Diffusion of a chain : concentration effects

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Diffusion of a chain : concentration effects

M. Daoud, G. Jannink

To cite this version:

M. Daoud, G. Jannink. Diffusion of a chain : concentration effects. Journal de Physique Lettres, Edp

sciences, 1980, 41 (9), pp.217-220. �10.1051/jphyslet:01980004109021700�. �jpa-00231763�

Diffusion of

a

chain : concentration

effects

M. Daoud

Boston _{University, Physics Department, Boston,} MA 02215, U.S.A.

and G. Jannink

C.E.N. _{Saclay, S.P.S.R.M.,} B.P. N° 2, 91190 Gif-sur-Yvette, France

(Re~u le 12 avril 1979, revise le 6 fevrier, accepte le 5 mars ₁₉₈₀₎

Résumé. 2014 La concentration de

cross-over entre solutions diluées et semi-diluées est _{la même pour les}

_propriétés

dynamiques que pour les

propriétés statiques

de

_polymères

flexibles en bon solvant. En solution _{semi-diluée,}

l’exposant 03B2

dans la variation du coefficient de diffusion du blob en fonction de la concentration en monomères Dc ~ C03B2 varie avec la _température et la concentration. La variation de cet _{exposant effectif n’est pas} monotone.

Cet effet peut se vérifier en _comparant les résultats des

_expériences

de diffusion

_élastique

des neutrons et

quasi-élastique de la lumière.

Abstract. 2014 It is shown that the

cross-over concentration between dilute and semi-dilute solutions is the same

for dynamical

properties

as for static

_properties

of flexible

_polymers

in a _good solvent. For semi-dilute _solutions, the _{exponent 03B2} in the concentration _dependence of the diffusion coefficient of a blob Dc ~ C03B2 is expected to _depend

on both temperature and concentration. The variation of this effective _exponent is non monotonous. A combined

neutron elastic and _hight

_quasielastic

_scattering

_experiment

is _proposed to observe this effect.

Classification

Physics Abstracts

05.20 - 36.20 - 61.40 - 64.90

Recently,

Weill and des Cloizeaux

_[1],

Akcasu and Han

_{[2] (A.H.W.C.)}

have

_proposed

a model for the

_dynamics

of a flexible

polymer

chain in a

_good

solvent

_[3].

This

_approach,

based on the blob model

_[10],

shows that the cross-over effects between theta and

_good

solvents are much more

_important

for the

_dynamical

_properties,

where one has to

calculate mean values of inverse distances than for

the radius of

_gyration

for

_{instance, where}

_{r 2 >}

is calculated. As a

_result,

the

experimental

studies

_[4]

measure an

effective

_exponent, as was

_{suggested by}

Stockmayer

and Albrecht

_[5]

some time _ago.

In this note, we wish to

clarify

the cross-over

between dilute and semi-dilute solutions in a

_good

solvent,

and extend the A.H.W.C.

_approach

to semi-dilute solutions.

The cross-over concentration C* is

usually

defined

via the

_(static)

radius of

_gyration

_RG

as

_[10]

where N is the number of statistical units in the

chain,

C the monomer

concentration,

and v the excluded

volume _exponent

_{[3], [6].}

If we consider now the diffusion coefficient

_Do

of a

_chain,

or the

viscosity

of a dilute

solution,

we

have

where

_RH

is called the

_hydrodynamic

radius

_[7]

and where the effective _exponent

_{[1 ], [2]}

v’ is less than v

_[4]

(usually v’ N

0.54).

where ~S

is the

_viscosity

of the

_solvent,

_M

the intrinsic

viscosity

[7]

and

_KH

the

_Huggins

_coefficient,

which is a constant

_{[3], [8].}

It can be shown that

_[9]

A

_straight

forward

_(but

_wrong)

dimensional

_analysis

would then lead to different laws for the cross-over

between dilute and semi-dilute

_{solutions, namely}

L-218 JOURNAL DE _{PHYSIQUE -} LETTRES

The flaw in the

analysis

lies in the fact that there is

not

_only

one, but two

_lengths

_[10]

in the

_problem

(none

of them

_being

_RH).

_RH

cannot be considered as a

characteristic

_length,

_except

in the

_asymptotic

_region

of _very

_high

N, which

_according

to refs.

_[1]

and

_[2]

seems

impossible

to reach in the

_present

state of the

art. In order to calculate

_{RH, [1], [2]}

one has to _separate

short from

_large

distances

_[10]

_namely

We may call the substructure associated to

_Nt

a

_flof.

This

_{analysis [1],}

_[2],

valid for dilute

solutions,

can be extended to semi-dilute

solutions,

as shown, I

below. So all the

_following

concerns semi-dilute

solutions.

Let us consider the diffusion

coefficient,

which is

more

_appropriate,

as relation

₍₃₎

does not hold for

high

concentrations

_[11].

In semi-dilute

solutions,

another

substructure,

called the blob

_[10],

_appears. It is associated to a number of

_{monomers g >}

_Nt,

and to an _average

_squared

length ~2

between successive

contacts. We know that inside a blob

_(i.e.

for short

distances)

the _segment has a

_single

chain

beha-viour

_[12].

Thus the blob _may be considered as a

single

chain,

with a flof substructure identical to that of a

_single

chain

_[15]

_(see

_Fig.

_1).

This substructure

has not been

_directly

checked

_{experimentally,}

but we

and we may define an effective

_exponent

_by

Notice that the last term in

₍₈₎

becomes dominant if

~ ~>

N~,

which is realized if the

_temperature

is _very

high

or the concentration _very low.

_(Usually

the limit _VH = v is not

_achieved.)

Of course this

_analysis

is valid above C*

(relation (1)).

When the monomer concentration

goes to

_{C*, g}

_goes to N and D crosses over

_smoothly

from relation

(7)

valid in semi-dilute solutions to

relation

₍₂₎

valid in dilute solutions. This can be

seen _very

_clearly

also in relation

_(8).

So this shows

that the cross-over concentration is C* and is the

Fig. 1. - a) The local behaviour of _a chain is theta-like. (See

notes _[10] and _[14].) For larger distances, the excluded volume interaction is _present. b) In a semi-dilute solution, inside the _blob, there is the same structure as for the _single chain.

consider that the

_discrepancy

between the static and

dynamic

measurements

_[13]

of the radius of the blob is an evidence for this substructure. The same

_analysis

as above can be done for the diffusion of the blob :

In order to

calculate

the diffusion coefficient

_Dc

one has to _separate the short distance behaviour

(inside

the

_flof)

from the

_larger

distance behaviour

The

_only

difference between

(7)

and

₍₅₎

is that one

has to

_replace

N

_by

the

_{number g}

of statistical units in the blob.

same for the diffusion coefficient as for the radius of

gyration.

The

_{preceding analysis}

can be

generalized

to _any other

_property.

One has

_just

to take care of

performing

the correct dimensional

analysis.

An

_interesting

effect is exhibited

by

this

analysis :

g

depends

on both the

_temperature

and the monomer

concentration C

_[15],

_[16]

where we know the

asymptotic

behaviour of the

On the other hand

_NT

_depends

on

_temperature

_only

(relation (6)).

So

_by

_varying

either the

_temperature

at a

_{concentration,}

or the reverse, one varies the ratio

g/NT,

and therefore the value of the _{exponent VH.} We can define effective _exponents for the

concen-tration and

_temperature

_dependence

of

_ÇH.

Leading

to

and

VH is

easily

determined with _eqs.

_(8),

and is a

decreas-ing

function of C. In order to determine the second factor of the

_right

hand side of

_(11),

we _may introduce

a self-consistent

equation for g :

where ~

is the static radius of the blob :

where x =

Nlg.

Combining

(12)

and

₍₁₃₎

leads to the self-consistent relation

and from

_{(8), (11)}

and

₍₁₄₎

where

_x(C/i)

is determined

_by

₍₁₄₎

_(we

take

_{~ ~}

_1:-2).

The

_{corresponding}

behaviour is shown on

_figure

2.

The main

_point

we wish to

_emphasize

here is the non monotonic behaviour of the

_exponent.

We discuss

this result. The effective

_exponents

are function of C

and T. Some recent

_light

_{scattering experiments}

_[17]

indicate a variation of the

_exponents

with

tempe-rature and concentration which may be

interpreted

this _way.

_Figure

2 shows first a decrease of the

_exponent

with

_increasing

C or

_decreasing

T. This decrease is

due to the

_competition

between the short scale ideal behaviour and the

_large

scale

_(blob)

excluded volume

Fig. 2. -

a) Variation of the _{exponent #} in the diffusion coefficient of the blob _{D. -} Ca versus _C/1:. Note that C* depends on N. For very high values of N, C* is very low, and the decreasing part

of the curve is easier to observe. b) Schematic variation of the

product

D(ç2)1/2

as a function of C/1:.

behaviour. As a result of this

competition,

the diffusion

coefficient

_depends

less

_strongly

on C than in a _pure

self

_avoiding

walk. When the concentration increases

(or

the

_temperature

decreases),

the size of the blob

decreases,

and the

_weight

of the flof increases. This is

true as

_long

_{as g >}

_Nt.

When the concentration is

sufficiently high, g

itself becomes a

rapidly varying

function of

C,

see relations

(9)

and

(10), leading

to

the increase in

_#.

The above

_approach

is valid in the

_good

solvent

semi-dilute

region.

Let us discuss

_briefly

the

cross-overs C* to the dilute

_regime

and C** to the theta solvent.

i)

In the dilute

_region,

the blob becomes the chain itself : There is no

dependence

on C of the diffusion

coefficient. So in the

_vicinity

of

_{C* ~i}

has to _go from

equation

(15)

to the value 0 in the dilute

_regime.

As C*

depends

on

_N,

so does the location of this

cross-over. Let us remark here that above this

cross-over, the

experimental

data

[4] give

an

_exponent

of

the order of 0.69

_showing

indeed a decrease. It would

be

_interesting

to _go both to

_higher

and lower molecular

weights.

The

_higher

molecular

_weights

should exhibit the

decreasing

part of the curve

_figure

2.

The lower molecular

_weights

are also

interesting

because

_they

can tell us how low the effective _exponent

can

_actually

be.

ii)

For concentrated

solutions,

the chain becomes

roughly

ideal at _every scale. So we _expect the

_exponent

to have the

_asymptotic

value 1 in this

_region.

A weakness of the above

_approach

is that the

_exponent

reaches its

_asymptotic

limit at C** ~ ~ whereas we know that this is

_just

a cross-over

concentration,

and that the _exponent reaches its

_asymptotic

limit for

higher

values of

_CIT.

This is due to our

model,

where

we _suppose a clear cut between

_gaussian

and excluded volume behaviours. In order to avoid this one has to

L-220 JOURNAL DE _{PHYSIQUE -} LETTRES

Let us

_finally

remark that relation

(15)

indicate that

the variation of the _exponent with

_CIT

is universal for C ~> C *

_(in

terms

_{of ~H}

or

Dc),

the coordinates

e-l ÇH

or

_D~

C vs.

_CIT

are the most

_appropriate.

Moreover, as we are

dealing

with local

_properties,

the

_{polydispersity}

is irrelevant. As a

_conclusion,

we extended the blob

_approach

to the diffusion of the blob in semi-dilute solutions. We showed that the

cross-over concentration C* is the same for static

and

_{dynamic properties}

_indeed,

when the correct

dimensional

_analysis

is

_performed.

For semi-dilute

solutions,

we could define an effective

_exponent

for

the variation of the diffusion coefficient of the blob with concentration. This

_exponent

has a non mono-tonous behaviour with

_increasing

C or

_{decreasing T.}

In

_particular,

it should decrease in the semi-dilute

regime.

This effect is due to the

_competition

between the short scale ideal behaviour and the

_large

scale

(blob)

excluded volume behaviour. It should be observed

_by

_{plotting experimental}

values of the

product

D(~2)1/2

against CIT (see Fig. 2b).

Diffusion coefficients D are obtained from the

_quasielastic

broadening

of an incident

_light

_beam,

and the blob

squared

length 1;2

is measured from the neutron

scattering experiment.

Acknowledgments.

- The authors wish to thank

G.

Benedek,

S.

Redner,

H. E.

_Stanley

and T. Witten for related

_discussions,

E. Geissler and A. M. Hecht for a _very

stimulating correspondence,

and one of our

referees for _very valuable comments. This work

was

_{partly supported by}

a

_grant

of A.F.O.S.R.

References

[1] WEILL, G., DES CLOIZEAUX, J., J. Physique 40 (1979) 99.

[2] AKCASU, A. Z., HAN, C. C., Macromolecules, in press.

[3] FLORY, P. J., Principles of Polymer Chemistry (Cornell) 1953.

[4] ADAM, M., DELSANTI, M., JANNINK, G., J. _Physique Lett. 37

(1976) L-53.

DELSANTI, M., Thèse, Université Paris-Sud Orsay. Available at C.E.N. Saclay, S.P.S.R.M.

MUNCH, J. P., CANDAU, S., HERZ, J., HILD, G., J. _Physique 38 ₍₁₉₇₇₎ 971.

[5] STOCKMAYER, W. _{H., ALBRECHT,} A. _C., J. _Polym. Sci. 32

(1958) 215.

[6] DE GENNES, P. G., Phys. Lett. 38A (1972) 339.

[7] STOCKMAYER, W. H., in Molecular Fluids, les Houches Summer

School, Balian, R. and Weill, G., ed. _(1973).

[8] MUTHUKUMAR, M., FREED, K. F., Macromolecules 10 ₍₁₉₇₇₎ 899.

[9] FERRY, J. D., Viscoelastic _properties of Polymers, J. Wiley ed. (1961).

[10] See for instance FARNOUX, B. et _al., J. _Physique 39 ₍₁₉₇₈₎ 77. See also note [14].

[11] For low concentrations, the analysis holds for ~.

[12] DAOUD, M. et al., Macromolecules 8 ₍₁₉₇₅₎ 804.

[13] As in the dilute case, the laws

for 03BE

are not the same _by static [12]

or _{dynamic [4] experiments,} and one can also define 03BEH in the same _way as in the dilute _regime.

[14] In this respect, the substructure in the single chain should have the same denomination, to _distinguish it from the blob which was _initially introduced for semi-dilute solutions _(see ref. _[12]).

[15] DAOUD, M., Thèse, Université Paris VI (1977). [16] COTTON, J. P. et al., J. Chem. _Phys. 65 ₍₁₉₇₆₎ 76.

[17] GEISSLER, E. and HECHT, A. M., J. _Physique Lett. 40 (1979)