The swedish physicist Anders Angstrom was the first person to study...
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Why
some
things
are
darker
when
wet
John
Lekner
and
Michael
C. Dorf
Angstrom
has proposed
that
rough
absorbing
materials
are darker
when
wet
because
their
diffuse
reflection
makes
possible
total
internal
reflection
in
the water
film
covering
them,
increasing
the likelihood
of the
absorption
of
light
by
the
surface.
His
model
is
extended
here
in
two
ways:
the
probability
of internal
reflection
is calculated
more
accurately,
and
the
effect
on
absorption
of
the
decrease
of the
relative
refractive
index
(liquid
to
material
instead
of air
to
material)
is estimated.
Both
extensions
decrease
the
albedo
of
the
wetted
surface,
bringing
the
model
into
good
agreement
with
experiment.
1.
Introduction
From
early
childhood
we
learn
to
distinguish
wet
from
dry,
not
just
by
touch,
but
also
by
sight.
Most
objects,
especially
those
with
rough
and
absorbing
sur-
faces,
are
darker
when
wet:
they
reflect
less
light.
In
a
model
for
this
phenomenon
proposed
by
Angstrom,1
the
surface
roughness
leads
to
diffuse
reflection,
and
thus
to
total
internal
reflection
at the
liquid-air
inter-
face
of
the
thin
film
of
liquid
covering
the
surface.
This
reflection
gives
an
increased
probability
of
the
absorption
of
light
by
the
surface,
and
thus
a
darker
surface.
Angstrom's
model
is
extended
in the
following
two
sections.
The
results
are
then
compared
with
experi-
ment.
In
the
final
section
we also
compare
Angstrom's
approach
to
that
of Bohren
2
and
Twomey
et
al.
3
(TBM),
in which
the
darkening
is taken
to be
due
to
the
increase
in
the
forward
scattering
on wetting.
II.
Diffuse
Scattering
and
Internal
Reflection
An
optically
smooth
surface
reflects
specularly,
whether
or not
the
surface
is covered
by
a
film
of liquid.
A ray
of
light
reflecting
at
a smooth
surface
will
not
be
totally
reflected
at the
liquid-air
interface.
If
the
surface
is
rough,
its diffuse
reflection
sends
some
light
out
obliquely
enough
to be
totally
internally
reflected
at
the
liquid-air
interface.
This
increases
the
chance
of
absorption
at the
surface.
The
authors
are with
Victoria
University
of Wellington,
Physics
Department,
Wellington,
New
Zealand.
Received
6 June
1987.
0003-6935/88/071278-03$02.00/0.
©
1988
Optical
Society
of America.
Consider
a
rough
absorbing
surface,
such
as
a
black-
board.
We
treat
the
surface
as
an
array
of
randomly
oriented
facets,
each
of
which
reflects
specularly.
When
wet,
a thin
liquid
layer
covers
the
surface.
Light
incident
on
this
layer
has
probability
1 -
R
1
of reaching
the
surface,
where
R
1
is
the
reflectance
at
the
air-liquid
interface.
Some
fraction
of
the
transmitted
light
is
absorbed
by
the
surface.
Call
this
fraction
a.
Of
the
light
which
is
reflected
by
the
surface,
let
p be
the
fraction
which
is then
reflected
back
at
the
liquid-air
interface,
so
that
it
is once
again
incident
on
the
sur-
face.
The
process
is
illustrated
in
Fig.
1.
Continuing
the
process
illustrated
ad
infinitum,
the
total
probability
of
absorption
by
the
rough
surface
is
A=
(1-Rl)[a
+
a(l-a)p
+
a(l-a)
2
p
2
+...]
(1
- R)a
1
- p(l-a)
(1)
Angstrom
evaluates
p
as follows:
all
the
light
with
an
angle
of
incidence
greater
than
the
critical
angle
,
(=
arcsinl/n)
onto
the
liquid-air
interface
will
be
to-
tally
reflected.
Thus
p
may
be
estimated
as
the
frac-
tion
of
diffusely
reflected
light
which
lies
outside
the
cone
generated
by
rays
whose
reflection
angle
is 60.
For
a
Lambertian
surface,
the
intensity
reflected
at
angle
is
proportional
to cos6.
Light
emerging
at
angles
to
+
d
subtends
a
solid
angle
27r
sinOdO.
Thus
i/2
27r
dO
sinG
cosO
P
p /2
= Cos
2
OC =
1-
1/n .
2,r I
dO
sinG
cosO
Equations
(1) and
(2) are
together
equivalent
to the
last
equation
in
Angstrom's
Sec.
4(ii),
except
that
he
omits
the
1 -
R
factor.
1278
APPLIED
OPTICS
/
Vol.
27,
No. 7
/ 1
April
1988
(2)
Fig.
1. Liquid
layer over
a
rough
surface.
The
coefficients
repre-
sent
the fraction
of the
incident
light
intensity
which
is transmitted
along
each
path.
Equation
(2) underestimates
p: the
reflectivities
for
both
polarizations
are
generally
small
but
not
zero
for
0
< 0,;
see,
for
example,
Ref.
4, Sec.
1.5,
or
Ref.
5,
Figs.
1-3 and
1-4.
For
a surface
which
reflects
diffuse-
ly according
to Lambert's
law,
the
probability
reflec-
tion at
the
liquid-air
interface
is
J r/2
| dO
sinG
cosOR(x,1/nj)
/2
X
dO
sinG
cosO
=
J dxR(x,
/ni),
where
x =
sin
2
o and
R(x,n)
is the
reflectance
at
an
interface
between
media
1 and
2, with
n =
n
2
/ni.
Here
n
= 1/ni <
1 and R
is unity
for x >
n
2
so
n2
p
1 - n
2
+
dxR(x,n).
(4)
For both
polarizations,
the reflectances
have
the
prop-
erty that
R(x,n)
= R(x/n
2
, i/n)
(5)
(this
was
noted
by
Stern
6
for
the
transmittances).
On
changing
to the
variable
y = x/n
2
,
the integral
in
Eq. (4)
may
be written as
n
2
j
dyR(n
2
yn)
=
n
2
J
dyR(y,i/n)
=
n
2
(/n).
(6)
The last
equality
expressed
the
integral
in terms
of
the
average
reflectance
of an
isotropically
illuminated
sur-
face,
r-l2
J
dO sinO cosOR(x,n)
I
R(n)
= °=
I
dxR(x,n)
(7,
j dO sinG cosO °
(in
correspondence
with
the
average
transmittance
de-
fined
by
Stern
6
). For
unpolarized
light,
Stern's
for-
mulas
(9a) and
(9b) lead
to
- 3n
2
+2n+i
2n
3
(n
2
+
2n-1)
n
2
(n2
+ 1)
3(n
+ 1)2
(n
2
+
i)
2
(n
2
- 1)
(n
2
- 1)2
n2- 1)2
n(n +
1)
(n
> 1). (8
(n
2
+ i)3
l o g
From
Eqs.
(4) and
(6) we
have
the result
p =
1- I [1-
R(nj)].
n
(9)
For
water,
this
formula
gives
a p
larger
by
-9%
than
the Angstrom
estimate
(0.475
instead
of 0.437).
Ill. Probability
of Absorption
by
a Wetted
Surface
The
parameter
a
in Eq. (1)
is the
fraction
of the
light
incident
on the
surface
which
is
absorbed.
(The above
refers
to a
single
interaction:
the
total
probability
of
absorption,
allowing
for
reflections
at
the
liquid-air
interface,
is A.)
Angstrom
takes
a to
have the
same
value
for
the wet
as
for the
dry surface.
We
expect
a,
(the
value
when
wet),
to
be greater
than
ad
(the
value
when
dry),
since
the absorbing
medium
will normally
have
the real
part
of its
refractive
index
greater
than
unity.
Since
reflection
is caused
by
wave
vector mis-
match,
and
since wave
vectors
are
determined
by
re-
fractive
indices,
covering
the surface
with
a layer
of
liquid
(with
nj
> 1) results
in less
reflection.
The
value
of
ad is
in principle
determined
by the
complex
refractive
index nr
+ ins of
the material,
and
by the roughness
of the surface
(which
influences
the
average
angle of
incidence
on its
randomly
oriented
facets,
and the probability
of multiple
interactions,
as
in a
crevice).
The value
of a,
is in addition
a
function
of nj,
the refractive
index
of the
liquid film
covering
it.
For
the purpose
of comparing
the albedos
1 - ad
and 1
-
A of the
dry
and wetted
surfaces,
we estimate
a, in
terms
of ad, nl,
and nr as
follows.
For
small
absorption
(ni <<
nr),
ad
1 - R(n)
where
R(n) is
the average
reflectance
of an isotropically
illu-
minated
surface,
defined
in Eq.
(7). The
assumption
made
here is
that the
angle of
incidence
on facets
of the
rough
surface
(for,
say, normal
illumination)
has the
same
distribution
as would
be obtained
for
a plane
surface illuminated
isotropically.
Similarly,
a, - 1 -
(nr/nl)
when
ni <<
n,.
Thus
when
the
absorption
is
small,
a,
ad[1 - R(nr/nl)]/[l
-
R(n_)] = a).
(10)
When
the
absorption
is
large, on
the other
hand,
and
ad
1,
we expect
aw ad
a(".
We
will
use an
interpola-
tion
formula
which
incorporates
these
limiting
forms
by giving
a(°) and
a(') the
weights
1 - ad
and ad:
aw
(1 - ad)a(°)
+ ada(l),
or
(11)
ad
~
~~ 1-R(n,1n
1
)
(I-d
-
+ ad-
ad
(1
ad)
-R(nl)
Because
R(n)
is
a monotonically
increasing
function
of
n (for
n > 1),
R(nr/nli)
<
R(nr)
for ni
< nr, and
so
aw > ad,
with the
greatest
percentage
increase
occurring
at
low
absorption.
For small
ad and
ni = 4/3
(water),
the
ratio
of aw/ad
takes
the values
1.07,
1.08, and
1.10 for
nr
= 1.5, 2,
and 2.5
(most minerals
have
refractive
indices
within
this range).
When we
put a,,
as defined
by
(11),
for a in
Eq. (1),
we can
find the
ratio of
the total
absorption
by a wetted
surface
to the
absorption
by the
dry surface,
A/ad. This
provides
one
measure
of how
much
darker
a given
surface
becomes
when
wetted.
1 April
1988 / Vol.
27, No.
7 / APPLIED
OPTICS
1279