Nonlocal dielectric response in dipolar polymers

(1)

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Nonlocal dielectric response in dipolar polymers

M. Warner, M. Cates

To cite this version:

M. Warner, M. Cates. Nonlocal dielectric response in dipolar polymers. Journal de Physique II, EDP

Sciences, 1993, 3 (4), pp.503-513. �10.1051/jp2:1993147�. �jpa-00247850�

(2)

Classification Physics Abstracts

41.10D 61.25H 77.20 77.30

Nonlocal dielectric _response in dipolar polymers

M. Warner and M. E. Gates

Cavendish Laboratory~ Madingley Road, Cambridge CB3 OHE, Great-Britain

(Received

15 October1992, accepted 5 January

1993)

Abstract. We calculate the nonlocal dielectric susceptibility

x(r r')

for _a fluid of rods carrying dipoles pointing along their _axes. At short distances, of order the rod diameter, _we get _a value of _x considerably smaller than that of

a fluid of individual _monomers. At distances comparable to the rod length, _x climbs to a value N times larger than _a fluid of _monomers, _as expected. In between these _t&vo limits dipolar rods _are predicted to exhibit _a form of Debye- Hickel screening. The relevance lo internal field calculations and to direct experiments is pointed

out. We also calculate the _response of dipolar Gaussian chains.

1 Introduction.

We wish to

explore

the

possibility

that the finite size of rods

carrying dipoles

_causes their dielectric _response to be nonlocal. Our motivation is two-fold: I) _many

dipolar

rods _are _very

long (and carrying

considerable

dipoles)

for instance the

synthetic protein

PBLG. The great

length

of the rods _means that the

length

scale for

nonlocality

in the

polarizability

_response

to an

applied

field will also be

appreciable

and _may have _a

signature

in

spectroscopic

and

reflectivity experiiuents, ii)

It has been

argued [1-4],

that

long dipolar

rods should have _a

large susceptibility, scaling

with the

length

of the rod. Related nonlinear

optics experiments predating

these

predictions

_[5] add

weight

to the arguiuent. ^Such _very

large susceptibilities might

lead _one to suspect ^that ^internal field efsects could be _very

large, leading eventually

to a ferro-electric

catastrophe

of the Clausius-Mossotti type. To

accurately

calculate the internal field _one has to evaluate the _response in the

region

about _an

appropriately shaped cavity.

Nonlocality

in the dielectric _response will be

important

in such

calculations~ tending

to

strongly

reduce any local field correction.

In this note we calculate the linear dielectric _response to _a

spatially varying

electric field

E(r).

We restrict attention to the

isotropic phase,

^and assume

throughout

that the orientation of rods and their

spatial positions

_are uncorrelated. We also present ^ideas on how such non-local efsects

might

be _seen

experimentally

and show their relevance in local field calculations. In addition

we demonstrate that the _response _on intermediate

length

scales is that of _an

electrolyte.

This will be

explained by solving

the

problem

of

screening

in _a gas of

charges strongly

correlated

(3)

504 JOURNAL DE PHYSIQUE lI N°4

in

pairs.

The

regime

of

Debye

Hiickel

screening interpolates

between the

(order unity)

relative

permittivity

at short distances and the much

larger

value at

long length

scales.

Finally

_we

calculate the _response of _a melt of Gaussian

dipolar

chains and _compare this with the

rigid

rod _case. Here the

screening

^effect is less marked because there is _no similar enhancement of

the dielectric _response at

long wavelengths.

In the present ^work we consider

only

the static response

function;

the related

time-dependent phenomena

could be _very

interesting

and remain open for future

study.

2. Nonlocal

susceptibility

of _a rod fluid.

2. I LENGTH _SCALES. In _a solution of rods there _are 3 separate

length scales,

the rod end- to-end

distance~ Lj

^the distance between

rods, f (the

correlation

length)

and the rod

diameter,

d

(see Fig. I).

d

Fig. I. The 3 length scales L, f and d for rods in solution. On _one rod the monomeric dipoles

(-)

are drawn in. They add to give _one large dipole, equivalent to the

charges

+ and separated by _a

distance L.

Because the molecules _are

rigid

and their

dipoles

_are

coherently arranged along

their

length,

L remains _a relevant

length

for the dielectric _response. If rods _arrange themselves _so _as to set up a local

polarisation charge,

it is the rod

length

that determines where the

corresponding

counter

charge

_appears. This is true _even in the melt where

f

_+~ d « L. Note that the

relevance of

lengths larger

than

f

is not

usually

^found in the static

properties

of conventional

polymer

systems [6].

2. 2 LINEAR RESPONSE. For _a uniform field the _response of _a rod is

large

x~~~ =

ip,~~(Np)2/31~T ₍₁₎

where _prod

+~

I/d~L

is the number

density

of rods under melt

conditions, considering

for

simplicity

_monomers to be isodiametric with linear dimension

d, #

is the volume fraction of

(4)

rods

(#

_< I if _we

are

dealing

with _a

solution),

and N is the number of _monomers in _a

rod,

each _monomer

carrying

_a

dipole

_p. Since N ₌

L/d

then prnon =

Nprod,

whence xrod becomes

Xrod "

4iPmonNfi~/3kBT

=

NXmon (2)

where _xmon is the

susceptibility

of the

equivalent

monomer system at volume fraction

#,

I.e.

of _a system where rods have been _cut into their constituent parts.

(In

what follows the volume fraction

dependence

will be contained in _xrnoni the results

presented

will then

apply

for both melts and semidilute

solutions.)

The _reason

why

_xrod is _so much

bigger

than _xmon is that when

considering

the bias between rods

pointing "up"

and "down" _an

applied field,

the energy

appearing

in the Boltzmann distribution

governing

the relative

probability

is

NpE,

rather than

pE,

since all the

dipoles

_on _one rod act

coherently

[1, 2, 5]. ^Note ^that

by

Gauss's

theorem,

each

dipolar

rod is

equivalent

to _an inert _one with

oppositely signed point charges

of

magnitude Np/L

at the ends of the rod. This

applies

at

length

scales

large compared

to the rod

diameter;

we

freely exploit

the

equivalence

in what follows.

2.3 NONLOCALITY. The enhancement in _x

by

_a factor of N

(Eq.(2))

_[1, _2]

applies only

if _we consider

lengths larger than,

_or _on the order of, L. Consider _a one-dimensional situation where the

polarization P(z)

increases from _zero to its full value Po _as _we _move

along

the _z axis

through

_some interfacial

region. By

Gauss's theorem V-P

= -ppoi

(the polarization charge density)

and hence the total

charge

_per unit _area associated with the interface is

Po,

but it is

spread through

_an interval Az _over which

dP/dz #

0. The

polarization charge

is furnished

by

the effective

charges

at the ends of each rod, and the full bias in the distribution of rod

ends associated with

polarization

of the rods

can

only

be built _up _over _a distance

+~

L This

argument was advanced in

[I]

to

explain why

internal field effects should be small in

polymer liquid crystal

systems

(see

also Sect. 2.5

below).

2.4 CALCULATION _OF NONLOCAL RESPONSE. Tile nonlocal _response is

Plr)

₌ _eo

/

di~'x(i~

r')E(r')

₁₃₎

creating

a

polarization

P at _i, in _response to _a field

E(i,') applied

at r'. The _response function must be

diagonal

in tensor indices for _an

isotropic

system ^and ^hence can be

represented by

the scalar function

x(r).

We restrict ourselves the _case &vhere

E(I)

varies

along

its _own

direction;

we shall below consider Fourier modes of the field. Given that _an

arbitrary potential V(r)

can be

synthesized

from Fourier components Vk

exp[ik.i,]

,

and since

E(r)

= -T7V

,

then field components are of the form

Ek(I)

₌ -ikvk

exp[ik.i,],

in which _case

E(r)

varies

only along

its

own direction _as assumed. The _case _we present ^is ^thus ^of ^sufficient

generality.

Consider in

figure

2 rods

contributing

to the

density (and polarisation)

at r and

being

_cen-

tred _at

position

x relative _to _r. Let the

angle

between _x and E

(E being

in the _z

direction)

be 0. In the

isotropic

state, for each such rod there is

(on average)

also _one at

angle

_~

0, #

+ ~

which

points along

_-x and contributes in the

opposite

_sense to the

polarisation

at r. Linear response consists

simply

of

biasing

the rods at

(0, #)

^with respect to those at

(~ @,#

+

~).

[Note

that _any other distortion to the rod distribution function _must be _at least

quadratic

in

E(r).]

If the

dipole

moment per unit

length

of rod is

a(e p/d)

then the distribution of rods

(5)

506 JOURNAL DE PHYSIQUE II N°4

~ ~ E(r+x+su)

/

u

W

r+x+su

9 x

Fig. 2. A rod centred at x relative to r contributing to the density at r. It has angle b with respect

to the field direction

E(r).

The azimuthal angle of this rod is #. The unit vector defining the rod direction is fi. A

general

_arc point _s along the rod, lying in the interval

[- §, _§],

_is _located _at

r +x + so

and accordingly has _a field

E(r

₊ _x ₊

sfi)

acting there.

centred at _x when _an E field is

applied

is

given by

the Boltzmann factor

deriving

from the rod energy:

fE lL/2

'~' exp

pa ~L/2 dsE(I

+

(x

+

s)fi).fi (4)

= l +

pa

_cm 0

/ _dsE(r

+

(x

+

s)fi)

₊

O(E~) (5)

2

where

fl

=

I/kBT,

and E

=

(E(.

In the linearized foriu

(5)

_one _sees that the

perturbation

to the

angular

distribution for rods _at

angle

0 is

equal

and

opposite

to that _at _~ 0

(hence f

remains normalized at linear

order).

The bias created at _r

by

the difference in energy of these two

configurations

in the

spatially varying

E fields is therefore

L/2

Al

₌

pa

_cos 0

~L dsE(i,

+

(x

+

s)fi) (6)

/2

To calculate the bias in the _monomers _at the

point

_i~

(and

hence the

polarization there)

we note in

figure

² ^that the _monomers with orientation

(0, #)

must

belong

to rods with their

centres

along

the line labelled

by

fi We _now average ^over all such rod

contributions,

that is

with centres at various distances _x _away from _r.

(Since

_we have taken

positive

and

negative

_x

we confine _our

subsequent

_sums _over unit vectors fi in the upper

hemisphere.)

The net bias in the distribution of _monomers at _i, with

angle

0 then becomes

L/2 L/2

A

trot

₌ dz

pa

_cos 0 ds

E(r

+

(z

+

s)fi) (7)

L

~L/2 ~L/2

The total

polarization

of the _monomers at _i, is

Pz

jr)

₌ _prnon _p

(cos 0)

₌ _prnon _p

/ _{d(cos 0)}

cm 0 A

ftot (8)

o

(6)

Since _we _are

calculating

the linear _response _we _can take

a

single

Fourier component of E

(and subsequently synthesise

the overall response to _a

general field).

^The E field variation is then

E = Ek

exp[ik. jr

+

(x

+

s)fi)] (9)

where k

= ki

(recall

that E indicates the

magnitude

of _a vector in the I

direction).

In terms of the Fourier components,

(3)

becomes the convolution Pz

(k)

₌

eox(k)E(k)

where-

upon we can

identify x(k)

_as

1 L/2 L/2

x(k)

₌ ^~~'°~~

pa. / _daa~ /

_dz

/

_ds

_exp[ikza

+

iksa] (10)

L 0 -L/2 -L /2

where _a

= cos 0. The

integrations

in

(10)

_can be

performed analytically

and

yield:

~~~~ ~~°~

~k2 ~ ~~~~~

~~~~

It is

interesting

_to

explore x(k)

in its liirits:

(I)

^kL « I. This

corresponds

to

large distances,

that is slow variation of

E(r)

_on _a

length

scale

compared

with L.

Expanding,

_we obtain

x(k)

_- _xrnon

j _(I

+

O(kL)~)

_e _xrnon N +

(12)

which is the

large

value

previously

discussed for

long

rods in

homogeneous

fields [1,

5].

(ii)

k

+~ I

Id,

that is kL

+~ L

Id

₌ N

corresponding

to distances of order _a _monomer dimension:

xik)

_- _xrnon

j

+ xrnon

6/N. i13)

The

susceptibility

_on short scales is much reduced

by

the

inability

of rods _to

respond

to

rapid spatial

variations. Given _a

periodic

variation of E

along

_a

rod,

half

cycles

of E _are nugatory ⁱⁿ ^their

alignment effect,

at most _one odd half

cycle giving

_an electrical _energy of

pE (characteristic

of _a

dipole's

_energy in the _monomer

fluid)

to the whole rod and hence _a

degree

of

alignment comparable

to that of the _monomer fluid itself. This argument ^alone ^would

give

x - xrnon, whereas in fact

(13)

is _even _a factor of N smaller than the monomeric value of _x.

The extra factor of

I/N

_comes from fluctuations in the

phase

of the extra half _wave

length

it is

randomly

+ and

only through

the _square of the accumulated electrical _energy

(the fdz

and

fds

in

(10))

does _an effect exist at all.

Finally,

from

equation (11)

_we _can calculate the nonlocal

susceptibility

in real _space,

x(r)

as defined in

equation (3),

which is then

~~~~~~~°~

2~)3 /

_~~

~~~~ ~~'~~

k~)2

~~

~)~~

=

j(I _r/L)8(1 _1,/L) ₍₁₄₎

with

8(1- r/L)

the Heaviside step function.

This,

_as

expected,

cuts off at _r ₌ L since the

nonlocality

is due _to the

propagation

of _a

dipolar

bias

along

rods themselves. For _a uniform

field, putting (14)

in

(3)

and

integrating yields

_xrod

obeying (2)

^for ^the _response; in contrast,

for _a field

acting

_over _a volume

+~

d3 one gets the

(d/L)~

reduction in _response relative to xrod

as described

by equation (13).

(7)

2.$ APPLICATIONS _OF _THE _ABOVE RESULTS.

2,$. I Ionization

equilibrium.

Consider

a 2witterionic teleclielic flexible

polymer carrying oppositely charged

functional _groups at each end of the chain. This has _an ionization

equilib-

rium between the closed

ring

and _open chain forms which is

clearly dependent

_on the dielectric

response of the solvent that surrounds it. If the

polarizability

is

large,

the energy ^cost of dissociation is much reduced. If the solvent in

question

is itself _a system of

dipolar rods,

then the dielectric _response will be _a function of the

length

scale _as described above. Chains with

a natural size

(in

the dissociated

state) larger

than the rod

length

L _can take

advantage

of the

larger

dielectric constant of the medium and _are _more

likely

to

ionise,

whereas those with

a smaller natural size have to pay a much

larger

ionisation cost. The dissociation _constant is therefore

strongly

chain

length dependent;

^the effect _can be examined

by scattering

_or fluores-

cence methods. This could

provide

_a

quantitative probe

of the nonlocal dielectric

properties

of the

dipolar

rod medium.

Any

other molecular-scale

probe

whose conformation is sensitive to the local dielectric constant of the medium will sho&v similar anomalies.

2.5.2 Internal fields efLects. In

[Ii

it _was

suggested

^that the

length

scale associated with

X(r)

could be

probed by suspending spheres

of low

polarizability

in _a rod fluid. Within these

spheres

would be material that is

spectroscopically

sensitive to the local E field.

For

spheres large compared

with the rod

length

L _one _recovers the standard result that the internal field Ei is uniform and takes the value

Ei

=

P/3eo (IS)

(see Fig. 3a).

If however the

sphere

is smaller than _or

comparable

to L then the

layer

that the

polarization charges

associated with

discontinuity

in P _are

spread through

is

significant

the field is

greatly

reduced below the value in

(17),

_see

figure

3b. Likewise the internal field

experienced by

_a rod in _a

rod-shaped cavity

_cut out of _a dielectric should be

strongly

reduced for similar _reasons. This efsect is _on top ^of ^the

geometrical

factor which _causes the local

correction to the field component

along

the

cavity

ac~is to be small [4].

Therefore,

local field corrections to the dielectric

susceptibility

^of

long dipolar

rods _are almost

certainly negligible.

2,5.3

Reflectivity.

With

large

^rods such _as PBLG which _possess coherent

dipoles

and

are also

nonlinearly optically active,

it

might

be

possible

to _see nonlocal _response at

planar

electrodes

by

reflection at, and

just beyond,

the critical

angle

for total internal reflection. The

angular

variation of the beam

intensity

reflected fi.om _a

glass /solution

interface _near the critical

angle

relates to the concentration

profile

of rods in the fluid

beyond

the surface. If the interface is also _a transparent

electrode,

then _an E field _can be

applied

to the

solution, biasing

the rods.

The full bias is built _up _over _a distance of order L. The second harmonic _wave built _up _as

a

result of the

x(~)

_response ⁱⁿ ^the

polarized

^rods ^will ^then

depend strongly

_on whether

or not the evanescent _wave penetrates ^of the order of L. The variation of

intensity

due _to the

simple

variation of rod volume fraction with

depth

in the fluid _can be extracted from the

knowledge

of its variation derived from the reflection of the fundaiuental _wave.

We have above considered three _cases &vhere the

nonlocality

of

x(r)

could

play

_a

significant

role. In fact for situations

involving

_a

sharp

interface between the

dipolar

rod fluid and _a solid

or vacuum, ^our

expression

^for

x(r)

will _not be

quantitatively applicable.

This is because the

geometry

imposes

_a correlation between the

spatial position

of _a rod and its

orientation,

which

we have

neglected,

but which could be accounted for in _a

more

complicated

calculation.

(8)

fi

1~ ~ '

+

+'

+ ' +

1+ ₊

r r

L

~ _~/~

~~

L ^~~

~

-~

a) b)

Fig. 3.

a)

A layer of polarization

charge

of thickness L much smaller than the sphere diameter

r

creates _a uniform internal field.

b)

The layer L is much thicker than the sphere size and the delocalized charge creates _a much smaller internal field.

3. Poisson-Boltzmann and

diagrammatic

methods.

An

illuminating

alternative to the calculation

presented

above

(which

considers

directly

the electric

field)

is based _on the _use of

potentials.

The

advantage gained

is that _one _can _see the

screening

effect of

dipolar

fluids at short distances in _a _more transparent manner. Also it allows

one to handle the _case of flexible

dipolar chains,

_as considered in section 4.

(The

avoidance of

dealing

with unit

tangent

vectors to define

polarisation

is of _course necessary when

dealing

with Gaussian chains where tangent vectors vary on

arbitrarily

small

length scales.)

Since

E(r)

_can be

expressed

_as -T7V then the field energy,

previously

taken _as _a

fds

_E-d,

can be written _as

U ₌

«lvlr

+

lx

₊

L/2)fi)

Vli~ +

lx L/2)fi)1 l16)

Equation (I)

reflects the fact that _we _can represent ^the ^effect ^{of the}

dipoles along

the chain

(a

_per unit

length) by point charges

of +a at each end of the rod. These

charges, by

virtue of their attachment _to the

rod,

_are of _course

highly

correlated. We _now evaluate the _response of such _a medium to _a

point charge

+e

placed

at 1, = 0.

Given overall

charge neutrality,

the

charge

at _a

general point

_i, is

given by

the imbalance between the

density

of rod ends

carrying

+a and those

carrying

_-a:

p(i,)

₌ _«prod

(e~P~+ ^e~P~- (17)

where the normalization of the

probabilities

in

(4)

is correct

only

for small U. The

potentials

U _are

U+ ₌

+alv(i~) V(r

₊

LA)] (18)

corresponding

to +a or -a ends _at _r

respectively

and vice _versa _at _i, + Lfi.

Linearising,

the

charge density

becomes

P(~)

_"

~~Prod p"~ i~(~) ~'i~

+

LA)] (~9)

(9)

Inserting

this and the

point charge

into the Poisson

equation,

_one obtains

T7~V =

~

b(r)

+

~°"~

/dr'g(r ^r')V(r') ⁽²⁰⁾

Eo Eo

where the structure factor

g(x)

for two

opposite point charges

at either end of _a rod has been

employed:

gjx>

₌ &jx>

&jx

+

Lo) j21)

~j~~

_~

~-;~.~~

_j~~~

(The

second form is the Fourier

transform).

Fourier

transforming (7)

leads to

k~

+

~°"~ (k.fiL)j

V(k)

₌

eleo. (23)

Eo

Now

taking angular

_averages _over fi, the average ^structure factor becomes J~:

J~ = 1-

)(

(24)

and _one then obtains for

V(k):

~~~~ 211~+

fib)

eok2er(k)

^~~~~

where in the modified Coulomb propagator

(26), er(k)

_e I +

x(k)

is the _wave _vector

dependent

relative

permittivity.

The

susceptibility x(k)

_can be extracted

by

identification within

(25).

It is

~j~) ^2Prodfla~ ^J~

e~

p

6J~

"

X~°~fi ^j26)

where,

_as

before,

_we

recognize

^that _prod _"

pmond/L

and _a ₌

p/d.

Since y~

obeys (24), equation (26)

coincides with

equation (II)

derived earlier. The

only subtlety

here is in the

taking

of the

angle

_average

(and, iiuplicitly,

_an _average _over the

positions

of the _centres of the

rods)

before rather than after

solving (23)

for V. This

corresponds

to _a standard random

phase approximation

used in

polymer theory

[6]; a

diagrammatic interpretation

is

given

in

figure

4.

Our results illustrate how

dipoles

of _a finite size _can either _screen the Coulomb interaction

or behave _as _a conventional dielectric

(depending

_on the

length scale).

To _see

this,

_one _can

rewrite

(12)

_as

VII)

₌

~~

j)~~ ¹²⁷⁾

with K~

= xrnon

A.

For kL » I

(length

scales inside _a

rod),

y~ ^becomes

featureless, J~

_+~ 1,

(10)

Fig. 4. A diagrammatic representation of the random phase approximation applied to the integral equation

(20).

The effective Coulomb propagator

(heavy line)

arises when the bare propagator

I/k~

(thin line)

is multiply scattered by charge pairs linked by the correlation function

g(x,u) (dotted line).

This multiple scattering series is averaged before summation with the assumption that each pair of scattering charges is independent of the others. Then each solid line represents ^the same,

angle-averaged correlator. In Fourier space the series is then geometric and _sums to give equation

(25).

and _K takes its usual

interpretation

_as the inverse of the

Debye-Hfickel screening length

1:

~ l/2

1 +~ d

(28)

6xmon

There arises _on these

length

scales _a screened Couloiub

potential, V(r)

_ci

fie~~/~.

_For

kL «

I,

_we _can instead

expand

_J~ to

give J~

ci

)(kL)~

+. hence in this

region

the

leading

k~

(Coulomb) dependence

in the denominator of

jib) picks

_up _an additional k~ contribution:

it becomes

k~(I

₊

K~L~/6)

₌

k~(I

₊

xrod).

The screened Couloiub

potential

then

crosses over

to

~~~~ "

4woli )

x,~d)i~

which describes _a medium of

high

relative

permittivity

but without free

charges.

Note that the

scaling

of the

Debye length

is I

+~

dN~/2 (since

_xrnon

+~

I).

Thus there is _a

length,

not

proportional

to the rod

length

L ₌

Nd,

which _governs the

screening

behaviour of the

dipoles.

This

length

is

determined,

_as in the usual

Debye-Hiickel screening problem, by

the inverse square root of the effective free

charge density,

which here varies _as

(# IN).

The average

charge separation length

scales like

N~/~,

much smaller than the

Debye length

I

+~

N~/~ (at

least for

long rods)

which confirms that _mean field

approximation

inherent in the

Debye-Hfickel analysis

is valid.

In

principle

the full form of

V(r)

_can be found

by taking

the Fourier transform of

(IS).

This function has

an infinite series of

poles

in the

complex plane

with finite real and

imaginary

parts, and _an exact treatment is intractable. However, _a convenient

analytical approximant

can be found

by replacing

the _correct y~ ₌

(I sin(kL)/kL) by

the

simple interpolating

form y~ =

)(kL)~/(l

₊

_)(kL)~) (which

has the _same asymptotes ^but ^is

monotonic).

The Fourier

transform _can then be

completed

and

yields

~'~~~ '~

4~e~r

Ii

+ xrod

~

(i

+

Xrod)

^~ ^~~~~

This form should become

increasingly

accurate at both

large

and small

distances, departing

from the true form

(by

^factors of order

unity) only

when _r _ci L. In this _crossover

region

the

(11)

true

potential

_may in

principle

show

slight departures

from monotonic behaviour

(reflecting

the character ofJ~ near k

= I

IL)

however the

practical

effects _are

likely

to be

small, especially

as

they

will be masked

by

_any

appreciable polydispersity

in rod

lengths.

4.

Dipolai

^Gaussian ^chains.

The above

picture

of _a

dipolar

molecule _as

consisting

of _two

point charges

+a

separated by

_an inert molecular _sequence _can be used for flexible

polymers

_as well. The result will describe the

physics

of flexible _or semiflexible

dipolar chains,

_as well _as that of _a dense system ^of zwitterionic telechelic chains of the type ^discussed

(as probe molecules)

in section 2.5. We need

only modify

the zero-field structure factor for the correlation bet~K,een the

charges

at the chain ends. We

assume

complete

dissociation

(which

^should be true at

high enough

chain densities _or for small

dipole

moments per unit

length)

in which _case the

charges

_are

separated by

_a

piece

of Gaussian

chain;

_as before _we _assume there is _no correlation between the

spatial position

of the chain and

the vector between its two ends.

To

generalize

_our

previous results,

_we need

only modify

the structure factor for the correlation between the end

points

of the molecule. For Gaussian coils it is:

y~ ₌

/

_dIL

_{P(IL, L)(I}

_e~~~

_~) ₍₃₀₎

where the distribution function is

P(R)

+~

exp(-3R~/21L)

for random coil

polymers

of _arc

length

L and effective step

length

I. The _mean _square end-to-end distance of _a chain is

<

[R[~

>= lL. The

averaging

in

(16) yields:

ji~ =

(I e~i~~~~) (31)

The

analysis proceeding

from

(12)

then continues _as

before,

but with this _new y~.

(I) Large

^k~lL

(length

scales within _a coil

radius)

the _structure factor tends to I and in

IS)

we have K~ ₌

xrnon6/d~N. Hence,

_as

above,

there is

Debye-lIiickel screening,

with

screening length

I

given by (28).

(ii)

^k~ <

I/lL (scales exceeding

the coil

size):

the structure factor tends _to

)k~lL.

^Now ^the

K~y~ term in

IS) simply

follo~vs the form of the bare Coulomb term

k~,

whence

by

identification

we obtain:

X ~ Xchwn "

Xmonl/d (32)

The

susceptibility

is

merely

that of the

equivalent

fluid of _monomers, enhanced

by

_a factor of

I/d specifying

the number of _monomers

acting

iii concert in _one effective step

length

of the chain.

(iii)

Very ^short ^distances ^k _+~

I/d. Returning

to

(14)

for

x(k)

_we obtain

x(k

+~ I

Id)

_+~

6xmon d/L.

As with stiff

rods,

the short scale _response is much smaller than that of _a fluid of

monomers.

Thus, although

the

long

distance behaviour of

a

polymer

returns

essentially

to that of _a fluid

composed

of its constituent parts

(as expected

for _monomers

along

_a chain

randomly disposed

with respect to each

other),

at short distances the

polymer

is constrained and cannot

respond strongly.

In between the two limits of

length

scale _a system ^of flexible

dipolar

chains will behave as an

electrolyte.

This weak _response of

dipolar

chains to

high

wavevector fields is counterintuitive at first

sight,

since _one is used _to

considering ^adjacent

^sections ^of a Gaussian chain _as

independently

oriented.

However,

at the monomeric level chain

connectivity imposes

_a strong correlation

(12)

between the relative orientation of

adjacent

_monomers and their

spatial separations,

and this inhibits the response ^to

high

wavevector fields. An extreme but instructive

example

is _a

freely jointed pair

of

just

two monomeric

dipoles

in one dimension. If _a field is

applied

with _a

wavelength

of

exactly

twice the _monomer

length,

the

energies

of the

(four) possible

states of the chain _are all unalsected

by

the field and the

polarization

_response is

strictly

_zero.

Over the full _range of

length scales,

the

potential

due to _a

point charge

_can

again

be modelled

by

~'~~~ ~

4~or ^l

₊

^ch~n

^~

_(l ^~~~~ain)

^~

^~~~

^~~~~

(compare (29)).

If _xch~n is

large,

this

gives

screened Coulomb behaviour at intermediate

length

scales.

Therefore,

until

lengths

are reached

comparable

to the chain dimensions where the correlation of the two

charges (enforcing

local

charge neutrality),

_a melt of Gaussian

dipolar

chains behaves _as _an

electrolyte.

However, if ycha;n is not

large (recall

there is _no enhancement factor N _as in the rod

case),

the first term is

comparable

to the second at short distances and this

analogy

is no

longer

correct. Note in any case that I

~w

N~/~

_is

now of order the chain size _so the

screening

^elsect is far less

spectacular

^than for rods.

5. Conclusions.

We have demonstrated that the

susceptibility

of _a system of

dipolar

rods is

nonlocal, having

a range

comparable

to the

length

of the rods. When fields _are

spatially varying

_on a scale

comparable

to the _monomer

length,

the fluid

susceptibility

is _very

small,

reduced

by

_a factor of N

compared

with _a fluid of

equivalent

_monomers. At the

opposite

extreme, ^the _response to

fields

slowly varying

_on the scale of the rod

length

is _i,ery

large:

enhanced

by

_a factor N with respect to disconnected _monomers, _as

predicted previously.

On intermediate

length scales, long dipolar

rods should exhibit _a strong

Debye-Hiickel screening behaviour,

with _a

Debye length

of order

N~/~ corresponding

to the fluid of elsective

polarization charges

which _can be

thought

of

as

residing

at the ends of each rod. This _very marked

length

scale

dependence

could be studied

by

_a

variety

of local

probes;

it should also influence various interfacial

phenomena.

For the

case of

dipolar

Gaussian

coils,

_a similar reduction in

polarizability

at

high

wavevectors is _seen,

but in _contrast with rods there is _no enhancement at low wavevectors. Thus the intermediate

(screening) regime

_may still present but is less dramatic.

Acknowledgements.

We thank Prof. Sir Sam

Edwards,

Dr David Wu, and Dr Aionica Olvera de la Cruz for

valuable

discussions,

and Dr E

Terentjev

for

communicating

to us as a

preprint

similar _concerns

regarding

the _response of

polymers

to electric fields.

References

[ii

Gunn J-M-F- and Warner M., Phys. Rev. Lefts. 58

(1987)

393.

[2] ^Warner M., MRS Proceedings 134

(1989)

61-71.

[3] Terentjev E., Phys. Rev. A46

(1992)

6564.

[4] Drye T. and Cates M. E., J. Chem. Phys.

(in

_press,

1993).

[5] ^Levine B. F. and Bethea C-G-, J. Chem. Phys. 65

(1976)

1989.

[6] ^de ^Gennes P-G., Scaling Concepts in Polj,mer Physics

(Cornell

U-P.,