Brillouin light-scattering from poly(acrylic acid) hydrogels

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Brillouin light-scattering from poly(acrylic acid) hydrogels

S. Ng, L. Gan, Y. Li, T. Chieng

To cite this version:

S. Ng, L. Gan, Y. Li, T. Chieng. Brillouin light-scattering from poly(acrylic acid) hydrogels. Journal

de Physique II, EDP Sciences, 1994, 4 (4), pp.715-722. �10.1051/jp2:1994158�. �jpa-00247994�

J. Phys. II IYance 4

(1994)

715-722 _APRIL 1994, PAGE 715

Classification Physics Abstracts

62.90 78.35 82.70

Brillouin light-scattering from poly(acrylic acid) hydrogels

S-C-

Ng,

L-M- Gan

(*),

Y. Li and T-H-

Chieng (*)

Department of Physics, National University of Singapore, Kent llidge, Singapore 0511

(Received

13 July 1993, received in final form 22 December 1993, accepted 3 January

1994)

Abstract. The Brillouin shift and width for

poly(acrylic acid)

hydrogels have been obtained

as a function of gel network volume fraction using _a five-pass Fabry-Pdrot interferometer. A

conspicuous peak has been observed in the Brillouin shift at high volume fraction. The results

are discussed in the light of

a theory for the Brillouin light-scattering from gels proposed by Marqusee and Deutch-

1. introduction.

In _a recent

study

_on Brillouin

light-scattering

from

polymer gels ill,

_our

previous

measurements of the

dependence

of the true Brillouin shift _uB and width

rB

_on the volume fraction

#

of the

gel

network for

poly(vinyl alcohol) (PVA)

_{[2] and}

poly(vinyl chloride) (PVC

_[3]

gels

_were

compared

with the

theory

of

Marqusee

and

Deutch(MD)

_[4],

incorporated

with _{a new}

assumption

that the bulk modulus of the

gels

scales _as

#.

It has been found that the measurements of _uB and

rB,

_over most of the _range of

#

_are

amazingly

consistent with the essential features of the

predictions

of the MD

theory

for the _case of weak

coupling

and strong frictional

damping [I].

However,

the weak

coupling

_case is _an abbreviated version of the

theory

and therefore the

experimental

data of PVA and PVC

gels

do not constitute _a

complete

test for the MD

theory.

There _are other features which _occur in the _case of strong

coupling

^but not in weak

coupling.

For instance, the MD

theory

for the _case of strong

coupling

and frictional

damping predicts

the presence of _a

peak

in _{uB at}

high #,

_a feature which is

missing

in the weak

coupling

case.

In order to

explore

this _more intricate

possibility

_we undertook _a

study

_on another

gel

system,

poly(acrylic acid) (PAA).

(*) Department of

Chemistry,

National University of

Singapore.

716 JOURNAL DE PHYSIQUE II N°4

2.

Experimental.

The materials used in this

work, acrylic acid(AA)

and

dibenzyl ketone(DBK),

_were obtained from

Tokyo Chemistry Industry.

AA of

purity higher

than 99% _was distilled under reduced pressure before _use. DBK of

purity higher

than 98% _was used _{as a}

photoinitiator

without further

purification. Samples

of the PAA

hydrogel

_were

prepared

in the

following

_manner.

Solutions of known _masses of

AA, doubly

distilled water and DBK

(0.3%

of the _mass of

AA)

_were introduced into clean

glass

tubes and

purged

with

purified nitrogen

_gas ^{for about}

10 minutes before

being

sealed. The sealed

samples

_were then

polymerized by

ultra-violet

light.

The conversion of

polymerization

_was ^determined to be

higher

than 97%.

Assuming

100% conversion, the

gel

network volume fraction

#

of the

resulting

PAA

hydrogels

_can be

determined

quite accurately by

the equation

~ l7ln~~~~ifPn

^~~~

where _pn and pf are the densities and _mn and _mf the _masses of PAA and water

respectively.

The PAA

hydrogels

_were transparent. ^{Those with}

#

less than 0.25 did not exhibit sufficient

strength

to be

self-supporting. However, they

became

rubbery

and

glassy

at

higher #

values.

Many

studies have been made _on the

preparation

and characterization of cross-linked PAA

hydrogels

[5, 6]. ^{It has been}

reported

that cross-linked PAA _can be

separated

from the _uncross- linked

PAA(atactic

and

stereoregular

forms of

PAA) by

extraction with dioxane followed

by

20% _water in dioxane [7]. ^Based on this

method,

_we have found that the

hydrogels

with

#

> 0.25 could not be solubilized in the solvents but formed

highly

swollen

gels.

These swollen PAA

gels

were further

disintegrated

into small

fragments

in hot I% Triton X-100

(non-ionic surfactant)

_aqueous solution. This indicates that the PAA

hydrogels

_may have cross-linked

network. The

gel

network _may be formed

by

strong interactions between PAA chains

through hydrogen bonding

via

carboxyl

_groups. In

addition,

the formation of _some chemical

(covalent)

cross-links _may also be

possible. However,

_we are unable to determine the cross-link

density quantitatively.

The Brillouin spectrum at _room temperature for each PAA

hydrogel

of different

#

_was ob- tained

by exciting

it

using

_a

Spectra-Physics argon-ion

laser

operated

at

single

mode

(linewidth

m 100

MHz)

_{at a}

wavelength

of ~ ₌ 514.5 _nm and _a power of 50 mW. The

light

^scattered

through

_an

angle

⁶ = 90° _was

analyzed using

_a

Burleigh

DAS-I five _pass

Fabry-Pdrot

inter-

ferometer

operated

with _an

optimized

free

spectral

_range of 33.8 GHz and finesse of 42, as

described in detail elsewhere [8]. ^The

optimized

free

spectral

_{range was} chosen to

provide

_a

sufficient

separation

of the Brillouin doublet and the strong central

Rayleigh peak

at low

#

and also _to prevent ^the

overlapping

of the Brillouin

peaks

at

high #. Any

measured Brillouin

spectrum is the result of _an unavoidable convolution between the true spectrum and the instru- mental response. The effects of the instrumental resolution _were removed and the true uB and rB _were obtained

using

_an iterative convolution and

least-squares fitting procedure, assuming

a Lorentzian line

shape

for the true spectrum _[9].

Marqusee

and Deutch

give

_an

analytic expression

for the

light-scattering

spectrum for

poly-

mer

gels

_[4]. It describes two Brillouin

peaks

and _a central diffusive

peak,

which _are all of Lorentzian line

shapes. However,

the non-Lorentzian contributions to the

light

spectrum _are

not included in their

expression,

_as

they

_are

expected

to be small [4].

Figure

I

depicts

the method

by

which three

Rayleigh-Brillouin

_spectra for PAA

hydrogel

of different

# 0.06,

0.65 and

1.0)

were computer

analysed.

The measured Brillouin

peaks,

shown _as full circles _on the

right-hand

side of the

figure,

were

least-squares

fitted with _a

profile given by convolving

a

Lorentzian line

shape, representing

the true Brillouin spectrum, with the measured instrumen- tal

resolution,

which _was obtained from the

Rayleigh peak,

_as shown _on the left-hand side of

N°4 BRILLOUIN LIGHT-SCATTERING IN PAA

H~DROGELS

717

#=0.06

~

- Rayleigh ^Brillouin

I fi

Peak

~~~kS ~ ~

© ~

~

~ ^j

+

~

~

_.j ^#=0.65

~ ,tf ].

O

£

# ^~

o 5 io is

Brilloujn Shjft (GHz)

Fig. I. The Rayleigh-Brillouin spectra for PAA hydrogels of #

= 0.06, 0.65 and 1.0. The figure is

fully explained the text.

the

figure.

A

sloping

linear

background

is also included in the fit to describe

approximately

any weak non-Lorentzian and diffusive contributions. The parameters of the Lorentzian and the

background

_are

optimized

to obtain the best fit to the

experimental

data. The fitted

profiles

_are shown in the

figure

_as fine lines

passing through

the

experimental points

and the

resulting

deconvolved _true spectrum as bold lines. The fitted _true Brillouin shift _uB is

equal

to

the _resonance

peak frequency

of the

optimized

Lorentzian

profile

and the fitted _true Brillouin

width rB is

equal

to its full width at half the maximum

intensity(FWHM).

Note

that,

while the measured and true spectra have

essentially

the _same

peak frequency,

their FWHMS differ

markedly

for _narrow spectra ^due to the

broadening

effect of the instruments.

The PAA

hydrogels

with

#

less than 0.56 _are rubberlike and their refractive indices at ~ ₌ 514.5 _{nm were} measured at _room temperature

using

_an Abbd refractometer.

3. Results.

The results for the Brillouin shift _uB at _room temperature for the PAA

hydrogels

_are

displayed

in

figure

2. For

increasing

#, _uB increases

slowly

at

first,

then at

#

_~ 0.4 it increases _more

rapidly reaching

_a maximum _near

#

~ 0.9 thereafter

dropping slightly

_as

# approaches

I. This is in contrast to _our

previous

^results _on PVA and PVC

gels

_{[2, 3], as} well _as those of Brown et al. _on solutions of

poly(methylmethacrylate) (PMMA)

in toluene

[10],

in which _uB increases

monotonously

with

#.

The measurements of the Brillouin width rB _are shown in

figure

3.

rB

increases

rapidly

with

increasing # reaching

a maximum _near # _~ 0.65 thereafter

dropping rapidly

_as

# approaches

I. This behaviour of rB is similar to those of

PVA,

PVC and PMMA

gels

[2, 3,

10].

Figure

4 shows the

dependence

of the measured refractive index _{n on}

#.

The value for PAA

(#=I)

is taken from

Polymer

Handbook [11]. These results show that _n

depends approximately linearly

_on

#

and the measurements for

n(#)

_were

parametrized

in accordance with

n(#)

₌

no +

ni#.

The values obtained for _no and _{ni are}

given

in table1.

718 ^' JOURNAL DE PHYSIQUE II N°4

$$

X

~ll

~

w

j_

^O

>

~

_o

I _~=

₀

C 8

5

O

~

ltl 4

0 0.2 0.4 0.6 0.8

Volume Fractjon a

Fig. 2. The Brillouin shift in PAA hydrogels _{as a} function of volume fraction. Open circles

are

experimental data. Lines _are theoretical _curves for ~ _m 0 and ~

= l

(see text).

_

4 N

X

~ll

~' 3

~tD

o O

i~

~

f

2 O ~

c O

5

o O

~

°

i5

CD

~0

O.2 O.4 O.6 O.8

Volume Fraction a

Fig. 3. The Brillouin width in PAA hydrogels _{as a} function of volume fraction. Open circles _are experimental data. The line is _a theoretical _curve (see text).

4.

Comparison

with

theory.

Marqusee

and Deutch consider _a

polymer gel

_{as a}

coupled polymer

network-fluid system. ^The

degree

of

coupling

between elastic _waves in the network and fluid is measured

by

a parameter 0 < ~ < l. Two

limiting

_{cases are} examined

by

them. The first _{case occurs} for small frictional

damping f

and

predicts

a twc-mode type of behaviour for the Brillouin spectrum. In this _case two

pairs

of Brillouin

peaks

should be observable, which for weak

coupling

I _t 0, _are close to the Brillouin

angular frequencies

+wo and +wn. Here _wo

=

Cok

and _wn

=

Cnk, Co

and

Cn

are

respectively

the

speed

of sound in the fluid and network and k is the sound _wave vector.

The other

limiting

_case is for strong ^frictional

damping.

It

corresponds

to _a one-mode type

N°4 BRILLOUIN LIGHT-SCATTERING IN PAA HYDROGELS 719

Table I. The bulk physical parameters of PAA hydrogel at _room temperature. All the symbols used _are defined in the _text.

Parameter Value Reference

998 1212 1475.5

3250 _ms This work

+

no 1.3358 This work

This work

.6

c

x 1.5 c

©

>

'iZ

#

I.4

li

+

CC

~'~O _{O.2 O.4 O.6 O.8}

Volume Fractjon

_4~

Fig. 4. The refractive index of PAA hydrogels _{as a} function of volume fraction. Open circles _are

experimental data. The line is _a linear least squares fit.

of behaviour which

predicts

_a

single pair

of Brillouin

peaks

at

angular frequencies

+17. Here 17 ₌ Ck and

©

_is the _average

speed

^{of sound} in the

gel. C(#

₌

0)

= Co and

C(#

₌

I)

=

Cn.

The present measurements of Brillouin spectra ^{of PAA}

hydrogel,

like all

previous

_measure-

ments _on

PVA,

PVC and PMMA

gels

[2, 3, 10], ^{show the} presence of

only

_a

single pair

^of

modes. It therefore _seems appropriate to compare the results with the

theory

for the limit of strong ^friction.

Marqusee

and Deutch

give

two equations ^for the $-

dependence

of the Brillouin shift _uB and Brillouin width rB

(FWHM):

uB "

©k/2x,

where

n2

~

~iPf

₊

<(~1Pn CiPf)

₊ ~~0Cn

~

~~~

Pf +

<(Pn Pf)

720 JOURNAL DE PHYSIQUE II N°4

and rB

"

rk~

lx,

where

~~

~

4~lS

/3

+ _~IB

~

PI PI (# #~)~

~

Pf +

#(Pn Pf) (Pf

+

#(Pn Pf))~ i©~

~

~~

~~

~ ~~~~

~~~ ~~~~ _{~ ~~~~}

~~~

~~~

Here _~s and ~B are

respectively

the shear and bulk viscosities in the fluid.

Originally

Cn in

equations (2)

^and

(3)

was assumed to be

only weakly dependent

of

#. However,

in

comparison

with the

experimental

data of PVA and PVC

gels,

it _was found that the essential features of the behaviour of the Brillouin shift and width _are well described

by

the

theory

_over most of the range of

#

if Cn is assumed _to

depend

_on

#

_as

Cn

=

C(#

=

I)fi. _[I].

This

<-dependence

of Cn

corresponds

to _an

assumption

that the bulk modulus of the

gels

scale _as

#,

which is valid for

gels

with _a fixed

degree

of

polymerization

_[12].

Analysis

of the present results is also based

on the _same

assumption.

The relevant values for the

physical

_parameters

required

for the calculations _are listed in table I. The

resulting predictions

for _uB have been

displayed

in

figure

2 _as lines for the

limiting

_cases of _no

coupling1

₌ 0 and maximum

coupling

I

= I. Note that

the value of

Cn

has been chosen _so that _most of the data points ^{fall within} these two

limiting

curves as

required by

the condition 0 < 1 < 1. It _can be _seen that for small and intermediate #

the theoretical _curve with 1

= 0 agrees

quite

well with the

experiment,

whereas for

large #,

the agreement ^with

experiment

_seems to be better for the theoretical _curve with I

= I.

Markedly,

the _curve with I

= I does

predict

_a

peak

in _uB at

large #, though

at a

slightly higher #

value and smaller in size than the

experiment.

The

origin

of this

peak

_comes from the

fi

_{term in}

equation (2). Obviously

the

peak

will

disappear

with 1 _- 0. This

comparison

shows that the PAA

hydrogel

system _{crosses over} from _{no or} weak

coupling

at small and intermediate

#

to maximum

coupling

at

large #.

This is in contrast to the inference from the results of PVA and PVC

gels

^{that I} ci 0

throughout

the whole _range of

# [I]. Treating

I _{as an}

adjustable

parameter, ^{it is} found that in order _to describe the

experimental

data of _uB, I has to vary

according

to

figure

5. This behaviour of ~ confirms the above observation that the

degree

of

coupling

between the elastic _waves in the network and fluid

changes

^from weak to strong

coupling

at

high #,

except that it

changes

back _to weak

coupling again

_as

# approaches

I.

The

prediction

of the

theory

for

rB

is shown in

figure

3 _{as a} line

using

the ~ values of

figure

5 and the

physical

_constants in table I. A constant value for the frictional

damping

^of

f

ct

1.6x10~~

kgm~~s~~

has been chosen. The

theory successfully predicts

the essential

qualitative

features of

rB,

in

particular

the _presence of _a

maximum, though

at _a somewhat

higher #

value than the experiment. In the _same

spirit

of the

approach

taken for _{uB, we may} also treat

f

_as

an

adjustable

parameter. A better agreement ^{is obtained if}

f

is to vary with

#

_as shown in

figure

5. It is to be noted that the behaviours of I and

f

_{as a} function of

#

_are rather

similar,

both exhibit _a maximum at about the _same # value which coincides with that _at which _uB is _a maximum. We know of _{no reason}

why

I and

f

should _vary

thus,

but believe this coincidence is not accidental. We _are

currently investigating

the

possibility

of

studying

other systems ^which may show similar behaviours.

However, gelation

systems with

good optical quality,

wide _range of

#

and

especially

with

large

I _are very difficult to find.

In the MD

theory,

Brillouin

light scattering

from

polymer gels

_are

analysed

in terms of _a model which treats the motions of the fluid and the network

separately

and _on

equal footing.

The merit of the

dynamical

set

adopted by Marqusee

and Deutch in their

hydrodynamic

equations

for

gels

is that the essential

physics

is contained

(I)

in the

simple

frictional

damping

term

proportional

to

f

that describes the

exchange

of momentum between the fluid and the

N°4 BRILLOUIN LIGHT-SCATTERING IN PAA HYDROGELS 721

cram

~

o o

j

o

QJ _o

o.2 o.4 o.6 o-S

VO(Ume Fraction _#~

O

~ O

y~

j

~

O~

f~ _~

m

~

~ O

C O o

~ ~ O

j© ~ O O °

o o.2 o.4 O.6 o-B I

Volume Fraction _4~

Fig. 5. The dependence of

~(#)

to reproduce the data of figure I and the dependence of

f(#)

to

reproduce the data of figure 3

(see text).

network and

(it)

the parameter ^{which describes the}

coupling

of the elastic _waves in the fluid and in the network [4]. It is these terms that _are

carrying

information about the

polymer

network into the

position

and width of the Brillouin

peak.

There _are

other, simpler

_{or more}

complicated, hydrodynamic descriptions

which _are consistent with the symmetry ^and

physics

of

gels.

For

instance,

the

hydrodynamic equations

introduced

by

Bacri et al.

[13,

14] ^{in their}

study

of ultrasonic attenuation in

gels

do not include the

coupling

of elastic _waves described

by

A. Nevertheless for

=

0,

MD

description

reduces to that of Bacri et al. Since all the known similar studies _on other

gels

_{[2, 3, 10]} do not exhibit any

peak

in _uB at

high # values, they

_may be described

by

the MD

theory

with ₌ o, _as we

reported

_on two of them

recently [I].

However

they

_are not

good

systems to

verify fully

the MD

theory.

The present

study

on PAA

hydrogel clearly

demonstrates the role

played by

the

coupling

_parameter in the behaviour of _uB. We believe this is the first observation of such _an effect.

According

to MD

theory

the frictional coefficient

f

is

likely

to be _a

complicated, nonanalytic

function of both the volume fraction

#

of the

gel

network and the

frequency

of the mechanical disturbance. We have found that not

only f

but also ~ _are

complicated

functions of

#.

Johnson [15] ^has

applied

the Biot

theory

_{[16, 17] of} acoustics in porous media _to the

gel problem.

In the work of Biot

722 JOURNAL DE PHYSIQUE II N°4

and that of Johnson, there is additional inertial

coupling

besides that introduced

by

Marqusee and Deutch. We have used the Johnson

theory

to

analyse

the PVC

gel

_[3].

However,

contrary

to

expectation,

it fails to account for the observations in the

gelation

system. In

particular,

the

tortuosity

parameter ^{calculated from} the

theory

is

nearly always

less than

unity

^which is

a

physical

unreasonable results [3]. ^{This is}

why

_no attempt has yet ^{been made} to

analyse

the present ^{results in accordance with the Johnson}

theory.

5, Conclusion,

Our measurements of the

dependence

of the Brillouin shift _uB and width rB _on volume fraction

#

for PAA

hydrogel

have revealed _new features which have not been observed in the

previous

similar studies _on other

gels

[2, 3, 10]. ^In

particular,

_a prominent

peak

has been observed in _uB at

high #

value. The results have been

compared

with the

theory

of

Marqusee

and Deutch for Brillouin

light-scattering

from

polymer gels, incorporated

with _{a new}

assumption

that the bulk modulus of the

gels

scales _as

#.

It is found that the measurements are

roughly

consistent with the essential features of the

predictions

of the

theory

^{for the} case of strong ^frictional

damping f.

However in order to obtain _a realistic

description

^of the

experimental

data in terms of this

theory

it is found that both the

coupling

parameter ^{and frictional}

damping f

must be forced to

depend strongly

_on # instead of

assuming

constant values _as we have done in _our

previous analysis

^{of PVA and PVC}

gels [I].

In

fact,

both I and

f

exhibit _a maximum at about the _same the

#

value where _uB is also maximum. It is

hoped

that the data obtained for I and

f

in this

paper will stimulate further theoretical

investigation

into the behaviour of these parameters as

functions of

#

in

polymer gels.

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(1993)

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