Dynamic light scattering from gels in a poor solvent

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Dynamic light scattering from gels in a poor solvent

E. Geissler, A.M. Hecht

To cite this version:

E. Geissler, A.M. Hecht. Dynamic light scattering from gels in a poor solvent. Journal de Physique,

1978, 39 (9), pp.955-960. �10.1051/jphys:01978003909095500�. �jpa-00208837�

DYNAMIC LIGHT SCATTERING FROM GELS IN A POOR SOLVENT

E. GEISSLER

Laboratoire de Spectroscopie Optique, ^Centre Universitaire de Savoie, B.P. 143, ⁷³⁰¹¹ Chambéry Cedex, ^France

and A. M. HECHT

Laboratoire de Spectrométrie Physique (*), Université Scientifique ^{et Médicale de Grenoble,}

B.P. 53, 38041 Grenoble Cedex, ^France (Reçu ^{le 23 mars} 1978, accepté ^{le 18 mai} 1978)

Résumé.

Le coefficient de diffusion coopératif Dc ^{et le module} d’élasticité Egel ^ont été mesurés par la spectroscopie par corrélation de photons, ^{dans des} gels ^de polyacrylamide ^{dans un mauvais}

solvant : un mélange, 3/1 ^en volume, ^{d’eau et} de méthanol. On montre que la présence ^de points

de réticulation chimiques gêne ^{le mouvement} des n0153uds physiques qui ^sont caractéristiques ^des

solutions 03B8. Le mouvement de respiration ^{ordinaire du} gel ^est observé ; les fonctions d’autocorrélation

qui ^{en résultent sont} exponentielles, ^le temps de relaxation étant inversement proportionnel ^au

carré de K, ^{vecteur d’onde de} diffusion.

Egel ^{suit une} loi d’échelle pour des concentrations ^c comprises ^{entre 0,07 et 0,3} g cm-3

Egel ~ C3,07±0,07

en accord avec les mesures par diffusion de ^neutrons de longueurs de cohérence statiques ^{dans des}

solutions 03B8.

La dépendance ^avec la concentration du coefficient de diffusion coopératif ^du gel Dc ^{est aussi une}

loi d’échelle, ^{mais il est} montré que ^ce résultat est fortuit puisque ^la longueur de cohérence ainsi que la viscosité du solvant qui ^entrent dans la définition de Dc ^{sont toutes deux} dépendantes ^{de la concen-} tration ; la viscosité du solvant obéit à une loi de volume libre.

Abstract.

Photon correlation spectroscopy measurements are reported ^{of the} collective motion diffusion coefficient Dc ^{and the} longitudinal ^{elastic modulus} Egel ^of polyacrylamide gels in the poor solvent region using ^{a water : methanol} (3 : ¹ by volume) ^{mixture as} solvent. It is shown that the presence of cross-links hinders the ^movement of polymer self-entanglements, ^{which are characte-}

ristic of 03B8 solutions, ^{and the} ordinary gel breathing mode is observed with exponential autocorrelation functions whose relaxation rate is proportional ^to the square of the scattering ^{vector K.}

Egel ^{obeys a} scaling ^{law for} concentrations c between 0.07 and 0.3 g cm-3

Egel ~ C3.07±0.07

in agreement ^{with neutron} scattering measurements of the static coherence length ^{in 03B8 solutions.}

The diffusion coefficient of the collective motion in the gel Dc also follows a power law ^concentra- tion dependence, but it is shown that this is accidental, since the coherence length ^{as well as the}

solvent viscosity, both of which enter into the definition of Dc, ^are concentration dependent; ⁱⁿ particular the solvent viscosity displays ^a free volume behaviour.

Classification

Physics ^Abstracts

36.20

61.40K - 78.35

1. Introduction.

The static behaviour of semi- dilute polymer solutions in 0 conditions has been

(*) Laboratoire associé au CNRS.

investigated by Daoud and Jannink [1] using ^{a tricri-}

tical expansion technique. ^The predictions ^{of this} theory ^have subsequently been confirmed by ^neutron scattering measurements [2] of the concentration

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003909095500

956

dependence ^{of the radius of} gyration ^{and of the static}

coherence length çst in solutions of polystyrene ^with cyclohexane. ^{In this} regime çst ^{was found to vary}

as c-1, ^{where c is the} polymer concentration in _g cm- 3.

The dynamic behaviour in 0 conditions, ^{on the other}

hand, has received less attention both theoretically

and experimentally ; observation of the gel-like ^concen-

tration fluctuation modes by photon correlation spectroscopy ^is hampered ^{in the} semidilute regime by

the presence of a range of relaxation modes giving

rise to non-exponential spectra [3, 4].

Recently, Brochard and de Gennes [5] proposed ^a scaling description ^of polymer dynamics in e condi- tions which concludes that the hydrodynamic mode, in which the polymer ^{diffuses as a whole, dominates}

the light scattering properties of the solution. Their results _may be summarized as follows : the maximum

quasi-elastic broadening imparted by ^the hydrodyna-

mic mode to the scattered light ^is Tr- 1, the inverse of the chain renewal time ; the elastic modulus associated with this mode, Eo, ^{varies as c3.} Coupled ^{with the} hydrodynamic ^{mode there} exists also a gel mode, which _may scatter light only weakly, ^{and whose} spectral ^{width is} proportional ^{to K 2 at} sufficiently large scattering ^{vectors K ; the} elastic modulus

corresponding ^{to the} gel mode, Egel, ^{varies as c2.}

A confrontation between these theoretical predictions

and experiment is made difficult by ^the complex

spectra mentioned above. Resolution of such spectra into their constituent parts ^{can be} performed only

at the _expense of loss of précision ; morever, the

assignment of the different components ^{is not necessa-}

rily unequivocal.

In order to avoid these shortcomings, ^{an investi-} gation ^was undertaken of the concentration fluctua- tions in chemically cross-linked gels ^{at, or near to the 6} temperature (defined with reference to uncross-linked

polymer solutions). ^{In such} samples, ^the presence of cross-links at a suitable density ^{limits the conforma-}

tional changes accessible to the polymer chain, ^and blocks the hydrodynamic ^mode by making T, ^{tend to} infinity. ^{In this case} only ^the gel ^mode contributes to the dynamic light scattering, and the autocorrelation spectra ^{are observed to be} exponential.

By ^{means of} techniques essentially ^{the same as}

described in reference [4] ^the collective diffusion constant Dc ^{and the} corresponding longitudinal

elastic modulus Egel ^{were obtained} by analysing ^the

relaxation rate and the intensity of the autocorrelation function of coherent light ^scattered by polyacrylamide gels. ^The measurement of a number of different

samples ^{at a fixed} temperature permits ^{the concentra-}

tion dependence ^{of the two} parameters Dc ^and Egel

to be compared with their theoretical behaviour.

Recent work by Tanaka, Ishiwata and Ishimoto [6],

and also by ^Tanaka [7] ^{on a dilute} polyacrylamide gel

have demonstrated that at a given gel concentration there exists a spinodal temperature, ^{at which the}

intensity ^of light ^scattered by ^the gel diverges. ^In

contrast, the measurements described in the present article were performed by exploring the concentration axis at a fixed température ; ^the temperature ^was

chosen to lie in the poor solvent ^zone, defined in references [1] ^and [2], ^{i.e. the} region, ^{in the} equivalent

uncross-linked polymer solution, lying ^{between the} phase separation temperature ^{and the} point ^located symmetrically ^{above the o} temperature.

2. Experimental procedure.

Polyacrylamide (PA) gels ^{were chosen for this} study ^{for a number} of reasons.

The pioneering ^work of Tanaka, ^{Hocker and Benedek}

on the light scattering properties ^of gels ^{was carried}

out on the PA-water system [8]. Recently ^the present authors investigated ^{the PA-water} system ^{at room} temperature ^{as a} function of concentration, ^and

concluded that the elastic modulus, ^{measured from} the intensity ^{of the} quasi-elastically ^scattered light,

is in acceptable agreement ^{with the} scaling prediction

for good ^solvents [9]. ^The samples ^{have the} advantage of being easily prepared from standard recipes [10, 1l],

and _any desired cross-linking density may be selected.

Since the usual solvent, water, does ^not form a 9 system ^{with PA close to room} temperature, ^{it is} desirable to introduce a binary ^{solvent to} bring ^{the e}

temperature ^{into an} easily accessible range. For this purpose, ^a water-methanol mixture (3 : ¹ by volume)

was used and the samples ^were prepared ⁱⁿ the absence of the usual buffer salts. The latter precaution ^ensured

that the resulting gels, ^as temary mixtures, possessed only ^two concentration fluctuation modes. The choice of two liquids ^{which are} mutually good ^{solvents and} possessing similar refractive indices, ^{reduces to} negli- gible proportions ^the scattering ^from the unwanted

high frequency mode, leaving ^{the desired low fre-}

quency mode alone visible ; ^{this mode} corresponds ^to

the two fluid components moving ⁱⁿ phase, acting

as a single ^solvent [12].

In the absence of precise ^{data on the 0} temperature of the system PA-water-methanol, ^{the choice of the} operating temperature ^{for the} light scattering ^measu-

rements was based on observations of the cloud points

of a series of uncross-linked samples of différent concentrations ranging between 0.06 and 0.2 _g cm- 3.

For a monodisperse ^solution, extrapolation ^{of these}

data back to zero concentration gives ^a value close to

the 9 température ; ⁱⁿ polydisperse systems, however, the result obtained is somewhat approximate. ^The highest observed cloud point (8.2 °C) represents ^a lower limit for the possible ⁹ température ; ^the intercept ^{of the cloud} point ^{curve at c}

^{0 occurred at}

about 12 °C. In order to be certain to be within the poor solvent region defined in references [1] ^and [2],

the working temperature ^{was chosen to} be 11.5 OC.

At this temperature ^the gels ^{could be} kept ^for long periods ^without showing signs ^of phase separa-

tion ; samples ^{left for a week in a} refrigerator ^{at 6 °C}

on the other hand, contracted, expelling ^a layer ^of

solvent in contact with the cell walls, ^{which was}

confirmed as a liquid of lower refractive index with

an Abbe refractometer.

The photon correlation technique ^{used to measure}

the intensity ^and decay ^{time of the} concentration fluctuations in the gel ^has previously been described in detail [4]. ^{As the} gels ^are prepared ^without filtering

the component solutions, dust and undissolved _par- ticles are trapped ^{in the} gel, ^{in addition to the structural} inhomogeneities ^{inherent in the} gel itself. These inho-

mogeneities ^scatter light strongly ⁱⁿ comparison ^with

the intensity ^{of the} light ^scattered by ^{the concentra-}

tion fluctuations, ^thus providing ^a local oscillator

to heterodyne ^the dynamic signal. ^The heterodyne

nature of the signal arriving ^{at the} correlator can be tested by varying ^the position ^{in the} sample ^from

which the light is collected : it is readily ^{seen that the} signal amplitude ^is proportional ^{to the} intensity ^{of the} bright spot ^{selected. Circular} diaphragms ^{were used}

to define the observation volume, arranged ^{so that a} single ^{coherence area was} presented ^{to the} photo- multiplier. ^Each experimental ^value reported ^{here is}

an

^{average over four or more} spectra ^{for each} sample.

For polarized ^incident light ^of intensity Io ^{at fixed} wavelength, ^the intensity ^{of the} light scattered in the

plane perpendicular ^{to the} polarization by ^concen-

tration fluctuations is given by [8]

provided ^{that the} wavelength of the incident light ^is

much greater ^{than the coherence} length ^{of the concen-}

tration fluctuations, ç. ^The constant g ^{at this wave-} length ^{and for a} given photomultiplier is obtained by measuring ^the scattering intensity ^{of a standard}

concentrated solution of polystyrene ⁱⁿ cyclohexane,

for which effective values of Egel have been obtained from ultracentrifugation measurements [13J.

The concentrations c were found by measuring ^the weight ^{of the} samples ^{in their} cells, ^then extracting ^the

gel ^and evaporating ^to dryness ^{for 14} days ^{in an oven}

at 50°C. The ratio of the dry weight ^{to wet} weight gives ^the polymer weight ^{fraction w. To} convert to the concentration ^c in _g cm- 3 it is _necessary to know the

density p of the gels, ^where

p ^was measured before extracting ^the gel ^{from the}

cell by filling ^{to the} top with the solvent and placing

a glass ^cover slip ^{over to define the enclosed volume,}

then weighing. The least _squares linear fit to the

expérimental points ^was

The refractive index n of the gels ^{was measured with}

an Abbe refractometer as a function of concentration for À

632.8 _nm, the laser wavelength ^{used in the}

LE JOURNAL DE

PHYSIQUE.

39,

9,

1978

light scattering measurements. The least squares fit to a cubic curve gave for 12°C

Using equations (3) ^and (4) ^{one can deduce}

for each sample concentration, ^{and the use of} equa- tion (1) ^allows Egel ^to be calculated.

3. Results and discussion.

We have shown _pre-

viously ^{that the} cross-linking density ^{is an} important

parameter ^{for the} transparency ^{and mechanical} elasticity ^{of PA-water} gels [9], although ^the dynamic light scattering properties ^are approximately ^inde- pendant ^{of this} parameter.

At low cross-linking ^{densities the} autocorrelation spectra of the scattered light ^{become non} exponential.

The cross-linking agent used is N-N’ methylene bisacrylamide (bis) ; bis/acrylamide (B/A ) ^{ratios of} 1/75 ^were selected for the less concentrated samples.

At values of B/A greater ^than 1 /50 multiple scattering

becomes significant ^{in the water : methanol} (3 : 1) solvent; ^for B/A ^{less than} 1/100 ^the spectra acquire long ^tails, which have to do with the movement of uncross-linked branches and the relaxation of entan-

glements. ^The compromise ^value adopted ^{for the} cross-linking concentration is sufficient to trap uncross-linked branches and self-entanglements ^and

hinders their conformational changes ^so that these movements do not contribute significantly ^{to the} spectral broadening ^{of the scattered} light. ^Provided

that B/A ^{is small} enough, ^the accompanying ^loss

in entropy ^{is not} sufficient to cause precipitation ^and clouding ^{in the} gel.

The concept ^{of a diffusion constant} Dc ^{for the}

collective motions is usually associated with move- ments subject ^{to an} equation ^{of the} type :

that is, giving ^{rise to a} quasi ^elastic broadening ^{r such}

that

where K is the scattering ^{vector. The model} proposed

in reference [5] ^{leads to a} dispersion relation in which, except ^for very small K, equation (6) ^{is not} obeyed.

Experimentally, the observed linear dependence ^{of r}

on K2, ^{shown in} figure 1, ^indicates that the low

frequency ^{centre of mass} modes associated with the

disentanglement ^{of the} polymer chains have been

effectively suppressed by ^the cross-linking operation,

and that the resultant motion is a simple gel breathing

65

958

FIG. 1.

Angular dependence ^{of the} decay ^{rate r of the corre-}

lation function plotted against ^sin’ 8/2, where 0 is the scattering angle. ^The sample temperature ^{is 11.5} °C and the concentration

mode, ^{i.e. one in} which the ^mean position ^{of the} polymer ^{chains is constant and the network is} locally

swollen or contracted by fluctuations in the solvent concentration [8].

In figure ² is shown the dependence ^of Dc upon

polymer concentration c at 11.5 OC, plotted ^{in a} log- log representation. ^Within experimental ^{error the} points ^{lie on a} straight ^line given by

Theoretically [14], Dc may be expressed in the form

where _’1s is the solvent viscosity and r the Stokes radius of the fluctuating entity ^{r is assumed} proportional

to ç, the coherence length ^{of the} fluctuations. ^{In a} good solvent, ç ^oc C-0.75, ^{so that} Dc ^should vary ^as

c1.71 [14] ; in practice, ^however, the observed exponents

are frequently smaller than 0.75 [3,15], ^a phenomenon

which has been attributed to dynamic scaling [16,17].

For 9 solutions, however, Brochard and de Gennes

predict ^{that the} characteristic length associated with the gel ^{mode is a, the} polymer step length ; ^thus

should be independent ^of concentration, ^{which is} clearly ^{not the} present ^{case. It} might ^be conjectured

that PA is a polyelectrolyte : ç ^{in such} systems ^is

FIG. 2.

Variation of Dc ^with gel concentration at 11.5 OC. The lower set of points ^{are those for} PA-water-methanol, and the continuous straight line shown has slope 0.507. The data on the upper line pertain ^{to the} system ^{PA-water at the same} temperature.

expected ^to vary ^as C-l/2, ⁱⁿ agreement ^with equa- tion (7) [18]. ^{Such an} explanation is however difficult.

to sustain, since with _pure water as a solvent PA gels

behave like a neutral (uncharged) polymer ^{in a} good

solvent [9] ; ^the addition of a less polar ^molecule (methanol) ^{is not} expected ^{to enhance} any electric

charge effects in the solution. Without information

on the concentration dependence ^of fis it ^{is difficult}

to draw _any definite conclusions conceming ^the

variation of ç.

Additional information is contained in the measu- rements of the longitudinal elastic modulus Egei(c)

at 11.5 °C, ^{shown in} figure ^{3 on a} log-log plot. ^For c 0.3 g cm- 3 ^the experimental points ^{lie on a} straight line obtained by ^a least squares fit

At higher ^{values of} concentration the gel ^becomes

stiffer than indicated by equation (10). ^{A similar} phenomenon ^was observed for the PA-water system, and _may be associated with the approach ^{to the} glassy

state in the _pure polymer. ^The slope ^of equation (10) ^is

too large ^to be accommodated by ^the scaling ^theories

for polyelectrolytes (1.5), good ^solutions (2.25), ^or

the high frequency ^branch in ç ^solutions (2) ^{as propos-}

ed in reference [5]. On the other hand, the result _agrees

with the value 3 obtained using ^{the usual} scaling

FIG. 3.

The elastic modulus Egel obtained from the light ^scat- tering measurements. The full circles relate to PA-water-methanol, and the continuous line is the least _squares fit to the data for

c 0.3 _g cm-3 (Eq. ^{(10) in the} text). ^The open circles are the data from the PA-water system ^{at the same} temperature (11.5 °C) ; ^the

least squares line through ^these points ^has slope ^2.4.

argument, assuming ⁰ conditions, ^{and in the absence}

of the ef’ects of self-knotting :

inverse osmotic compressibility in dilute solutions :

semidilute solution :

cross-over concentration :

where Ro - ^{M 1/2.}

The condition of continuity between the dilute and semidilute regimes together with relations (11) ^to (13), imposes m

^3.

This simple ^result may be understood ^as follows.

When the gel ^forms, self knots are trapped by ^the cross-linking process, which in ^tum considerably ^slow

the conformational changes ^{of all} entanglements. ^The remaining free chain segments ^have insufficient stored

length ^to generate ^new entanglements in the time scale of the concentration fluctuations of the gel.

In the absence of this complication, ^the polymer dynamic response is govemed by ^the elastic modulus

(A being ^a constant), which has the same fonn as in the static _case, i.e. with a screening length ç ^oc c- 1 [2].

In the light ^{of the} above result it is clear that the apparent scaling behaviour of Dc ^is purely fortuitous, and is the result of the variation of other parameters in addition to ç.

Attention should be drawn to the fact _{that equa-} tion (14) differs from the expression ^{for the longitu-}

dinal elastic modulus developed by ^Tanaka f7] ^from mode-coupling theory, in which for Kç 1,

This expression is similar to the gel elasticity ^control- ling ^the high frequency ^mode described in reference f5].

The corresponding relation for the collective mode diffusion ^constant in the mode-coupling picture ^is, however, ^not equation (9), but instead

which leads to a friction constant of the form il,

instead of the more usual _’1s ç- 2 [5, 14].

Assuming ^{that the} temperature dependence of ’1s

arises only ^from changes ^{in the} viscosity ^{of the bulk}

solvent (i.e. neglecting possible influences of the

polymer ^on 11s)’ ^Tanaka obtains excellent agreement with the equations (15) ^and (16) ^from the measured temperature dependence ^of Egel ^and Dc. ^{If, on the}

other hand, ^the hypothesis conceming ^’1s is relaxed

on account of its simplification ^{of the} polymer-water interaction, ^the light scattering data alone are not sufficient to indicate which of the equations (14) ^or (15)

is operative ^{in the} present ^case.

In what follows we shall adopt ^{the form} kB T/ç3 ^for

the _reason, stated above, ^that experimental ^observa-

tions by ^neutron spectroscopy ⁱⁿ polymer ^solutions

close to the 9 temperature [2] ^{show that the static}

coherence length çst ^oc c-’ ; identification of çst

with ç, together ^{with the} present observation ^that

Egel oc C3, ^{leads to} equation (14).

Expression (8) ^{allows one to consider the local} viscosity fis ^{as a} supplementary concentration depen-

dent parameter. Assuming ^{that the volume of the} fluctuating entity ^is

where r is the radius of the equivalent ^Stokes’ sphere,

one obtains from equations (9), (14) ^and (17), ^where

the constant A has been set equal ^to unity,

960

FIG. 4.

The apparent viscosity ^{of the} solvent 11s obtained from the relation (18) ^{in the text} (in ^c Poises), ^{as a} function of concen-

tration. a) PA-water-methanol at 11.5 °C. The bracketed asterisk

gives ^the macroscopically ^measured viscosity of the water-methanol mixture at the same temperature. b) ^{PA-water at 11.5 °C. The lower}

asterisk indicates the macroscopic viscosity ^{of water at this tem-} perature.

In figure ^{4 relation} (18) ^is plotted against polymer

concentration in a semilogarithmic scale. The straight

line dependence ^is strongly suggestive of a free volume behaviour, ^as has been observed previously ^{with PA-}

water at a higher temperature [9]. ^It is notable that the apparent ^solvent viscosity extrapolated ^{to zero concen-}

tration lies some 60 % higher ^{than the measured} macroscopic ^value for the methanol-water mixture at the same temperature. ^We suggest ^{that this is} caused, ^{at least in} part, by ^a dynamic coupling ^between

the two concentration fluctuation modes present ^{in the}

ternary system, ^the effect of which is to reduce the relaxation rate of the low frequency ^mode [12]. ^In support ^{of this} assertion, values for _{11 s} are shown for

seven samples ^{of the} binary system PA-water, ^also

at 11.5 OC. In this case the extrapolated value 11s ^at

zero concentration is in excellent agreement ^{with the} measured macroscopic ^{value. It is} equally ^remarkable that, ^{within the} experimental error, the two curves

have the same slope. ^It is therefore clear that the

viscosity increment is independent of the solvent.

The temperature dependence ^{of this} parameter ^is currently being investigated.

4. Conclusions.

The dynamic light scattering

measurements of concentration fluctuations in a

polyacrylamide gel ^{in a} poor solvent show that when the movement of self-entanglements ^{is hindered} by ^the

presence of chemical cross-links, ^the ordinary gel breathing ^{mode is} observed, ^and exponential ^correla-

tion functions are generated ^with relaxation rate

proportional ^to the square of the scattering ^{vector K.}

The longitudinal elastic modulus Egel, deduced from the intensity ^{of the} concentration fluctuations, obeys ^a scaling ^{law for} concentrations c in the range 0.07

to0.3gcm-3

in agreement with the results of static scaling ^{in 0}

solutions. This result differs from recent theoretical

predictions by Brochard and de Gennes for 0 systems, but since these authors specifically consider the effects of relaxing self-entanglements, ^{which are} suppressed

in the present experiments by ^the permanent ^cross- links, ^the comparison ^{is not} properly ^valid.

The diffusion coefficient of the collective gel ^motion Dc also follows a power law dependence upon ^concen-

tration, although ^{it is shown} that this is accidental,

since in the concentration _range studied both the solvent viscosity and the correlation length ^are

concentration dependent.

Acknowledgments.

The authors would like to thank M. Adam and P. G. de Gennes for helpful

comments.

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