Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit

HAL Id: jpa-00212514

https://hal.archives-ouvertes.fr/jpa-00212514

Submitted on 1 Jan 1990

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

the multiple scattering limit

D.J. Pine, D.A. Weitz, J.X. Zhu, E. Herbolzheimer

To cite this version:

Diffusing-wave

spectroscopy: dynamic

light scattering

in the

multiple scattering

limit

D. J.

Pine,

D. A.

Weitz,

J. X. Zhu and E. Herbolzheimer

Exxon Research and

Engineering,

Route 22 East, Annandale, New

_Jersey

08801

(Received

on _January 3, 1990,

accepted

in

final form

on _{May 16,}

₁₉₉₀₎

Abstract. 2014

Dynamic

light scattering

is extended to

_optically

thick

(opaque)

media which exhibit

a very

high degree

of

_{multiple scattering.}

This new

_technique,

called

diffusing-wave

spectroscopy

(DWS),

exploits

the diffusive nature of the transport of

_light

in

strongly scattering

media to relate the

_temporal

fluctuations of the

_multiply

scattered

light

to the motion of the scatterers. A

_simple

theory

of DWS, based on the diffusion

approximation

for the _transport of

_light,

is

_developed

to

calculate the

_temporal

electric field autocorrelation functions of the

multiply

scattered

light.

Two

important scattering

geometries

are treated : transmission and

_{backscattering.}

The

theory

is

compared

to

_experimental

measurements of Brownian motion of submicron-diameter

_polystyrene

spheres

in _aqueous

_suspension.

The _agreement between

theory

and

experiment

is excellent. The limitations of the

_photon

diffusion

_{approximation}

and the

_{polarization dependence}

of the autocorrelation functions are discussed for the

backscattering

measurements. The effects of

absorption

of

_light

and

_{particle polydispersity}

are also

incorporated

into the

theory

and verified

experimentally.

It is also shown how DWS can be used to obtain information about the mean size of the

particles

which scatter

_light.

Classification

Physics

Abstracts 05.40

1. Introduction.

Dynamic light scattering (DLS)

has proven to be a

powerful

tool for the

study

of

dynamical

processes in

simple

and

_complex

fluids.

_{Using photon}

correlation

_techniques

to

_analyze

the

temporal

fluctuations of scattered

_light,

DLS

_provides

detailed information about the

dynamics

of the

scattering

medium

_[1,

_2,

_3].

A

_significant

limitation of DLS

_(also

called

quasielastic light scattering

or

_photon

correlation

spectroscopy)

has been the lack of a

_general

scheme for

_applying

it to

_systems

which exhibit

_strong

_{multiple scattering.}

However,

in a

series of recent

_experiments,

DLS has been extended to

_{study optically}

thick media which exhibit a _very

_{high degree}

of

multiple scattering

_[4,

5,

6].

This new

_technique,

called

_diffusing

wave

_spectroscopy

_{(DWS), exploits}

the diffusive nature of the

_transport

of

_light

in

_strongly

scattering

media to relate the

_temporal

fluctuations

_{of multiply}

scattered

_light

to the motion of the scatterers.

_Thus,

DWS can be used to

_{study particle}

motion in concentrated fluids such as

colloids, microemulsions,

and other

_systems

which are characterized

_by

_strong

multiple

scattering [6, 7].

The basic features of DWS have been outlined

_previously

in _{brief reports}

_by

several groups

[4,

5,

6].

Here,

we

_present

a more extensive _report on the theoretical and

experimental

details of DWS.

Before the

_development

of DWS there were numerous

_attempts

to account for the effects of

_{multiple scattering}

in DLS

_experiments.

Most focused on

_systems

which exhibit

_relatively

weak

_{multiple scattering.}

The most successful of these

_{techniques, developed by}

Phillies

_[8]

and Dhont et al.

_{[9] rely}

on elaborate schemes which discriminate

_{against multiply}

scattered

light

so that

_only

the fluctuations of

_singly

scattered

_light

are

_analyzed.

These

_techniques

are

practical,

however,

only

when a

significant

fraction of the scattered

_light

which is collected has been scattered

_just

once.

_By

contrast, DWS relates the

_temporal

fluctuations of

_multiply

scattered

_light

to the

_{underlying dynamics}

of the

_scattering

medium.

_Thus,

DWS extends the

application

of conventional DLS to

_{strongly scattering}

_systems

which are

_completely

dominated

_{by multiple scattering,}

that

is,

to

_systems

which are

_essentially

opaque.

One

_important

consequence of

multiple scattering

is that DWS

probes

particle

motion over

length

scales much shorter than the

_wavelength

of

_light

in the

_scattering

medium,

A. This is to

be contrasted with DLS in the

_{single scattering}

limit,

where the characteristic time

dependence

of the fluctuations is determined

_by

_particle

motion over a

_length

scale set

_by

the inverse wavevector

q - 1 -,A,

_{where q}

=

(4

7r/A )

sin

(0 /2)

and 0 is the

scattering angle.

In

the

_{multiple scattering regime,}

the characteristic time

_dependence

is determined

_by

the cumulative effect of many

scattering

events

_{and, thus,}

_{by particle}

motion over

_length

scales

much less than A.

Therefore,

the characteristic time scales are much faster and the

corresponding

characteristic

length

scales are much

shorter than

for conventional DLS. In the

_following

section we review the

_theory

of DWS based on the diffusion

_{approximation}

for the

_transport

of

_light.

We then

_apply

this

_theory

to two

_important

_experimental

geometries, backscattering

and

_{transmission,}

_{and compare the}

_predictions

of the

_theory

to

experimental

measurements on _{aqueous colloidal}

_suspensions

of

_{polystyrene spheres.}

We also discuss the limitations of the

_photon

diffusion

_{approximation}

for

_describing

the

_transport

of

_light

within the context of our

_{backscattering measurements.}

In

_subsequent

section,

we

show how DWS can be used for

_{particle sizing,}

and discuss the effects of

_absorption

of

light

and

_{particle polydispersity.}

2.

_Theory.

The

_key

to the

_theory

of DWS is the

_description

of the

_propagation

of

_light

in a

multiply-scattering

medium in terms of the diffusion

_{approximation.}

Within this

_approach,

the

_phase

correlations of the scattered waves within the medium ar

_ignored,

and

_only

the scattered intensities are considered

_[10].

Thus,

the

_path

followed

_by

an individual

photon

can be described as a random walk. Whether the diffusion

approximation

is valid in a

_given

_system

depends

on the

_length

scales over which the

_transport

of

_light

is described. For

_length

scales less than the mean free

_{path (or}

extinction

_length),

f,

of the coherent

field,

the wave

_equation

for the electric field must be used. For

_length

scales

_longer

than

_l,

the

_phase

correlation of the

waves must be considered

_only

for

limited,

specialized

cases. The most

_commonly

studied

example

is that of enhanced

_{backscattering}

where the

_phase

coherence of the scattered fields

plays

an essential role

_{[11, 12].}

More

_generally,

it is

_only

in the case of

_extremely

_strong

scattering,

where the

_Ioffe-Regel

limit is

_approached,

and

_light

localization is

_predicted

to occur, that the use of the full wave

_equation

is essential

_[13].

_However,

achieving

this

criterion has proven elusive

and,

for

_nearly

all cases of

_practical

_interest,

the

_transport

of

light

can be described in terms of the

_intensity

or _energy

_density.

over a distance

f.

In this case the

_transport

of the

_light

_energy

density, U(r, t ),

can be

described over

_length

scales

_longer

than

f

_by

the diffusion

equation

where

_De

is the diffusion coefficient of the

_light.

_{Then, Dl}

=

cf/3

where c is the

_speed

that the

light

travels

_through

the effective

_background

medium. For

_particles

which are not small

compared

to

_A,

the

scattering

is

_anisotropic,

and the mean number of

_scattering

events, no,

required

to randomize the direction of

_{light propagation}

is

_greater

than one. The

_length

scale over which this randomization occurs is the

_transport

mean free

path, l *,

defined

_by

e * = no f.

Thus,

for

_{no > 1,}

the diffusion

_equation

is valid

_only

over

_length

scales

_longer

than

f*

and

_Dl

=

cf*/3.

In

practice,

this _means that the

theory

for DWS based

solely

on the

diffusion

_{approximation}

will

_provide

an accurate

_description

of the

_temporal

fluctuations of the scattered

_light

when the

sample

dimensions are

_greater

than f*

and the distance that

protons

travel

_through

the

_sample

is much

greater

than

l*.

Two theoretical

_approaches

have been used to describe the fluctuations in the scattered electric field. The first

_approach

starts from the wave

_equation

for the

_propagation

of

_light

and uses a

_{diagrammatic expansion}

to describe the

_scattering

from random fluctuations in the dielectric constant. The lowest order term in this

_{expansion corresponds}

to the diffusion

approximation.

The consequances of motion of the scatterers were first considered

_by

Golubentsev

_[14]

and

_temporal

autocorrelation functions for different

_{scattering geometries}

were calculated

_{by Stephen [ 15]. Subsequently,}

MacKintosh and John

[16]

refined this

approach by considering

the effects of

_anisotropic

scatterers and non-diffusive modes of

propagation

of the

_light.

The second theoretical

_approach

differs from the first

_by

_considering

the diffusive

_transport

of individual

_{photons directly,}

rather than

_starting

from the wave

_equation.

The

_path

of each

photon

is determined

by

random,

multiple scattering

from a _sequence of

particles.

The

fluctuations in the

phase

of the electric field due to the motion of the scatterers is calculated for each

_light

_path.

The contributions of all

_{paths, appropriately weighted}

within the

_photon

diffusion

_{approximation,}

are then summed to obtain the

temporal

autocorrelation function. It is

_{mathematically simpler}

and

_physically

more

_transparent

than

_diagrammatic

methods. It can

also be more

_{easily generalized}

to

_{larger particles}

which scatter

_{light anisotropically.}

This

approach

was first

_{developed by}

Maret and Wolf

_[4]

to

_{analyze temporal}

corrélation functions of

_light

backscattered from dense colloidal

_suspensions.

Pine et al.

_[6]

extended their

_analysis

to transmission

_geometries

and calculated

_{explicit expressions}

for the correlation functions. This work was

_recently

reviewed

_by

Pine et al.

_[17].

In section

_2.A,

we follow the earlier

works of Maret and Wolf

[4]

and Pine et al.

_[17]

and review the

_theory

which leads to a

general expression

for the

temporal

correlation function of

_multiply

scattered

_light.

In

section

2.B,

we discuss the calculation of the distribution of

_{photon path lengths through}

the

sample using

the

_photon

diffusion

_{approximation}

and in section 2.C we discuss the method we use to calculate

_{explicit expressions}

for the

_temporal

autocorrelation functions we measure.

2.A. CORRELATION FUNCTION FOR MULTIPLY-SCATTERED LIGHT. - For

_simplicity,

we

consider a

_suspension

of identical

_{spherical particles randomly dispersed}

in a fluid and

illuminated with laser

_light

of

_wavelength

À. For

_specificity,

we assume a slab

_geometry,

as

shown in

figure

la ;

other

_geometries

will be discussed later. The

_suspension

must be

_optically

thick so that all the

_{light entering}

the

_sample

is scattered many times before

exiting.

Scattered

light exiting

the

_sample

is collected from a small area near the

sample

surface

(within

Fig.

1. - Transmission

geometry for a

_point

source.

_(a)

A

_point

source of

_light

is incident on the

_sample

on the left side and is collected from a

_{point directly opposite}

the

_input

point

on the

_right

side. Pinhole

apertures

(A)

collimate the scattered

light. (b) 1 (t)

vs t for transmission of a delta function

_pulse,

& (t), through

a slab of thickness L The

_plot

can also be

interpreted

as

_{P (s) vs s}

since the fraction of

photons travelling

a distance s

through

the

sample

in a time t =

s/c,

where c is the

velocity

of

_light

in the

sample.

Here, L = 1

mm, f * =

100 _im, and c = 2.25 _x

10’° cm/s (the speed

of

light

in

water).

Note that

_{P (s ) decays exponentially}

for

_large

s.

the fluctuations of the scattered

_light.

In DLS

_experiments,

these fluctuations are

_usually

characterized

_by

the normalized

_temporal

autocorrelation

_function,

where

_{E(T )}

is the electric field of the scattered

_light

collected

_by

the detector and T is the

delay

time. To calculate

_g1(,r),

we follow the

_approach

of Maret and

Wolf [4]

and first

consider a

single light path through

the

_sample.

For a

_given

_{sequence of}

_scattering

events at

positions

rl, ..., r 11 and with

corresponding

successive wavevector transfers qi =

ki - ki _ 1,

the

product

of the scattered electric field at time T with that at time 0 will be

where

_{Ar i (,r )}

=

ri ( T) - r i (0)

is the distance the i-th

particle

_moves in a time T. The

_argument

of the

_exponential,

represents

the

_aggregate

_phase

shift of the scattered

_light

due to the motion of all n scatterers

Here

_{P (n)}

is the fraction of the total scattered

_{intensity, Io, in}

the n-th order

paths

and

( )

denotes averages over both the

_{particle displacement, Or ( T ),}

and the distribution of

wavevectors _{q. For}

_large

_n, that

_{is n » n (),}

the

_transport

_{of light through}

a

_sample

is

_accurately

described within the diffusion

_{approximation.}

This allows

_P(n)

to be

_easily

evaluated.

Furthermore,

the

_large

number of

scattering

events in each

_{path greatly}

_simplifies

the evaluation of the averages in

equation

(2.5).

For

_large

n, we can relax the condition that the sum of the intermediate

_scattering

vectors

n

must

_equal

the difference between the incident and

_exiting

_{wavevectors, ¿}

_{qi =}

kn - ko.

In

i

addition,

we also assume that successive

scattering

events are uncorrelated.

Thus,

the

distribution of the qi is determined

_{solely by}

the

_scattering

form factor of a

single sphere.

Then,

the

_right

hand side of

_{équation (2.5)}

becomes

_{P (n) (exp [- iq -Ar (T ) ] > n.}

For Brownian

_motion,

the distribution of Ar

_{(T )}

is _{Gaussian and the average} over

particle

motion

is

_{readily performed yielding}

where >q

denotes the form-factor

_weighted

average over _q. For

_simple

_diffusion,

(âr2( T) =

6 DT where D is the diffusion coefficient of the

_particles.

The average over q can be

performed explicitly

for

point-like

scatterers which scatter

_light

isotropically ;

this

_gives

where

_{To - (Dk))}

and

_A;o=27r/A.

However,

for

_{larger particles}

which scatter

_light

anisotropically,

the _{average over q} is more difficult to

_perform.

_Thus,

we restrict ourselves to

small

_delay

_times,

r _{T o,} and retain

_only

the first term in a cumulant

expansion,

We can _compare this more

_{general approximation}

with the exact

_{expression, equation (2.7),}

for

_point-like

scatterers

_{by evaluating}

_(q2)

for a flat

_{distribution,}

where dn denotes the solid

angle.

This

gives,

Expanding

the exact

_expression

in

_equation

_(2.7)

_gives

so that the cumulant

_{expansion gives}

the correct results to

_leading

order in

_{T / T o.}

Furthermore,

the effective

_decay

time is

_{To/2 n,}

so that for

_large

n,

(E(n)(T)E(n)*(O»

Nevertheless,

for

_large

_{T / To}

the cumulant

_expansion

must fail since the exact

_expression

in

equation (2.7) decays

as a power

law,

( T / T o )- n,

due to the contributions

_{of scattering}

at small

q. We note,

however,

that at these

_long

times low order

_scattering

events dominate. For small

n

n, the distribution

of ql

is not random but must

_satisfy

the condition

_{that £}

_{qi =}

_{kn - ko.}

This

i

will

_modify

_{the average in}

_{equation (2.8)}

and could have the effect of at least

_partially

offsetting

the error introduced

_by

the cumulate

_expansion.

Equation

(2.8)

can be

_{readily generalized}

to treat the

_anisotropic

scatterers

_typically

used in

experiments.

Here,

the

_{single particle scattering intensity}

is

_peaked

in the forward direction

and q2>

is less than

_{2 ko.}

_Using

the

_{relation 1 - cos}

_{0 ),}

_[10,

_18]

we have

With this result and the cumulant

_expansion

in

_{equation (2.8),}

we can evaluate the average over q in

equation (2.6)

and obtain a

_{simple expression}

for the contribution of the

n-th order

_{scattering paths}

to the autocorrelation

_function,

For

_isotropic

scatterers,

f

=

l*,

and we recover

_{equation (2.10),}

the

approximate

result for

point-like

scatterers. We note that the accuracy of

equation (2.13) improves

as the

scattering

becomes more

_anisotropic.

The second cumulant of

e- Dq 2 r) q is

e - ! 2 (Dr )2(

q4) - q2)

2

. For

typical single-particle

form

_factors,

_{F (q ),}

we

expect q4) -- q2) 2 --

(f 1 f * )2,

so that the

leading

order correction to the cumulant

_expansion

in

_equation

_(2.8)

is

_{0 [( TITo)}

_{(f If}

_{* )2].}

Thus,

for

_large

scatterers, where

_{$ * > Q,}

_{the range}

_{of validity}

of

_{equation (2.13)}

may extend up to T --

T o.

The total field autocorrelation

function,

g1(,r),

is obtained

_{by summing}

over

_scattering

paths

of all orders

If we rescale n

by defining

n * - n

/no

=

(f/f*)

_n, the

exponential

term in this

equation

is

identical to

_{equation (2.10)}

for

_isotropic

scatterers.

_Furthermore,

for _{n > n 0,}

_{P (n )}

can be

written as a function of n* rather than n.

Physically,

this reflects the fact that over

_length

scales

_longer

than

_l*,

the

_scattering

appears to be

_isotropic

and the

_photon

diffusion

_equation

should be

_adequate

to describe the

_transport

of

_light.

As a

result,

in this

approximation,

This reflects a renormalization of the mean free

_path

for

_anisotropic

scatterers, so that the

light

may be viewed as

_undergoing

an

_isotropic

random walk with an _average

_step

_length

f*.

We note that the

_decay

rate of a

_{given path depends critically}

on its

_{length. Long paths}

reflect the

_aggregate

_{contribution of many}

_scattering

events.

_Thus,

each

_particle

need move

_only

a

small distance for the total

_{path length}

to

_{change by}

a

_wavelength.

This occurs in a short

_time,

leading

to a

_{rapid decay}

rate.

_By

_{contrast, short}

_paths

reflect the contribution of a small number of

_scattering

events.

_Thus,

each

_particle

must

_undergo

substantial motion for the total

The

_key

to the solution of

_{equation (2.15)}

is the determination of

_P(n*).

This task is

greatly simplified

if we _pass to the continuum limit.

Thus,

we

_approximate

the sum over n*

_by

an

_integral

over the

_{path lengths,}

s = n * f * ( = ni ),

and obtain

In this case, P

(s)

is the fraction of

_photons

which travel a

_path

of

_{length s through}

the

scattering

medium.

_{Physically, equation (2.16)}

reflects the fact that a

_{light path}

of

_{length s}

corresponds

to a random walk of

slf*

_steps

and that

_{g1( T ) decays,}

on _average,

exp (-

2

_{T /To)}

_per

_step.

Now we can use the diffusion

_{approximation}

to describe the random walk of the

_light,

and

_{P (s)}

can be obtained from the solution of the diffusion

equation

for the

appropriate experimental

geometry.

We

_emphasize,

_however,

that the continuum

approxi-mation to the discrete

_scattering

events will break down whenever there are

_significant

contributions from short

_{light paths.}

2.B. BOUNDARY CONDITIONS AND CALCULATION OF

_{gl ( T )}

FROM

_{P (s).}

- Our method for

determining P (s)

can be illustrated

_{by considering}

a

_{simple experiment.}

An instantaneous

pulse

of

_light

is incident on the face of a

_sample

which

_multiply

scatters the

_light.

The scattered

_photons

will execute a random walk until

_they

_escape the

_sample.

_Here,

we focus on

those

_photons

which exit the

_sample

at some

specified point,

_rd, on the

_boundary

where

_they

are detected. The flux of

_{photons reaching}

rd is zero at t =

0,

then increases to a

maximum,

and

_finally

decreases back to zero at

_long

times when all the

_photons

have left the

_sample

_(see

Fig.

1 b).

At time t, the

photons arriving

at

_point

_rd are those that have traveled a

_{path length}

s = ct

through

the

sample.

Thus,

the flux of

photons arriving

at

point

_rd at a

_given

time is

directly proportional

to

_P(s).

The details of this time

_dependence,

and hence the

shape

of

P(s),

will

_depend

on the

_experimental

_geometry.

To obtain

_{P (s)}

for a

_{given experimental}

_geometry

we use the diffusion

_equation

for

light

equation (2.1),

to determine the

_dispersion

induced in a delta function

_pulse

as it traverses the

scattering

medium. We denote the

density

of

_{diffusing photons}

within the medium

_by

U(r ).

For initial

_conditions,

we

_exploit

the fact that the direction of the incident

light

is

randomized over a

_length

scale

_comparable

to

f*

and therefore take the source of

_diffusing

light intensity

to be an instantaneous

_pulse

at a distance z =

zo inside the illuminated

face,

where we

_expect

that _{zo -}

f *,

that is

In sections 3 and

_4,

where we calculate

_{explicit expressions}

for the

temporal

autocorrelation functions for transmission and

_{backscattering geometries,}

we discuss the consequences of

approximating

the source term with a delta function. ’

In addition to the initial

conditions,

the

_boundary

conditions must ensure that there is no

flux of

_{diffusing photons entering}

the

_sample

from the

boundary.

Thus,

outside a

_sample

consisting

of a slab of thickness L between z = 0 and _z =

L,

where J+

(z)

and J

(z)

are the fluxes of

_{diffusing photons}

in the + z and - z

directions,

respectively.

In the

_appendix

we show

_that,

within the diffusion

_{approximation,}

the fluxes of

where J±

are taken to be

_{always positive.}

The

Uc/4

term will dominate the total flux in a

_given

direction in the interior of the

_sample

where the

_density

of

photons

is

_high.

However,

the net

flux in a

_given

direction will be dominated

_by

the

_gradient

_term, consistent with Fick’s law

To obtain the

_boundary

conditions,

we consider the

sample

interface at z = 0 where

J+ =

0 and J is

_{given by equation (2.19).}

Hence,

J(z

=

0) = 2013

J_

(0)

and the _net flux is

directed out of the

_sample.

Inside the

_sample,

a

distance £*

_away from the

_boundary,

the total

flux is

_{given by equation (2.20).}

Since

Î*

is the minimum distance over which the

_intensity

can

change

within the diffusion

_{approximation,}

these two

_expressions

for the flux near the

boundary

must be

_{equal :}

Simplifying

this

_expression,

we obtain the

_boundary

condition,

Alternatively,

this

_boundary

condition can be obtained

directly by setting

the

expression

for

J+

in

_{equation (2.19) equal}

to zero

_[19].

Similar

procedures

can be used to derive a

_boundary

condition at z = L. In

general,

the

boundary

conditions at either interface are

_{given by}

where n is the outward normal

_[10].

We

_emphasize

the

_approximate

nature of these

boundary

conditions which were obtained

_by

requiring

that the net inward flux of

_{diffusing photons}

be

zero

_[20].

More

_generally,

the flux must be zero for all

_incoming

directions.

However,

this

more

_stringent

condition can

_only

be met be

_{going beyond}

the diffusion

_{approximation}

to a more

_general

_transport

_{theory [10, 21].}

Using

the

_boundary

conditions in

_{equation (2.23),}

we can solve the diffusion

_equation

for

U(r, t )

for a

_{given experimental}

_geometry

_[22]

and then calculate the

_{time-dependent}

flux of

light emerging

from the

_sample

at time t from

_{equation (2.19)}

evaluated at the exit

_point.

All

light emerging

from the

_sample

at time t has traveled a distance s = _ct.

Thus,

the fraction of

light, P(s),

which travels a distance s

_through

the

_sample

is

_{simply proportional}

to the flux of

light emerging

from the

_sample

at time t =

sic:

where rd denotes the exit

point

of the

_light

and,

again,

n is the outward normal. The last

equality

in

_{equation (2.24)}

follows from the

_boundary

condition,

equation (2.23). Having

determined

_{P (s )}

for a

_given

_geometry,

_{équation (2.16)}

can be used to calculate

_{g1( T ) .}

2.C. DIRECT CALCULATION OF

_{g1( T ) .}

- The calculation of

_g1(T)

can be

greatly simplified

by noting

that

_{equation (2.16)}

is

essentially

the

_Laplace

transform of

P (s).

Thus,

instead of

diffusion

_equation

and obtain

_{g1(T) directly.}

To this

end,

we introduce a

simple change

of

variables in

_{équation (2.1)}

and let

_{t = s/c.}

Recalling

that

_Dt

=

cl*/3,

equation (2.1)

becomes,

Multiplying

both sides

_{by exp (- ps )}

and

_integrating

with resect to s from 0 to _oo, we obtain

the

_Laplace

transform of the diffusion

_equation,

where

Û

=

U(r,

p )

is the

Laplace

transform of

U(r, s ),

and

_{Uo(r )}

= lim

Uo(r,

t )

= à

(z -

zo, t ) [22]. Equation (2.26)

can be solved

_subject

to the

t -> 0

Laplace

transform of the same

boundary

conditions used

previously equation (2.23), using

Green’s function

_techniques.

This method is

_carefully

discussed

_by

Carslaw and

_Jaeger

[22]

where

_explicit

solutions to the diffusion

_equation

and its

_Laplace

transform are

given

for most

geometries

of

experimental

interest. The solution to

_{equation (2.26)}

can be related

_directly

to

g1(,r) by taking

the

Laplace

transform

of P (s )

and

_{using equation (2.24)}

to write

Comparing equations

(2.16)

and

(2.28),

we see that

where we have made the

_{identification, p}

=

(2

T

/To)/l*,

and

U(r,p)

has been normalized

so that

_g1(0)

= 1.

Thus,

by solving

the

Laplace

transform of the diffusion

equation

we can

obtain the desired autocorrelation function

_directly.

We illustrate this method of solution in sections 3 and 4 with

_explicit

_examples

for transmission and

_{backscattering geometries.}

Finally,

for

_comparison

to our

experiments,

we note that we

usually

measure normalized

intensity

autocorrelation

_{functions, (/(T) /(0) / (/(0)2 =}

1 +

_/3g2(T),

where

_/3

is deter-mined

_{primarily by}

the collection

_optics

and is defined so that

_g2(o)

= 1. For _most

_systems

of

experimental

interest,

the scattered electric field is a

complex

Gaussian random variable and

the

_intensity

autocorrelation function is related to the field autocorrelation function calculated above

_by

the

_Siegert

relation,

Wolf et al.

_[18]

have

_{experimentally}

verified that there is no observable deviation from Gaussian statistics for

_strong

_{multiple scattering}

in

_samples

similar to ours. In

_addition,

_by

measuring g1(T) directly

in

_{heterodyne experiments,}

we have verified the

_validity

of

3. Transmission.

The first

_experimental

_geometry

we consider is transmission

_through

a slab of thickness L and

infinite extent

_[6,

_15].

The

_light

is incident on one side of the

slab,

and is detected after it has diffused a distance L across the

_sample.

The incident

_light

beam can either be focused to a

point

on the

face,

as illustrated in

_figure

_la,

or

_expanded

to

_uniformly

fill the face of the

_slab,

as illustrated in

_figure

2. The autocorrelation functions for these two

_{experimental geometries}

are different and can be calculated

_{by solving}

the

_Laplace

transform of the diffusion

_equation

as

_previously

discussed. In both cases,

wc

take the source of

_{diffusing intensity}

to be a

distance z = _zo inside the illuminated face where we

_expect

that

_{zo - 1 *.}

Fig.

2. - Transmission

geometry for an extended source. An extended source of

light (plane wave)

is incident on the

_sample

on the left side and is collected from a

_{point directly opposite}

the center of the

input

beam on the

right

side. A -

pinhole

apertures.

We first consider the case of uniform illumination of one side

by

an extended source so that

U(x, y, z, t ) = & (z - zo, t ).

The solution of the diffusion

_{equation using}

the

_Laplace

transform is

_given

in Carslaw and

_Jaeger

_{[22] (§ 14.3)}

for this

geometry

and for these

boundary

and initial conditions.

_They

obtain

where

_{a2 ~ 3 p /Q * = 6}

_{" 1"0 f *2 and}

_Di

=

cf*/3.

The coefficients A and B are chosen so

that the

_boundary

_conditions,

_{equation (2.23),}

are satisfied at z = 0 and z = L. After

tedious,

but

_{straightforward algebra,}

we obtain

where the second

_expression

holds for r « _{T o.} The characteristic time scale in these

For the second

_geometry,

we consider

_light

incident from a

_point

source on axis with the

detector so that

_U(x,

_{y, x,}

_{t )}

= 8

_(x,

_y, z -

z0, t).

The solution for this

_geometry

and these

boundary

and initial conditions can

_again

be found in Carslaw and

_Jaeger

_[22]

_{(§ 14.10) :}

where q

_ 2 + a 2,

a 2 == 3 p / f *

= 6

’T / ’T 0 f *2,

and R = 0 when the source is on axis with

the detector. The coefficients A and B are

again

found

by

applying

the

boundary

conditions. We

_obtain,

where

and e =

2 f*/3

L.

Again,

the characteristic time scale for the

decay

of the autocorrelation

function in

_{equation (3.4)}

is

_{To(f * / L )2.}

This time scale reflects the diffusive nature of the

transport

of the

_light,

which introduces a characteristic

path length,

sc = n, n c * i *,

where

n c

(L/f*)2

is the mean number of

_steps

for a random walk of end-to-end distance L and average

step

length

f*.

Physically,

this characteristic time scale

_corresponds

to the time taken for the characteristic

_{path length}

to

_{change by - k,}

so that the total

_phase

shift is

roughly

unity.

Thus,

we can estimate the

_typical

distance an individual

particle

has moved from nc

(q2) (âr2)

.--

1,

which

gives âr rms

=

k f

*

/2

TT’ L. Since

f

*IL «

1,

the

typical length

scale

over which

_particle

motion is

_{probed by}

D W,S in transmission is much smaller than the

wavelength.

This reflects the fact that the

_decay

of the autocorrelation function is due to the cumulative effect of many

scattering

events, so that the contribution of individual

_particles

to

the total

_decay

is small. This is in

_striking

contrast to

_{ordinary dynamic light}

_scattering

where,

by

varying

q,

length

scales greater than or

roughly equal

to the

_wavelength

are

probed.

Similarly,

the time scale over which

_particle

motion is

_probed

with DWS is much shorter than with DLS.

_Thus,

_{by adjusting}

the

_sample

thickness L in transmission measurements, DWS

provides

a means for

_varying

the _range of

_length

and time scales over which

_particle

motion

can

be measured

using dynamic light scattering techniques.

Another

_significant

difference between DLS and DWS is the

_dependence

on

_scattering

angle.

In

_DLS,

_varying

the

_{angle changes}

the

_scattering

vector, q, thus

varying

the

length

scale over which motion is

_{probed. By}

_contrast, in

_DWS,

the diffusive nature of the

_transport

ensures that the transmitted

_light

has no

_{appreciable dependence}

on

_angle

and we do indeed find that the measured autocorrelation functions are

_independent

of detection

_angle

in

transmission.

Autocorrelation functions measured in transmission are shown in

figure

3. The

sample

consisted of an _aqueous

_suspension

of

0.605-jjLm-diam.

polystyrene

latex

_spheres

at a volume

fraction

_{of .0 =}

0.012 in a 1.0 mm thick cuvette. There is no unscattered

light

transmitted

through

the

_{sample, insuring}

that the

strong

multiple scattering

limit is achieved. The data in the lower curve were obtained when the

_sample

was illuminated

_{uniformly by}

al-cm diameter beam from an _argon ion laser with À o = 488 nm.

_{Imaging optics}

collected the transmitted

Fig.

3.

_{- Intensity}

autocorrelation functions vs time for transmission

through

1-mm thick cells with

0.605-fJ.m-diameter polystyrene spheres and 0

= 0.012. The _{upper curve} is for _a

point

_source

(see

Fig. la)

and the lower curve is for an extended

_plane

wave source

_{(see Fig. 2).}

Smooth lines are fits to

the data

_{by equations (3.2)}

and

_(3.4)

with f * = 167 ktm for the extended source and f* = 166 ktm for

the

point

source.

beam _was focused to a

_point

on one side of the

sample

and transmitted

_light

was collected

from a

point

on axis with the incident

_spot.

These data

clearly decay

more

_slowly

than the data obtained with an extended source illumination.

Physically,

this difference reflects the fact that

for the extended source there is a

_larger

contribution from

_{long paths, resulting}

in a faster

decay

than for the

_point

source.

To compare these data to the

_expression

for the autocorrelation function derived

above,

we

take zo =

(4/3)

f*,

but note that the solutions _are insensitive to the exact value used

since,

in

general,

zo - f *

and

_f*/Z.

1. _{The value chosen for zo affects}

_only

the first few

_steps

of a

random walk which

_typically

consists of a

_great

number of

_steps.

Thus,

the relative contribution of the first few

_steps

is small. In

_figure

_3,

the solid lines

_through

the data are fits

to the

_{appropriate equations}

above. The time constant _{To -}

(Dkô)-1

1 is set

_equal

to 4.52 ms

where D was obtained from a DLS measurement in the

_{single scattering}

limit

_using

.0 =

10-

5.

The diffusion coefficient D remains

_independent

of concentration for the

_particle

concentrations used in these measurements. For both _cases, excellent

agreement

is found between the data and the

_predicted

forms

_{of g2 ( T ) .}

The

_only

_fitting

parameter

is f*

and values

of f*

= 167 _)JLm for the extended source and

f*

= 166 _tm for the

point

source are obtained.

This excellent

_consistency

for

the fitted

values of

f*

confirms that the different

_decay

rates in

figure

3 are due

_solely

to

_geometric

effects. The values obtained from the fit to the data are in reasonable

_agreement

with Mie

_scattering

_theory,

f*

=

180 ktm. Nevertheless,

there is a

discrepancy

of about 7

_%,

with the measured values lower than

_predicted.

This in not

experimental

error, as a similar

_discrepancy

is found for other

_polystyrene

latex

_samples

with different sizes and different volume fractions. The values obtained from DWS measurements

in transmission are

consistently

lower than the

predicted

values,

_{typically by}

about 10 %. This

discrepancy

is

_probably

not due to _{the consequences} of

_particle

correlation on the value of

f*.

The

_particle

interactions in these

_{polystyrene samples}

can be

_approximated

as hard

_sphere

repulsion, allowing

the

_particle

correlations to be determined

_[4].

These are

_relatively

weak

for çb 0.1,

leading

to

_negligible

effect on

f*. Instead,

we believe that the

_discrepancy

arises from the

_approximate

nature of the

_boundary

condition that we use,

equation (2.23).

In

condition and will result in a renormalization of the value of

f*

obtained from DWS transmission measurements. If this effect is not

_included,

the

_{apparent f*}

measured will be less than the correct value.

While the autocorrelation functions shown in

_figure

3 are

_clearly

not

_{single exponentials,}

their curvature in the

_{semi-logarithmic plot}

is not

_large.

This reflects the existence of a

dominant characteristic

_{path length}

and time scale for these

_data,

as

_expected.

This also

suggests

the

_suitability

of

_analyzing

the data

_by

means of the first cumulant

_FI,

or the

logarithmic

derivative at zero time

_delay.

For

_{equation (3.2),}

the first cumulant is

_{given by}

For

_{equation (3.4),}

the first cumulant must be evaluated

_numerically.

Measurements of

fl

as a function of

_sample

thickness are shown in

figure

4 for several volume fractions.

These data were obtained

_{using polystyrene}

latex

_spheres

with a diameter of 0.497 jjbm. The incident beam was focused onto the face of a

_wedge-shaped

_cell,

_allowing

the thickness to be varied

_{by translating}

the cell in the beam. The

_angle

of the

wedge

was about

3 ;

so the cell

shape

had very little effect on the form of the autocorrelation function. The data in

_figure

4 show the

_expected

linear

_dependence

of

T

1 over a broad range of concentrations

confirming

the diffusive nature of the

_transport

of

_light.

It is

_apparent

that the data do not

extrapolate through

the

_origin

but that the

_L-intercepts

are a

_{monotonically decreasing}

function

_{of particle}

concentration and consistent with

_{equation (3.5),}

to within

_experimental

uncertainty.

The different

_slopes

are due

_primarily

to the concentration

_dependence

_{of f*,}

which is

_expected

to scale

_inversely

with

_{particle density}

for the volume fractions used here.

Fig.

4. -

Square

root of the first cumulant

_Fi

1 of the

intensity

autocorrelation function vs

sample

thickness L. The data were obtained

using

the

point-source

transmission geometry for 0.497-fJ.m-diameter

_spheres

with volumes fractions of 1 %, 2 %, 5 _%, and 10 %. The linear

dependence

shown in the

_plot

confirms the diffusive nature of the _transport of

_light.

4.

_{Backscattering.}

The second

_geometry

we consider is that of

_{backscattering [4,}

_{5, 6,}

_15].

_Here,

the

light

is

from the same

_face,

as illustrated in

figure

5a. In

deriving

a functional form for the

autocorrelation

_function,

we would like to maintain the

_simplicity

and

_elegance

of the

_Laplace

transform

_approach

used in the case of transmission.

However,

we must be

_extremely

cautious in

_using

this

_approach

for

_{backscattering.}

In

_obtaining

the

_Laplace

transform in

equation (2.16),

we have used a continuum

approximation

to

_change

the summation over

paths

in

_equation

_(2.15)

to an

_integral.

_However,

since s =

nf

and we must have at least one

scattering

event, we

_{require n -- 1 ;}

_thus,

the lower bound on s must

be f

rather than 0.

Equivalently,

for

_isotropic

scatterers, where

f

_{= f*,}

the

_decay

time for a

_path

of

_{length s}

is

exp[-

(2 7-/70) sli].

Allowing

s Q

leads to

_{unphysically long decay}

_times,

since

_To/4

represents

the shortest

_decay

time

_{possible, corresponding}

to

_{light singly}

scattered

_through

180°.

_Similarly,

for

_anisotropic

scatterers, where

_{f * > f,}

we

_require

s >1*

to obtain

physically meaningful decay

times.

However,

in order to maintain the

_simplicity

of the

Laplace

transform,

the lower bound on the

_integral

must be zero. For

transmission,

this does

not

_present

a

_problem

since the shortest

possible paths

are s =

L,

_so that n > 1.

By

_contrast,

for

_{backscattering,}

short

_paths

do contribute and

thus,

in

_using

the diffusion

_{approximation,}

we must ensure that the contribution of the

unphysically

short

paths

is

suppressed.

A

_simple

_way of

_achieving

this is to solve the diffusion

_{equation using}

an initial condition of a source a fixed

distance,

zo, in

from the illuminated face at z = 0. This ensures that there are no contributions from

_paths

shorter than _zo.

_Physically

we can

_regard

this as the source of the

diffusing intensity,

which we

_expect

to be

_peaked

at

_{zo - f *.}

_Thus,

for the initial condition we

take

_{U(r, t = 0)}

= 5

_{(x, y,}

z -

_{Z 0, t)}

and for the

_boundary

conditions

_again

we use

equation (2.23).

The solution for this

_geometry

is the same as for the case of transmission with

an extended source, but here we evaluate it at the incident face and

obtain,

For a

_sample

of infinite

_thickness,

_{equation (4.1) simplifies,}

We note that the initial condition of a source at _{zo appears}

_explicitly

in the

solution,

reflecting

the

_importance

of the contribution of the short

paths.

In

_fact,

we

_expect

there to be a distribution in the

_position

of the

_apparent

_{source, zo.} Thus we must

_{integrate equation (4.2)}

over a distribution of sources,

f (zo).

While the exact form

_{of f (zo)}

is

_unknown,

we do know

that f (zo)

goes to zero as

zo «-c f *

and for _{zo >}

f *. Furthermore,

we

_{expect f (zo)}

to be

_peaked

near

zo - f *.

_Thus,

to

_leading

order in

_{-B/ 7- / 7- 01}

_{g 1 (r)}

is

_{given by equation (4.2),}

with

Zo/f*

replaced by

(zo)

/f*,

the _average over the source distribution

_{f (zo).}

The first order

correction is

_{[(zJ) / (zo) 2 - 1] TITO,}

which is small for a narrow distribution

_{f (zo).}

Physically,

we can think of

(zo)

as _{the average}

_position

of the source of

_{diffusing intensity,}

and we

_expect

that the exact form of the distribution

_{f (zo),}

and therefore

_(zo),

will

_depend

on the

_anisotropy

of the

scattering

as reflected

_{by f */Î.}

In contrast to

_{transmission,}

in

_{backscattering}

there is no well-defined characteristic

_path

length

set

_by

the

_sample

thickness. In

_fact,

since

_paths

of all

_lengths

contribute in

Fig.

5. -

Backscattering

geometry for an extended source.

_(a)

An extended

_plane

wave source of

_light

is incident on the

_sample

from the left side and is collected from a

_point

near the center of the illuminated area.

_{(b) 1 ( t ) vs t}

for

backscattering

of a delta function

pulse,

8

_{( t ),}

from a semi-infinite slab of thickness L. The

_plot

can also be

_{interpreted as P (s)}

vs s since the fraction of

_{photons travelling}

a

distance s

through

the

sample

in a time t =

s/c,

where c is the

velocity

of

light

in the

sample.

Here,

L = 1

mm, f*=100

_JLm, and c = 2.25 _x

10 10 cm /s

(the speed

of

light

in

water).

Note that

,P (s)

decays

as

s-3/2

for

_large

s.

multiple scattering.

As a _consequence, there is a much broader distribution of time scales in the

decay.

The

_{longer paths}

consist of a

_larger

number of

_scattering

events and thus

_decay

more

_{rapidly, probing}

the motion of individual

_particles

over shorter

_length

and time scales.

By

contrast, the shorter

_paths

consist of a smaller number of

_scattering

events thus

_decaying

more

_slowly

and

_probing

the motion of individual

_particles

over

_{longer length}

and time scales.

This feature is

_{particularly advantageous}

for

_exploring

the

_dynamics

of

_interacting

_systems,

which can have a broad distribution of relaxation rates associated with motion over different

lengths

scales.

An autocorrelation function

measured

in

_{backscattering}

is shown in

_figure

6. The

_sample

consisted of an aqueous

suspension

of

_{0.412-J-Lm-diam.}

_polystyrene

latex

_spheres

at a volume

fraction

_{of 0}

= 0.05 in _a 5 mm thick cuvette. It was illuminated

_by

a uniform

beam,

1 cm in

diameter ;

light

from a 50 _tm diameter

_spot

near the center of the illuminated area was

imaged

onto the detector. The

_{scattering angle}

was -

175°,

although

we found

virtually

no

dependence

of the results on

_{scattering angle}

when it was varied - 20° from that used

here,

as

expected

for

_{diffusing light.}

After

_subracting

the

baseline,

the

_logarithm

of the normalized autocorrelation

_function,

is

_plotted

as a function of

time,

in

_figure

6a. The

_large

amount of

curvature exhibited

_by

the data reflects the contributions of

_paths

with a wide range of

_length

scales,

leading

to a wide _range of

_decay

times. As

_{suggested by equation (4.2),}

the

expression

for

_{g 1 (,r )}

for DWS in

_{backscattering,}

we

_replot

the data in

_figure

6b as a function of the

square root of

time,

in units of _To. We use _To = 3.01 _ms, as measured

experimentally

in the

single scattering

limit,

and consistent with the value calculated from the Stokes-Einstein relation. The solid line is a fit to the functional form

_{given by equation (4.2),}

and is in

reasonably good

agreement

with the data.

However,

to within the

_precision

of

_experiment,

the data shown in

_figure

6 is linear over

_nearly

three decades of

_decay

when

plotted

Fig.

6.

- Intensity

autocorrelation function for

_{backscattering}

from a 5-mm thick cell

_containing

0.412->m-diameter polystyrene spheres with *

= 0.05.

(a)

Data is

plotted

_{as a} function of reduced time. A

large

degree

of curvature is evident.

(b)

Data is

_plotted

as a function of the square root of reduce time. The dotted line

_through

the data is a fit to the more exact form in

_{equation (4.2).}

The dashed line

through

the data is a fit to the

_simpler

form of

_{equation (4.3).}

The residuals are shown above as a

function of the square root of the reduced time. The

_{simpler exponential decay}

with square root of time

provides

an excellent

_description

of the data.

where y =

(zo > /f *

₊

2/3.

This result _was obtained

previously using

the

_simpler,

but less

rigorous, boundary

condition,

U(z

=

0)

= 0

[6, 23].

The behavior of the autocorrelation function shown in

_figure

6 is in fact

_{quite general.}

For

particles

with diameters

_comparable

to or

_greater

than

À,

where

_{i * / f »}

_1,

the

_simple

exponential

form

_given

in

_{equation (4.3)}

is observed. For small

_particles,

where f

*

/i -

1,

there is some curvature for data

_{plotted logarithmically}

vs _square root of time. The data curves

_upward

or

_downward,

_depending

on whether

_polarized

or

_{depolarized light}

is detected

_[24].

_However,

the

_theory

we

_{develop, expressed}

in

_{equation (4.2), always}

exhibits a subtle

_upward

curvature when

_{plotted logarithmically vs}

_square root of time. MacKintosh and John have obtained similar results

_{using diagrammatic techniques [16].}

_Therefore,

since the