornea. Indeed, in a clinical
across the thin, aqueous film located between study (3), patients with chronic conjunctival
the eyelid and the ocular globe and thus inflammation were observed to have an abredistributes and smoothens the mucous layer normally short breakup time ( ~ 3 to 5 s) of
on the corneal epithelium. Immediately after the tear film, despite the normal amounts of
its structural integrity is restored, this mucous aqueous tears. Conjunctival biopsy revealed
layer is responsible for effectively masking a marked decrease in the population of conany lipid contamination as well as the basic junctival goblet cells which are responsible
hydrophobic nature of the epithelium. At for the production of conjunctival mucous.
this stage, the overlying aqueous film is com- The crucial role of the mucus in the tearplete and is slowly thinning due to the evap- film stability and thus in determination of
oration and the osmotic transfer across the BUT, has also correlated well in other clinical
findings. The deficiency of vitamin A has
cornea.
(ii) Upon its restoration, the small inhom- been observed to catalyze the disappearance
ogeneities of the mucous layer begin to am- of conjunctival goblet cells and the conseplify at several places under the influence of quent appearance of the dry-eye syndromes
the dispersion forces. If this process is not (2). Some other conditions such as ocular
reversed by an intermediate blink, the grow- pemphigoid, Stevens-Johnson syndrome,
ing interfacial perturbations cause the rupture trachoma, chemical burns, and certain forms
of the mucous layer in about 15 to 50 s in a of drug-induced diseases also decrease the
2x10 -15
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
TEAR FILM RUPTURE
goblet cell population, resulting in decreased
BUT, even in the presence of normal tear
volume (2, 33).
An important consequence of the stability
of the tear film is in the normal wearing and
functionality of the contact lenses (12, 13,
34). A well-fitted contact lens rests on a
continuous tear film sandwiched between the
epithelium and the lens and is also coated
with a continuous tear film on the outside.
The stability of both the prelens (in-front-ofthe-lens) and the postlens (behind-the-lens)
tear film is important for good contact lens
wearing performance. Most of the silicone
and hydrogel lenses being used presently are
not completely water wettable. Thus, it is
most likely that the conjunctival mucus soon
coats the anterior surface of a lens placed in
the eye and contributes to the wettability of
the prelens film. The premature rupture of
this mucous layer would result in the prelens
film breakup and the accumulation of denaturated proteins contaminated with the lipids
which are observed on the lens surface (13).
This arises due to the high interracial tension
between the aqueous layer and the exposed
(without the mucous coating) prelens surface
and results in a deposition of proteins and
lipids because they tend to lower the interfacial tension at the aqueous-lens interface.
The strong interactions between the lens surface and the proteins give rise to multiple
site adsorption, resulting in less exposure of
hydrophilic groups of the proteins to the
aqueous tears and consequently, their denaturation. While the interfacial tension is
somewhat lowered by this process, it may
not be sufficient to stabilize the aqueous tear
film due to the unavailability of enough
hydrophilic groups which face the tear film.
Indeed, some successful attempts have been
made to correlate the length of deposit-free
wear time with the tear-film breakup time
(35), which show that a rapid deposit formation is associated with a short BUT. This
is expected from the proposed mechanism,
because both the BUT and the prelens film
breakup time depend on the thickness of the
mucous layer coating the epithelium and the
23
lens surface, respectively. The mucus deficiency would thus decrease both the BUT
and the prelens breakup time and the latter
would encourage deposit formation. The
cause for a condition called "giant papillary
conjunctiviries" has been linked with an
immune-type reaction to antigenic, denaturated proteins accumulated on the lens surface
(36). Thus the stabilization of the prelens
tear film by coating the lens surface with
mucus or mucus-like substances seems a
logical step for the prevention of lens surface
contamination.
According to our mechanism, the stability
of postlens tear film is determined by the
structural integrity of the epithelial mucous
layer. As the epithelial mucous layer ruptures,
the postlens film would also break if the lens
surface is not wettable and the adhesion of
the contact lens to cornea can take place.
This may result in excessive ocular discomfort, irritation, or even epithelial damage. In
fact, a clinical study (12) shows that five of
the six cases of corneal erosion observed
occurred under the stationary lens, which
results due to the premature rupture of postlens film. In view of this, wearing of contact
lens in a normally mucus-deficient eye may
result in these complications. In addition, in
the presence of contact lens, even a marginally
mucus-deficient eye may be affected by these
conditions because of two reasons: (i) contact
lenses adversely affect the lid-globe congruity
(13) which is important for the formation
and renewal of an evenly distributed mucous
layer; (ii) contact lens wearers often become
lazy blinkers and in addition the blinking
becomes incomplete (13), resulting in incomplete restoration of mucous layer. Both of
these factors may eventually contribute toward a poor lens tolerance and various conditions of a pathological eye, because they
encourage a thinner than the normal mucous
layer and hence reduce its time of rupture.
The only objective clinical measurement
which reflects the relative stability of the tear
film is the measurement of BUT under properly controlled environmental conditions. The
proposed mechanism actually correlates BUT
Journal of Colloid and Interface Science, Vol. 106, No. 1, July 1985
24
SHARMA AND RUCKENSTEIN
with the thickness of mucous layer and the
mucous-aqueous interfacial tension. Tests
such as the lack of mucus in inferior fornix
and the absence of goblet cells as determined
by conjunctival biopsy (3), may be readily
used to diagnose the mucus deficiencies.
Thus, in the absence of other factors such as
lipid abnormalities, large aqueous-tear deficiency, impaired lid functions, and gross
irregularities in the corneal epithelium, the
replacement of mucus-like substances in the
eye seems to be the logical starting point for
the therapy of short BUT and its consequences.
Finally, the model proposed does not account for the non-Newtonian rheology of the
mucous layer, since its rheological properties
have never been measured. It seems likely
that a better characterization of both the
Hamaker constant (for epithelium-tear and
contact lens-tear systems) and rheological
behavior of mucous material would result in
improved knowledge of the kinetics of rupture. Thus there is a need for both in vivo
and in vitro experiments on the rupture of
"thin" mucous layers, for this indeed appears
to play a central role in the pathological
states associated with a dry eye.
CONCLUSIONS
We have derived a nonlinear evolution
equation for the interface, a dispersion relation, and an estimate for the time of rupture
of a thin film. The formalism developed
incorporates the effects of van der Waals
dispersion forces, the convective motion of
the film and the two-dimensional perturbations arising at the interface. Application of
this model to the tear film, considered as a
several-micrometer-thick homogeneous film,
fails to predict the observed kinetics of rupture. Based on many clinical and experimental observations about the role of mucus in
determining the stability characteristics of
the entire tear film, we propose a new mechanisms for the formation of dry spots on the
cornea. Immediately after blinking, a thin
(200-500 A) mucous layer is restored on the
Journal of Colloid and Interface Science. VoL 106, No. 1, July 1985
corneal epithelium. The van der Waals dispersion forces acting on the mucous layer
act to destabilize this layer, resulting in its
eventual rupture. This, in turn, exposes the
hydrophobic epithelium to the aqueous film
and to the omnipresent lipids. It is at these
sites that the aqueous tear film breaks rather
rapidly causing the dewetting of the cornea.
The time of rupture computed by this mechanism is in agreement with the clinically
observed tear-film breakup times of about 15
to 50 s for a healthy eye. This rupture does
not occur in a normal eye because the interblink time is small compared to the time of
rupture of the tear film.
The mechanism proposed is also consistent
with other observed characteristics of the
tear-film rupture, such as the almost spontaneous rupture of the tear film after it thins
from an initial thickness of about 7 #m to a
thickness of 4 um, an abnormal decrease of
BUT in mucus-deficient eye and the adhesion
of the contact lens to the cornea in mucusdeficient patients or lazy blinkers. The mechanism proposed thus have implications for
the diagnosis and treatment of pathological
eye, as well as in determining the contact
lens tolerance. It also lends support to a
number of clinical and experimental observations about the crucial role of mucus in
determining the stability characteristics of
the tear film.
APPENDIX A
Making use of the set of transformations
[14], [15] and the perturbation expansions
[16a]-[16e], the set of governing Eqs. [3] to
[ 12] may be manipulated to yield a hierarchy
of equations.
Equating the terms of order ~ gives the
first nontrivial set of equations and the corresponding boundary conditions as
=0,
02VOl
Ox 2 = 0 ,
02Vzl
Ox 2
0 [la]
and
O/)rl
Ox
- 0,
[lb]
TEAR FILM RUPTURE
25
ch = 1 + EChl+ ' ' ' ,
together with
Vrl=Vol=Vzt=O
at
x=0
[lc]
and
Vrl ~ 0 7
OVzl
Ovol _ 0
2hl + ~
= O, Ox
[ld]
and
1
OJ hi
Pl --
V2hl
x = 1.
at
[le]
The solution to this first approximation is
immediately obtained as
Vrl = vo1= O and
Vzl = - 2 h l X
then we would have terms ~(O4)l/Oy) and
rt(O4)l/OZl), instead of ~cl(o(o/Oy) and ncl(o(o/
OZl), respectively, in the second-order equations. Such an expansion would, however,
destroy the dispersion-force nonlinearities
which we wish to retain in the first-order
equation of evolution for the interface. The
boundary conditions for the second-order
problem are
at
at
x=
x = 0,
1,
OVo2 _ o,
2h2+Ovz2
-~x = 0 '
Ox
[2a]
(3' - 1) Oh~
3zl
and
1
OJ hi - V2hl -1- nqSlx=l
nq5 -t- P l --
= r/q~(y, zl) + PI(Y, zl),
[3d]
Vr2 = V02 = 'Oz2 = 0
Vr2 = 0.
[3e]
The solutions to the set of Eqs. [3a]-[3e] are
Vr2 = X 20hl
OZl '
[2b]
where
[4a]
1
Vo2 = -- -~ Re x(x - 2)
As shown by Eq. [2b], the further hydrodynamic calculations require only the interaction potential at the interface, viz. at x = 1.
The second approximation is obtained by
equating the terms of order e2, giving
X I 1~-5 Ohl
-~y + V2 ~Oh1
y
0hi(1
vz2=Reoz---~l
X (x - 2)
O~)r2
--
+
Ox
_ Op~ _
Oy
0q5
1
O'Ozl
Ozl
= 0,
2x3
I--~ Ohl
1R
-'~
Oz~
,e
2)hl - 2Xh2,
Od
[3b]
OVzl
3' Ozz + 2(x - 1)l)r2 q- X ( X -- 2) OVz~
OZl
OZ 1
O(])
~ E - I -OZl
~-
1 02~)z2
Re
1
OVzl
-Ox
- 2 -~- -aRe
_Ox
_
[3c1
In the derivation of [3b] and [3c], we take
~ O(1), in order
to retain the dispersion force nonlinearities
in the lower-order equations. This point is
better understood by noting that if the potential, qS, is expanded in a power series as
rle-l(Od~/Oy) and , e l(O~9/OZl)
ex
-1 3.~121
]
[4c]
and finally,
3' = 2.
Opl
[4b]
[3a]
1
- 0
4)
+sx
~2 Ohl
az---~l+
and
--
-5
+-x(x-
021)02
rte-l Ov + Re ~3x
4
gx
~/C1 O~yy] ,
[4d]
Equation [4d] reflects the well-known fact
that the wave velocity for a falling film on a
flat vertical plane is twice the velocity of the
free interface.
Going to the terms of order E3, the kinematic condition gives the following equation
of evolution for the interface:
_Ohl
_ + 2 Oh~ _ 2hl
OT
OZ 1
-]-
(
Ohl
OZ 1
l)zllx=l
Oh2
Vr3[x= 1 _ _ _
OZl
')3hl
-- G
~
= 0.
Journal of Colloid and Interface Science, Vol. 106, No.
[5]
1, July 1985
26
SHARMA AND RUCKENSTEIN
The velocity component, Vr3, appearing in
the above equation is determined by equating
the terms of order ~3 in the continuity equation, to obtain
Oi)r3
1
Ovo2 O'l)z2
OX "[- -~ l)r2 -[- ~ y "~ --OZ1 = 0.
2 Oh1 4hl Oh---!l+ 8 Re 02hi
3a Ozl
Ozl
15 Oz2
Ohl
aT
Re5
+ 3a----
xTZh1 + Re
-~- ~74hl - Re~le-lxTZffa = O.
[81
[6a]
For a nondimensional potential of the form
With the help of boundary condition
vr3=O at x = 0 ,
4, = 1/h c,
[6b]
2(~x3_
)Oh1
-Ozl
1
2X2
a
+--
1
2
F
× LCV2hl - E2h-l(1 +
X Re(3x3-x2)(-.-~V2hl-}-V4hl-t-V2q~)
--
Re
(~0
x5
1X42)
- -6
02hl Ohzx2"
Ozl
+ 3 x30z---~l +
Making use of the expressions [2a] and [7]
for vz~ and Vr3, respectively, the kinematic
condition [5] is manipulated to yield the
following closed-form equation of evolution
for the deviation in the interface from its
equilibrium value:
~effkho [ (sinh( kho)cosh( kho) 2#h0
APPENDIX
B
In order to evaluate the effects of viscous
aqueous layer on the stability of thin mucous
film, we employ the results of Jain and
Ruckenstein (37). Their final dispersion relation for the linear stability of a thin film
on a solid support and bounded by a semiinfinite viscous fluid is
kho) + R ( sinh 2( kho) -
(kho)2)]
L ~5--~¥-(~-~sh(kh-~+g-~i--~o~
where ~eff = ~ + (Odf/Oh')h=ho(1/k2) and R is
the ratio of the viscosities of the aqueous and
mucous layers.
Since the mucous film is thin and the
critical wavelength is orders of magnitude
higher than the film thickness, Eq. [1] may
be simplified by noting that
sinh(kho) ~
oh11=l
C)~
oh1 + OzlJ ] " [91
(Oy
Equation [8] is the desired nonlinear equation
of evolution which retains both the convective
and the dispersion-forces nonlinearities and
may be solved numerically for a given initial
disturbance.
[7]
/3-
(kho) 3
(kho) + - -
These estimates simplify Eq. [1] to
(kho)2 I _d
/3 - 3#ho
]
2 ~ oz - ¢(kh°)2 (1 +
The wavenumber of the fastest growing perturbation is the solution of dfl/dk = 0, viz.,
RoA
2
Journalof Colloidand InterfaceScience,Vol.
(kho) < 0.3.
106, No. 1, July 1985
1
2R(kho))
[3]
- 27rh----~km
(kho) 2
for
[1]
3'
(3aRha)k3m + (2ah02)k~
6
and
cosh(kho) ~ 1 + -
ch-(l+c)
V2q~ = (02/Oy 2 + 02/0Z2)~ =
the solution to Eq. [6a] is given by
l)r3 = - - __
7z~b in Eq. [8] is determined as
[2]
A
2~rh----~- 0,
[4]
which in the event of an inviscid semiinfinite fluid gives the well-known result
TEAR FILM RUPTURE
lim kern -
R~O
4~rah 4 "
[5]
The other limit of infinitely viscous, semiinfinite fluid is easily shown to be
lim k 2 - - R~oo
6~r~h4"
[6]
Thus the fastest growing wavenumber is
bounded by
0-41 ( A
_ _)
ha ~ J
1/2 ~ < k m ~ <0"5
--
(~zl)
__ 1/2
ho2 ~ /
for all R E (0, ~ ) .
For the tear film, R < 1, and its inclusion
in Eq. [3] makes negligible contribution to
the final result.
REFERENCES
1. Holly, F. J., "Wetting, Spreading and Adhesion"
(J. F. Paddy, Ed.), p. 439. American Press, New
York, 1978.
2. Holly, F. J., and Lemp, M. A., Surv. Ophthalmol.
22, 69 (1977).
3. Lemp, M. A., Dohlman, C. H., and Holly, F. J.,
Ann. Ophthalmol. 2, 259 (1970).
4. Norn, M. S., Acta Ophthalmol. 47, 865 (1969).
5. Lin, S. P., and Brenner, H., J. Colloid Interface Sci.
85, 59 (1982).
6. Lin, S. P., and Brenner, H., J. Colloid Interface Sci.
89, 226 (1982).
7. Norn, M. S., Acta Ophthalmol. 57, 766 (1979).
8. Dohlman, C. H., Friend, J., Kalevar, V., Yadoga,
D., and Balazs, E., Exp. EyeRes. 22, 359 (1976).
9. Norn, M. S., Acta Ophthalmol. 41, 531 (1963).
10. Maurice, D. N., Invest. Ophthalmol. 4, 464, (1967).
11. Ehlers, N., Acta Ophthalmol. Suppl. 81, 1 (1965).
12. Fanti, P., and Holly, F. J., Contact Intraocular Lens
Med. J. 6(2), 111 (1980).
13. Holly, F. J., Amer. J. Optom. Physiol. Opt. 58(4),
331 (1981).
27
14. Holly, F. J., Exp. EyeRes. 15, 515 (1973).
15. Holly, F. J., and Lemp, M. A., Exp. Eye Res. 11,
239 (1971).
16. Holly, F. J., Int. Ophthalmol. Clinics 3, 171 (1980).
17. Doughman, D. J., Holly, F. J., and Dohlman,
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Journalof ColloidandInterfaceScience,Vol. 106,No. 1, July 1985