A zipper model approach to the thermoreversible gel-sol transition

HAL Id: jpa-00212485

https://hal.archives-ouvertes.fr/jpa-00212485

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.

A zipper model approach to the thermoreversible gel-sol

transition

K. Nishinari, S. Koide, P.A. Williams, G.O. Phillips

To cite this version:

1759

A

_zipper

model

_approach

to

the thermoreversible

_gel-sol

transition

K. Nishinari

_(1,

_*),

S. Koide

_(2),

P. A. Williams

₍₁₎

and G. O.

_Phillips

₍₁₎

(1)

Polymer

and Colloid

_{Chemistry Group,}

The North East Wales _Institute, Connah’s

_Quay,

Deeside,

Clwyd

CH5 4BR, G.B.

(2)

President, Yamanashi

_University,

Takeda 4, Kofu, Yamanashi,

Japan

(Received

on _{January 19, 1990,} revised on

_April

_{24, 1990,}

_accepted

on

_April

26,

₁₉₉₀₎

Résumé. 2014 La transition

gel-sol

est assimilée à un processus

simple

d’ouverture des zones de

jonction

macromoléculaires semblable à l’ouverture d’une « fermeture éclair ». Cette

analogie

a

déjà

été

_proposée

pour

expliquer

la transition

_{hélice-pelote}

ou la fusion de l’acide

désoxyribonu-cléique.

Le

_{profil expérimental}

des

_{pics endothermiques qui}

accompagnent la transition

_gel-sol

lors d’une

_{analyse enthalpique}

différentielle a _pu être

_ajusté

en utilisant ce modèle.

Abstract. 2014

The

gel-sol

transition of thermo-reversible

_gels

is examined

_{theoretically by using}

a

zipper

model which has been used for

_explaining

the helix-coil transition and the

_melting

of

deoxyribonucleic

acid

(DNA).

The endothermic

peaks

in a

_heating

curve of differential

scanning

calorimetry

for various thermoreversible

_gels

are

_{analysed by}

this treatment.

J.

_Phys.

France 51

₍₁₉₉₀₎

1759-1768 15 AOÛT _1990,

Classification

Physics

Abstracts 44.50 - 61.90 - 82.70

Introduction.

The dissolution of thermo-reversible

_gels

has been studied over many years not

_only

because of its scientific interest but also because of its

_{practical importance}

in food

_processing

and in many other industrial

applications [1]. Eldridge

and

Ferry [2]

found that the

_plot

of the

logarithm

of the concentration

_log

c versus the inverse of the

_melting

_temperature

1 T

( Tm

in

_{degrees Kelvin),}

and the

_plot

of

_logarithm

of the molecular

_weight,

log MW

versus

_1/Tm

were both almost linear for

_{gelatin gels. Using}

van’t Hoffs

_law,

_they

proposed

a method to determine the heat absorbed on

_forming

a mole

_{of junction}

zones.

Tobolsky proposed

a

_theory

of dissolution of

_gels

based on the

_highly

swollen network

model

[3].

It seems,

however,

difficult to _{compare his}

_theory

with

_experiments.

Differential

_{scanning calorimetry (DSC)}

has been used to

_study

the

_gel-sol

transition for

thermo-reversible

_gels.

Endothermic

_{peaks arising}

from the

_gel-to-sol

transition as well as

exothermic

_{peaks arising}

from the

_sol-to-gel

transition have been observed in

_heating

and

cooling

DSC curves

_{respectively by}

many

investigators [4-11].

The heat _{absorbed per unit} time

_{dQ /dt}

is

_{given by}

C

_dT/dt,

where C is the heat

_capacity.

In the

_heating

DSC

measurements,

_dT/dt

is

_positive,

then the

endothermic

peak

is

equivalent

to the maximum of the heat

_capacity.

In the

_cooling

DSC measurements,

_dT/dt

is

negative,

then the exothermic

peak

is

_{again equivalent}

to the maximum of C.

In the

_present

work,

we _propose a method of

_{phenomenological analysis}

of the

_gel-sol

transition

_by

means of a

_zipper

model which has been

_{previously applied}

to

_explain

a

helix-coil transition

_[12-14].

Zipper

model and

_junction

zones.

It has been

_{widely accepted}

that most thermo-reversible

_gels

consist of somewhat

_crystalline

regions,

called

_junction

zones, and somewhat

amorphous regions.

The structure of

_junction

zones

_depends

on the molecules which form the

_{gels ;}

e.g.

junction

zones of

_{gelatin gels}

are

thought

to consist of

_triple

helices

[15]

and those of _agarose

_gels

are believed to be a

substantial association of double helices

_[16].

As a model for these _structures, we _may assume

such

_junction

zones as the association of molecular

_{zippers standing}

for a

_rigid

ordered

molecular structure such as helices or extended molecules. A

_{single zipper}

with N links is shown in

_figure

_1,

where broken lines

_represent

_{secondary bondings}

such as

_hydrogen

bonds.

Fig.

1. - A

single zipper

with N links. Links can be

opened

from both ends. If the links 1, 2, ...,

p are all open, the energy

required

to _open

_{the p}

+ 1 st link is E. Helices may not

_necessarily

be double

helices ;

they

may be

single

helices or

_triple

helices. Here, double helices _represent the association of somewhat ordered structure

_{symbolically.}

Now,

we assume that a

_junction

zone consists of either an association of helices

(single,

double or

_triple)

or that of rod-like molecules

_{(Fig. 2a).}

As in the case of the mixture of

galactomannan (such

as locust bean

_gum)

and xanthan

_solutions,

where each

_component

does

not form a

_gel

when

_they

are not

_mixed,

the structure of

_junction

zones is

supposed

to be the

association of xanthan helical molecules and mannan backbones

_[17,

_{18] (Fig.}

2b).

Both of

these

_gels

can be

_regarded

as

_{gels consisting}

of

_zippers

of one kind whether molecules which

construct

_junction

zones are the same or not. The molecular forces which make these helices

or rods

_aggregate

are

_generally

believed to be

_secondary

forces such as

_hydrogen

bonds rather

than covalent bonds because the

_disruption

of the covalent bond needs much

_higher

energy

than

_{experimentally}

observed values. We can,

therefore,

simulate the dissolution of

gels

as an

opening

process of molecular

zippers.

We treat a molecular

_zipper

of N

_{parallel links}

that can

be

_opened

from both ends. When the links

1,

2,

..., p are all open, the energy

required

to

open the p + 1 st link is assumed to be E. _{We suppose that each open link} can assume

G

_{orientations, i.e.,}

the open state of a link is G-fold

degenerate,

corresponding

to the rotational freedom of a link. These

assumptions

are the same as those used for the

1761

Fig.

2.

_{- (a)}

Gels

_consisting

of

_zippers

of one kind. Gels consist of

microcrystalline regions

and

amorphous regions.

The former is called

_junction

zones, and consist of association of molecules with

ordered structure.

(b)

Gels

consisting

of

_zippers

of one kind. Schematic

_{representation}

of a mixed

_gel

of

galactomannan

whose backbone is shown

_by

a

_zigzag

and xanthan shown

_by

a double helix

(proposed by

Dea).

The

_partition

function for such a

_{single zipper}

is

_{given by}

where T = kT. If the chain is

long enough,

the terminal contribution _may be

ignored

and

is

_easily

obtained from

_{equation (1).}

Now,

we consider a

_system

_consisting

of N’

_{single zippers}

each of which has

N links. The

_partition

function Z of this

_system

is

_{given by}

Since the heat

_capacity

C of this

_system

is

_{given by}

the substitution of

_{(2), (3)}

into

₍₄₎

leads to

When the interaction _among different

_components

is not

_strong

as in the case of blend

gels

of agarose and

gelatin,

the

_gels

formed on

_cooling

_may be

_thought

as a

_{phase separated gel}

as

shown in

_figure

3. We consider here the case in which

_junction

zones consist of associations

Fig.

3. - Gels

_consisting

of various

zippers (phase separated).

When two

_gel-forming

macromolecules

are mixed, and form a

_gel,

the

_gel

consists of

_separated

two

_regions

if these two macromolecules do not

interact so

_strongly.

number of molecular

_species.

Let

_{N1, N2,...,}

_{Nn be}

the number of links of

_{single zippers}

of various kinds

_{1, 2,}

..., n

respectively.

We assume that a

gel

consists of

X

single zippers

of the

kind 1 and

_{X 2}

_{single zippers}

of the second kind

2,

and so on. The heat

capacity

of such a

system

is written as

where

_Ci

is

_{given by equation (5).}

Some numerical results and discussion.

As stated

above,

the order of

_magnitude

of

_bonding

energy may be taken as that of a

hydrogen

bond,

i.e.

_{E1 =}

2 000 K. As for the

_parameter

_Gi,

it will decrease with

_increasing

concentration because the

_mobility

of an

_opened

link will be more constrained at

_higher

concentrations. When the

_temperature

is

_raised,

the volume of the

_gel

increases,

and G will also increase because the restriction on the motion of the links will be reduced.

However,

it is

_hopeless

to derive the functional form of G

_{(c, T) by using microscopic}

model

because of the

_{extremely complicated}

structures of thermo-reversible

_{gels. Strictly speaking,}

moreover, the value of G may

change

in course of successive

_openings

for a

_{single zipper.}

These

_points

will be discussed later.

The heat

_capacity

C

_{given by equation (5)}

as a function of

_temperature

is

_{expressed by}

a

1763

N,

examples

are shown in

_figures

4a-4c. For the sake of

comparison,

the curve

_{corresponding}

to G =

1 000, E

= 2 500 K and N = 100 is

superimposed

in each of these

figures. Figure

4a shows the G

_dependence

of C with

_{El =}

2 000

K,

E2

= 2 500

K,

N

₁ =

N2

=

100,

G 2

= 1 000 and various

_{JY’ and}

_N2.

_G

_{1 is changed}

from 500 to 1 000

_{(from Fig. 4a(l)}

to

_{Fig. 4a(4))}

and

the other

_parameters

are fixed. The

_higher

_temperature

_{peak position}

is

_mainly

determined

by G2( G1)

and not shifted

_{by G 1.}

The lower

_temperature

_peak

is determined

_by

G 1 and

is shifted to

_higher

_temperatures

with

_{decreasing G 1.}

Since the value of G

_represents

the rotational freedom of a

_link,

this is

_quite

reasonable.

_Figure

4b shows the

E

_dependence

of C as a function of

_temperature

with

_E2

= 2 500

_K,

_G

=

G 2

=

1 000,

N1 = N2 =

100 and various

_n1

and

_X2.

_El

is

_changed

from 2 200 K to 2 460 K and the other

parameters

are fixed. The

_higher

_temperature

_{peak position}

is

mainly

determined

by

E2

( > E1)

and not shifted

_by

_El.

As is

_expected,

the lower

_temperature

_peak

shifts to

_higher

temperatures

with

_increasing

_El.

_Figure

4c shows the N

_dependence

of C as a function of

temperature

with

_El

= 2 000

K,

E2

= 2 500

K,

N2

=

100,

G

1 =

G 2

=

1 000,

and various

NI

and

_{JY’2. NI}

is

_changed

from 50 to 400. The

_peak

temperature

is

mainly

determined

by

Ei

and

_G

_i and is shifted

_{only slightly by}

_Ni.

The lower

_temperature

_peak

shifts to

_higher

temperatures

with

_{increasing N}

_which

is

_expected

from

_empirical

formulae of

_{Eldridge-Ferry}

(2) :

higher

the molecular

_weight,

the

_higher

the

_{melting point}

of

_gels.

These numerical results can

_{explain experimental}

results accumulated for many years

(19).

When the concentration of

_gels

_increases,

the

_mobility

of chain molecules

_decreases,

and then the

_degeneracy

G will decrease as mentioned above. As a

_result,

the

_peak

of the heat

_capacity

shifts to

_higher

_{temperatures.}

This is in accordance with

_{Eldridge-Ferry’s empirical}

formulae.

It is well known that the

_melting

_temperature

_T.

of a thermo-reversible

_gel

in its

_heating

DSC curve _appears at a

higher

_temperature

than the

_setting

_temperature

_Ts

in its

_cooling

DSC curve. When the

_temperature

is

raised,

G would start from the lower value

_{G g}

_{corresponding}

to the

_gel

state.

_Then,

the

_gel

will

expand,

which will

_give

rise to an increase in the rotational freedom. On the

_contrary,

when the

_temperature

is lowered from

higher

temperatures

than the

_melting

_point,

G will start from the

_higher

value

_G

s

corresponding

to the sol state.

Therefore,

the

_opening

of molecular

_{zippers begins}

to occur at small G values in

_heating

process, while

gelation by cooling

will take

place

with

_decreasing

G which starts from

_large

G values at

_higher

_{temperatures.}

_Then,

_{the average effective value of G is small in}

_heating

and is

_large

in

_cooling.

As a first

_{approximation,}

therefore,

we can _say that the

_{melting peak}

temperature

T.

is determined

_by

_{certain average}

_{G g}

of G for

_gel

state and the

_setting

temperature

T,

is determined

_by

an _average

Gs

_S of G for sol state.

_Apparently,

G g

G S,

hence, T.

is

_expected

to be

_higher

than

_Ts.

In the case of

_heating,

the transition caused

_by

the

_opening

of the

_zippers

will start as soon as the

_temperature

arrives at the tail of the C - T curve

_{corresponding}

to G = G

g. In

cooling,

on the

_contrary,

the

_{pair-wise coupling}

cannot start so

_easily

because of the

_difficulty

for a

long

molecule to find its

_partner

in

_{appropriate positions}

for the

_zipper

construction. Hence a

state like

_supercooling

may take

place

in the course of

_cooling.

It

is,

therefore,

reasonable

that the transition is

_sharper

in

_cooling

than in

_heating

as has been observed for agarose

_gels

[10].

Comparison

with

_experiments.

1. KAPPA-CARRAGEENAN GELS. - The solid curve in

_figure

5 shows an

_{experimental heating}

DSC curve for 0.6 %

_{kappa-carrageenan}

_{gel [20].}

Since the endothermic

_peak

is very

sharp,

1765

Fig.

4. -

(a)

G

_dependence

of the heat

capacity

as a function of the temperature calculated

by

equation (6)

with n = 2.

Only G

is changed. El

= 2 000 K,

E2

= 2 500 K,

N

=

N 2

= 100,

G 2

= 1 000

are fixed for all the cases. a :

_N1

= 450,

JY’2

= 50 ; b :

N1

= 300,

JY’2

= 200 ; _{c :}

N1

= 150,

JY’2

= 350.

(b) E dependence

of the heat

capacity

as a function of the temperature calculated

by

equation (8)

with n = 2.

Only El

is

changed. E2

= 2 500 K,

G

1 =

G 2

= 1 000,

NI

=

N2

= 100 are fixed for all the cases. a :

_N1

= 450,

IV2

= 50 ; b :

N1

= 300, JY’2 = 200 ; _{c :}

N’1

= 150,

JY’2

= 350.

(c)

N

_dependence

of the heat

capacity

as a function of the temperature calculated

_{by equation (6)}

with

n = 2.

Only N

is changed. El

= 2 000 K,

E2

= 2 500 K,

N2

= 100, G

1 =

G 2

= 1 000 _are fixed for all the

cases

₍₁₎

a :

_N1

= 450,

JY’2

= 50 ; b :

N1

= 300,

JY’2

= 200 ; _{c :}

N1

= 150,

JY’2

= 350.

(2)

_a, b, _c the

same as in

_{figure 4c(l). (3)}

a :

_N1

= 1 800,

.N’2

= 200 ; b :

N1

= 1 200,

JY’2

= 800 ; c :

_N’1

= 600,

JY’2

= 1 400,

(4)

a :

_Xl

₌ 3 600,

JY’2

= 400 ; b : _{NI = 2 400,}

_JY’2

= 1 600 ; c : _NI = 1 200,

JY’2

= 2 800.

N =

1 850,

N =

100,

and G = 1 000 in

equation (5).

The calculated result is shown

by

the dotted curve in

_figure

5.

2. POLY _{(VINYL ALCOHOL)} GELS. - The

_{experimentally}

obtained

_heating

DSC curves for

poly(vinyl alcohol) (PVA) gels, prepared by repeating

freeze-thaw

_{cycles [7],}

are shown in

figure

6

_by

solid curves. The lower curve is for PVA with the

_degree

of

polymerisation

(DP)

=

1 000,

and the upper _curve is for PVA with DP = 2

400,

respectively. Choosing

n = 2 and

substituting

values of _parameters

_{El = E2}

= 2 300

K,

N1 = N2

=

100,

G

=

820,

G 2

= 1

100, NI

1 = 3

900,

IV2

= 1 200 into

equation (6),

we obtain the lower dot-and-dash curve in

_figure

6 which is slimmer than the observed curve shown

by

the lower solid curve in

Fig.

5.

_{Fig. 6.}

Fig.

5. - The solid _curve

represents the observed

heating

DSC curve of a 0.6 %

_{kappa-carrageenan}

_gel. The dotted curve is obtained

_by

using equation (5)

with E = 2 290 K, .N’ = 1 850, N = 100,

G = 1 000.

Fig.

6. - Solid curves _represent the observed

_heating

DSC curves of

_{poly(vinyl alcohol) gels (the}

upper

curve : DP = 2 400, the lower curve : DP =

1 000).

A lower dot and line curve is obtained

by using

equation (6)

with n = 2

choosing

_parameters,

E1

=

E2

= 2 300 K,

NI = N2 =

100,

G

= 820,

G2

= 1 100,

NI

= 3 900,

IV2

= 1 200. An

upper dot and line curve is obtained

by using equation (6)

with n = 2

choosing E1

= 2 265 K,

E2

= 2 365 K,

NI

= 1 750,

X2

= 3 250,

N1

=

N2

= 240,

G

= 980,

G2

= 950. A dotted curve for a lower curve is obtained

by using equation (6)

with n = 8 and

substituting

El

= .. =

Eg

= 2 300 K,

G

= 820,

G 2

= 900,

G

= 960,

G 4

= 1 030,

G 5

= 1 100,

G 6

= 1 200,

G7

= 1 300,

Gg

= 1 400,

N1

= 3 600,

IV2

= 600,

.N3

= 450,

.N’4

= 500,

.N’s

= 750,

.N’6

= 350,

.N7

= 150,

JY’g

= 100,

N

= .. =

N

8 = 100, into

equation (6).

As is seen in

_figure

_4c,

the endothermic

_peak

caused

_by

the

_gel-to-sol

transition shifts to

higher

temperatures

with

_{increasing degree}

of

_{polymerisation.}

The number of links

Ni

must increase and the rotational freedom of the

_constituting

_{links may decrease with}

increasing degree

of

_{polymerisation.}

Based on such

consideration,

another

_fitting

has been tried

_{by choosing El =}

2 265

K,

E2

= 2 365

K,

_N1

=

1 750, JY’2

= 3

250,

NI

=

N2

=

240,

G

=

980,

G 2

= 950. The result is

given by

the upper dot-and-dash _curve, which is

again

slimmer than the

_experimental

curve shown

by

the upper solid curve in

_figure

6.

As stated

_earlier,

the G value _may not be fixed even for a

_{single zipper}

and it

_might

have certain distribution around a mean value even at a fixed

_temperature.

In order to take these situations into account, n has to be further increased

_anyhow

in the

_{fitting, though}

it is certain that the

_parameters,

as well as their

_numbers,

cannot be determined

_uniquely.

As an

example,

the result of calculation is shown in

_figure

6 as the dotted curve which is obtained

_by

choosing n

= 8 and

substituting E1 = ..

=

Es

= 2 300

K,

G

=

820,

G 2

=

900,

G 3

=

960,

G 4

=

1 030,

G 5

= 1

100,

G 6

=

1 200,

G7

= 1

300,

G g

=

1 400,

Xl

1 = 3

600, IV2

=

600,

X3

= 450, X4

=

500, JW5

=

750,

JY’6

=

350,

X7

=

150,

X8

=

250,

N1 =

..

= Ng

=

100,

into

equation (6),

for a PVA

_{gel (DP}

=

1 000).

The

fitting

is

fairly good.

1767

lower

_energies

and others should be released at

_{higher energies.}

Such situations are often

observed in

_{polymeric crystals,}

_i.e.,

two different

_{crystallites of poly}

(y-benzyl-l-g1utamate)

in

benzyl

alcohol were observed

_{by light microscopy,}

and

_{they disappear}

at two different

temperatures

[21].

3. KONJAC GLUCOMANNAN-KAPPA-CARRAGEENAN GELS. - The solid

curve in

figure

7

shows a

_heating

DSC curve for a 0.42 %

_{kappa-carrageenan-0.18}

%

_{konjac glucomannan}

blend

_{gel (20).}

The endotherm is _{very broad. It may be attributed} to the formation of

_zippers

with different

_energies

for

_link-opening

or to a distribution of the

_degeneracy

G.

Choosing

n = 8

again

and

substituting El = E2 = .. = Es = 2000

K,

G

=

370,

G 2

=

390,

G 3

=

410,

G 4 = 440, G 5 = 470, G 6 = 490, G7=520, G8=550, N1 = 1 000, N2 = 250, N3=150,

JY’4 = 250,

IV5

=

150,

X6

= 50,

X7

= 75,

Y8

=

50,

N1 =

..

= N 8

=

100,

into

equation

(6),

we can

_get

_{fairly good}

_agreement

of the calculated curve with the observed one.

Fig.

7. - The solid

curve _represents the observed

heating

DSC curve of 0.42 %

kappa-carrageenan-0.18 %

konjac glucomannan gel.

The dotted curve is obtained

_{by using equation (6)}

with n = 8 and

substituting El = E2 = .. = Es = 2 000 K, G

1 = 370, G 2 = 390, G 3

= 410,

G4

= 440,

G 5

= 470,

G6

= 490,

G7

= 520,

Gg

= 550,

JY1

= 1 000,

JY’2

= 250,

JY3

= 150,

JY’4

= 250,

JY’S

= 150,

JY’6

= 50,

JY’7

= 75,

X8

= 50,

N1

= ..

= Ng

= 100, into

equation (6).

Experimental.

-The DSC curve for

_{kappa-carrageenan gels}

in

_figure

5 and that for

_konjac

glucomannan-kappa-carrageenan gels

in

_figure

7 were obtained

_by

a Setaram micro DSC « batch and flow »

calorimeter.

_{Approximately}

1 ml of the

_sample

was

_{accurately weighed}

into the DSC pan and the reference pan filled with

exactly

the same amount of distilled water. The two _pans were

then

_placed

inside the

calorimeter,

heated to 85 °C to ensure the

_sample

was in the form of a

homogeneous

solution and cooled to the

_starting

_temperature.

_Then,

the

temperature

was

raised at 1

_°C/min.

The details of the

_{experimental procedure}

will be described elsewhere

[20].

The DSC curves for PVA

gels

in

figure

6 were obtained

_by

a sensitive DSC SSC 560U from

Seiko Instruments & Electronics Ltd. 41.5 ± 0.1 mg of PVA

gel sample

was sealed into a

70

_jjbl

silver pan. Distilled water was used as a reference material. The

_temperature

was raised

at 2

_"C/min

from 5 °C. The details of the

_{experimental procedure}

were described in the reference

[7].

Conclusion.

It is shown that the observed DSC endothermic

_peaks

of the thermoreversible

_gels

are well

simulated

by simple

calculations based on the

_zipper

_model,

if the

rotational

freedom of

_links,

the

_bonding

energy, and the number of links in thermo-reversible

gels

are chosen

appropriately. Experimental

data

_suggest

that

_bonding

energy in many thermo-reversible

gels

energy, we can estimate the rotational freedom of links and the number of links

_by

curve

fitting.

Therefore,

we can obtain a clue to the structure

_{of junction}

zones in thermoreversible

gels.

Needless to _say, the number of

_fitting

_parameters

should be as small as

_possible.

Otherwise one cannot determine their values

_uniquely.

In some cases,

however,

the observed

peaks

are too broad to simulate

_by

an

assembly

of a

few kinds of

_zippers.

_{This may be attributed} to the

_variety

of the

parameter

values,

which

might

be

_{expressed by}

continuous distributions. In this

_respect,

it is shown that such a

distribution can be well

_{replaced by}

a finite number of

_species

of

_zippers.

Since the choice of

parameter

values is

_by

no means

_{unique, they}

should not be taken too

_literally.

In order to

testify

these

results, therefore,

much more informations are

_urgently

awaited as for the

structure

_{of junction}

zones.

Acknowledgements.

One of the authors

_(K.N.)

expresses his sincere thanks to Prof. P. Gautier for his

_stimulating

discussion. He is also

_grateful

to Profs. K.

_Ogino

and E. Fukada for their continuous

_guidance

and

_{encouragement.}

References

[1]

CLARK A. H. and Ross-MURPHY S. B., Adv.

Polym.

Sci. 83

₍₁₉₈₇₎

57.

[2]

ELDRIDGE J. E. and FERRY J. D., J.

_Phys.

Chem. 58

₍₁₉₅₄₎

992.

[3]

TOBOLSKY A., J. Chem.

Phys.

11

₍₁₉₄₃₎

290.

[4]

SNOEREN T. H. M. and PAYENS T. A. _J., Biochim.

_Biophys.

Acta 437

₍₁₉₇₆₎

264.

[5]

MORRIS E. R., REES D. A., NORTON I. T. and GOODALL D. M.,

Carbohydr.

Res. 80

₍₁₉₈₀₎

317.

[6]

ROCHAS C. and MAZET _J.,

_Biopolymers

23

₍₁₉₈₄₎

2825.

[7]

WATASE M. and NISHINARI K., Makromol. Chem. 189

₍₁₉₈₈₎

871.

[8]

WATASE and NISHINARI K.,

Polym.

J. 18

₍₁₉₈₆₎

1017.

[9]

NISHINARI K. and WATASE M.,

Agric.

Biol. Chem. 51

₍₁₉₈₇₎

3231.

[10]

WATASE M., NISHINARI K., CLARK A. H. and ROSS-MURPHY S. B., Macromolecules 22

₍₁₉₈₉₎

1196.

[11]

FRANÇOIS J., GAN J. Y. S. and GUENET J.-M., Macromolecules 19

₍₁₉₈₆₎

2755.

[12]

GIBBS J. H. and DIMARZIO E. _A., J. Chem.

Phys.

30

₍₁₉₅₉₎

271.

[13]

CROTHERS D. M., KALLENBACH N. R. and ZIMM B. H., J. Mol. Biol. 11

₍₁₉₆₅₎

802.

[14]

KITTEL _C., Am. J.

_Phys.

37

₍₁₉₆₉₎

917.

[15]

DJABOUROV M., LEBLOND J. and PAPON _P., J.

_Phys.

France 49

₍₁₉₈₈₎

319.

[16]

DJABOUROV M., CLARK A. H., ROWLANDS D. W. and ROSS-MURPHY S. B., Macromolecules 22

(1989)

180.

[17]

MORRIS V. J., Gums and Stabilisers for the Food

Industry

3, Eds. G. O.

Phillips,

D. J. Wedlock and P. A. Williams

_(London

& New

_York)

_1986,

_{pp. 87-99.}

[18]

DEA I. C. M., Solution

Properties

of

Polysaccharides,

Ed. D. A. Brant, ACS

_Symp.

Ser. 150

(1981) pp. 439-454.

[19]

NISHINARI K. and WATASE M., Netsu Sokutei

_(Calorimetry

and Themal

Analysis)

15

₍₁₉₈₈₎

172

_(in

Japanese

with

_English

summary and

figure captions).

[20]

CLEGG

_S.,

DAY D., NISHINARI K., WILLIAMS P. A. and PHILLIPS G. O.,

Manuscript

in

preparation.