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Michigan State
University

b-‘F

’ 13“ C (w

  

 

   

This is to certify that the

thesis entitled

A LIGHT BEATING STUDY OF THE THERMAL
EXCITATIONS OF A BILIPID MEMBRANE

presented by

Edward Felix Grabowski

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Phy31cs

 

 

 

Major professor

3A m

November 7, 1977

 

Date

0-7639

A LIGHT BEATING STUDY OF THE
THERMAL EXCITATIONS OF
A BILIPID MEMBRANE

BY

Edward Felix Grabowski

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics

1977

(’J

ABSTRACT
A LIGHT BEATING STUDY OF THE

THERMAL EXCITATIONS OF
A BILIPID MEMBRANE

BY
Edward Felix Grabowski

Random thermal motion excites propagating
capillary waves (Ripplons) on the surface of bilayer
lipid membranes. The dispersion law for excitations of
an oxidized cholesterol membrane was measured using light
beating spectroscopy. The dependence of w on q is
consistent with a model of a fluid film of interfacial
tension 0 surrounded by an aqueous medium having den-

3 and viscosity n = 1.0 x 10-2 P.

sity p = 1 gm cm-
The interfacial tension was measured for cholesterol in
two different states of oxidation giving values of
o = 2.5 dyn cm"1 and o = 1.9 dyn cm-l.
This simple model adequately describes both the
dispersion law and the shape of the power spectral den-
sity curve when the effects of imperfect sample and in-
strument broadening are included. Experimental methods

are presented for:

1. making 0.6 cm diameter membranes.

Edward Felix Grabowski

detecting the scattered light using heterodyne
detection.
accurately determining values of the wave

number q of the excitation.

This thesis is dedicated to my father,
my mother and the Kalamazoo Polish

American Society.

ii

ACKNOWLEDGMENTS

I would like to acknowledge the support and
guidance of Dr. J.A. Cowen during the course of this
research. I wish to thank Mr. Dan Edmunds for keeping
the computer running; Dr. Phil Gaubis for many fruitful
discussions and assistance with electronic design; and
Dr. H.T. Tien for providing the cholesterol samples used
in this thesis. I would also like to thank the men in
the machine shop for their good humor and craftmanship.
Finally, I would like to thank my wife, Mary, for her

support and encouragement.

iii

Chapter
I

II

III

IV

TABLE OF CONTENTS

INTRODUCTION
LIGHT SCATTERING

Geometrical Considerations
Derivation of Scattering Formula
Average Ripplon Amplitude

Final Form of the Scattering Formula

LIGHT BEATING SPECTROSCOPY

Ripplons on Water
Necessary Resolving Power
Light Beating

Coherence Considerations
Experimental Approach

MEMBRANES

“ Physical Description

Membrane Formation
Size Effects

An Observation
Holder Technology
Optical Properties

EXPERIMENT

Membrane Formation

Noise

Illumination
Determination of q
Surface Variation
Photodetector

Noise Filter

Spectrum Analysis

Summary of Experiment
Technique for Better q Resolution
Local Oscillator Strength
Locating Sources of Noise

iv

12
18
21

22

22
23
24
26
26

28

28
29
32
33
34
38

39

39
40
41
43
45
46
48
49

51
52
54

Chapter Page

VI DATA 56
The Signals 57
Date Acquisition 57
Confirmation of Ripplons 60
VII HYDRODYNAMIC THEORY 65
Hydrodynamic Equations 65
Power Spectral Density 66
A Model 67
VIII COMPUTATIONS AND RESULTS 70
Physical Considerations 70
Approximation of Instrument Function 72
Comparison of Theory and Data 73
Other Membrane Materials 75
1x FUTURE DIRECTIONS 77
Better Membranes 77
More Data Points 77
Measuring q 78
Recording Signals 79
More Complex Model 79
Supression of q = 0 beam 80
Elimination of Destructive Inter-
ference 81_
Stimulated Scattering 81
Absence of Forward Scattering 82
x CONCLUSION 84
REFERENCES 87
APPENDICES
A High Pass Filter 90
B Computer Programs 98
C Correction for the Effects of the
Sample Cell on q 114

LIST OF TABLES

Listing for SPECGEN.FT
Listing for DSP20.FT
Subroutine Listings

Flow Chart for DSP20.FT

vi

Page
101
102
107

108

Figure

10

11.
12
13
14
15

16

17

18

LIST OF FIGURES

Geometry of a light scattering experi-
ment

Relationship between the projections of
the wave vectors onto the interface

The two possible scattering conditions
for ¢ = 0 and the corresponding
projections on the y axis
Relationship for reflected and re-
fracted rays and the n'th order
diffracted rays

Membrane holder cross section

Membrane holder edge profiles

Membrane holder and sample cell

Top View of equipment on table

Method of determining q

Power Spectral Density for three g
values

Data as a function of x

Data as a function of q

Theoretical fit to PSD curve
Dispersion law for aged cholesterol
Schematic Diagram for high pass filter

Functional diagram for high pass
filter

High pass filter frequency response

Top View of cell

vii

Page

10

14
30
31
37
42

44

58
61
62
74
76
91

92
96

115

CHAPTER I

INTRODUCTION

In 1908, M. Schmoluchowskil suggested that the
free surface of a liquid would be constantly disturbed
by thermal motion. In 1907, Rayleigh2 had calculated the
intensity of light scattered by static surface roughness
in terms of the square of the amplitude of the Fourier
coefficients describing the surface. In 1913, L.
Mandelstam3 computed the mean square amplitude of a
thermal fluctuation of wavelength A by computing
the increase in surface energy and applying the equi-
partition theorem. He then computed the intensity of
the light scattered from these fluctuations using
Rayleigh's results.

The mean square amplitude and frequency
spectrum of thermal excitations were computed by Levich4
while M. Papoular5 considered the spectra that should be
observed from these fluctuations in a light scattering
experiment. For high viscosity liquids, where there
are nonpropagating excitations, they predict a single
lorentzian component centered at the frequency of the
incident light. For low viscosity liquids with pro-
pagating modes which doppler shift the frequency of the

-l

incident light, they predict two lorentzian lines
symmetrically located about the frequency of the incident
light.

Katyl and Ingard6 observed the effects of surface
scattering from thermal excitations using a Fabry-Perot
interferometer but could not resolve spectral shapes.
Papoular's predictions were confirmed by Bouchait and
Meunier7 who were able to resolve the spectral lines
using light beating techniques.

The development of light beating spectroscopy
began in 1956 with experiments of Hanbury Brown and
Twiss8 who were seeking funds to build the stellar inter-
ferometer at Narrabri, Australia but were enmeshed in a
theoretical argument with members of the physics
community over the existence of time coherence between
photons emitted by a single source at different times.9
They not only demonstrated the existence of the correla-
tion on which the interferometer depends, but showed
that it was preserved in the process of photodetection.

In 1947, Forresterlo

had suggested that beat
frequencies should be observable between two light waves
with slightly different frequencies. He observed the
beat frequencies in 1955 using two Zeeman split lines of

11

the 5461 mercury line. He later proposed the light

beating technique as a spectroscopic tool and derived

expressions for the expected output of the phototube for

different spectral line shapes of the incident light.12

13 had observed beats be-

In 1961, Javin et a1
tween different intercavity modes of a single Helium
Neon Laser. The technique was also used to determine
the stability of a laser by measuring the frequency dif-
ference between two different HeNe lasers.14

Extension of the light beating technique into
other areas rapidly occurred. These included:

1) Measuring velocity profiles in flowing
fluids by studying the doppler shifts of
light scattered from 0.5 micron polystyrene

spheres suspended in the fluid.15

2) Measuring the Rayleigh linewidth of light
scattered from nonpropagating entropy
fluctuations in toluene.16

3) Studying the properties of the interface
between liquid SF6 and its vapor near a
critical point.17

4) Determination of the diameter of particles
sUspended in a liquid from the spectral
width of Rayleigh scattered light from the

particles.18 .

5) Studies of turbulence at high Reynolds

number.19

The light beating technique has become an every-
day tool for the study of flow and is one of the few
methods available to study liquid-vapor interfaces near
a critical point. The interested reader is directed to
several general references.20’21'22'23

The facilities for light beating spectroscopy
were developed at Michigan State University during 1973
and 1974. Preliminary studies of liquid surfaces, for
which results were known, were made to develop measuring
techniques. Light beating spectroscopy was applied to
study bilipid membranes for several reasons:

1. Although bilipid systems were and are of

much current interest, there are very few
measurements one can make on single membranes -
many are performed on lipid vesicles.

2. Professor H. T. Tien of the MSU Biophysics
Department was willing to discuss and demon-
strate techniques for the preparation of
bilipid membranes The cholesterol samples
used for this thesis were provided by
Professor Tien.

3. The lipid molecules in the membrane undergo
a phase transition._ Engelman24 has used
x-ray diffraction techniques to show the
phase transition corresponds to a change in

the structure of the hydrocarbon chains in

the lipid molecules. It was hoped that
information about phase transitions could
be obtained.

4. Unlike many "biological systems", repro-
ducible membranes can be made from the same
stock solution.

5. A physical system 100 Angstroms thick is
very difficult to probe. The light
scattering experiments presented the
possibility of studying membrane properties,
such as interfacial tenSion, under equilib-
rium conditions.

6. The bilipid membrane, sometimes referred to
as black lipid membrane, is only about 100
Angstroms thick so light reflected from the
front and back surfaces undergoes almost
total cancellation. Since the signal in-
tensity in a heterodyne light beating
experiment depends on the product of the
scattered light intensity and local
oscillator intensity, it is an ideal techni-
que for detecting the light scattered by
thermal excitations of the membrane surface.

It was hoped that using light beating spectro-

scopy to determine the dispersion law of the thermal

excitations on the bilipid membrane surface would give

information about both the interfacial tension and the
viscosity and thus would lead to some new information
about the phase transition.

There is one interesting comment to offer in
passing. The bilipid membrane is an interesting physical
system but it is often asked if any of the measurements
being made actually have any biological significance.
Melchior et a1,25 using highly sensitive differential
calorimetry techniques, have studied phase transitions
in the lipid material of liVing organisms grown at dif-
ferent temperatures. It was found that the temperature
interval over which the rather broad phase transition
occurred contained the temperature at which the organisms
were grown. The lipid phase transition appears to be a

property of living cells.

CHAPTER II

LIGHT SCATTERING

Thermal motion produces fluctuations on a fluid
interface that propagate as small amplitude capillary
waves called ripplons. These waves will scatter an
incident laser beam, doppler shifting the frequency of
the light. Let us consider the geometry of the experi-
ment and derive an expression for the intensity of the

scattered light.

Geometrical Considerations

 

Consider the geometry for a typical light
scattering experiment, Figure l . The interface to be
studied lies in the X-Y plane. Light, of wavelength
A, with wave number ki (k1 = 2n/l) is incident on the
interface in the Y-Z plane making an angle 91 with
respect to the Z axis. The direction of the scattered
light, with wave vector Hg, is specified by the angles
0 and :9. The interaction takes place in the x-Y
plane. Conservation of wave vector, k, in this plane
_implies that the projection of the incident and scattered
wave_vectors (on the X-Y plane) must be connected by
q, the wave vector of the excitation, Figure 2 .

7

 

 

z
I \
I
I ~ —.
| RI k, l
' I
l I
I 9I as :
' I
' I
: I
Y I
k; 8"“09 ¢ :
I
. l
010
X (‘31 :

Figure 1. Geometry of a light scattering experiment.

k;sin(0,

 

)
4’ q
kssin (as)

Figure 2. Relationship between the projections of the
wave vectors onto the interface.

Applying the law of cosines to the triangle in

Figure 2 gives:

2 _ 2 . 2 2 . 2 _
q - ki Sln 9i + k8 Sin 6i

2 kiks Sln 6i Sln 68 cos w (l)

The ripplon wavelength is much larger than the wave-
length of the incident light so there is only a small
difference between ki and ks. Let '9 = 0 and

ki z ks; equation (1) becomes, to first order:

2 2 . . 2
q — ki (Sin 6s - Sin 6i) (2)
For 65 = 9i + 66 one has:
Sln as = Sln 6i cos 66 + cos 6i Sln 66 (3)

The assumption ki z kS is equivalent to the condition:

59 << 1,

sin 66 = 66. cos 66 = 1

I Equation (2) becomes:

2_2 2 2
q - ki (cos 61) (66)

q = |ki cos(61) 66] (4)

Figure 3 shows two possible conditions. The incident
wave vector and a can add to scatter light at an angle

61 + 66. This corresponds to creating a ripplon

10

 

 

 

 

It; sln‘Oi) > 9 a
k! sln( 9:) >
It; SIMOI)

>
k,sln(9,) q

 

><

Figure 3. The two possible scattering conditions for
¢ = 0 and the corresponding projections on
the Y axis.

11

propagating in the negative Y direction or annihilating
one propagating in the positive Y direction. The
second possibility is that the incident wave vector and

q add to scatter light at an angle 61 - 66. This
corresponds to creating a ripplon propagating in the
positive Y direction or annihilating one propagating

in the negative Y direction. Conservation of energy

implies

hwi : m = hws (5)

where h is Plank's constant divided by 2n, mi and
ms the angular frequencies of the incident and scattered
light, and Q the angular frequency of the ripplon.

The ripplon of wave number q represents a

surface displacement given by:
Z = A cos (qY) cos (Ht) (6)

This ripplon can be thought of as a sinusoidal
diffraction grating that is propagating along the sur-
face. Equation (2) is equivalent to the grating equa-
tion and equation (5) is equivalent to the expression
for the doppler shift in the frequency of light dif-

fracted from a translating grating.

12

Derivation of Scattering Formula

Rayleigh first derived the expression for the
intensity of a sound wave scattered from a surface
variation of periodicity L and then transformed it
into an expression for light waves.2 Let us rederive
the expression for the scattering of plane electro-
magnetic waves from spatially periodic variations in the
height of a horizontal surface. Let the equation of the
surface be Z = £(x), periodic in x with period L and
uniform in the Y direction with average value zero.

The surface can be Fourier decomposed into:

€(x) a1 cos(px) + b1 sin(px) + a2 cos(2px) +

b2 sin(2px) +... (7)

2 C empx
m

m=-w

C = 1/2 (am - lbm) for m > 0

= 1/2 (am + lbm) for m < 0

= 0 for m = 0

f-T

p = 2n/L, i

The wave may also have a time dependence emt but
since we do no operations involving time, this term will
be common to all expressions and will not be written

out explicitly.

13

Consider the geometry of Figure 4 . An incident

wave E, with unit amplitude, is given by:

E exp(iE-I)

E

exp[i k(Z cos 6 + x sin 6)] (8)

where exp(x) is the exponential function, k is 2n/l
where A is the wavelength of light in medium 1, and

i = 7- . The reflected wave E0 is given by:

E0 = A0 exp [i k(-Z cos 6 + x sin 6)] (9)

where A0 is a reflection coefficient. There will be
waves diffracted from the surface at angles 6n given

by the diffraction formula:

L sin 6n - L sin 6 = n l (10)
The equations for these waves are given by:
En = An exp [i k(-Z cos 6n + x sin 6n)] (11)

There is also a refracted wave at an angle ‘9 'given by

Snell's law:
k sin 6 = k' sintp ' (12)

where k' is the wave number of the light in medium 2,

k' = Zn/A'. The equation for this wave is:

E0' = Bo exp [i k'(Z cosco + x sin:p)]' (13)

l4

a-lnl

\

 

4

TIM

 

¢-—ln|

Figure 4. Relationship for reflected and refracted
light rays and the n'th order diffracted
rays.

15

The diffracted waves in medium 2 come out at angles (on

given by the diffraction grating equation:

L sin(pn P L sinrp = n 1' (14)
and the waves corresponding to these angles are given by:
En' = Bn exp [i k'(Z COqun + x sin(pn)] (15)

Using the grating equations (10) and (14), one replaces

the terms sin 6 and sin.¢ with:
n n

sin 6n sin 6 + np/k

sin @n - sincp +-np/k'
These equations and Snell's law, equation (12), imply:

E = a exp (ikZ cos 6)

E0 = a Ao exp (-IkZ cos 6)

En = a An exp (-IkZ cos 6n) exp (Inxp)
E0' = a B0 exp (ik' Z cos q)
I _ ’ I ‘
En - a Bn exp (1k Z coscpn) exp (Inxp)

a = exp (ik x sin 6)

Specific boundary conditions must now be applied. Let
the electric field be polarized perpendicular to the
plane of incidence. At the interface the following

equations must hold26:

.. I =
E + 2 En 2 En 0
n=—oo n=-—oo
k(E - 2 En) cos - k' ( Z En')COS(p = 0
nz—oo n=-°°

Divide both equations by the common term a,

express the exponential functions in a series expansion
and replace Z by the Fourier series representing the
surface. The functions exp (inpx) form a complete
set of orthogonal functions so the coefficients of each
term, exp (inpx) must be zero.

Consider the first boundary condition. From the

constant term (n = 0) one obtains:
l + A = B , (16)
From the non zero n terms one obtains:
- =.° \ — ' ,
An Bn 1Cn[k(Ao 1) cos 6 + k Bo cos p] (17)
The second boundary condition gives:
- = I .-
k (1 A0) cos 6 k Bo cos p (18)
k A cos 6 + k'B cos = i C [k2(l + a ) c0326 -
n n n ‘pn n o
2 2
k' Bo cos cp] - (19)

Equations (17) and (18) give:

A = B' (20)

17
Equations (16) and (18) give:

B0 = 2k cos 6 / (k cos 6 + k' COScp) (21)

Using (16) and (20) in (19):

i B C [k2 c0326 - k'2 cos2 T)

_ on
An _ E cos 6n + k' COSQpn (22)

 

Using (12) and (11) in (22):

21k Cn cos 6 sin (3 --6)

 

n sincp cos 6n + COSfpn STE 6

 

An = 21k Cn f(6, en) (23)
Where:
f(6, 61) = cos 6 Sln (9 - 6) (24)

in cos + s. sin
3 r9 On CD ‘pn 9

where f depends only on 6 and 6n since '0 and ‘Dn
can be expressed in terms of 6 and en. One sees that
the scattering coefficient for the direction 6n depends
only on the amplitude of the n'th Fourier component of

the surface. Note that equation (10) can be written as
Sln 6n - Sln 6 = : A/Ln (25)

where Ln = L/n, the wavelength of the n'th Fourier com-
ponent. Each Fourier component acts as a sinusoidal
diffraction grating scattering light only into the first

order in the direction given by the grating equation (25).

18

The intensity in the n'th beam is given by:

2 _ 2 2 2
| — 4k cn f (e,en)

H
II

[A
n

_ 2 2 2 2
I — k (an + bn )f (e,en) (26)

Average Ripplon Amplitude

Mandelstam3 considered the problem of light

 

scattered from a square surface element with sides of
length 2L' using the geometry of Figure 1 . The sur-
face was decomposed with a two dimensional Fourier trans-

form into terms like:

_ ZflBX 2n
EBY - CBY COS(-2—L—.—) COS(—§%¥) (27)

The diffraction grating equation gives:

° , ;— ' =fl_
Sln eBY cos ”BY Sin 6i 2L' (28)

sin 6 sin

._ BA

For light scattered in the Y-Z plane,cp = 0 and

B = 0. Using equation (26) the intensity of the scattered

light is:

_ 2 2 2
IO'Y - k COY f (ei'eOY)

Let B and y go over to continuous variables. The
fraction of the incident intensity scattered by the 7

component of Z, dIy, becomes

19

_ 2 2 2
dIY - k cOY f (91,90Y) dde (30)

From equations (28) and (29), we obtain to first order

in asap:-

dB dy = (ELL)2 sin 9 cos 9 d6 do (31)
A Oy 0y

Equation (30) becomes:

2 4 2
I
L R Co

2
cs 6 f 6. 6 d0
d1 = Y C OY ( ll )
Y

OY

 

2
n

where d9 = sin eOYde dm. ~To evaluate C0y2’ compute the
change in surface energy due to the excitation. In the

' Y-Z plane the surface is given by:
E = CY cos (2nyY/2L')

An element of length along this line is given by:

 

dL' =\/hY2 + dz2

=\/& + (3%)2 dY

The derivative is small so expand the square root and

 

integrate over the Y axis

L' n . 11X 2
f_L. {1 + 1/2I{-.- cY sm( Ln] } dY

Length

2L' + 1/2 qZCY2

Ll

where q = Zny/L', the wave number of the ripplon.

20

The increase in surface area is just 2L' times

the increase in length 1/2 q2CY2L'. Neglecting gravity,

the increase in surface energy is:

= O qzc 2 L,2

where o is the surface tension. Equating this energy

with 1/2 kBT, where. kB is Boltzman's constant and T

the absolute temperature, gives:

k T
c 2 = __§___§ (32)

Y ZquL'
The equation for the scattered intensity becomes:

4
d1 - kBT R

5'52"-

cos 6 £2(ei.e)

ano q2

 

where the index y has been dropped. This is the in-
tensity integrated over all frequencies. The excitations
' will have some frequency spectrum specified by the

normalized power spectral density function S(q,m):

I (q,w) = Iq S(qrw)

with
I: sum) den = 1

This power spectral density will come from a more de-

tailed hydrodynamic study of the interface.

21

Final Form of the Scattering Formula
The final form for the intensity of light
scattered from a ripplon of wave number q at an angular

frequency w is:

2 2
k T k cos(6)f (6..6) S(q.w)
dI (g,w) = B 1 (33)

d9 2n2 o 02

J

This expression was derived with many assumptions that
were chosen to correspond to the actual experimental
situation whenever possible. With the addition of
viscosity terms, the ripplon would not have constant
amplitude and the expression for the surface energy
would change. The qualitative features of the expression
do not change, however. The scattering depends on the
inverse fourth power of the wavelength of the incident
light. Shorter wavelengths should give larger signals.
The scattering intensity falls off as the square of the
wave number of the ripplon so only small q scattering
may be considered. The scattering intensity depends on
the inverse of the surface tension so interfaces with

small surface tension should give more signal.

CHAPTER III

LIGHT BEATING SPECTROSCOPY

Before attempting the study of thermal excitations
on a membrane surface with light scattering, one should
recall what is known about similar physical systems and
decide what sort of apparatus will be needed to observe
the scattered light. The surface of a simple liquid,

like water, is a good place to start.

Ripplons on Water

The dispersion law, the relationship between the
angular frequency 0 and wave number q, for ripplons
was derived for the free liquid surface by Levich4 using
linearized hydrodynamic theory. The ripplons have
finite lifetimes and arise from random thermal fluctua-
tions so the ripplon frequency spectrum will be cen-
tered at 0(q), given by the dispersion law, with some
finite width. The details of the spectral shape were
worked out by Papoular5 for the interface between two
fluids and for the free surface of a fluid. At the free
surface of water, the spectrum consists of a pair of
lorentzian shaped lines centered at frequencies mo : Qq
where w is the angular frequency of the incident light

0

22

23

and Hg, the angular frequency of the ripplon, is given
by

0 3
0 = —

q p q
with o the surface tension and p the density of the

1

water. For water with o = 72 dyn cm- and

p = 1.0 qm cm?% a q value of 600 cm.1 corresponds to

a frequency shift of 20,000 Hertz. The half width of

these lines, AG, is given by:
_ 2
A9 — an /p

where n is the viscosity of the water. For water with

2 1

n = 1.01 x 10‘ poise at a q of 600 cm- , the width

is 1100 Hertz.

Necessary Resolving Power

 

The incident light has a frequency on the order

15

of 10 Hertz so the resolving power needed to observe

3 15 or one part in 1012.

the spectral shape is 10 in 10
Since the amount of light scattered depends on l/qz,
one would like to take data at small q values where
even more resolving power is needed.

It is interesting to compute the physical
dimensions of a Michelson interferometer needed to re-

13

solve one part in 10 in a fast Fourier transform ex-

periment. The frequency separation of the analyzed

24

spectrum, Av, is given by:27

Av = c/d

where c is the speed of light and d is the change in
the Optical path length of one arm of the interferometer.

For an incident frequency of 1015

, one needs a path
length change of 3 x 106 meters.

Katyl and Ingard6 were able to detect the light
scattered from ripplons with large q values using a
Fabry-Perot etalon that could resolve a part in 108.
Bouchait and Meunier7 used the technique of light beat-
ing spectroscopy which can resolve a part in 1013 to
obtain detailed experimental information about the

spectral shape derived by Papoular.

Light Beating

 

The technique of light beating, sometimes called
optical mixing, is the optical analog of a heterodyne
radio receiver. Consider a signal which differs from

the 1015

Hertz incident light frequency by an amount

6v, directed to the active surface of a photomultiplier
tube (PMT) along with a "local oscillator", some of the
incident light. The PMT is a square law device which
responds to the square of the incident field. If the
signal and local oscillator light are coherent, one adds

the two electric fields and squares to obtain the photo-

_ current. There will be terms that oscillate at

25

frequencies which are the sum and difference of the
frequencies of the incident light beams. Using an audio
spectrum analyzer on the photocurrent allows the detec-
tion of 6v's in the 0 to 20,000 Hertz range. The photo-
current arising when the signal of interest has a power

12 When the

spectrum Ps(w) can be rigorously derived.
local oscillator is much more intense than the signal,

the power spectrum of the photocurrent, Pi(w), is given
20

 

by:
e<i > <i ><i >
P<w> = —2§£2— + <iLo>2 6(w) + L2“ 8 Ps(w)

where e is the charge on an electron; <iLo> the average
current that would arise from just the local oscillator;
<is> the average current that would arise from just the
signal and 6(m) the Dirac delta function Of w.

The first term is the constant shot noise that
arises from the flow of current. The second is the D.C.
level of the photocurrent and the third term is the light
beating signal which depends on the product of the in-
tensities of the signal and local oscillator and has the
same functional form as the power spectral density of the
scattered light wave. This power spectral density in'

turn has the same functional form as the power spectral

density of the fluctuations of the surface.

26

Coherence Considerations

This analysis of the photocurrent requires the
local oscillator and signal to be coherent across the
illuminated surface of the PMT. An idea of the area
allowed on the photocathode comes from diffraction theory.
The reflected light from a spot of diameter d will

spread into an area A at a distance D given by:

A: (.221)202

Since the light in this area is the superposition of
light reflected from each element of the surface of.dia-
meter d, the light from all those surface elements must
have maintained coherence over the area A. Forrester12
cites the results of a more formal calculation involving
the correlation between light emitted from different
surface elements and finds the light will be coherent over

an area A given by:

2
A= (g-Dfl; -) =$—

s
where as is the solid angle subtended by the source at

the detector.

Experimental Approach

 

The experimental approach for this thesis was to
use the heterodyne detection scheme with an audio spectrum

analyzer to obtain the poWer spectral density P(w) for

27

ripplons of different wave numbers q on
bilipid membranes. This will be referred
The dispersion law obtained from P(q,w)

with the various theories and, hopefully,

information about the physical parameters

the surface of
to as P(q,m).
was compared

used to obtain

of the membrane.

CHAPTER IV

MEMBRANES

The bilipid membrane (BLM) consists of a bimol-
ecular layer of lipid molecules formed in an aperture in
a thin holder. These membranes serve as model cell walls
in many biological experiments but will be considered in

this thesis as only an interesting physical system.

Physical Description

 

The cholesterol molecule has a polar head and a
long carbon chain tail. A thin film of cholesterol
molecules can be formed by dissolving some oxidized
cholesterol in normal octane and floating several drops
,Of this solution on the surface of a beaker of water
(a polar solvent). As a thin teflon sheet with a hole
in the center is dipped down through the cholesterol
solution, a thin film is drawn across the hole. If the
cholesterol was properly oxidized, this film will
spontaneously thin to two parallel monolayers of choles-
terol molecules with the polar heads oriented outward
into the polar solvent and the carbon tails pointing

inward.

28

29

The method for properly oxidizing the cholesterol

28 The bilayer is about 100 Angstroms

is given by Tien.
thick across the membrane surface and expands at the
edge into the Plateau-Gibbs border (P-G border), Figure
5 . Excess cholesterol solution is contained in this
border and forms a reservoir from which new surface area
is generated if the membrane is stressed. Applying
pressure to one side of the membrane will cause the
membrane to expand outward like a soap bubble, the in-

crease in surface area will come from the creation of

new surface rather than a stretching of the old surface.

Membrane Formation

 

The membrane holders were thin sheets of teflon
or polycarbonate with an aperture. The edge of the
aperture was given various shapes, Figure 6 . The first
membranes were made on polished polycarbonate forms with
'profile b. The membrane holder was immersed in a
rectangular cell containing a 0.1 molar KCl solution.
Using 0.1 cm diameter holes, the cholesterol solution
was brushed across the hole with a fine camel hair brush.
This technique allowed one to make 5 to 10 membranes
before cleaning of the cell was required. The dip
method requires a cleaning of the sample cell every time
as the octane evaporates from the surface layer and

small particles of aggregated cholesterol are created

    

3????3?
ééééééé

 
   
 

 

 

 

 

 

 

E
O
m .
O
' e 5 holder-cross section.

31

 

 

 

 

 

 

 

\/

 

 

 

 

 

 

 

 

 

 

 

/\
3 V

 

 

 

 

 

Figure 6. Membrane holder edge profiles.

 

32

that get into the membranes causing early rupture.

These membranes were stable, lasting several hours, but
one could not obtain a collimated specular reflected beam
from their surface with a collimated incident beam. The
next step was to make larger membranes with more surface
area. Membrane holders with 2 mm diameter holes and pro-
‘file b did not make stable membranes. They ruptured
after several minutes. The basic problem with membrane
lifetime is the diffusion of border material into the
surrounding water. When the border is gone, the membrane
ruptures. A simple hole, profile a, Figure 6 , gave
longer lifetimes but the membranes were not forced to
orient themselves in a single plane. In an experiment,
the plane of the membrane holder is vertical. The addi-
tion of some curvature to the edge, profile c, caused the
membranes to orient vertically, Profile d, Figure 6 ,
also forces the membrane to orient vertically and traps
more of the border material. Holes with this edge pro-
file (profile d) allowed 0.6 cm diameter membranes that

lasted for many hours.

Size Effects

 

The large membranes form differently than the
smaller ones. Cholesterol films on 1 mm holes take 1
to 5 minutes to thin to several wavelengths of visible
light. Bands of color appear from the interference of

light reflected from the front and back surface of the

33

membrane. In another 2 to 10 minutes black spots appear
near the edge and expand until the entire membrane looks
black. There are usually 3 or 4 places where the black
spots start. The spreading takes place quickly with the
entire membrane going black in 10 to 60 seconds. The
entire membrane surface thins at the same time, the
colored bands becoming very wide just before the black
spots appear. Tien shows photographs of the thinning
process.29

The 0.6 cm membranes have the same initial
thinning and appearance of colored bands but the first
black spot always appears at the very bottom of the mem-
brane (the octane is less dense than the water so the
film drains up). This black spot slowly grows but there
are many colored bands in the rest of the membrane show-
ing it has not uniformly thinned. The black spot expands
but unlike the expanding black regions in the small mem-
branes, which remain circular, the transition between
black membrane and thick membrane is a straight horizontal
line. The transition line slowly rises as more of the
membrane becomes black, taking 10 to 30 minutes to reach

the top.

An Observation

 

An interesting observation can be made by focussing
a 1 mm diameter beam from a 5 milliwatt Helium Neon laser

onto the thinning surface with a 15 cm focal length lens

34

and observing the reflected beam. As the border between
the colored and black regions passes through the beam,
one can see interference bands in the light reflected
from the thick portion. These bands are parallel to the
line of transition and uniformly spaced. As many as 15
of these lines have been observed indicating that at the
transition from thick membrane to bilayer, the membrane
thins uniformly.

One can actually see the surface by directing
collimated white light or an expanded laser beam at near
normal incidence and looking at the specularly reflected
light. The surface is never flat but the curvature in
the large membranes was small enough that a spot could
always be found that would reflect a 0.2 mm diameter
laser beam. The article by Pagano3o has a picture

illustrating this curvature.

Holder Technology

 

The teflon holders were made_from 1 inch diameter
teflon rod. The rod is held in a lathe and a 3 to 5 mm
hole is drilled down the center. The face of the rod is
machined smooth and the edge of the hole is cut with a
regular lathe bit ground to the shape of the desired
edge profile. -A 2 mm thick disk is then cut from the end
of the rod with a sharp cutoff tool. Any roughness is

polished out using plastic polish31 on Q--tips® cotton

35

swabs that are attached to a motor which rotates at
several hundred RPM. This technique of cutting thin
disks from a solid rod works well. The rod adds con-
siderable structural rigidity while the edge profile is
being cut and gives a uniform edge about the entire
periphery of the holder.

The polycarbonate holders were made from 1/16
inch sheets of Lexan ®.32 It is difficult to cut a
uniform edge profile in these thin sheets but the follow-
ing technique works well. The polycarbonate sheet is
attached to a block of aluminum with double sided carpet
tape (glue on both sides). The aluminum block is mounted
on a milling machine with the polycarbonate sheet 3
horizontal in a rotating holder that allows one to rotate
the block about a vertical axis. A new l/16 inch dia-
meter straight milling tool is used to cut the hole. The
axis of rotation of the table is displaced from the axis
of rotation of the milling tool by the radius of the
hole less 1/32 inch. The cutting tool is lowered into
the polycarbonate sheet. The sheet is then rotated,
producing the hole. One must go slowly as polycarbonate
has low thermal conductivity and is easily melted by
heat generated by cutting too fast. The rigid aluminum
block holds the polycarbonate sheet firmly in place giving
a uniform hole. The milling tool is then replaced with

a ball shaped cutting tool of the kind made by the Dremel

36

Corporation, available in any hobby shop. These come in
many diameters and a 3/32 inch diameter tool was used
with the 1/16 inch polycarbonate. The tool is carefully
centered in the vertical direction and the axis of table
rotation is displaced until the tool is cutting into the
edge of the sheet. Rotate the holder to cut the edge
profile into the hole. Several small cuts are better
than one large cut. The edge will not be smooth but can
be polished with jeweler's rouge or plastic polish on
the rotating Q-tip. This technique should work with
teflon but has not been tried.

It is essential to have a uniform, highly
polished border on the membrane holder. The membranes
tend to stick to any rough spot. Several hours of
polishing are needed. Visual inspection with a 10 power
eyepiece is helpful in determining prOgress.

The membrane holder is cut down to a l/2 by 3/4
inch rectangle with the hole centered. It is fastened to
a 1/4 inch diameter teflon rod with a 2-56 nylon screw,
Figure 7 . The teflon rod fits snugly in a hole in the
plexiglass top of the sample cell and allows one to rotate
the plane of the membrane relative to the walls of the
cell so that light reflected frOm the membrane does not
go in the same direction as that reflected from one of
' the cell walls. The sample cell is a rectangular 1 cm

by 2 cm quartz cell 5 cm high. The cell walls are optical

37

 

 

 

 

 

 

F_“‘fi

I o I

L____J

fi I I 1

I I I r
_

 

 

Holder

C) 01

 

 

 

Cell

H"

Cfll

 

 

 

 

 

 

 

.__l

 

Figure 7. Membrane holder and sample cell.

38

quality quartz. The entire sample cell is mounted on a
table which can be rotated about two orthogonal hori-
zontal axes. This allows one to shift the plane of the
membrane to direct light reflected frOm the surface into
a microscope used to visually examine the membrane sur-
face. The membrane is illuminated with white light from
a microsc0pe illuminator. The adjustable table is
essential as the membrane can only be seen on reflection.
If the membrane is not flat, it must be moved about to

view different parts of the surface.

Optical Properties

 

The membrane has a Brewster angle. For an angle
of incidence of about 45 degrees, laser light polarized
perpendicular to the plane of incidence is reflected but
light polarized in the plane of incidence is not reflected.
Since the tangent of the Brewster angle is the ratio of
the index of refraction of the membrane to the index of
the water, the value of about 45 degrees implies the
membrane-water interface behaves like the interface be-
tween media with almost identical indices of refraction.
Two optical properties were verified:

l. The intensity of the light reflected from

the surface of the membrane increases as
[the angle of incidence is increased.

2. The fraction of light reflected from the

membrane surface is greater for light of

shorter wavelength.

CHAPTER V

EXPERIMENT

The experiment consisted of measuring the power
spectral density of the light scattered from thermal
excitations, ripplons, of the surface of bilipid mem-

branes made with oxidized cholesterol.

Membrane Formation

 

The experimental data used in this thesis were
taken from oxidized cholesterol membranes formed on a
teflon holder with edge profile d, Figure 6 . The dia-
meter of the hole was 0.6 centimeters. The membranes
were surrounded by a 0.1 molar solution of KCl in water.
The laser light can be Rayleigh scattered by particles
suspended in the liquid so this solution was passed
through a 0.5 micron Millipore33 filter. The membrane
holder was suspended in the quartz sample cell as des-
cribed in the section on membranes. The membranes were
formed by placing two drops of cholesterol-octane solu-
tion on the surface of the KCl solution and slowly lower-
ing the teflon form through this layer, drawing a film
of cholesterol solution across the hole. The film is

inspected with a 10 power microscope. One must be very

39

40

careful to keep small bubbles out of the edge of the
membrane. The membrane will form around a bubble but
will not last long. One also checks to make sure no
particles of dried cholesterol are in the membrane sur-
face. Cleaning the cell well after each membrane avoids
this problem but one ends up doing a lot of cleaning.
The teflon and plastic parts were cleaned by placing
them in a beaker of methanol and placing the beaker in
an ultrasonic cleaner for 5 minutes. The quartz cell is
washed repeatedly with methanol.

If any bubbles or foreign particles were present,
the cell was cleaned and another membrane made.

The membrane thins to blackness in about 1/2
hour as described in the membrane section. The membrane
surface tension is about 2 dyne/cm and great care must
be taken to isolate the membrane from mechanical and

acoustical noise.

Noise
The experimental apparatus is mounted on a 2,000
Kilogram granite table suspended on 4 commercial vibra-

tion isolation supports.34

This system forms a mechanical
low pass filter with a l Hertz corner frequency that
effectively isolates the experiment from the 25 Hertz
resonant vibrations of the building.

The membrane can also be driven by acoustic noise.

All noise generating equipment (such as oscilliscopes

41

with cooling fans) is mounted in an enclosed equipment

rack lined with sound absorbing foam.

Illumination

The beam from a Coherent Radiation CR500K Krypton
laser, mounted on one edge of the granite table, is
reflected twice at 90 degree angles by mirrors M2 and M3,
Figure 8 and directed back up the table. The 1 mm
diameter laser beam is passed through a spatial filter
consisting of a 10 cm focal length lens L1 and a 50
micron diameter pinhole. The light is then focused
with a second 10 cm focal length lens, L2. The beam is
directed into the sample holder with a movable mirror,
Ml, which allows one to illuminate any spot on the mem-
brane. The size of the beam on the membrane is adjusted
with the focussing lens L2.

The beam is moved around on the membrane surface
until a reflected beam of light is observed. A well
defined reflected beam could only be obtained for laser
beam diameters of 0.02 cm or less and then only from
selected spots on the surface. This difficulty in ob-
taining good reflections for large laser beam diameters
is not unique to the cholesterol membrane. A 50-50 solu-
tion of liquid dish soap and glycerine will make black
films in air that last several hours. The same difficulty
in obtaining good reflections were observed in these

films.

Figure 8. Top view of equipment on table.

42

m2

HHOO oaaemm

macaque

 

82m IFII
6.3. 62.na

 

HA maoccwm NA

fl _ _

 

 

 

 

 

 

 

 

 

 

 

:. L

 

 

 

 

 

Momma covemum Moommo pawnmsoo

 

 

 

43

Once a flat spot is found the experiment begins.
The specular reflected beam corresponds to light
scattered from ripplons with a wave number q of zero.
The light scattered in the plane of incidence by ripplons
of non zero q comes out at an angle 66, relative to

the specular q = 0 beam, given by:
66 = q lo/Zn cos 91

where 10 is the wavelength of the laser light and 9i

is the angle of incidence of the light on the membrane.

Determination of q

 

Figure 9 shows the method used to determine
the direction of the scattered light and q. A pinhole
of diameter d is placed a distance D from the sur-
face and translated in the plane of incidence, perpen-
dicular to the reflected beam, a small distance x. The

angle 66 is:

66 = x/D

and q can be determined from:
(%1) 66 cos (6.)
o I

Note that an angle 66 on either side of the specular
reflected beam, which corresponds to q = 0 scattering,
corresponds to a ripplon with wave number q. Data
taken at the same angle on opposite sides of the q = 0

beam should be identical.

44

\‘24< d

9+8 9

 

Figure 9. Method of determining q.

45

The pinhole is mounted on a precision transla-
tion stage with a micrometer drive to measure the dis-
tance x. The pinhole moves parallel to the surface of
the granite table. The laser beam is initially adjusted
to lie in a horizontal plane parallel to the table top.
The pinhole is placed at this same level and the sample
cell positioned so the laser beam strikes a flat spot
on the membrane giving a good specular reflection. The
membrane holder is tipped until this reflected beam
strikes the pinhole. This can be done when the membrane
has just been formed as there is still a lot of excess
solution in the border and the membrane is hard to break.

In this configuration, the incident and reflected
beam lie in the same horizontal plane, the plane of
incidence. The surface normal for the part of the mem-
brane surface reflecting the light must also lie in this
plane.

Small corrections are sometimes required for the
effects of the sample cell. These corrections are dis-

’cussed in Appendix C.

Surface Variation

 

The membrane is initially constrained by the
holder to lie in a vertical plane but it is often
necessary to tip the sample holder by as much as 10

degrees to have the reflected beam come out in the

46

horizontal plane. This difference between the normal to
the plane of the membrane and the local surface normal
causes considerable experimental difficulty as the direc-
tion of the local surface normal shifts with time. The
reflected beam undergoes sudden shifts in direction of
S to 10 degrees on a time scale of 15 minutes to 2 hours.
These shifts are rapid reorientations of the surface.
rather than gradual drifts. It is easy to tell when this
happens because a shift in direction of more than 1 de-
gree corresponds to a wave number q of more than 2,000
cm"1 for which no signals have been observed. When the
signals go away the membrane has shifted or ruptured.
When the membrane shifts, one is tempted to move
the membrane holder to keep all the beams in the
horizontal plane. This, however, introduces enough
vibration to break membranes more than an hour old. The
reflected beam usually comes close to lying in the
horizontal plane so the pinhole is moved until it is
centered on the reflected beam. This often requires a
change in the height of the pinhole, shifting the plane
of incidence a little. By moving the pinhole instead
of the membrane, lifetimes were extended to periods of

up to 48 hours.

Photodetector

 

The requirement to frequently move the pinhole

placed restraints on the photodetector. In light

  
 

-. A . ' - ’9 _ u '
I I I I .
El,'vwfim /

Ju' . I n! J. 'n'
\nr—l _Z']I—Ir;9

 

47

scattering experiments on liquid surfaces, where surface
normals are well defined and constant in time, one
places the photomultiplier tube (PMT) behind the pinhole.
In the membrane experiments, the pinhole had to be moved
and moved quickly as one had to finish taking a set of
data before the membrane shifted again. In order to
save time, the pinhole was coupled to the PMT with a one
foot long, 1/8 inch diameter glass fiber optic light
pipe.35 This results in the loss of half the light
collected by the pinhole through attenuation in the glass
fibers and coupling loss at the ends of the pipe. It
does allow the PMT to remain fixed while the pinhole is
moved about.

The light gathered by the pinhole and guided to
the active surface of the PMT consists of the light in-
elastically scattered by the ripplon of wave number q
and light elastically scattered by imperfections in the
cell. This elastically scattered light serves as the
local oscillator for heterodyne detection. The resulting
photocurrent is passed through a high pass filter,
Appendix A, to a SAICOR 51A Real Time Analyzer/Digital
Integrator. The intensity of the light reaching the PMT
varies by several orders of magnitude so the gain of the
PMT must be controlled. Since the gain is a function of
the applied voltage, a variable power supply (0 to 2,000

volts) is used. The voltage to the PMT is slowly raised

48

until enough photocurrent flows to give a signal to the
SAICOR. The EMI 6094B PMT was run at voltages less than
1,400 volts. The PMT voltage is returned to zero when
not taking data as the PMT can be damaged by excess
current flow if the pinhole passes through the reflected

beam with more than 1,000 volts applied.

Noise Filter

 

One serious experimental problem was the presence
of low frequency (less than 1,000 Hertz) noise in the
spectrum, often more than an order of magnitude larger
than the signal of interest. In order to minimize the
noise with as little pertubation of the signal as possible,
a high pass filter with an adjustable corner frequency
was inserted before the SAICOR 51A. A filter using the
standard "state variable" design, found in any opera-

tional amplifier textbook,36

built by the Physics
Department Electronic Shop proved to be inadequate for
two reasons:
1) The filter actually attenuated some of the
high frequencies (above 10,000 Hertz).
2) The low frequency response was very broad.
As one decreased the frequency, the filter
started to fall off at 20 db per decade and
eventually fell off at 40 db per decade.

One could adjust the filter to eliminate the

low frequency noise but the attenuation

49

would extend into the frequency range
occupied by the signals of interest.

A high pass filter with a corner frequency ad-
justable from O to 2,000 Hertz was constructed, Appendix
A. The high frequency gain was unity out to 30,000
Hertz and the low frequency response fell off at 40 db
per decade. This filter is an essential part of the
experimental setup. The corner frequency is adjusted
until the low frequency noise is smaller than the signal
of interest but still observable. This keeps the corner
of the filter as far from the actual signal as possible.
With the 40 db per decade fall off, the corner frequency
was typically set at 500 Hertz. Since one rarely takes
data at frequencies below 1,000 Hertz, there was little

attenuation of the signals of interest.

Spectrum Analysis

 

The SAICOR 51A divides the frequency range from
0 to some maximum, fm Hertz, into equal intervals,
measures the amplitude of the signal in each interval,
and stores it in a 200 word storage register represent-
ing the real time spectral density. These 200 numbers
are then added to the contents of another 200 word
storage register that represents the averaged spectrum.
By repeatedly analyzing and adding, one generates a 200

point average spectral density. The values for the

50

maximum frequency fm are 20, 50, 200, 500, 1000, 2000,
10000 and 20000 Hertz. The advantage of this type
spectrum analyzer is that one can observe the spectrum
over the entire frequency range in real time. This real
time spectrum does have a lot of noise but it allows one
to see the approximate position of signal peaks and shows
signal to noise ratios. The SAICOR displays the real
time signal and summed signal while the averaged spectrum
is being taken. Completed spectra are recorded on an

X—Y recorder.

Summary of Experiment

To summarize the experiment:

1. Make a cholesterol membrane.

2.. Adjust the membrane so the incident laser
beam and the reflected beam all lie in a
horizontal plane.

3. Place the pinhole a distance D from the
membrane surface, centered on the reflected
beam. Define this position to be q = 0.

4. Mbve the pinhole a distance x perpendicular
to the reflected beam in the horizontal plan
of incidence. This corresponds to a

scattering angle 66 of:
66 = x/D

The light scattered in this direction comes

51
from ripplons of wave vector q given by:

q = (él) 66 cos 6i
O

5. The resulting PMT signal is frequency
analyzed to generate the power spectral
density function P(q,w) for this particular
value of q.

. 6. Move the pinhole to a new position corres-

ponding wave number q' and Obtain P(q',w).

7. Continue until enough g values have been
sampled to generate a good picture of the

two variable function P(q,w).

Technique for Better q Resolution

One practical experimental difficulty involves
measuring the position of q = 0. For a membrane-to-
pinhole separation of 30 cm and 10 = 476.2 nm, the
largest pinhole displacement x is about 0.3 cm while
the reflected beam is 0.05 to 0.15 cm in diameter. The
distance x can be measured to 0.001 cm but the actual
position of q = 0 is measurable to only about 0.025 cm
if you center the pinhole (typically 200 um in diameter)
in the reflected beam by eye. A method for improving
the accuracy of the location of q = O has been developed
but requires stable membranes or rapid data taking. Re-
call that ripplons of wave number q scatter light at

an angle 66 = q/ko on both sides of q = 0. The pinhole

52

is visually centered on the q = 0 beam and data are
taken on both sides in the plane of incidence. After
determining the value for mm, the value for which
P(q,w) is a maximum, for all x values (the exact x
corresponding to q = O is not known yet so q cannot
be computed) graph mm versus x. This graph should be
symmetric about the x value corresponding to q = 0.
Determine this x point of symmetry, x0, and determine

q values using the distance from xo to x.
q = kolx - xol/D

Graphing the values of mm versus q should give a
single curve with data from both sides falling together.
If the data do not lie on the same line, the membrane
shifted while data was being taken. This technique
requires taking twice as many data points in order to
define the dispersion law on both sides of q = 0 but
results in a much more accurate determination of q.

The consistency of data from both sides of q = 0 shows
that the location of q = 0 was well determined and
that the direction of the q = 0 beam remained fixed

during the experiment.

Local Oscillator Strength

 

There is considerable scatter in the data which
has been traced to small amplitude oscillations (about

2 milliradians) in the direction of q = 0 on a time

53

scale of l to 5 minutes. This can be observed by leaving
the pinhole fixed and taking data as a function of time.
The position of mm shifts about some average value.

It takes 2 to 5 minutes to do a spectrum analysis on the
SAICOR and 2 minutes to record it on the X-Y recorder
so we have a random sampling of the deviations in the
direction of q = 0. Since the precise location of

q = 0 cannot be measured, no attempt was made to correct
for these oscillations. If the distance from g = 0 to
the position corresponding to. q could be rapidly de-
termined for each point, there would be considerable
reduction in the scatter of the data.

The heterodyne detection of the scattered light
requires the presence of a "local oscillator" generated
by elastically scattered light from some defect in the
cell. At times there were good reflections but no
signals. No imperfections happened to be in the beam.
Moving the beam a little usually increased the signals.
If not, another spot on the membrane was tried. It was
always possible to find a spot that gave signals. This
empirical approach to finding signals is easy using a
real time analyzer like the SAICOR 51A which displays
the entire frequency spectrum in real time. One sits
on a signal and moves things around a bit to peak the

local oscillator strength.

54

Data equivalent to the power spectral density
could also be obtained by measuring its Fourier time
transform, the autocorrelation function, with a SAICOR
42 Correlation and Probability Analyzer or the auto-
correlation program on the Solid State Group PDP8/e mini-
computer. These devices do not have a real time display
making it difficult to know if you have improved the
signal by moving something. The following may be of some
help to anyone caught without a real time analyzer.
Listen to your spectrum. These are audio signals and
the ear is a sensitive device capable of evaluating

signal strength and signal-to-noise ratios.

Locating Sources of Noise

At times, strong lines will appear in the spectra
from mechanical resonant vibrations of the membrane
driven by some noise in the building. It is easy to
identify the source of the driving noise by using the
membrane as an accelerometer. Center the pinhole on the
reflected beam and turn up the PMT voltage until a
signal appears. The mechanical vibrations will cause
the reflected beam to oscillate. Since the laser beam
has a Gaussian intensity profile, as the beam moves the
light going into the pinhole changes intensity. The
q = 0 beam is much more intense than light scattered
from q # 0 excitations so one sees only the frequency

spectrum of the mechanical vibrations. Once the real

55

time spectrum of the noise is obtained, turn off pieces
of equipment one at a time to find which is generating
the driving noise. It is sometimes helpful to change
the resonant frequency of the membrane holder by placing
small weights on theztable that holds the sample cell.
Using this method, troublesome vibrations were traced to
circulating air currents from an air conditioner. The
addition of air deflectors changed the flow patterns in

the room and eliminated the noise.

CHAPTER VI

DATA

The data consist of a set of power spectral
density (PSD) functions, P(q,w), taken for various values
of q. The dispersion law for the ripplons is obtained
from a graph of the w value for which P(q,w) has a
maximum value, mm, versus q. The conclusions reached
in this thesis stem from the fact that "once" there was
a membrane.

For most membranes, the large scale shifting of
the membrane surface normal discussed in the experiment
section prevented the acquisition of enough data to
accurately determine the dispersion law. The membrane
would shift before an adequate number of points could
be taken and many broke in l to 4 hours.

Once, however, a membrane lasted 48 hours.

There was the same trouble with shifting surface normals
for the first 12 hours but as the membrane aged, the time
between shifts increased and it was possible to take
many (30 to 50) data points. By the second day, the
surface was fairly stable. This allowed the acquisition

of sets of data for different incident angles 6i and

56

57

for different values of the membrane-to-pinhole separa-
tion D. Since all data came from the same membrane,

one could be reasonably certain that the physical pro-
perties of the membrane and surrounding KCl solution were
the same for all sets of data. Data sets taken from dif-
ferent membranes, made from the same stock solution,

were consistent with data taken on this one stable mem-
brane. The greater instability of these membranes, how-
ever, caused more scatter in the data and did not allow

the acquisition of a large number of points.

The Signals

 

Figure 10 shows the first experimental evidence
for ripplons on the BLM surface. Only the relative
magnitude of the three q values is known as this mem-
brane broke before q = 0 could be determined but the
characteristics of the PSD function are illustrated. As
q increases, the peak in the PSD curve shifts to higher
frequencies and the width increases. The corner fre-
quency of the high pass filter was adjusted for equal

noise and signal amplitudes to minimize signal distortion.

Data Acqpisition

 

Data was taken at several angles of incidence
between 6 and 25 degrees with the pinhole placed at dif-
ferent distances from the membrane surface. The pinhole

is positioned a distance x from the q = 0 reflected

Figure 10. Power Spectral Density for Three q Values.

58

.ano_c

_

n
_

>ozmnommm

_ a q _

nevacv .c

S,

ALISNBINI

(SIINn A 8V3 IIGHV)

 

59

beam to gather light scattered from ripplons of wave
number q(x) and the resulting photocurrent is spectrum
analyzed to generate P(q,m). The value of w for
which P(q,w) has a maximum, mm, is determined. A graph
of mm versus q represents the dispersion law for the
ripplons on the interface. Any experimental distortion
of the spectrum must be removed before determining mm.

Data were taken by going to the largest q value
for which signals could be seen and taking points for
progressively smaller q values until the q = 0 central
spot was reached. Data were then taken on the other side
of q = 0 for progressively larger q values until
signals could no longer be obtained. Several data points
on the first side were then retaken to provide a check
that the membrane had not shifted.

The data were plotted as a function of x and
folded about the x axis point of symmetry, x0. The
q values corresponding to other values of x were de-

termined using x as the location of q = 0. Since

0
q only depends on the distance from x0, the data

points taken on both sides of q = 0 should lie on the
same curve. The check points that were taken at the

end of the run should agree with the corresponding points

taken at the start of the run. Data sets having the

points from both sides of q = 0 on the same curve with

60

reasonable agreement of the check points were accepted as
meaningful. Data sets not meeting these criteria were
not used.

Figure 11 shows a typical set of data taken with
10 = 476.2 nm, ai = 6° and T = 18°C. Figure 12 shows
this same set of data after folding about the symmetry
point. Data could not be obtained for q values

smaller than 500 cm-1

because the light from the specular
reflected beam (q = 0) began entering the pinhole,
saturating the PMT.

Data sets taken on the one stable membrane were

similar to the set shown. Data from other membranes had

more scatter and fewer data points.

Confirmation of Ripplons

Bargeron37 et a1. had mistakenly identified a
signal from light scattered by convection currents in
a solution containing erythrocyte (red blood cell)
ghosts as signals arising from the active transport of
ions across the cell membrane. In order to verify that
the observed signals actually came from the membrane,
the following experiments were performed:

1. The incident laser power was changed. If
the signals arose from convection currents
due to heating of the liquid, changing the
power should change the amount of heat

absorbed and change the rate of convection.

61

.X NO COHflocsm 0 mm MHMQ

.HH museum

 

Eons: .
com; com 00¢ o
_ _ _ _ o
. l
o
o o i m
w.
o
. M
. '
0 AW.
0 I on"
. O
u.
z
. (
C 1
o

 

Figure 12. ~Data as a function of q.

62

 

 

63

If the signals were doppler shifts, the fre-
quency should change with the rate of flow.
No change in the frequency distribution was
observed for a factor of five change in
incident power.

The membrane was removed and the signals
went away. This shows that either the sig-
nals came from the membrane or the membrane
provided the local oscillator light for
heterodyne detection.

The membrane was replaced with a plastic
microscope cover slip. If the membrane just
provided a local oscillator, so would the
cover slip. No signals were observed.
Identical signals were obtained on either
side of the q = 0 reflected beam. This
puts a considerable limitation on the
direction of flow of any convection current.
The value of q for ripplons depends on

incident angle 61 through:

q = (Zn/lo) 66 cos ei

Data taken for incident angles between 6 and
25 degrees fell on the same curve when g

was determined with the ripplon formula.

64

There was still a faint possibility that the signals
arose from convection currents in the membrane surface
itself. If the effective resistance in the membrane
surface were high enough, the molecules might reach
some terminal velocity and not change speed with
changing laser power. This final possibility was
eliminated by increasing the viscosity of the KCl solu-
tion by a factor of 2.75 by adding glycerine. In the
various theoretical treatments, prOpagating modes with
a given wave number q only exist for small values of
the viscosity. As the viscosity increases, the ripplons
decay more rapidly giving a broader frequency spectrum.
The increase in viscosity of the surrounding liquid
should have little effect on the signals if they came
from convection currents in the membrane surface. No
signals displaced from w = 0 were observed with the
higher viscosity liquid. The increase in viscosity had

eliminated all detectable propagating ripplons.

CHAPTER VII

HYDRODYNAMIC THEORY

The data are to be compared with the predictions
of a theoretical model. The power spectral density was
obtained for frequencies between 0 and 10,000 Hertz for

1 to 2,000 cm-l. The

a range of q values from 500 cm-
wavelength of the surface variation is, therefore, much
larger than molecular dimensions. The time scales in-

volved are long compared to the time between molecular

collisions so hydrodynamic theory will be used.

tHydrodynamic Eqpations

 

The fluids are assumed to be incompressible so
the equation of continuity becomes:
v . 6 = o

where 3 is the velocity of the fluid. One also has the

Navier-Stokes equation:

pi
p at

2

-> + + ->
+ p v V - v + VP - nV v = p q

where p is the density of the fluid, n the viscosity,
P the pressure and 3 the acceleration of gravity. One

is interested in wave-like solutions that vary as

65

66

exp[i(kx - mt)]. The velocity V is of the order am
where a is the amplitude of the wave. The following

order of magnitude comparisons can be made:

+

lav ~ ~

'a—EI (0V am

1* v - ll ~ 2/1 ~ 2 2/1 ~ 9 (ii)
V V V a (.0 A at

The wave amplitude, a, is much smaller than the

wavelength for ripplons so the term 3V - V is small
ii
at

amplitude is small, the gravity term will also be

compared to and may be dropped. Since the wave
neglected. The interface will be vertical in the ex-
periments so gravity terms would not enter the equa-

tions anyway. The Navier-Stokes equation reduces to:

+ VP - n V26 = o

3621

One looks for solutions to the hydrodynamic equations
that satisfy the boundary conditions of the physical

system.

Power Spectral Density

The power spectral density of the fluctuations
is obtained by calculating the autocorrelation function
for the q'th component of the Fourier decomposition of

the surface:

_ . l
<€q(o)€q(t)> — rim T

T-HIO

T
f0 €q(o) €q(t + T)dT

67

The autocorrelation function is related to the power

spectral density through the Wiener-Khintchine theorem:

P(q,w) = % f:<€q(o)€q(t)> eiwt dt
A Model

The selection of a physical model must be done
with great care. If one includes too many details, the
theory may not be solvable. The solution for the free
surface of a liquid gives some insight to the membrane
problem. The solutions for the velocity vary as
exp(mz) where m is a complex constant with magnitude
the order of q.

One sees that the wave penetrates into the fluid
a distance on the order of the ripplon wave length,
three to four orders of magnitude larger than the mem-
brane thickness. The majority of the molecules involved
are water molecules. One would expect the physics to
be dominated by the mass and viscosity of the KCl solu-
tion with the membrane providing an interfacial tension.

As a first approximation, treat the membrane
system as the interface between two identical fluids of
density' p and viscosity n with an interfacial tension

l
O. Assume a surface, uniform in fihe y direction,
I

given by:

2= 60:)

68

The dispersion law for this model was derived by M.

Papoular5 who assumed solutions exponential in x and

applied the boundary conditions:

1.

The velocity is continuous across the inter-
face.
The xz component of the stress tensor

Oxz' must be continuous across the interface.

avX avz
Oxz = n (62 + 6x

The 22 component of the stress tensor,

022, must be continuous across the interface.

3v 2
22 62 ax2

-where P is the pressure in the fluid.

Papoular also allowed one of the fluid layers
to have a finite thickness h and showed
that the oscillations of the interface with
wave number g were decoupled from those

at the surface if qh was much larger than

one o

The dispersion law can be determined by finding

the roots of the dispersion equation. In order to correct

the experimental power spectra for instrument and sample

effects, one must know the theoretical power spectral

density, P(q,m). Herpin and Meunier38 give the results

of a calculation of P(q,w) as:

69

P(q,w) = (y kBT/o qznw) ImIl/D(iwro)] (34)

2

To = o/Zn q

2
y = 0 p/8 n q

D(s) = y +%{[s $1 + 25][l + $1 + 23]}

where O is the interfacial tension, kB is Boltzman's
constant, T is the absolute temperature, p the density
of water, n the viscosity of water, D(s) is the dis-
persion law in terms of a dimensionless parameter 5,

Im signifies the imaginary part and y~—— is the square

root with positive real part. Propagating solutions

exist for values of y greater than 0.16.

CHAPTER VIII

COMPUTATIONS AND RESULTS

In order to compare the theory with the data,

one must correct for instrumental effects and imperfect

samples.

Physical Considerations

Angular divergence of the scattered beam is

responsible for the pinhole gathering light from ripplons

of different q values. The angular divergence arises

from:

The pinhole at a distance D from the
sample, with diameter d, gathers light

over an angle 66S given by:

668 = d/D

If the incident light beam has wavelength
lo and diameter R, there will be an

angular variation, 660, in the reflected

light from diffraction effects given by:

660 = 1.22 AO/R

Normally, the surface has some curvature

with an angular variation in surface normals
70

71

of 66C so the pinhole will gather light
from the q values corresponding to 66c.

4. If the incident beam is not collimated, there
will be a variation in gathered q values
corresponding to the angular divergence of
the beam.

Since the pinhole gathers light from many q values, the
experimental power Spectral density will be the convolu-
tion of the theoretical PSD function with an instrument
function that accounts for the variation in q. In
principle, determination of the instrument function
involves computing the angular spread in the light caused
by the various physical processes and properly adding

the effects together to produce some total angular
variation. This total angular variation is then re-
.lated to a spread of q values. In practice, one is

not able to add the effects together and treats the
instrument function as a variable parameter, requiring
the same instrument function for all points.

Improper consideration of these effects lead
early experimenters to erroneous conclusions about the
viscosity of a liquid near the surface. A proper treat-
ment of instrument effects lead to values for the
viscosity consistent with those measured by conventional

means . 3 9

72

Approximation of Instrument Function

 

A simple rectangular instrument function was
used to analyze the data. This is equivalent to assuming
that the pinhole gathered light over some range of g
values Aq, centered on the measured value of q. In
reality, some of the light scattered by q values near
the edge of the pinhole would not get through but one
starts with some "equivalent" rectangular function.

The noise in the data and the presence of low frequency
components from noise prevented the deconvolution of the
data.

The approach taken was to convolute the theory
with the instrument function and compare the convoluted
theory to the data. Since the density and viscosity of
water are known, there were only two free parameters,
the interfacial tension and the width of the instrument
function. Extensive computation of theoretical spectra
on the MSU Control Data 6500 computer showed the effects
of the surface tension were only weakly coupled to the
effects of the width of the instrument function. The
surface tension changed the slope of the dispersion law
curve while increasing the width tended to shift the

curve uniformly to lower m values.

73

Comparison of Theory and Data

 

The interfacial tension was adjusted to obtain a
theoretical curve which was parallel to the data and then
q was varied to shift the convoluted dispersion law
until it passed through the data points. Figure 12 shows
the results of these computations. The dashed line is
the unconvoluted dispersion law for o = 2.5 dyn cm-1
and the solid line is the convolution of this dispersion
law with an instrument function of width 500 cm-1.

The circles in Figure 13 represent an experimental
PSD curve for q = 813 cm-1, 61 = 6°, T = 18°C and
10 = 476.2 nm. The solid line is the convoluted
theoretical expression using the above parameters and

the values 0 = 2.5 dyn cm.1 and Ag = 500 cm".1

obtained
from the fit to the dispersion law.

Within the scatter of the data points, one can
say that this analysis of the membrane is certainly con-
sistent with the data. The experimental power spectra
are qualitatively described by the theoretical result.

A different instrument function might give rise to better
agreement for the spectral shape but would take a lot of
computer time to obtain results. Since the dispersion
-law is in excellent agreement with the data, no other
instrument functions were tried.

Note that the experimental PSD curve is not

meaningful for frequencies below 500 Hertz. The high

Figure 13. Theoretical fit to PSD curve.

l

'74

L L A

" Isrmn wvauaaw ALISNBINI

IO

1

 

l

.7

 

'JO
0

(IOSHz)

FREQUENCY

75

pass filter has been adjusted to minimize the low fre-

quency noise in this region.

Other Membrane Materials
One would have liked to obtain data from lipids
. with different interfacial tensions but large membranes
could not be produced with egg lecethin or glycerol
monooleate, two common lipids used for BLM's. The inter-
facial tension of oxidized cholesterol does depend on
the state of oxidation, however, and experiments were
run on a cholesterol sample that had aged in the
refrigerator for a year.
The circles in Figure 14 were taken at T = 18°C,
6i = 8°, and 10 = 520.8 nm. The dashed line is the best
fit convoluted dispersion law for o = 2.5 dyn cm-1

and Ag = 500 cm-1, which was used for fresh cholesterol.

The solid line is the convoluted theory for o==l.9 dyn cm-

with Aq = 500 cm-1, which is a best fit for'this data.
Tien4o gives a value for o = 1.9 i 0.5 dyn cm"1
depending on the state of oxidation. The state of
oxidation does change with age and the fact that the
same instrument function successfully describes the dif-
ferent data sets leads one to conclude that this simple

picture is adequate.

l

Figure 14. Dispersion law for aged cholesterol.

76

 

(gm-Icon 12h

 

I .000

q hm")

CHAPTER IX

FUTURE DIRECTIONS

One always asks, where do we go from here.

Several possibilities exist.

Better Membranes

 

The most obvious need is a better membrane.
Oxidized cholesterol was used because it was available
and one could obtain data.

If a different material could be found that made
flat membranes, one could use larger diameter laser
beams increasing the signal intensity and decreasing the

spread in g values gathered by the pinhole.

More Data Points

 

The problem of not obtaining enough data points
before the membrane shifts can be reduced by interfacing
the spectrum analyzer with a computer. It takes two
minutes to record a power spectral density curve. Pre-
liminary experiments on interfacing the SAICOR 51A to a
PDPB/e minicomputer were done. It is possible to send
the 200 point PSD function serially over a single coaxial

cable in 40 milliseconds. One can store 20 curves for

77

78

each available 4,096 word block of memory. This results

in a factor of two increase in the rate of data taking.

Measuring g

 

The problem of measuring the displacement of the
pinhole from q = 0 when the direction of the q = 0
beam changes with time might be solved with a two axis
translation stage that would move at a rapid rate, have
good spatial resolution and be computer controllable.
The computer could direct the pinhole to the center of
the reflected q = 0 spot by measuring intensity. Once
the brightest Spot, assumed to be the center of the
Gaussian intensity profile, is found, the pinhole is dis-
placed some known distance. A spectrum is taken and the
pinhole returned to zero to make sure the q = 0 spot
has not shifted. A system of this type would be in-
dependent of membrane Shifts as q = 0 is measured for

every point. Such technology does exist. The InchwormTM

41

translator will move 2.5 cm with a resolution of 60

Angstroms. It will exert 20 Newtons of force and slews
at rates between 3 X 10.5 and 2.0 cm per minute. The
pinhole moves no more than 0.5 cm for a 30 cm pinhole-
to-membrane separation which would take 16 seconds. A
less expensive possibility would be to drive a pre-
cision X-Y stage with stepping motors. Layer42 has

shown how to reduce the step size of a conventional 1.8

79

degree per step motor to 0.06 degrees per step while

retaining the conventional slew speed and accuracy.

Recording Signals

 

The signals have frequencies in the audio range.
Preliminary experiments show that the audio signals can
be recorded on a conventional tape recorder and then
analyzed at a later time with the spectrum analyzer. An
array of light pipes, multiple phototubes and a 16 track
analog data recorder would allow the simultaneous
acquisition of 16 different signals with g values de-
termined by the positions of the light pipes. The re-

corded spectra could later be analyzed one at a time.

More Complex Model

 

The next step in building a theoretical model
would be treating the membrane as a fluid layer with
density p' and viscosity n' that had some thickness
h. Reasonable estimates can be made for p' and h
but what to use for the "viscosity" of a bilayer is open
to question. The dispersion law for this model, which
involves the solution of a set of 8 simultaneous linear

homogeneous equations has been derived.43

The equation
for the power spectral density of the fluctuations has
not, as yet, been published. One could numerically
solve for the roots of the dispersion equation but could

not compare the dispersion law with the data without a

knowledge of the spectral shape.

80

The value of the interfacial tension in this
model should be compared to one half the value measured
by Tien. The membrane actually has two interface layers
so the membrane “interfacial tension" that was used in
the zero thickness model was actually twice the inter-

facial tension between cholesterol and water.

Suppression of q = 0 Beam

 

M. Papoular5 made an interesting point related
to the inability to take data near q = 0 caused by the
angular divergence of the q = 0 beam. At the Brewster
angle, light polarized in the plane of incidence is trans-
mitted but the value of the reflection function f in

equation (33) takes on the value:

nm - nW
nm + nw

where nw is the index of refraction of the water and

nm the index for the membrane. There is usually some
scattered light so one could polarize the laser beam
perpendicular to the plane of incidence, find the q = 0
location and then rotate the polarization to eliminate

the q = 0 beam. One must note, however, that q values'
smaller than 300 cm"1 correspond to ripplon wavelengths
that are larger than the diameter of the incident laser
beam so the diffraction grating formula used in deriving

the scattering intensity is no longer valid.

81

Elimination of Destructive Interference

 

It is theoretically possible to increase the in-
tensity of the scattered light. Visualizing the membrane
as an interface between two fluids with both surface
layers oscillating in phase leads to the prediction of
almost total cancellation between the light reflected
from the front and back surfaces. The Brewster angle ex-
periment implies the membrane index of refraction is
close to that of water but there must be some difference
as light is reflected from the surface.

If one could arrange the fluids so one index was
larger and the other smaller than that of the membrane,
the light reflected from front and back interfaces would
add constructively. The dispersion law would change as
the fluids would no longer be identical but the formulas
for the expected power spectral density and the dis-
persion equation are given by Herpin and Meunier.38
The increased reflectivity might enable one to take
data at higher q values where the effects of membrane

density and viscosity should enter.

Stimulated Scattering

 

Considerable progress could be made if one could
stimulate ripplons with a single q value by some method

similar to that used on phonons in stimulated Rayleigh

44

scattering. The value of the average amplitude of the

82

ripplon could be increased two to three orders of
magnitude and still satisfy the condition for lineariz-
ing the hydrodynamic equations. An increase of two
orders of magnitude would increase the scattered light
from that particular ripplon by four orders of magnitude.
Signals from the other ripplons would be down in the
noise.

The power spectral density obtained would then
correspond to a single q value. There would be no
instrument effects to compensate for and the experimental
dispersion law could be compared directly with theoretical
expressions. Since there are no instrument effects to
take out, the functional form of the spectral density
curve is not needed and one could compare data with

theories having only a dispersion relation.

Absence of Forward Scattering

 

The formula for the intensity of light scattered
in the forward direction can be derived. Since there is
no phase change for transmitted light, one would expect
the light scattered from the two faces of the membrane
to add constructively and be much stronger than those
from reflection.

Attempts have been made to observe these signals
with null results. The failure to observe signals in the

forward direction may be due to the difficulty in

83

obtaining local oscillator light. A more detailed in-

vestigation might reveal some new information.

CHAPTER X

CONCLUSION

Thermal excitations of the surface of a bilipid

-l
membrane were observed for wave numbers between 500 cm

and 1800 cmfil. The frequency corresponding to the
maximum in the power spectral density curve falls between
0 and 10,000 Hertz. For this range of q values, the
surface capillary waves (ripplons) extend into the
surrounding fluid a distance three to four orders of
magnitude larger than the membrane thickness.

The experimental dispersion law is adequately
described by the model of a zero thickness interface
between two identical fluids of density p and
viscosity n with the membrane providing an inter-
facial tension.

Measurements on two different cholesterol
samples, 0 = 1.9 dyn cm.1 and o = 2.5 cyn cm-l,
demonstrated the ability to prove the membrane inter-
facial tension but the original intention of probing
other membrane parameters was not achieved.

The rectangular instrument function was rather

simplistic but there was reasonable agreement between

84

85

the theoretical and experimental spectral shape. The
fact that the fit to the dispersion law for a cholesterol
sample with a different interfacial tension required the
same 500 cm.1 instrument function supports the assumption
that the actual effects of imperfect sample and instru-
ment distortion are well represented by this equivalent
rectangular instrument function.

It appears that the interfacial tension is the
only membrane parameter that can be probed at this time.
The effects of membrane density and viscosity might
perturb the shape of the dispersion law for large q
values but the present level of scatter in the data
eliminates the possibility of observing small changes in
the shape of the dispersion law curve. If one could
reduce the scatter by measuring q for every data point
or stimulating a particular ripplon, it should be
possible to detect the small changes in the shape of
the dispersion law and relate these changes to the other
membrane parameters.

The possibility of stimulated scattering is
particularly promising as this method would reduce the
effects of the instrument on spectral shape. One should
also be able to detect signals for much larger values of
q. Assuming a two order of magnitude increase in the
amplitude of the ripplon gives a four order of magnitude

increase in the scattered light intensity. If one assumes

86

that the scattered intensity falls off as q2 and that
everything else scales, one should be able to observe
signals for q values 100 times larger than the present
1,800 cm"l limit.

The technique sounds promising but one problem
remains; By what method does one stimulate a ripplon

without rupturing the membrane?

REFERENCES

10.

11.

12.

13.

14.

15.

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181, 558 (1973).

Cleaning and Polishing Compound Plastic Type I,
FSN 7930-634-5340, Clarkson Labs, Inc., Camden,
N.J.

32.
33.

34.

35.

36.

37.

38.

39.
40.
41.
42.

43.

44.

89

Lexan is a product of the General Electric Company.
Millipore Corp., Bedford, MA.

Type A, NeWport Research Corp., Fountain Valley,
CA.

Number 40,641 Flexible Fiber Optic Light Guide,
Edmund Scientific Co., Barrington, N.J.

See for example: G. Tobey and L. Huelsman,
Operational Amplifier Design and Applications,

 

McGraw-Hill, New York.

C.B. Bargeron, R.L. McCally, S.M. Cannon and
R.W. Hart, Phys. Rev. Lett., 30, 205 (1973).

J.C. Herpin and J. Meunier, J. de Physique, 35,
841 (1974) N.B.: In eq. (1), the expression of
D(S) must be corrected: in the S term, m'(m-l)
and m(m'-l) should be replaced by m'(m+1) and
m(m'+l).

D. Langevin, Faraday Soc. Trans., 1212! 95 (1974).
H.T. Tien, op cit., p. 40.

Burleigh Instruments, Inc., East Rochester, NY.
H.P. Layer, Rev. Sci. Inst., 11, 480 (1976).

J. Lucassen, M. Van Den Tempel, A. Vrij and

F. TH. Hesselink, Proc. K. Ned. Akad. Wet., B-73,
109 (1970).

A. Virj, F. Th. Hesselink, J. Lucassen and

M. Van Den Tempel, Proc. K. Ned. Akad. Wet., B-73,
124 (1970).

D.W. Pohl, S.E. Schwarz and V. Irniger, Phys. Rev.
Lett., 31, 32 (1973).

APPENDICES

APPENDIX A

HIGH PASS FILTER DESIGN

In order to eliminate low frequency noise while dis-
turbing the signal as little as possible, a filter with
the following characteristics was needed:

1) High frequency gain of unity up to 30 Kilohertz.

2) Adjustable corner frequencies from 0 to 2,000 Hz.

3) Low frequency rolloff of 40 db per decade.

The following circuit, designed by Mr. Phil Gaubis
meets these specifications. It looks simple but has some
subtle features. Since this filter was an important part
of the experimental setup, it is presented in detail.

Figure 15 shows the electrical schematic and
Figure 16 shows a diagram of the functional blocks.
Operational amplifier (op amp) A4 is an integrator with
time constant l/RC. Potentiometer RSb scales the out-
put of this integrator by the factor A which takes on
values from 0.0 to 1.0 as R5 is varied. Op amp A3 is
an identical integrator with output scaled by the same
factor A by R5a. The scaled output of A3 is multiplied
by -2 by Al. Op amp A2 serves as an inverting adder

whose output is:

90

Figure 15. Schematic diagram for high pass filter.

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93

Output = -(Input + D0 + B04) where D = Rl/R3,

l
1 is the scaled output of Al and 04

B = Rl/R4, 0

is the scaled output of A4.

To analyze the response of this filter, consider
the functional diagram, Figure 16 . Starting at the
output X, compute the output of the two integrators.
The output of the inverting adder must be the same x
fed into the input of the first integrator. The equa-
tion expressing this self consistency in the time domain

has several integrals. In order to avoid calculus, one

Laplace transforms both sides of the equation by multi-

Ist (i = /:I) and integrating

plying both sides by e
over s from 0 to m. In the 5 domain, the integrals
are replaced by algebraic operations.

The integral over time becomes multiplication by
-l/RCs. Starting at the output x, integrator A3 gives
-X/RCs and RSa scales this to ~AX/RCs. Integrators A4
gives AX/chzs2 and RSb scales this to AZX/chzsz. The
inverting adder A2 then adds the input I, B times the
scaled output of the integrator A4 and -2D times the
scaled output of integrator A3 and yields the negative

of this sum. Since we are back where we started, the sum

must equal -X.

~X(s) = I(s) + BA2X(s)/R2C2s2 + 2ADX(s)/RCs

Collecting terms in X(s) gives:

94

1(3) = -X(s) (1 + BAZ/RZCZSZ + 2AD/RDs)

Solving for the ratio of the output voltage to the input
voltage, defined to be the gain G(s), gives:
X(s)/I(s) = G(s) = -sz/(s2 + ZsAD/RC + BAz/RZCZ)

If B = D2 the denominator is a perfect square:

G(s) = -sz/(s + AD/RC)2

To recover the steady state frequency response let:

iwt
Eoe

I(t)

I(s) Eo/(s - iw)
The ouptut X(s) becomes:

X(s) = -EOG(s)/(s - iw)

Taking the inverse Laplace transform of both sides:

X(t) -Eoe1th(s)|s = iw

-I (t) G(iw)

The gain of the filter is just the negative of G(s)
evaluated for s = im.
Ignoring phase, the magnitude of the gain be-

comes:

Gain = wz/(wz + (AD/RD)2)

95

When w is small compared to AD/RC
Gain = wZ/(AD/RC)2

and the gain falls off as the square of the frequency
(40 db per decade). When w is large compared to AD/RC
the gain has magnitude 1. This is a high pass filter
that falls off at 40 db per decade with a corner fre-

quency fC of:

fc = AD/ZWRC

Note that making B = D2 was the key to obtaining this
40 db per decade.

4 ohms, c = 10"8 farads,

For this filter, R = 10
D = 1.26 and the dual potentiometer R5 varies A from
.0 to 2,000 hertz. Figure 17 shows the response of the
filter for 3 different settings of R5 as recorded on a
Hewlett Packard 3580 spectrum analyzer.

Since this filter was designed to work with a
photomultiplier tube (PMT), which is a current source,
one might wonder why the resistor R1 is in the circuit.
This resistor provides some protection for the PMT if
there is ever an excess amount of light striking the
photo cathode. When the maximum recommended current of
3 milliamps flows in the tube, the PMT anode is raised
30 volts above ground by the voltage across R1. This

tends to reduce the current flowing in the tube. In

normal operation with about three microamps of current,

96

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97

there is only 30 millivolts developed across R1. The
presence of R1 also allows one to use the filter with

other amplifiers which are voltage sources.

APPENDIX B

COMPUTER PROGRAMS

An interactive computer program was developed
for the PDP8/e minicomputer that allowed the operator to
put an experimental spectrum on a cathode ray tube (CRT)
display, compute a theoretical spectrum and display the
theory along with the experimental spectrum. After
visually comparing the results, the operator can list
numerical values Of q and w for the spectra, plot the
Spectra on an X-Y recorder, or go back to generate a
theoretical curve with different parameters. Since the
computation of the theoretical spectrum requries numerous
‘evaluations of the complex PSD function, equation (34),
the computation process was divided into two parts.

The membrane surface tension and_the viscosity
and density of the surrounding liquid were selected and
the PSD function for a given g was evaluated for 170
evenly spaced m values (500 to 85,000). Two hundred
.PSD functions were generated for evenly spaced q values
from 10 to 2,000. These PSD computations were stored in
a 200 by 170 point array. .One generates theoretical

spectra by summing appropriately scaled values from the

98

99

array. For a given value of q, q = 10 n, the PSD func-
tion P(q,m) is the n'th column of the array.

The effect of the instrument function is to in-
clude some fraction of the PSD function for other q
values. One defines an m point instrument function and
a starting value n. The 170 rows of the array are
summed from n to (n + m - 1), each element of the sum
being multiplied by the instrument function value corres-
ponding to the element's column. To obtain the con-
voluted PSD, the 170 values of the sum are multiplied by
10, the separation of g values. For a symmetric instru-
ment function, this convoluted PSD corresponds to a q

value of
q = 10(n + m/2)

Using an array of computed PSD functions and
summing the array to generate convoluted spectra allows
one to compute a spectrum in 10 to 60 seconds. The array,
which requires 102 thousand words of storage, was kept
in a direct access disk file as the PDP8/e had only 16k
of core memory. Experimental spectra were stored in
groups of 10. The program READIN.FT allows one to read
10 experimental spectra into a data file. The program
SPECGEN.FT generates the 200 by 170 power spectral density
array. Subroutine PSDLPD.FT computes the value of the
PSD function for a given q and w while subroutine

RMAX.FT finds the maximum element in a set of numbers.

100

The program DSP20.FT is the interactive program that
uses a PSD array generated by SPECGEN and data files
generated by READIN. All programs are written in 058
Fortran IV. All other subroutines are system routines
in the 088 Fortran IV library.

The program must be used with care. The avail-
able core limited the program size so that jumps from
one part of the program to another are accomplished by
entering unphysical parameters. Parameters that are 3
used to compute disk read operations will not be accepted

if they fall outside allowed limits.

101

Table 1. Listing for SPECGEN.FT

SPECGEN.FTvPABE 1

50

10
100

.PROGRAH TO GENERATE POWER SPECTRAL DENSITY FUNCTIONS

FOR LIPID MEMBRANES, USES SUDROUTINES FSD(OvU)
AND RHAX(A7N9L)
COMMON SIGMAvRHOvETAvFSCALE
DEFINE FILE 1(20091709U7I1)
DIMENSION F1(170)
THE FOLLOUING ARE SYSTEM PARAMETERS CHANGED FOR EACH RUN
SURFACE TENSION!DENSITYIVISCOSITY95CALE FACTOR
¥*******
SIGMA=2.5
RHO=1 o 0
ETAxloOIE-Q
FSCALEzloEIO
*****¥*
URITE(4920)SIGMAIRHOIETA
FORMAT(’ SIGHA3’961206" RH03'961206,’ ETA=’VGIB.6)
URITE(4925)FSCALE
FORMAT(’ FSCALE=’9GIQ.5)
URITE(4930)
FORMAT(’ ’)
URITE(4135)
FORMAT(’ '96X1’0’95X9’UMAX’76Xv’F(OvUMOX)')
URITE<4730)
DO 100 1:1!200
O=IOOII
DO 50 J=19170
=JOOXJ
F1(J)=FSD(OvU)
CONTINUE
STORE THIS FIECE IN FILE 1
URITE(1’1)F1
TEMF=RMAX(F171709L)
M35OOXL
URITE(4!10)G:H7TEHF
FORMAT(11031109512.6)
CONTINUE
CALL EXIT
END

102

Table 2. Listing for DSP20.FT

DSP20.FT9PAGE 1

C
46

48
SO
55

60

100

110

120

130

140

PROGRAM TO DISPLAY AND INTEGRATE PSD FUNCTIONS FROM LIPID
MEMBRANE THEORY AND EXPERIMENT.

LOGICAL CONvURTEvUTEvPLOvPLT

INTEGER FvaFNZrSKP198KP2

DIMENSION RINSl(200)9RINSZ(200)vPT(170)vP(340)vPE(170)
DIMENSION X(340)!DUFFER(230)

DEFINE FILE 1(20091709Uv11)v2(1091709UvI2)

NBUF=O

URITE(4!10)

MAXU IS THE LARGEST U TO USE IN COMPUTING AND DTSFLAY
FORMAT(' ENTER MAXU9#OF PLOTSvURITE T DR FvPLUT T OR F 'yi?
READ(4912)MAXU9NPLOTS!URTE9PLO
FORMAT(I49137L19L1)

FORMAT(GIZ.6)

ANY RESPONSE OTHER THAN 1 GIVES 2 PLOTS
IF(NPLOTS-1)15920115

NPSU=-1

GO TO 21

NPSU=1

CHECK LIMITS ON I

IF(MAXU)25v?5

IF(170’MAXU)25

GO TO 45

URITE(493O)

GO TO 5 '.

FORMAT(’ ERROR')

IF(.NOT.PLO)GO TO 46

THERE WILL BE A GRAPHleITIALIZE PLOT ROUTINFS
URITE(4!33)

FORMAT(’ SPEED= '93)

READ(4737)SPEED

CALL PLOTS(SPEED)

Z=9.99/1o3

CALL FACTOR(Z)

FILL X DISPLAY BUFFER WITH 2 SETS OF X COORDINATE?
DO 48 I=19MAXU

TEMP=MAXU

TMP3I

X(I)=(TMP/TEMP)*1.3

X(MAXU+I)=X(I)

CONTINUE

URITE(4755)

FORMAT(’ #1-FILEVSKIPISUM ’vi)
READ(4960)FleSKPviU1

FORMAT(314)

RESPONSE OTHER THAN 1 GIUFS FILE 2
IF(1-FN1)10071109100

FN1=2

CHECK LIMITS ON PARAMETERS

IF(SKP1)120

IF(170-SKP1)120

GO TO 130

URITE(493O)

GO TO 50

IF(KU1)1409140

IF(200-KU1)140

GO TO 143

URITE(4730)

GO TO 50

103

Table 2 (cont'd.)

DSP20.FT9PAGE 2

C
143
145
147
C
C
149

C
150

165

175
170
C

178

180
183

185
C
190

210

280
C

(

.0

t
.l

ASK IF UE USE INSTRUMENT FUNCTION

URITE(49145)

FORMAT(’ INS FN T OR F ’73)

READ(49147)CON

FORMAT(L1)

IF(CON) GO TO 150

FILL INSTRUMENT FUNCTION UITH 1.0’8

DO 149 L=19KU1

THIS PUTS IN THE UIDTH OF THE BINS FOR THE INTEGRAL
RINSl(L)=10.

CONTINUE

GO TO 180

HERE HE ENTER ENST FUNCTION. 10 NUMBERS AT A TIME
JSN=<KU1/10)+1

D0 170 J=10JSU

URITE(49165)J

FORMAT(’ ENTER 10 IFiS’vIE)

LSU=10¥(J"1)

REAB(49175)(RINSI(LSU+L)vLL]110)

FORMAT(10812.6)

CONTINUE

PUT IN UIDTH OF BIN

DO 178 J=1rKU1

RINSl(J)=10.*RINSl(J)

CONTINUE

GET STARTING PARAMETERS FOR INTEGRAL. NEGATIVE SCALE
SENDS PROGRAM BACK FOR A NEW FILE AND INTEGRAL PARAMETERS.
POSITIVE SCALE UILL BECOME THE LARGEST Y VALUE OF THF
DISPLAY. ZERO SCALE HILL CAUSE THE MAX Y VALUE OF THE
INTEGRAL TO BE THE LARGEST T VALUE OF THE DISPLAY.
NEGATIVE KSTART SENDS THE PROGRAM BACK TO THE VERY START.
URITE<49183>

FORMAT(’ ENTER KSTARTvSCALE ’73)
READ(49185)KST1vSCALE1

FORMAT(I49612.6)

CHECK LIMITS

IF(KST1)200.210

IF(201*NST1~KN1)210

GO TO 220

GO TO 5

URITE(4930)

GO TO 180

NSUSI

IF(SCALE1)23092509260

GO TO 50

NSU=0

PNORM=SCALE1

ZERO THE BUFFER

DO 280 J=17MAXN

P(J)3O.

CONTINUE

HERE WE SUM THE PSD TABLE FOR K VALUES FROM NOTART TH
KSTART+SUMvUE SUM THE N VALUES FROM SHIPJ+1 TO MAX“
FOR EACH K, THE SET OF U VALUES IS MULTTFLIFH HT ifl
APPROPRIATE INSTRUMENT FUNCTION VALUE.

DO 400 MrviUI

JRD=M-1+KST1

READ<FN1’JRD)PT

DO 350 N=SKP1+19MAXU

P(N):P(N)+RINSI(M)*PT(N‘

104

Table 2 (cont'd.)

DSP20.FTvPAGE 3

350
400
C
410
420
450
C

430
C

435

437

438
439

436

440
453

C'C'.’
JJ

445

470
460

490
495

C
500
$310

600
610

CONTINUE

CONTINUE

SCALE THE BUFFER

IF(NSU)4109410.420
PNORM=RMAX(P.MAXU9ITEHP1)

DO 450 N=1vMAXU

PE(N)=P(N)

P(N)wP(N)/PNORM

CONTINUE

IF(NPSU)490

URITE INTEGRAL PARAMETERS
URITE(4943O)PNORM91TEMP1

FORMAT(’ NORMtl» ’r612.6r’LOC— '913)
CLEAR PLOT BUFFER AND DISPLAY PJINTS
CALL CLRPLT(2309BUFFER)

CALL PLOT(MAXU9X7P)

IF(.NOT.URTE)GO TO 455

URITE<4943S)

FORMAT(’ URITE T OR F ’v$)
READ(49147)UTE

IF(.NOT.UTE)GO TO 455

URITE OUT VALUES FOR THE TNTEGRAL
ITEMP1=1OXKST1
ITEMP2210*(KST1+KU1-1)
URITE(4v437)ITEMP19ITEMP2

FORMAT(’ INTEGRAL FOR Ku’vI4v’ TO ’9I4)
PMAX=RMAX(PE9MAXU9MAXF)
MAXFKSOOXMAXF

URITE<49438)MAXF9PMAX

FORMAT(’ MAX U=’v612.59' PMAX='7612.5)
URITE(49439)

FORMAT(’ ’16X9’U’913X7’P’916Xy'N'913X9’P')
URITE(49436)

FORMAT(’ ’)

DO 453 J=SKP1+19MAXU72

TEMPJflSOOXJ

TEMP2$500*(J+1)
URITE(49440)TEMP17PE(J)7TEMP2!PE(J+I)
FORMAT(’ ’7612.5.2X9612.575Ky81?.5.5X.H12.3)
CONTINUE

IF(.NOT.PLO)GO TO 470

URITE(49445)

FORMAT(’ PLOT T OR F ’r$)
READ(4v147)PLT

IF(.NOT.PLT)GO TO 470

GO TO 1000

URITE(49460)

FORMAT(’ KSTART! SCALE ',$)
READ(4.185)RST1.SCALE1

GO TO 190

URITE(4943O)PNORMrITEMP1
IF(NBUF)500v500

GO TO 965

THERE ARE TUO BUFFERS#**$#***
URITE(49510)

FORMAT(’ #2wFILErSKIPvSUM ’95)
READ(4960)FN293KP2.KU2
IF(l-FN2)60096109600

FN2=2

IF(SKP?)620

IF(170wSNP2)6?0

GO TO 630

1105

Table 2 (cont'd.)

DSP20.FT7PAGE 4

620

630

640

645

(549

C
C350

675
(570
680
685

700
710

730
750
760

780

790

uRITE<4.30)

GO TO 500

IF(KU2)64OI64O

GO TO 645

URITE<4930)

GO TO 500

URITE(49145)

READ(49147)CON

IF(CON)GO TO 650

FILL INST FN UITH BIN WIDTH
DO 649 L317KU2

RINSZ(L)$10.

CONTINUE

GO TO 680

ENTER INST FN 10 NUMBERS AT A TIME
JSU2(KU2/10)+1

DO 670 JxleSU
URITE(4v165)J

LSU=10*(J-1)
READ(49175)(RINSZ(LSU+L}9L—1710)
DO 675 J=19KU2
RIN82(J)=10.*RINSZ(J)
CONTINUE

CONTINUE

URITE(4v183)
READ(49185)KST29SCALE2
IF(KST2)7001710
IF(201*KST2-KU2)710

GO TO 720

GO TO 5

URITE(4930)

GO TO 680

NSU=1

IF(SCALE2)73077509760

GO TO 500

NSUSO

PNORM=SCALE2

DO 780 JxlvMAXU

CONTINUE

DO 790 J=1+MAXU72*MAXU
P<J>=o.

CONTINUE

DO 900 M=viU2

JRD=M~1+KST2
READ(FN2’JRD)PT

DO 850 NzSKPQ+1vMAXU
PE(N)=PE(N)+RINS?(M)kPTfN)
CONTINUE

CONTINUE.
IF(NSU)91099107920
PNORMxRMAX(PErMAXUvITEMPI)
DO 950 N=SKP2+17MAXU

FILL SECOND HALF OF PLOT EUFFFR
P(MAXU+N):PE(N)/PNORM
CONTINUE
URITE<49960)PNORM91TEMP1
FORMAT(’ NORM #2: '9612.év‘LOC~ “.13)
DISPLAY BOTH CURVES .
CALL CLRPLT(230.BUFFFR)
FORMAT(’ N! KSTARTv SCALE 'v$)
CALL PLOTfQXMAXUvaP)

1106

Table 2 (cont'd.)

DSP20.FT.PAGE S

978

975

980

990

1000

1010

1020

1100

IF(.NOT.PLO)GO TO 978
URITE(4.445)

READ(4.147)PLT
IF(.NGT.PLT)G0 T0 978

Go TO 1000

URITE(4.970)
READ(4.975)NBUF.NST.SCALE
F0RHAT<12.I4.G12.6)
IF(1-NGUF)980.990.980

UE CHANGE BUFFER 2

NGUFzz

KST2=KST

SCALE2=SCALE

GO TO 685

we CHANGE BUFFER 1
KST1=KST

SCALE1=SCALE

GO TO 190

URITE(4.IOIO)

FGRHAT(' x.Y.I= '.s)
READ(4.1020)X1.Y.I1
F0RHAT<2G12.6.I1>
IF(X1)1100

CALL XYPLOT(x1.Y.I1)

GO TO 1000

CALL PENUP

CALL XYPLOT(O..O..3)

A PAUSE T0 ALLGU CHANGING PEN GR PAPER
PAUSE 1

CALL XYPLOT(X(1).P(1).3)
no-1200 JzzynAxu

TEHP1=X(J)

TEHP2=P<J>

CALL XYPLGT(TEMP1.TEHP2.11>
IF(.NOT.<II.EG.3)>GG TO 1200
CALL PENDN

CALL PENUP

CONTINUE

CALL PENUP

IF(NPSU.EG.1>GG T0 470
THERE ARE Two CURVES

CALL XYPLOT(O..0..3)

A PAUSE TO ALLOW CHANGING PEN 0R PAPER
PAUSE 2

TEMP1=X(1+MAXU)
TEMP22P(1+MAXU)

CALL XYPLOT(TEMP1.TEMP2.3)
no 1300 J=2+MAXU.2*MAXU
CALL XYPLoT(x<J).P(J).II>

107

Table 3. Subroutine Listings

RMAX.FT:PAGE 1

100

PROGRAM TO RETURN MAXIMUM VALUE OF A REAL ARRAY X.
RETURNS LOCATION OF MAX IN LvLOOKS AT N ELEMENTS.
FUNCTION RMAX(XvaL)

DIMENSION X(N)

TMAX=X(1)

DO 100 J=19N

IF(X(J)~TMAX)1009100

TMAX=X(J)

L=J

CONTINUE

RMAXfiTMAX ;

RETURN 1

END

PSDLPD.FTvPAGE 1

C

PROGRAM TO CALCULATE POWER SPECTRAL DENSITYFUNCTIONS FOR
LIPID MEMBRANESPPSCALEaKT/PI=1.EIO IN 6500 RUNS.
FUNCTION PSD(Q'U)

COMMON SIGMAvRHOFETAvPSCALE

COMPLEX SvRvaOIrOlvDZvDSvD4
YlgRHO{(8*ETA*ETA*G)

Y=Y1¥SIGMA

TAU=RHO/(2*ETA*O*G)

SQz-UtTAU

S=CMPLX(Oo752)

R=CMPLX(Y70o)

D1=CNPLX(1 o '00)

I|2=CMPLX(2. 700)

U3$CMPLX(0.590.)

D4=CSQRT(D1+UQ*S)

O=R+D3KS¥D4*(D1+D4)

OI=OIIU

PSO=(PSCALE*Y1*AIMAG(DI))/(U*OXG)

RETURN

END

108

 

 

 

 

 

 

 

Table 4. Flow chart for DSP20.FT
lnitiolze the
Pa r ameters
if
MAXW, WRTE
<NPLOTS, PLO

 

 

 

 

 

 

No
Aflowed
Initiolze
PLO T Plot
Routines

 

 

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FNI, SKPI
KWI _

Allowed .

 

 

 

Read Inst F"

T RINSI in
groups of
ten

 

 

 

 

Fill RINSI
with Lo's

 

 

 

I
F

 

 

@

109

Table 4 (cont'd.)

IBO
KSTI , SCALE!

 

 

 

 

 

 

 

 

 

 

PN OW=SCALEI
Zero _l Meqrol

buffer PIJ)
I

 

 

Do Sums

 

 

 

>0

 

PNORM.=
RMAXIPI

 

 

 

 

 

 

110

Table 4 (cont'd.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1420
Save P values
in PE ,
Normalize P
<0
NPSW I
Write integral
List short statistics
statistics
>0
@ NBUF
List the
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111

Table 4 (cont'd.)

 

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112

Table 4 (cont'd.)

 

 

 

 

 

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Table 4 (cont'd.)

113

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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' 978

APPENDIX C

CORRECTIONS FOR THE EFFECTS OF
THE SAMPLE CELL ON q.

The formulas used to relate the wave number q
of the ripplon to the change in direction, 69, relative
to the reflected q = 0 beam do not contain parameters
describing the KCl solution surrounding the membrane.
Let us derive the relationship between the incident and
scattered light rays. Figure 18 shows a top view of
the sample cell. Light (in medium 1) is incident on the
cell, passes through the cell wall (medium 3), passes
through the KCl solution (medium 2), is scattered by a
ripplon of wave number q and passes back out through

the cell. Snell's law gives:

nl sin e = n3 Sin w = n2 Sln ¢
111 sin 9' = n3 sin w' = n2 sin ¢'

The scattered light leaves the membrane at an angle

8 = a + 6 where 6 is given by:
6 = q Az/(Zn cos(a))

One has:

114

115

 

 

 

 

 

 

 

Figure 18. Top view of cell.

 

 

116

11‘“2

where y is the angle which the normal to membrane makes

with the normal to the cell wall. Consider the angle ¢':

sin ¢' sin (8 ' Y)

sin (a + 6 - Y)

sin (¢ + 6 - 27)
Using this expression in Snell's Law gives:

n1 sin 6' = n2 sin (¢ + 6 - 27)

Since 6 is small this becomes:

n1 sin 6' = n2[sin (¢ - 27) + 6 cos (¢ - 2y)]

For y = 0 one has:

nl sin 6' = n2 sin ¢ + 112 (q lz/(Zn cos (¢)))cos e
= n:l sin 9+ (nlq 1.1/21!)
= n1 sin e + n1 lq ll/(Zn cos (6)))cos a
sin (6 + 66)

where:
66 = q ll/(Zn cos 9)

When y = O, the angular difference between q = O
and q f 0 scattering is determined by q, the incident

angle 9 and the wavelength of light in medium 1.

117

In practice, one must orient the holder so the
membrane is not parallel to the face of the cell. The
light reflected at the various interfaces would saturate
the PMT so the membrane is rotated by a small angle 7
relative to the plane of the cell wall. For 7 i 0, one

has:

cos(¢ - 27) cos a]
coalB - 7)

 

nl sin 6' = n2 sin (¢ - 27) + 69[
For q = 0:
. , _ . _
Sin 6° - (nZ/nl) Sin ( 27)

For q # 0:

cos(¢ - 27) cos 9
cosl¢ - Y)

 

sin 6% = (nZ/nl) sin (¢ - 27) + [ I

Let q = 1000 cm‘l, A = 500 nm, and let us calculate the

error in (9% - 90) for a worst case value of 7 = 3°,

n1 = 1, n2 = 1.33 and o = 2.5 dyn cm-l:
69 = 8.78 x 10'3 radians
¢ = 18.5°
e' = 16.7°
C
e' = 17.2°
q
9' = e' = 3.43 x 10’3 radians
q 0

which is a 4% error. For 6 = 10°, all other parameters

the same:

118

3

69 = 8.70 x 10‘ rad

¢ = 7.5°

C
e' = 2.49°

q 3

l _ I = -
eq 80 8.68 x 10 rad

which is a 1% error. Most of the data was taken with
6 near 10°. Since the scatter in the data was at least

5%, it is reasonable to ignore this correction.