Laser Light Scattering. Basic Principles and Practice

342 Pages·1991·5.125 MB·English

by  Benjamin Chu (Auth.)

LASER
LIGHT
SCATTERING
Basic Principles and
Practice
Second Edition
Benjamin Chu
Deportment of Chemistry
State University of New York
Stony Brook, New York
ACADEMIC PRESS, INC.
Harcourt Brace Jovanovich, Publishers
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Copyright © 1991, 1974 by Academic Press, Inc.
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United Kingdom Edition published by
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Library of Congress Cataloging-in-Publication Data
Chu, Benjamin.
Laser light scattering : basic principles and practice / Benjamin
Chu — 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-12-174551-1 (alk. paper)
1. Laser beams—Scattering. I. Title.
QC446.2.C49 1991 90-21194
621.36'6-dc20 CIP
Printed in the United States of America
91 92 93 94 9 8 7 6 5 4 3 2 1
PREFACE
More than fifteen years have passed since I wrote the first edition of Laser Light
Scattering in 1974. The technique has matured, and dynamic light scattering has
become one of the routine analytical tools used in particle sizing. There have been
diversified and important developments on its applications to many fields in
biology, chemistry, engineering, medicine, and physics. However, the basic princi­
ples of intensity correlation-function measurements have remained essentially
unchanged. The original aims of the second edition were to try to delete portions
of the book, such as Chapter 2 in the first edition, which were difficult to follow
for the readers without formal training in electrodynamics, and to update the
optical arrangements and correlator schemes commonly used in laser light scat­
tering. This exercise turned out to be much more difficult than I originally
anticipated. There was always a conflict of approach: whether to present a
qualitative but less rigorous answer or a quantitative but more difficult one.
The subject matter concentrates exclusively on the technical aspects of laser
light scattering, including the basic principles and practice. Only the application
to dilute polymer solution characterizations is included in some detail, leaving the
diverse developments on applications to many fields for other texts (see Chapter
1) and selected papers (see Appendix in Chapter 8). In introducing the Laplace
inversion in Chapter 7, the level of discussions unavoidably becomes more mathe­
matical. Thus, the reader who is not interested in the details should skip some of
the sections.
I am grateful to many of my coworkers and colleagues who have contributed
to this second edition, and to M. Adam, S. H. Chen, T. Lodge, H. S. Dhadwal, and
G. Patterson for reading the manuscript and making valuable comments.
The thoughtful criticisms by Patterson and Dhadwal were particularly appreciated.
Benjamin Chu
ix
I
INTRODUCTION
Successful applications of scattering techniques, such as light scattering and
neutron scattering, often involve a considerable amount of interdisciplinary
knowledge. Chemists and physicists have utilized light scattering, small angle
x-ray scattering (SAXS) and small-angle neutron scattering (SANS) to study
the size and shape of macromolecules in solution as well as a whole range of
materials including colloidal suspensions and solid polymers. Meteorologists
have used microwaves to observe the scattering by rain, snow, hail, and other
objects in the atmosphere, while astrophysicists have been interested in the
scattering of starlight by interplanetary and interstellar dust. The same basic
scattering principles govern all such phenomena. Wave interference yields
information on particle size whenever the wavelength of the electromagnetic
radiation (or the de Broglie wavelength of neutrons) is of the same order
of magnitude as the size of the scatterer. Before the advent of lasers, most
scattering studies, as we have discussed above, were concerned with the time-
averaged scattered intensity.
Historically there have been two different approaches to light-scattering
theory: scattering theory based on solving the Maxwell equations of a single
particle of well-defined geometry, and that based on fluctuation phenomena.
In particle scattering, we can consult the classical text by Van de Hulst (1957)
and the comprehensive book by Kerker (1969) on "The Scattering of Light
and Other Electromagnetic Radiation." Another book, edited by Huglin
1
2 INTRODUCTION CHAPTER I
(1972), contains extensive descriptions of many aspects of light scattering from
polymer solutions. Huglin (1977) has since reviewed the practice of light
scattering intensity measurements.
In condensed media or whenever the scatterers are close to one another, a
detailed computation of the induced electromagnetic field surrounding a
particle becomes very complex because intermolecular interactions have to be
taken into account. Einstein (1910) was able to bypass the difficulties inherent
in particle-scattering analysis in the presence of interactions. He assumed that
local density fluctuations in neighboring volume elements could be indepen­
dent of one another, and carried out a quantitative calculation of the mean
squared amplitude of those fluctuations. Although Einstein's theory was able
to explain the scattering from pure liquids and to predict the enormous in­
crease in the scattering as the liquid-gas critical point was approached (the
so-called critical opalescence), it failed to account for the angular dissymmetry
of the strong scattered intensity in critically opalescent systems. Later,
Ornstein and Zernike (1914, 1915, 1916, 1926) tried to include the effects of
correlation between fluctuations of neighboring volume elements. Again,
there have been extensive reviews (e.g. see Chu, 1967) on the fluctuation
theory.
A typical light-scattering geometry is shown in Fig. 1.1(a), where the
incident beam (7 ) impinges on the scattering medium (dots), with most of
I N C
the radiation being transmitted (I ) and a small portion being scattered (7 ).
t S
In laser light scattering (Chu, 1968), we study not only the changes in the
number (intensity) and the direction (momentum) of each type of photon in
the incident and the emerging light beams, but also the corresponding fre­
quency (energy) changes. Whereas the scattered intensity can be related to the
structure of the particles, the optical spectra reveal the dynamical motions of
particles. The word "laser" is included in the title in order to emphasize its
essential role in obtaining effective measurements of optical spectra where
wavefront matching of scattered light must be accomplished. Thus, a laser
light source is needed in dynamic light scattering. Furthermore, "laser light
scattering" is retained as the title so as to remind us of the importance of time-
averaged light-scattering intensity measurements.
The scattering process that will be discussed in this book can be described
classically and is confined to using lasers as a light source. Quantum-
mechanical phenomena, such as the Raman effect, are excluded. There now
exist many reviews, books, and proceedings, all of which cover one or more
aspects of topics related to laser light scattering. A listing is provided in
Appendix 1. A of this chapter.
In view of the extensive range of books available on laser light scattering, we
shall give only pertinent references that are intended as guides to the interested
reader or that have historical significance. The problem with the available
CHAPTER 1 INTRODUCTION 3
FIG. 1.1. (a) Scattering geometry. 7 I ,N 7S,C and It are, respectively, incident, scattered, and
transmitted intensity, θ is the scattering angle. The magnitude of the scattering vector (k) for
the incident beam is kx = 2π/λι and λλ ~ λ8 = λ0/η with λ0 and η being the wavelength in vacuo
and the refractive index of the scattering medium, respectively, (b) Conservation of momentum
2 2 2 2 2 2 2
(Eq. (1.1)) in the scattering process. From the law of cosines and |k,| ~ |k s| = (2πη/λ0), we have
K = |k, - ks| - k + k - 2k, · ks - 2k - 2/cf cos0 = 4fc, sin(0/2), or Κ = {4π/λ^ίη{θ/2).
This diagram represents a simple derivation for the magnitude of K.
books and proceedings, many of which are excellent, is that it remains some­
what difficult for uninitiated readers, especially those who are less physics-
oriented, to utilize laser light scattering for their applications. Thus, in this
book, I hope to cover some aspects of basic light-scattering theory and prac­
tice, leaving the advanced discussions to the monographs, proceedings, and
research articles. More specifically, this book is directed primarily toward
the experimental and technical aspects of laser light scattering, with emphasis
on how such experiments can be performed (e.g., see Chapter VI), while the
books by Berne and Pecora (1976) and Schmitz (1990) deal with the relations
between the measured parameters and the molecular quantities of interest.
INTRODUCTION CHAPTER I
In the remaining portion of this introduction, the inter relationship between
laser light scattering and other types of scattering techniques which use x-rays
and neutrons will be discussed. We want to get a general idea of how to find
appropriate scattering techniques for studying structural and dynamical
properties of the system of interest. It is clear that each physical technique has
its limitations. The important point is to learn when it becomes useful and how
it works.
1.1. MOMENTUM AND ENERGY TRANSFERS
In using thermal neutrons or photons, the probing radiation is usually
weakly coupled to the system of interest. Interaction of radiation with matter
can be expressed in terms of two characteristic parameters, namely, the mo­
mentum transfer (hK) obeying
hK = fi(k, - k), (1.1.1)
s
and the energy transfer (h Δω) obeying
hAco = h((o — co), (1.1.2)
x s
where k, (k) and ω! (ω ) are, respectively, the incident (scattered) wave vectors
s 8
and angular frequencies as shown in Fig. 1.1. The incident and scattered wave
vectors k, and k are in the directions of the incident and scattered beams,
s
respectively. Equations (1.1.1) and (1.1.2) are known, respectively, as the
momentum and energy conservation equations. They also imply that the
scattered light from a well-defined incident beam (ftk,, fta>,) has a distribution
of wave vectors and that, at each wave vector k , there can be a distribution of
s
frequencies depending on the dynamics in the scattering medium. As the
energy transfers in terms of frequency changes (Δω) depend on the scattering
dynamics, the scattered intensity fluctuates as a function of time.
The transfer of momentum (hK) correspond1s to probing the structures of
the system with a spatial resolution R ~ K . Measurements of the time-
averaged scattered intensity </(K)> can be related to structural studies and are
known as classical light-scattering measurements. Measurements of the time
(or frequency) dependence of the scattered intensity I(K, Δω) as a function of Κ
constitute dynamic light-scattering measurements. The subject matter of this
book includes both classical and dynamic light scattering, with emphasis on
the latter. Appendix l.B of this chapter lists some typical spatial and temporal
correlation ranges based on Eqs. (1.1.1) and (1.1.2).
The ranges covered by different complementary scattering techniques are
summarized in Fig. 1.1.1. The figure shows typical regions of momentum-
and energy-transfer space covered by different types of scattering techniques.
Using visible light, it clearly illustrates that optical scattering (i.e., light scat-
SECTION 1.1 MOMENTUM AND ENERGY TRANSFERS 5
1 1 2 S 4 5
10 10° 10" 10" 1(T 10" 10"
I 1 1 1 1 1 1
1
Κ (nm" )
FIG. 1.1.1. Momentum- and energy-transfer space diagram showing regions covered by scat­
tering techniques. The operating energy domains of optical mixing spectrometers are illustrated
more explicitly in Fig. 1.1.2. G: grating instrument; FP; Fabry-Perot interferometry; TOA: time-
of-arrival photoelectron statistics; PCS: photon correlation spectroscopy, which is the pre­
ferred technique for optical mixing spectroscopy .The ends of the arrows suggest approximate
limits of energy transfer by various techniques.
tering) experiments follow the energy-transfer axis and involve only small
momentum transfers (i.e., small K-values). Extension of light scattering mea­
surements to cover larger momentum transfers will be limited even if the
incident wavelength is extended to the ultraviolet region. On the other
hand, optical mixing spectroscopy (see Fig. 1.1.2) enables us to observe energy
changes that are many orders of magnitude smaller than those detectable by
other spectroscopic methods.
By introducing the time-of-arrival (TOA) photoelectron statistics (Dhadwal
et α/., 1987), the experimental gap along the energy-transfer axis between
photon correlation spectroscopy (PCS) and optical Fabry-Perot inter­
ferometry (FP) was filled. Energy changes that should be within reach, in
principle, between PCS and FP were difficult to achieve in practice because it
was difficult to perform PCS experiments with sample (or delay) time
increments < 50 nsec or FP interferometry with a resolution ~ 1 MHz. Here,
the uninitiated reader may feel uncomfortable with abbreviations such as
TOA, PCS, and FP or expressions such as "time scales of 50 nsec" and
frequency resolutions of 1 MHz. All such terms will be explained in
6 INTRODUCTION CHAPTER I
Energy Transfer
Au — Vi - Vs L_
1 Hz 1 KHz 1 MHz 1 GHz
Photon Correlation
Av = Av/C _ j_
1-100 -7 -4 -1 1 02 15 0-l 1 c0 m1 0 1 0
» Fabry-Perot Interferometry
E = hAv -J I I I I l_
y
1-140 -11 -8 -5 1 0-2 1 0 1 0 11 0 0e
l 1
τ = Av~ ι • 1 1 1
1 sec 1 msec 1 μsec 1 nsec 1 psec
Resolution Gratings
14 11 8 5 2
vjjAv ι ι ι ι ι 1
5 χ 10 5 χ 10 5 χ 10 5 χ 10 5 χ 10 0.5
FIG. 1.1.2. Comparison of energy-transfer ranges for photon correlation spectroscopy (PCS),
Fabry-Perot interferometry (FP), and grating spectrometry. The ends of the arrows suggest
approximate limits of energy transfer. For PCS, which is an extremely high-resolution spectro­
scopic technique, we are concerned with the shortest delay-time increment achievable (on the
order of nanoseconds). Longer delay-time increments test the patience of the experimenter; e.g.,
G. Patterson has collected good data out to delay-time increments of the order of 100 sec. For
FP, the best interferometers can search the megahertz (or even kilohertz) range, while grating
_1
spectrometers usually have much lower resolutions (see Section 3.2.1). In comparing spectral
-1 _1
frequencies and correlation times, τ = Δν was used. More precisely, one could use τ =
Δω = (2πΔν) with Δω in radians per second, because in a Lorentzian curve for the power
spectrum, the characteristic line width is expressed in radians per second. Thus, there is a factor of
In which needs to be taken into account.
subsequent chapters. It is sufficient to realize at this time that the frequency
ranges covered by PCS (low frequency range) and FP (high frequency range)
can be connected using TOA. Furthermore, the reader should know that
Fabry-Perot interferometers and grating spectrometers are used to measure
</(K Δω)>, a long-time average of the intensity, while PCS and TOA measure
/<K,i)> integrated over all frequencies. In classical light scattering, the
quantity of interest is the time-averaged scattered intensity </ (K, f)> = 7 (K).
s S
In the scattering of x-rays it is usually the time-averaged intensity for x-rays
within a narrow energy range. Then, by means of near-ultraviolet radiation
for light scattering and extension of x-ray scattering to very small angles,
the gap in the momentum transfer for time-averaged scattered intensities can
be filled (Chu, 1967), as demonstrated in Appendix l.B. A striking feature of
Fig. 1.1.1, as has been pointed out by Egelstaff (1967), is the extent of blank
spaces that are not accessible by present-day techniques. Thus, it becomes
important to explore the complementary nature of various radiation scatter­
ing processes.
SECTION 1.2 TIME-AVERAGED AND TIME-DEPENDENT SCATTERED INTENSITY 7
A deeper understanding of the dynamical properties of systems often
requires theoretical and experimental examinations of the scattering pheno­
mena over wide ranges of momentum and energy changes and the com­
bination of these results with measurements using other techniques such as
ultrasonics, dielectric relaxation, and magnetic resonance.
1.2. TIME-AVERAGED AND TIME-DEPENDENT
SCATTERED INTENSITY
In laser light scattering, we consider two aspects of the scattered intensity:
(1) Angular distribution of time-averaged scattering intensity, / (K). From
S
this quantity we can determine equilibrium properties such as the molecular
weight, the size, and (at times) the shape of macromolecules in solution or
colloidal particles in suspension. For 7 (K), we can use different probing
S
radiations, resulting in the complementary techniques of light scattering
(intensity), SAXS, and SANS.
(2) Spectral distribution of scattered intensity at different scattering angles,
7(K, Δω). Here we are more concerned with energy transfers that can be
S
detected using an optical spectrometer (i.e., a grating spectrometer or an inter­
ferometer) or an optical mixing spectrometer. In the optical spectrometer,
an optical dispersing element, such as a grating, is used. In the optical mix­
ing spectrometer, we use the photomultiplier tube directly as a square-law
detector (see also Fig. 3.2.1), The frequency ranges covered by PCS (includ­
ing TOA), FP, and grating spectrometers are schematically represented by
Fig. 1.1.2.
In Chapter II, the fundamental principles of light scattering, dealing with
time-averaged scattered intensity and the light scattering spectrum, are dis­
cussed. The approach from the point of view of fluctuation phenomena is
used in order to provide the reader with a quantitative derivation of how
wavefront matching needs to be considered in optical mixing spectroscopy in
the scattering volume, a very important practical problem for dynamic light
scattering experiments. Chapter III deals with optical mixing spectroscopy,
and Chapter IV with photon correlation spectroscopy. For completeness, dis­
cussions of Fabry-Perot interferometry are included in Chapter V. The prac­
tice of light scattering is summarized in Chapter VI, and methods of data
analysis surveyed in Chapter VII. This basic book makes no attempt to in­
clude all the diverse applications of PCS, but it does single out the character­
ization of intractable polymers in dilute solution as a special demonstration
of particle sizing in Chapter VIII. Additional summary on the topics covered
in each chapter (I-VII) is presented in the introduction to Chapter VIII. The
scattering behavior of complex fluids, as exemplified by Brillouin scattering,