New method of viscosity measurement near the gelatin sol-gel transition

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New method of viscosity measurement near the gelatin

sol-gel transition

J. Dumas, J.-C. Bacri

To cite this version:

J. Dumas,

J.-C. Bacri.

New method of viscosity measurement near the gelatin

sol-gel transition.

Journal de Physique Lettres, Edp sciences, 1980, 41 (12), pp.279-282.

�10.1051/jphyslet:019800041012027900�. �jpa-00231779�

New

method of

_viscosity

measurement

near

the

gelatin

sol-gel

transition

J. Dumas and J.-C. Bacri

Laboratoire d’Ultrasons (*), Université Pierre-et-Marie-Curie, Tour 13, 4, place Jussieu, 75230 Paris Cedex 05, France

(Reçu le _{28 janvier 1980, accepte} le 28 avril ₁₉₈₀₎

Résumé. 2014 La

mesure d’un temps caractéristique de relaxation de rotation de

_petites

_particules

_{ferromagnétiques}

dispersées dans une solution à 7 _% de _{gélatine, permet} l’observation de la viscosité au _voisinage de la transition

sol-gel.

L’exposant

critique de la viscosité s = _0,95

± 0,10, valable sur 3 décades en _{température,} est en accord avec des calculs numériques effectués dans le cadre d’une

_{approximation}

de

_champ

effectif en théorie de la _percolation. Abstract. 2014 The

viscosity of a 7 _%

_gelatin

solution has been measured in the very close vicinity of the gelation transition, by rotational relaxation of _{ferromagnetic}

_particles

imbedded in the mixture.

The viscosity critical _{exponent s} = 0.95 _± 0.10 which is obtained _over 3 decades in

temperature, is well accounted for _by numerical calculations in the effective field treatment of

_percolation

_theory.

Classification

Physics Abstracts

46.30J - 64.70E - 64.80

Gelatin is the water-soluble

_product

of the

disso-lution,

disorganization

or

_degradation

of

water-insoluble

_collagen

fibers

_[1].

The

_{resulting peptide}

chains take _up random

_{configurations}

and at

suffi-ciently high

temperature

the mixture is a blend of

monomers in a form of

coils,

and solvent. As the

temperature

is

decreased,

polymerization

and reorga-nization of some helical structure occur with the

formation of n-mers and

subsequent

growing

of the

gyration

radius of the macromolecular clusters. An infinite

_polymer

_{appears in the}

_bath,

at the

_temperature

Tg

of the reversible

_liquid-gel

transition,

but there are

still a

non-negligible

amount of small n-mers in the

system. For T

_Tg,

the

_gel

fraction

_{800 -}

the infinite

polymer

-

grows

by

linkage

of the residual n-mers.

When the

_gel

fraction is

_completed

_by

all monomers

incorporation,

curing

of

_the

_gel

occurs

_by

the

linkage

of

_pending

chains to form an

unique

infinite

cross-linked network at T

_Tg.

The

_{macroscopic viscosity ~}

of the

_{sol-phase depends}

of the size of the

_polymer

clusters : ~

grows when the

temperature

is lowered to attain an infinite value

when T =

Tg.

In this

letter,

we _present a _very

_simple

method for

observing

the

_growing

of the

_{macroscopic viscosity}

in the

_sol-phase

when

_Tg

is

_approached

and for

mea-suring

the

_{corresponding}

critical _exponent.

(*) Associated with the Centre National de la Recherche

Scien-tifique.

1.

_Experimental

method. - The method of

mea-surement is based on the

_study

of rotational relaxation time of small _aggregates of

_{magnetic particles}

imbedded in the

gelatin

solution.

In a uniform

_magnetic

field the

_{particles experience}

a _torque and line _up with the field : the solution shows a

linear

_{birefringence [2].}

We observe the

decreasing

birefringence signal

when the field is turned off. We use a

_commercially

available

product,

A05

ferrofluid

_(from

Ferrofluidic

_Corporation,

_U.S.A.)

which is a colloidal

_suspension

of monodomain

ferromagnetic

Fe304

particles, roughly spherically

shaped,

of dimension around

_{125 A,}

coated with a

monomolecular

_layer

that acts as an elastic cushion

preventing agglomeration.

In fact when the solution is submitted once to a small

magnetic

field

( ~

20

_G),

the ferrofluid

particles

in

water-base

_suspension

form

_{needle-shaped}

agglo-merates which are _very difficult to

_destroy

_[3].

The size distribution of these needles is not

known,

but observations from electronic

_microscope

pic-tures

_[4]

_suggests that the number of individual

_grain

is _very

low,

the main structure

_being

_composed

of chains

_ranging

from

_{0.1 ~}

to more than

_{10 ~.}

1.1 SAMPLES PREPARATION. - A

_gelatin

solution in water was

_prepared

with the

_{following composition :}

7 g of

_{gelatin powder}

_(from

Prolabo,

Paris)

are

dissolved in 100

cm3

of hot water and

_kept

at 70 °C

during

15 min.

L-280 JOURNAL DE _{PHYSIQUE -} LETTRES

A

_drop

of water-based ferrofluid is introduced in

the

_gelatin

solution to obtain a mixture with 1013

magnetic

particles/cm3.

Reference solutions of the

same concentration of ferrofluid in pure water have

also been made to measure the size distribution of the

magnetic

agglomerates

in a

liquid

of known

viscosity.

The

_liquid

mixture is introduced in a

(10

x 3 x

₃₎

mm

brass tank in which

_optical

windows allows a laser

beam to _go

_through

the

_sample.

A

_platinum

sensor

is immersed

_directly

in the

_liquid

to measure the

temperature.

The tank is fixed on a Peltier effect

_temperature

regulation

system

giving

a stabilization of the tank

temperature

of 5 x

10 - 3 K.

1.2 BIREFRINGENCE MEASUREMENTS.

_{- Figure}

I

shows the sensitive

_set-up

used in these measurements.

The

_principle

of

_operation

is described elsewhere

_[5].

The

_voltage

obtained at the _output of the lock-in detector is

_{directly proportional}

to the

_{birefringence}

of the

_sample.

In our _system, we cannot observe any

decay

rate of the

_{birefringence signal}

which is more

rapid

than 0.5 ms.

Fig. 1. -

Experimental set-up. Laser L, analyser and polarizer A and _{P, photoelastic phase} modulator _{PEM, magnetic} coils MC and

pulsed alimentation H, temperature regulated sample RS, photocell

PC, lock-in _amplifier LA, trigger TR and _scope S.

1. 3 MAGNETIC COILS. - The

sample

tank is

sur-rounded

_by

two

_magnetic

Helmholtz coils. The alimentation of the coils delivers

adjustable

current

pulses

of 5

A,

4 s

_long

and

_{100 ps}

fall time. The

magnetic

field available at the tank location was near

100 G. The current

_pulse

is

_{triggered manually}

or at a

very slow rate

_(10-2

_Hz)

to

_prevent

_{parasitic heating}

of the tank.

1.4 EXPERIMENT METHOD. - The

temperature of the

_sample

is maintained stable within

10-3

K and a current

_pulse

is

_applied

on the

_magnetic

coils : the

birefringence

signal

grows, shows a constant value

during

the main duration of the

_pulse

and decreases after the cut-off of the field. This decrease is recorded

as a function of time on an

oscilloscope

and then

photographed.

In this

_{preliminary experiment}

the

analysis

has been done on the

photograph

serie.

The measurement

_begins

in the

_sol-phase

of the

gelatin

solution

_just

after the

_sample

_preparation,

near 38 °C. Then for _any

_temperature

stabilized

during

the

_cooling,

the

_{relaxing birefringence signal}

is recorded. The

_{thermodynamic}

_equilibrium

is expect-ed to be attained in less than 5 min. for a 1 K variation as can be deduced from

_quenching

_experiments

on

gelatin

of closed

_composition

_(6.5

_%)

_[6].

1. 5 EXPERIMENTAL RESULTS.

~ The main observed feature is the increase of the

total time needed for the

_{birefringence signal}

to

relax

_g(t)

when the

_temperature

decreases in the

sol-phase :

it varies in a ratio

103

between 34 °C and the

gel

transition

_temperature

_Tg

= 23.8 °C.

~ The

_relaxing

_signal

does not follow an

expo-nential variation as function of time : the best fit is

obtained with

as is shown

_by

the

_semi-log

_drawing

on

_figure

2. The

time _to

_corresponds

to the

_beginning

of the

experi-ment. In

fact,

to cannot be taken smaller than 0.5 ms

due to the bandwidth of the lock-in

_amplifier

_out-put.

tm is the time needed

_{by g}

to attain

_practically

the noise level.

Figure

2 shows

_only

the

_analysis

of the

_relaxing

signal

near 29 °C. The

_logarithmic

variation is observed in the whole range of

temperature

attained.

Fig. 2. -

Analysis of the _{birefringence} relaxation at 29 °C as a

function of time on a _{linear-log plot;}

The meaning of to and tm is explained in the text.

In

_spite

_{of the fact that the variation is}

non-expo-nential,

the time At needed to divide the

_{birefringence}

signal g

by

a

factor e,

has been measured and is

given

versus

(T -

_Tg)

on a linear

(Fig.

3)

and

_log-log

aligned

for 3 decades of variation in AT = T -

Tg

and defined a

_slope

(Fig. 4)

~ The

_sol-gel

transition is determined _very

_sharply

( ~

10 - 2

_K)

as the

_temperature

at which the

bire-fringence signal begins

decreasing

as a _consequence

of the

_locking

of the ferrofluid chains

by

the infinite network. The absolute value

of s,

measured over 10

K,

is

_practically

not affected

_by

this

_uncertainty.

Fig. 3. - Characteristic time At of

birefringence relaxation as a

function of the temperature. It is shown in the text that At is

propor-tional

to the macroscopic viscosity if.

Fig. 4. -

Log-log plot of the characteristic time At of the

bire-fringence as a function of the _temperature difference AT = _{T -}

Tg

where _Tg = _{26.8 oC, is the}

gelation point. The slope ~=0.95±0.10

is the critical exponent of the _{macroscopic viscosity ~.}

2.

_Theory.

- The

relaxing birefringence signal

g(t)

is

_proportional

to the number of

_optically

_anisotropic

ferrofluid

_{particles :}

where G is a constant

_depending

on the

_{birefringence}

of all the

_particles,

_{D f}

the rotation diffusion constant

of a chain of

_f

_magnetic

_particles,

and _{cP f} the number concentration of chains formed

_by

_{f particles.}

F is the number of

_magnetic

_{grains belonging}

to the

longest

chain.

In the case of

_{long prolate ellipsoids}

the rotation-diffusion constant

_{D f}

is related

_mainly

to the

_{length a}

of the

_{elongated particles}

and shows _very little

depen-dence on _{any other} dimensions b of the

_ellipsoid

_{[7] :}

We then take the coefficient

_{D f}

_{corresponding}

to an

equivalent sphere

of

_{radius r f ~ a}

which takes into

account all

_important

_{parameters :}

We _{suppose r f}

_{= a 1 f a}

where 6 is an unknown

parameter

and a1 the ferrofluid

one-particle

radius.

The most

_important

terms in the

_integral

_correspond

to values of

_f

and t for which the

_exponential

argu-ment :

this

_{expression defining}

the lower limit of

_integration.

We

_expect

that for a

_given

value of

_tlq

the

_{birefringence}

signal

is

_mainly

due to

_particles

_ranging

from

_{f *}

to F. A

_rough

calculation of _{r f*,}

_taking

the exact

value of

_{D f,}

with

_{a/b ~}

50 and 6 =

1,

gives

_a value

of

_{0.1 ~ :}

it is

_probable

that all

_{birefringent particles}

are taken into account in this

_{approximation.}

As is

_pointed

in

_[8]

the value of the

_viscosity

mea-sured

_depends

on the

_probe

dimension _{r f,} and the

macroscopic viscosity ~

is obtained

_only

for

rf

=

rN*,

with ~

the characteristic

length

of the

largest

clusters

_{of gelatine :}

with a monomer coil radius

100

_A,

the correlation

_length

of

_{10 Jl}

is obtained for

Ap -

10 ‘ 3.

Thus a

_viscosity

correction should have

perhaps

been made for the last decade in our mea-surement.

The

_experimental

variation

_g(t)

is

_logarithmic

(Fig.

2)

and this

_suggests

that the size

_distribution

L-282 JOURNAL DE _{PHYSIQUE -} LETTRES

Then,

the

_{birefringence}

_{relaxing signal}

is :

We measure a certain interval of time At on the

relaxing

signal

defined

_{by :}

In our case we take _to ~ 0.5 ms

_and,

as At varies

from 8 ms to 12 _{s, to} t is a _very

good

approximation.

In these

conditions,

we find that At is

_proportional

to the

_{macroscopic viscosity ~ : ~ ~}

At

_noticing

that

_to/r~

has a constant

_value,

which is a _consequence

of the

_{experimentally}

observed constant level of

g(to).

The

_viscosity

critical

_{exponent s}

is defined

_{by :}

in which

_Ap

=

/? 2013 ~ where p is the fraction of

reacted bonds and Pc the

gelation

threshold. 10 is the

viscosity

of the

_original

solution of monomers at

high

temperature.

Supposing

a linear relation between

Ap

and AT = T -

Tg

in the

_vicinity

of

_Tg,

the

_slope

of the

log-log drawing

of

_At(T)

_(Fig.

₄₎

_gives

the critical

_exponent

s.

It can be remarked that the result

At - $

would

have been found for any distribution~ ~ 1/f ~, E

> 1 :

this

_{approximation}

is not critical in the calculation. 3. Discussion. - In

spite

of the fact that the classical

_theory

of

_gelation

_[9]

does not

_attempt

to

predict

the

_viscosity

behaviour,

three values of the

critical _exponents are available in the literature

_[10,

11, 12]. Among

them,

the effective medium

approxi-mation of the

_percolation

theory

[12]

gives

a value s =

1,

which is very well verified in _our

experiment.

This _exponent has

_already

been found

_{experimentally}

in _styrene

_{divinylbenzene}

_gelating

_system

_[13]

and

governs the

viscosity

behaviour far from the transition

point,

whereas s ~ 0.78 _very close to the

transition;

this last value has been

numerically

calculated and found for the electrical conductance in random mixtures of normal and

_{superconducting}

elements

_[11]

_]

which as certain

_arguments

_suggest

_[10]

follow the

same behaviour as the

_gelating

_system

_viscosity.

We do not find _any indications of such a crossover

in the

_gelatin

_system.

Some remarks can be made at this

_stage.

- The effective medium

approximation

is a mean

field treatment and

_applies

_only

outside a critical

region,

which is difficult to

_appreciate

in the

_gelatin

case as the connection is not known between p and

T,

in

_spite

of certain theoretical _arguments

_[14].

We can

only

claim that

_Ap

is

_proportional

to ~T in the

immediate

_vicinity

of

_Tg,

and this

_{approximation}

is used here.

- The calculations and theories that we

just

mentioned are concerned with chemical

gelation

which supposes strong and not reversible

_crosslinking

between

_polymers.

It is not obvious that the same

kind of theoretical treatment can be

_applied

to the

gelatin

system.

Aknowledgments.

- The authors thank P. G. de

Gennes and F. Brochard for a

_helpful

and

_stimulating

discussion.

References

[1] VEIS, A., The macromolecular _{chemistry of gelatin (Academic}

Press) 1964.

[2] MARTINET, A., Rheol. Acta 13 (1974) 260.

[3] HAYES, C. F., HWANG, S. R., J. Colloid _Interface Sci. 60

(1977) 443.

[4] HORN CHANDESRIS, D., Thèse de 3e cycle, Orsay (mars 1977). [5] KEMP, J. _C., J. Opt. Soc. Am. 59 ₍₁₉₆₉₎ 950.

[6] EAGLAND, D., PILLING, G. and WHEELER, R. G., Farad. Discuss. Chem. Soc. 57 (1974) 181.

[7] TANFORD, C., Physical Chemistry of Macromolecules _(John

Wiley and Sons) 1965.

[8] DE GENNES, P. G., J. Physique Lett. 40 (1979) L-197.

[9] FLORY, P. J., Principles of polymer chemistry (Cornell Univ.

Press, Ithaca) 1953.

[10] DE GENNES, P. G., C.R. Hebd. Séan. Acad. Sci. Paris 286B

(1978) 131.

[11] STRALEY, J. P., J. Phys. C (Sol. St. Physi.) 9 (1976) 783.

[12] KIRKPATRICK, S., Rev. Mod. Phys. 45 (1973) 574.

[13] ADAM, M., DELSANTI, M., OKASHA, R., HILD, G., J. Physique

Lett. 40 (1979) L-539.

[14] KASTEKEYN, P. W., FORTUIN, C. M., J. _Phys. Soc. _Japan 26