Critical behaviour of the elastic constant and the friction coefficient in the gel phase of gelatin

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Critical behaviour of the elastic constant and the friction coefficient in the gel phase of gelatin

J.-C. Bacri, D. Gorse

To cite this version:

J.-C. Bacri, D. Gorse. Critical behaviour of the elastic constant and the friction coefficient in the gel phase of gelatin. Journal de Physique, 1983, 44 (8), pp.985-991. �10.1051/jphys:01983004408098500�.

�jpa-00209682�

Critical behaviour of the elastic constant and the friction coefficient in the gel phase ^of gelatin

J.-C. Bacri and D. Gorse

(*)

Laboratoire d’Ultrasons (**), Université Pierre et Marie Curie, Tour 13, ⁴ place Jussieu, 75230 Paris Cedex 05, ^France (Rep ^le 23 février ^1983, révisé le 24 mars, accepté le 28 avril 1983)

Résumé. ²⁰¹⁴ Nous avons étudié le comportement critique ^{de la constante} élastique ^{et du} coefficient de friction dans la phase gel ^{de la} gélatine. ^Les exposants critiques trouvés, 1,8 pour la ^constante élastique ^et 1 pour le coef- ficient de friction, ^{sont en} bon accord avec une théorie de 3d-percolation.

Abstract. ²⁰¹⁴ The critical behaviour of the elastic constant and the friction coefficient have been measured in the

gel phase ^of gelatin. The critical exponents, ^1.8 for the elastic constant, and 1 for the friction coefficient ^are in

good agreement ^with 3d-percolation theory.

Classification

Physics ^Abstracts

46.30J ^- 64.70M - 64.80

1. Introduction

The aim of this _paper is to describe the critical beha- viour of a

typical physical gel, gelatin,

^{near the}

sol-gel

phase transition. At

high

temperatures, ^{in the sol} phase,

gelatin

appears ^{as a} viscous

polymeric

^solution

of random coils. As the temperature decreases,

gela-

tion occurs because of two

competing phenomena :

one is an intramolecular

phenomena

^{where the}

triple

helicoidal structure is recovered [1] from the initial random coil, the other is an intermolecular

phenomena

which is a random

bonding

^between

adjacent poly-

meric chains.

It is assumed that

going

from the sol to the

gel

phase ^{involves a critical}

point.

At the critical tempe- rature,

bonding

^has

spread

^{from one}

edge

^{to the other}

of the sample; ^the

gelatin

^{is now} considered, ^{in the} gel

phase,

^{to be a} 3d-elastic medium

(collagen

^net-

work)

immersed in the solvent

(water).

Two different theories describe the

sol-gel phase

transition : in the classical

gelation

theory of Flory [2],

the

gel

is viewed as

growing

up like a

spanning

^tree

(Cayley

tree). Then excluded volume effects and

cyclic ^{bonds are} omitted The classical

expectation

for the exponent t of the network elastic constant is

(*) Present address : C.E.N. Saclay, S.P.A.S., ^F-91191

Gif sur Yvette Cedex.

(**) Associated with the Centre National de la Recherche

Scientifique.

t ⁼ 3. The deviations from this idealistic tree model

stipulated

^{’above are} taken into account in the _perco- lation model

[3].

^The

prediction

^{is 1.7 t 1.8 in}

3d-percolation theory.

The

gel phase

^is

investigated

^here

by

^{the method}

of

magnetic probes (ferrofluids [14])

immersed in the

polymeric

solution. Under a static

magnetic

^field

(H

¹⁰⁰ G) the ferrofluid

particles

^{rotate and induce}

a shear deformation in the network

(the

^solvent

remaining

motionless). The motion equation ^{of the} gel network is

given by

u is the transverse

displacement

^{vector of} the elastic medium from its

equilibrium

position, ^{p the}

density

of the network, ^G the shear elastic constant ot the network and

f

^{is defined as the} friction coefficient between the network and the solvent. We deduce from

equation

(1) that the ferrofluid

particle

^relaxation time, when the field is cut-off, ^{will be}

proportional

^to

fr2/G

where r is a characteristic

length

^{of the} magnetic particle (cf. § 6.1). The elastic _energy of the ferrofluid

particle ^is

equal

^to

2 I C02

(cf. § 3.2) where 0 is the rotation

angle

^{of the} ferrofluid

particle

^under H, and C is the

product

of the shear elastic constant G and the

perturbed

volume. So we are able to obtain,

independently,

the variations of these two characte- ristic coefficients of the

gel,

^G

and f,

^around

Tg,

as functions of the temperature.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004408098500

986

2.

Experimental

^method.

The

experiment

^{uses a} method which has been described elsewhere [4]. ^A small number of magnetic particles (ferrofluids) ^{are added to the}

gelatin

sample.

When

applying

^{a uniform}

magnetic

^{field H}

(by

^means

of two coils

surrounding

^the

sample)

^these particles

tend to line _up

along

^the field ; ^a

macroscopic

^bire-

fringence

An is induced in the

sample.

In the sol

phase,

^we obtain the critical variation of the

viscosity il by measuring

^the

birefringence

^relaxa-

tion time, ^at the instant the field H is switched off.

In the

gel phase,

^the measurement of the

birefringence

level leads to the critical variation of the elastic constant; ^the

birefringence

relaxation time

gives

^the

friction

coefficient f.

Our ferrofluids are obtained

by

^{a new chemical}

method [5]. Every elementary

magnetic grain (100

^A

diameter)

^{is a} macroanion immersed ⁱⁿ an aqueous solution. The balance between the

repulsive

^electro-

static _energy and the attractive

magnetic dipole- dipole

interaction _energy leads to an

agglomeration

of few grains. ^The

resulting

agglomerate ^{has the}

shape

^{of a}

prolate ellipsoid

^of

major

axis 2 a N 1 400 A

and minor axis 2 b - 300 A

[6].

3. Theoretical expression ^{of the}

birefringence

^induced by ^the ferrofluid particles.

We _suppose that each grain ^{has its own internal} birefringence. ^An

assembly

^of

grains

^{in a}

liquid

^is

optically

isotropic because of the brownian motion,

but in the _presence of a

magnetic

^{field the}

suspension

becomes birefringent. ^This

optical birefringence

^is

related to the anisotropy of the electrical

susceptibility

tensor x. This tensor for an uniaxial dielectric

particle

located

by

0, (p is :

where

and

G 1 is the dielectric constant of the

liquid,

_8jj and 81 the

principal

dielectric constant of the

particle,

v its volume and L its

depolarizing

factor, ^with

The

anisotropic

part of the electrical

susceptibility

of a

suspension

^{of these}

particles

^{in a}

liquid

^is

Xan = r(X I I - Xi)(( f(0, qJ) )ø,qJ

where r is the volume fraction of the

particles (r

^{= N v}

where N is the number of

particles

per unit

volume)

and

f( 0, qJ))

represents ^the average value

of/(0, T)

with respect ^{to 0 and} T. The

birefringence

(which

we shall call An from now on) ^is

proportional

^{to the}

anisotropic

part ^{of the}

susceptibility

tensor, ^{An oc} xan.

3.1 APPLICATIONS.

3.1.1 In the sol

phase.

^{- For} temperatures ^above the

gelatin

threshold, ^the

gelatin sample

^{is a}

liquid

The ferrofluid

particles

^{are submitted to thermal}

motion. If we

apply

^a

magnetic

^{field H}

along

^the

Z-axis

(the

^Y-axis

being along

^{the laser}

beam),

^the magnetic

particles

^line up in the field

(Fig. 1); they

make the

sample

uniaxial. We ^can write

In this case the distribution function

P(O, qJ)

^{is not} w

Fig. ^{1. - (J and} 9 angles definition in sol phase.

dependent

because of the revolution symmetry ^around Z-axis. Thus

with

where J-lH ^{cos 0 is the}

magnetic

energy

and fl

⁼

1 /k T.

After

integration

^{over 0 and} T ^we get

(with fllth

⁼ a) :

Experimentally,

^{we measure :}

with A = Xil - ^XI.

3.1.2 In the gel

phase.

^{- For} theoretical and

experi-

mental

simplifications,

^the

magnetic particles, quenched

^{in the}

polymeric

^network

by

^a strong

magnetic

held, ^{are all}

aligned

in the direction of the laser beam

(Z-axis).

^A

magnetic

^{field H is}

applied perpendicular

^to the Z-axis

(X-axis).

The measured

birefringence

is the result of an

equilibrium

^between

the

magnetic

energy - ,uH ^{sin 0 cos} qJ and the elastic

energy 2

^{C02 due to the}

gel

network : 0 and rp describe the

position

^{of the}

magnetic

^moment in the field H

(Fig. 2) ;

^{C is the}

product

^{of the} shear elastic constant G times the volume

perturbed by

^the

particle

^rotation.

In this geometry

(Fig.

2) ^the

birefringence

^{is obtained,}

Fig. ^{2. -} Definition of the angles 0 ^and T. X and Z are

respectively the directions of the applied magnetic ^{field H}

and the light propagation direction which is also the initial

position ^of magnetic dipole p ^{in the} gel phase.

making

^{a thermal} average ^over all the contributions of the ferrofluids,

taking

^{into account} their individual orientations. The distribution function is here :

Experimentally

^{we measure}

For the critical determination of the elastic constant, it is useful to make a limited

expansion

of On around

Tg

^when

^PC

^{1 with a low magnetic field}

^{(f3 p,H}

^1).

We find :

with

f(flC) = - 0.33(pC)2

^{+ 0.15}

j8C

^{+ 1.}

We note that,

surprisingly,

^{An has a} maximum for

pC =j:.

^0. This maximum must appear ^{at a} temperature smaller than

Tg ^{(where pC}

^{= 0).}

4. Experimental conditions.

Gelatin is an

example

^{of weak} gel [7]; the crosslinks between the

neighbouring peptide

chains fluctuate with time, ^{at thermal}

equilibrium.

Here, the evolution of the parameters ^{of the}

gelatin

is studied as a function of temperature. ^We avoid the evolution in time of the

gelatin ^{with a}

good

choice of the temperature ^variation

rate. To this end, ^a temperature

cycle

^{is made on a}

gelatin

sample (with concentration 1.5

% by weight) prepared

with ferrofluids. We record the

birefringence

level versus T -

Tg. Tg

^is determined

by

^{the sudden}

988

Fig. ^{3. -} Birefringence ^function of T -

T g

^{for two tem-}

perature ^{variation rates.}

variation (or stabilization) ^{of the}

birefringence

^level

starting

from the sol

(or

^the

gel) phase. By

examination of

figure

3, the critical variations of the

birefringence

presents ^no detectable difference

(exact overlap)

when the temperature is lowered or increased at a

rate smaller or equal ^{to 3 x 10-4}

K/s.

However, with such a rate, the

Tg

value determination varies

by

^{5 K}

depending

^{on which is} the initial

phase.

^This

is due to the fact that the

gelation

process

depends

on the whole

history

^{of the}

sample

^{from the} origin

(sol

^or

gel).

5. Critical variation of the elastic constant,

To get the critical variation of the elastic constant,

we

proceed

^{in the}

following

way.

5 .1 EXPERIMENT. - The

sample (gelatin

^{+ ferro-} fluids) ^is

quenched

^{at T =} 5 OC under a

high

magnetic

field

(Ho

^{= 5 000} G) ^{in order to line} up all the ferro- fluid

particles

ⁱⁿ the direction of the laser beam

(Fig.

2) ;

then this field is turned off. We impose ^a rotation of the

magnetic particle by applying

^a

magnetic

^{field H}

perpendicular

^to the initial orientation. For each temperature, ^we

adjust

the value of the magnetic field in order to

keep

^the

birefringence

^{level On}

constant The variation of the field H is

given in figure

^{4 versus} temperature. ^{When H} is switched off,

An relaxes because of the

restoring

^{force due to the! I}

Fig. ^{4. - In the} gel phase, ^we adjust ^the value of the magnetic field H(G) ^{in order to} keep ^the birefringence ^{level constant} (An ^{is here} equal ^{to 0.11 x} 10-2) ^{for each} temperature.

When increasing T, ^{the network comes} loose, ^{and the field}

necessary ^to get ^{the same} birefringence level becomes smaller and smaller.

gel elasticity; ^the relaxing

birefringence signal

^is

also recorded The

advantage

^{of this}

experimental

procedure ^is that, ^{close to}

Tg,

^{H is so} small that the

magnetic energy J-lH and the elastic

energy 2

^{Cø2 are of}

the same order of

magnitude

^as kT and therefore the gel ^{is not}

destroyed by

the small rotation of the

magnetic

probes.

Another important

point

^{is that}

Tg

^{cannot be}

deduced from this

experiment.

^{In the}

magnetic

^field

variation observed there is no great ^{variation near}

Tg.

Thus we need another

experiment

^{to determine}

Tg.

5.2 Tg DETERMINATION. -Experiment : ^The

following experiment

^is performed; ^the

sample

^is

prepared

ⁱⁿ

the same way ^as in 5. l, ^{but we}

keep

^the magnetic

field constant

during

^the

experiment

^{We record the}

birefringence

level On and its relaxation time as

function of temperature.

Observation :

Figure

5a shows the

typical experi-

mental variation of An as a function of temperature.

The remarkable feature is that we observe near

Tg

a maximum in the

birefringence

level. This maximum is

predicted by

the limited

expansion

of the theoretical value of An

(Eq.

3), ^{but the}

physical explanation

^is

hot simple.

At the field cut off, ^the

birefringence

^relaxation

time

T(T)

^is

plotted

^{versus T}

(Fig. 5a2).

^{We see that}

the maximum of the relaxation curve coincides with

a temperature T1 for which the

birefringence

^level

Fig. 5at. ^- Experiment « An(T), ^H = constant ». Bire-

fringence level variation as a function of temperature, H remaining ^constant. The maximum is predicted by ^the

theoretical value of An (Eq. 3).

Fig. 5a2. ^- Experimental decrease of the relaxation time

as a function of temperature, ^{when H = constant} Fig. ^{5b. -} Experiment ^{« An = constant,} H(T»). ^Decrease

of the magnetic ^field H(T) ^{as a} function of temperature.

measured is

equal

^{to that of}

Tg ; ^ðn(T1)

⁼

An(Tg).

We found

experimentally

^that

Tg -

^{T1 =} 0.5 ± 0.1 K with quite

good reproductibility.

^{Thus we have}

obtained

^an

interesting

method for the determination of

Tg ;

^{in the case}

^(5.2),

^{where, in order to get the}

critical variation of the elastic constant, ^{we measure}

simultaneously

^{H and the}

birefringence

^relaxation

time as a function of temperature, ^we

just

^{need to}

make a translation of 0.5 K from the maximum of the

birefringence

relaxation time curve to get

Tg. Tg

^is

located 0.5 K above

T1,

^which

corresponds

^{to the}

maximum of the curve T(7).

5. 3 EXPERIMENTAL RESULTS. ^- Now we look at

figure

3 giving

H(T)

^{for An = cost} The theoretical

expression

^{of the}

birefringence

^{level is}

given

by equation ^{2. For} each value of the field

(H)

^we get ^the

corresponding

^{elastic constant C.} Figure ^{6 shows a} log-log plot of elastic constant as a function of T -

Tg.

We obtain almost a

straight

^{line over one decade in}

T -

Tg.

^{We have} fitted the curve

C(T - Tg)

^for

three different values of the magnetic ^{moment of the}

agglomerate (the

three kinds of

symbols

^observed

on

Fig. 6).

^{If we} consider that the mean value of the moment is about 5.0 ^x 10-16

erg/G,

^{we have to notice}

that a relative

dispersion

in the value of that moment is irrelevant; ^no variation in the

slope

^{of the curve}

can be observed The mean value of this slope

give

the exponent t,

corresponding

^to C -

Tg - ^Tit,

t ⁼ 1.7 but if we consider the

asymptotic

slope, ^the

value of t is near 2. So a reasonable value of the elastic critical exponent ^{is t = 1.8} ± ^0.2.

6. Critical variation of the friction coefficient f.

Now, ^we try ^to

explain

^the

divergence

of the bire-

fringence

relaxation time -r obtained in the

experiments

described above in the gel

phase.

Fig. ^{6. - Elastic constant variation as a} function of T - T g

in a log-log plot (+) J.l ⁼ 10-is erg/G; (A) It ^{= 2.5 x} 10-16 erg/G; (0) u ^{= 10-16} erg/G. ^The straightline ^repre-

sents t ⁼ 1.7 and the dash-line t ⁼ 2’0.

6.1 RELAXATION PROCESSES. - Two different relaxa- tion _processes are known in a two medium system

(network

⁺

solvent)

^{of this} type. ^We consider ^the shear movement

produced

by the rotation of the

probe.

We introduce two transverse

displacement

vectors from this

equilibrium

position, ^{one for the} network, u(r, t), the other for the solvent,

v(r,

t). ^The

coupled

motion of the

gel

network and the solvent is described by

p, G and

f

^are

respectively

^the

density

^{of the} network,

its shear elastic constant and the friction coefficient between the network and the solvent. There are two

simple

solutions of

equation

^{4. If the} network and the solvent are moving ⁱⁿ

phase,

^{u =} v, and the return to

equilibrium

^{is due to a}

damped (because

^{of viscous}

terms not considered in

Eq.

4) sinusoidal motion [8]

we did not observe

experimentally.

If the network is

moving alone, ^{v =} 0; ^{at low}

frequencies, equation

⁴

becomes V’u ⁼

f/Gu.

And the relaxation time -r is found to be

proportional to fr2 / G (or fr2 / C

^{as C oc} G)

where r refers to the

hydrodynamical

radius of the ferrofluid agglomerate.

6.2 EXPERIMENTAL RESULTS. - We define a critical exponent ^{for r}

(T

^-

(Tg - T)’").

^Thus, the critical, variation of

f

is obtained : it _goes to zero at

T g

^like

C-r. From a

log-log plot (Fig. 7)

^{we find x = 0.8} + ^0.1.

7. Discussion.

Two kinds of results have been obtained for the elastic constant and the relaxation time, ^near

T g’

ⁱⁿ

the

gel phase.

The

experimental

exponent of the shear elastic constant C is centred on the value t = 1.8 + ^{0.2. This} value is in

good

agreement with the theoretical value

using

^the

analogy

between the

macroscopic

^conduc-

tivity

^of 3d-dimensional lattice of random

conducting

bonds

[9]

and the elastic constant. On the other hand,

it is _very different from the classical result t ⁼ 3

[10].

Only

^{a few}

experiments

^{have been}

performed

^{in rela-}

Fig. ^{7. -} Birefringence relaxation time in the gel phase ^as

a function of T - Tg ^{on a log-log plot..}

990

tion to the measurement of the elastic constant critical behaviour in

gelatin.

^{As far as we know, there is one,}

due to Peniche-Covas et al. [11] (the ^initial purpose of which was to find the correct

multiplicity

^{of the}

helical crosslinked zone) which leads to the same

exponent ^{value t = 1.7}

[3].

^{We note that our}

experi-

ment was

performed

^{with a 1.5}

% gelatin

system whereas the Peniche-Covas

experiment

^{was done with}

a 5.7

% quenched

sample.

(In

^{their case, the}

experi-

mental set-up ^{was a}

sphere rheometer). Comparing

these results with those in chemical

gels,

^{we find that}

the exponents t ^{obtained are all}

higher

than those of the

gelatin

system. ^{As an}

example,

^{for a} system ^made

by

^radical

copolymerization

^of styrene-meta

divinyl

benzene with a solvent, t ^{was found}

equal

^{to 2.1}

[12].

In another system ^obtained

by polycondensation

^of

hexamethyl di-isocyanate, t

^{was found}

equal

^{to 3.2}

[13].

Theoretically,

the critical variations of the friction coefficient is not well established But one

possibility

is that

f

^can be related to the

permeability

^{of the}

gel

as

’1/,2 (Darcy’s

law) where, is the correlation

length

of the system (the ^mean distance between two reticu- lation

points)

and ?I ^{is the}

viscosity

of the fluid

phase

(solvant and finite clusters) [14, 15].

If we assume

that q

^{has a critical behaviour} r ~ DT

I-s’,

^{we obtain}

f -

I AT

I2v-s’.

In our

experiment

the relaxation time in the

gel phase

^{is T _}

fr2/G.

^{We can} suppose that the elastic constant G in a

dynamic experiment

^{does not have}

the same critical exponent ^as that measured in a static

experiment.

G =

I AT It’.

^The

entanglements

^increase

the value of the static elastic constant (t’

t).

^The

critical form of the relaxation time is : s =

I AT I-x

with x = s’ + t’ - 2 v. The

experimental

^{value of x}

is 0.8 ± ^{0.1. In}

3d-percolation theory

^{v = 0.88 and}

we deduce that s’ + t’ ⁼ 2.56 ± 0.1. ^One

hypothesis

is that the

viscosity

^{has the same critical} exponent ⁱⁿ the sol and

gel phase

(s ⁼

s’).

^{Thus we need s. An}

experiment quite

^{similar to that}

previously reported [4]

^was

performed

^{in the sol}

phase

where the birefrin- gence relaxation time i is

proportional

^{to the}

viscosity

of the sample. ^{i was recorded as a} function of T. A

log-log plot

is shown in

figure

^{8. The}

slope

^is

directly

the exponent s ^{= 0.72} ± 0.1. ^Then

according

^{to the}

previous hypothesis t’

^{= 1.84 ± 0.2. This} value of the critical exponent of the elastic constant obtained from the relaxation time is _very near to the value obtained

Fig. ^{8. -} Log-log plot ^{of the} birefringence relaxation time

as a function of the temperature difference T - T g ^{in the}

sol phase. ^The slope ^{is s = 0.72} ± 0.1. ^T is proportional ^to

the viscosity ^{of the} polymeric ^{solution T oc n} oc 111 T I-s.

in the static

experiment

^{t =} 1.8 + 0.2. This last feature shows that the

entanglements

^{do not} affect the elastic constant in the relaxation

experiment.

8 Conclusion

In

gelatin

(1.5 % ⁱⁿ

weight)

the critical behaviour of the shear elastic constant, the friction coefficient in the

gel phase,

^{and the}

divergence

^{of the}

viscosity

ⁱⁿ

the sol

phase

have been measured. We have used an

original experimental

method which enables the shear elastic constant to be measured _very close to

Tg ;

^it

is

possible

^{to have near}

Tg

^{an elastic energy}

^(induced by

^the

magnetic field)

^{of the same order as kT. The}

time

dependence phenomena

^{are very}

important

ⁱⁿ

gelatin

^{but we} have found a

reproducibility

^criterion

for a defined temperature ^rate. With these

experimen-

tal conditions, the critical exponents ^{found are} very close to those of

3d-percolation theory.

References

[1] VEIS, A., Macromolecular chemistry of gelatin (Acade-

mic Press, ^New York) ^1964.

[2] ^{FLORY, P. J., J. Am.} Chem. Soc. 63 (1941) 3083, 3091, 3096.

[3] STAUFFER, D., Physica ^106A (1981) ^177.

[4] ^{DUMAS, J. and} BACRI, J.-C., J. Physique ^{Lett. 41} (1980)

L-279.

[5] MASSART, R., ^{C. R.} Hebd. Séan. Acad. Sci. 291 (1980).

[6] JOLIVET, J.-C., MASSART, R., FRUCHART, ^J. M., ^Nouv.

J. Chim. (1983).

[7] ^DE GENNES, ^P. G., Scaling concepts ⁱⁿ polymer physics (Cornell University Press, Ithaca, ^N. Y.) ^1979.

[8] TANAKA, T., KOCKER, ^L. U., BENEDEK, ^G. B., ^J. Chem.

Phys. ⁵⁹ (1973) ^5151.

[9] ^DE GENNES, ^P. G., ^J. Physique ^{Lett. 37} (1976) ^L-1.

[10] STAUFFER, D., ^J. Chem. Soc. Faraday Trans. II 72

(1976) ^1354.

[11] PENICHE-COVACS, C., DEV, S., GORDON, M., JUDD, M., and KAJIWARA, K., Faraday ^{Dics. 57} (1974).

[12] GAUTHIER-MANUEL, B., ^and GUYON, E., ^J. Physique

Lett. 41 (1980) ^L-509.

[13] ADAM, M., DELSANTI, M., DURAND, D., HILD, G., ^and MUCH, ^J. P., ^Pure Appl. ^{Chem. 53} (1981) ^1489.

[14] STAUFFER, D., CONOGLIO, A., ^and ADAM, M., ^Adv. Poly.

Sci. 44 (1982) ^103.

[15] JOUHIER, B., ALLAIN, C., GAUTHIER-MANUEL, ^{B. and} GUYON, E., ^The sol-gel transition (preprint).