Flory exponents from a self-consistent renormalization group

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Flory exponents from a self-consistent renormalization group

Randall Kamien

To cite this version:

Randall Kamien. Flory exponents from a self-consistent renormalization group. Journal de Physique

I, EDP Sciences, 1993, 3 (8), pp.1663-1670. �10.1051/jp1:1993208�. �jpa-00246824�

Classification Physics Abstracts

61.25H 64.60A

Short Communication

Flory exponents IFom

_a

self-consistent renormalization _group

Randall D. Kamien

(*)

School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, ^U-S-A-

(Recejved

7 April 1993, accepted 3 June

1993)

Abstract. The wandering exponent _u for _an isotropic polymer is predicted remarkably

well by _a simple _argument due to Flory. By considering oriented polymers living in _{a one-} parameter farniy of background tangent fields, _{we are} able to relate the wandering exponent to the exponent in the background field through _an e-expansion. We then choose the background

field to have the _same correlations _as the individual polymer, thus self-consistently solving for

u. We find _u

=

3/(d

₊ 2) for d _< 4 and _u

=

1/2

for d > 4, which is exactly the Flory result.

1 Introductiono

The

theory

of

polymer

conformations is at _once

elegant

^and

confounding. Flory theory,

which is based _on dimensional

analysis,

does _a remarkable

job

in

predicting

the

wandering

exponent _v

defined

by [r(L) r(0)]~

+~ L~" [1]. Nonetheless, ^controlled

approximations

which attempt to

improve

_on

Flory theory

for

polymers

_are not

nearly

_as

good. Flory theory predicts

_vF

=

1/2

for d > 4 and _vF

"

3/(d+2)

for d <

4,

which is exact in and above the critical dimension d~ = 4,

where the

polymer

acts as a random walk [2]. It has been shown _[3] that _vF is exact in d

= 2

dimensions _as well. In three-dimensions the resummed

e-expansion gives

_{v =} 0.5880 + 0.oo10 [4],

Flory theory gives

vF =

0.6,

and the most recent numerical simulations cannot

distinguish

between the two [5].

In this note _we rederive the

Flory

exponent. We do this

by considering

oriented

polymers interacting

with _a

background directing

field.

Flory theory

would

predict

the _same value of _v for oriented _or non-oriented

polymers.

For _a

family

of

directing fields, pararneterized by A,

_we

can derive _an exact result for _v in terms of A. Unlike other self-consistent

analyses,

_we find the self-consistent exponent

by matching

exponents, not correlation function

prefactors

_[6]. We then choose A _so that the tangent-tangent correlation function of the

polymer

scales with the

same exponent _as the

background

field

correlation,

thus

self-consistently choosing

A. We find

(*)

email:

kamien@guinness.ids.edu

1664 JOURNAL DE PHYSIQUE I N°8

the

Flory value, namely

_{v =}

1/z

₌ 3

lid

+

2),

where _z is the

dynamical

exponent, _{as we} discuss below. It is

amusing

that the

Flory

result _comes from _an

e-expansion

which

happens

to

be,

in this case, exact. This _may suggest

why Flory theory

is _so

good. Additionally,

it suggests how

one

might study

tethered

surfaces,

where

Flory theory

is not _so

good

and

e-expansions

_are not so easy.

20 Formulationo

We consider _a d dimensional system ^with a

background

tangent ^field _u,

zu_{~r Si =}

fit/llllll ^ran

^exP

^- £~

^d8~

(~il"~ >U<r<S') ')I j

ii)

~vhere _s labels the _monomer

along

the

polymer.

Then the annealed

partition

function is

Zanir, s)

=

/~dujZujr,s)P~uj. ^j2)

A factor of I has been introduced to

help

to

organize

the

perturbation expansion,

and

will,

in the

end,

be set to

unity.

In

addition,

it is needed to make the units correct, I-e-

)]

=

LS~~

The vector field

u(z, s)

is _a function of both _space and the _monomer label _{s, as} if each

monomer interacts with _a different _vector

field,

_even

though they

could be _near each other in

space. Note that in _mean field

theory dr(s) Ids

₌

lu(r(s), s).

Consider the tangent correlations

of two _monomers at two

nearby points

in space, A and B. If the

polymer

takes _a short _route

from A _to

B,

then the tangent vectors

along

that

length

will be

strongly correlated,

since

they

must

mostly

lie

along

the line

connecting

the two

points.

On the other

hand,

if the

polymer

takes _a

long

circuitous

path

while

getting

from A _to

B,

the _two tangent vectors will _not be very correlated. Since

u(z, s)

is the self-consistent tangent

field,

it must reflect this

simple geometric

argument. If _u did not

depend

on s, the

tangent-tangent

correlations

along

the

polymer

would

only depend

_on their distance in space, and would not respect the constraint

relating

the

path

to the tangent vectors.

We _now view the functional

integral

_over

r(s)

_{as a} quantum mechanical propagator ⁱⁿ

imagi-

nary time. That is,

Zu

will

satisfy

_a Fokker-Planck equation with the initial condition

r(0)

= 0.

In this _case the Euclidean

Lagrangian

is

just

_our Hamiltonian. The Euclidean Hamiltonian is the

Legendre

transform of the

Lagrangian. However,

_as with the

theory

of _a

point particle

in _an

electromagnetic field,

there is _an

ordering ambiguity.

The functional

integral produces

the symmetric Hamiltonian.

Because,

in statistical

mechanics,

the Hamiltonian _comes from _a

transfer matrix, _we must take the

symmetric ordering

of the momentum and the

velocity field,

as is done in

electrodynamics

of _a quantum ^mechanical particle [7].

The Fokker-Planck

(Euclidean Schr6dinger)

equation obtained is:

~~j~'

^~~ =

DV~

_Zu

jr, s) )I _(V ~u(r, s)Zu jr, s)]

+

u(r, s) VZU(r, s)) (3)

s

If _we write

ifi(r, s)

₌

9(s)Zu(r, s)

and Fourier transform in space and

time,

_we get:

(-1t4

₊

Dk~)lllk>t4)

₌

llolk) )(ii) / ) / ) _{U;lk q)lk'}

+

q')1l(q> n) (4)

where _~v is the Fourier variable

conjugate

to _s, k is the Fourier variable

conjugate

to _z and _~bo is the Fourier transform of the initial conditions. The

boundary

condition

r(0)

₌ 0

corresponds

to _~bo + I.

Equation (4)

_can be solved

recursively

in _powers of I.

Now _we must consider the random field _u. If _u is to describe the self-consistent

background, then,

if the

polymers

have _no ends

(by

either

being cyclic

_or

spanning

the

system),

V.u ₌ 0.

With this

constraint,

_we take

u;(k,~v)uj(k',~v')

=

j~

_~

~')/(~

_{b(~v +} ~v')

b~(k

+

k/) (5)

We will

self-consistently

choose A at the end of the calculation. We have chosen the w~

dependence

in the denominator for

simplicity.

If _we had chosen _a

dependence

other than

quadratic

_we could

always

capture the _same relative

scaling

of _~v and k

by

_an

appropriate

choice of A in

(5).

We note an

important simplification

due to the constraint V.u ₌

0,

_[8]

namely

that there is _no difference between

taking

_u to be

quenched

_or annealed. The

quenched probability

distribution for _r is

just:

However, by integrating (3)

over space, we find that

) / _{d~r Zu(r, s)}

=

/ _d~r

V

~DVZU(r, s) lu(r)Zu(r, _s)] (7)

s

where _we have used the fact that V.u

= 0.

By

Gauss's

law,

the

integral

on the

right

hand side will vanish since

Zu(r,s)

will fall to 0 at

infinity.

Thus the normalization of Zu is _s-

independent.

Since

fd4~Zu(r,0)

is _a constant,

independent

of _{u, so} is

fd~rZu(r,s),

and it factors out of the functional

integrand

in

(6).

But then _{we see} that

p~~~~

_{~~ ~}

Jldulzu(r, _s)Plul

'

j

ddr

j~dujz~(r, s)P~uj

~

JldUlZu(r, S)P~UI (8)

J

^d~r

Zu(r,

_S)

JldUlPlUl

=

Pqu(r, s).

Because of the directed nature of the propagator, ^{it is}

possible

to argue that z* ₌ 2 _e to all orders. Since this

problem

_can be viewed _as

quenched disorder,

_u will not suffer _any nontrivial

rescalings,

^and so A and B will

only

rescale

trivially.

In

addition,

_{as we} will _argue in

appendix A,

I does not get renormalized at any order of

perturbation theory.

3. Perturbation

theory

and the renormalization _group.

We

analyze

this model within the context of the

dynamical

renormalization _group,

along

the lines of [9]. ^For a renormalization group with parameter

t,

_we rescale

lengths according

to k'

=

e~k _and w'

=

e?(°~v,

where

7(t)

is _an

arbitrary

function oft to be determined later. When

integrating

out

high-momentum

modes in _{a momentum} shell

Ae~~

_{< k <}

A,

_we

integrate

_over

1666 JOURNAL DE PHYSIQUE I N°8

all values of _w. We _can choose the field _u to have dimension 0. We find differential recursion relations:

~

~/~

=

D(t)

2 + z +

£(1

()Ad) ^(9a)

~

()~

=

l(t)

I +

z) (9b)

~~~~~

=

B(t) (d

2A +

z) (9c)

where

z(t)

₌

nt'(f)

and

Ad

₌

2(4~r)~/~/r(d/2)

is _a

geometrical

factor.

Putting

^these

together,

we can get a recursion relation for _g

=

l~(I j)Ad/2BD

((

"

9(~ 9) (10)

where _e

= 2A d. Thus in d > 2A dimensions there is _a stable fixed

point

at g = 0, whereas if d _< 2A then there is _a stable fixed

point

at g = e. In the details of the

calculation,

it is

essential that A _< 2.

Succinctly

_put, this is because the

pole

in _w which is used to evaluate the

self-energy

correction (w =

@@k~)

_{must dominate} the diffusive part ^of the propagator

(Dk~)

at small k. We will check that this constraint holds when

finding

the self-consistent value of A.

4. Self-consistent value of A.

Choosing

D to be fixed

simplifies

calculations

(though

does not

change

the

results),

_{so we}

choose

z(t)

= 2

g(t).

As _we show in

appendix A,

at the nontrivial fixed

point,

_{z =} 2 _e

exactly. Using

^this we find the

following

two

scaling

relations for the

position

and

velocity

correlations:

~r(L) r(°)i~

=

2dDL~/~ (iii

Now we _{would like} to

~d~ _~~ ~ik.~(~)~-<w~(d 1)

~'~~~~~'~~~'~~'~~ /

(2~r)d ^{2~r A~v2} + Bk2A ~~~~

Since

iris))

₌ 0, we take

r(s)

= 0 in

(13)

and then check that the corrections _are themselves consistent. We find

~~~~(~)>

8)U,(0, 0)

=

(~ l)Adr(I e)A(d-2A>

^/2A 28~/2~A

L~~~~l/A

_~

Matching

the

scaling

exponents, we have

A _z 2-2A+d ^~~~~

Solving

for

A,

_we find A

=

d/2

or A

=

id

+

2)/3.

The former

gives

_{e =}

2(d/2)

d

=

0,

and

hence _{z =} 2. In this _case

(12)

is incorrect, since the second derivative of

ill)

vanishes.

Thus,

we must choose A

=

id

+

2)/3.

We note that this solution satisfies both A < 2 and 2A > d for d <

4,

^and _{so we can say} ^{that the critical dimension} of this model is d~ = 4. Above

this,

_e is

negative,

^and _we return to the

simple

random walk fixed

point,

_{z =} 2.

Finally,

_we compute

z = 2 2A + d

=

id

+

2)/3,

in

complete agreement

^with

Flory theory.

We

point

out that the

matching

of exponents ^{breaks down when d} = 4, for if

ill

is described

solely by

_{an exponent,} without

logarithms,

there is _no

matching

to do that

is,

^the exponent in

(12)

would be 0.

Thus,

_we do not expect, and in fact do not,

reproduce

^the correct

logarithmic

corrections to

scaling.

If _we

expand

the

complex

exponent in the

integrand

of

(13)

in powers of

r(s)

_we would find

an

expansion

of the form

~~

~~~

~~~~~~~~~~

_~~~~

J=

where _we have

suppressed

the indices. This _sum

only

contains _{even powers} since the

integration

over k will eliminate odd powers of k

by

rotational invariance. Since

jr _(s)~ )i

+~

s~J/~

_{and each}

power of k~ _will

produce

_a factor of

s~~/~,

we find that the

higher

order corrections scale the

same way as the

leading

term if A

= z,

which,

in

fact,

it does. Thus the

approximation

is

truly

self-consistent.

5 Conclusions.

One

might imagine adding

to this model in _a number of _ways. One

possibility

is to add _an

explicit self-avoiding

term to

(I).

This would

only

_cause the _same

complications

present in

Flory theory.

Another

possibility

would be to add _a small

divergence

to the field _u,

corresponding

to

polymer

heads and tails ~10]. ^We would then have

u,(k,~v)uj(k',~v')

₌ ^~

j

~

~'~~~~~

d(~v + ~v')

d~(k

+

k') (17)

where _a is close to, but not

equal

to I. In this _{case we} find that _new

graphs

arise which

spoil

the exact argument in

appendix

A. Moreover, some of these _new

graphs

will

diverge logarithmically

when A

= AF

"

(d

+

2)/3.

One

might

add _a small correction d to AF and do

a d expansion around d

= 0.

Unfortunately,

_now, there is _{a new} fixed point, and the self-consistent correction d is not small.

It would have been _more natural to consider _a walk in _a random

potentjal

and then self-

consistently

choose the

two-point

correlation of the

potential

to match the

density-density

correlation. It

is, unfortunately, notoriously

difficult to

study polymers

in _a random

potential

in _an

arbitrary

dimension

~ll].

Here _we have

exploited

the

solubility

^of a random-walker in _a random

velocity

field.

It is

perhaps

_a

curiosity

^that self-avoidance _was not involved in this calculation. In

fact, by mapping

the system to quantum

mechanics,

_we have

actually mapped

the system to _a directed walk in _an external _vector

field, ignoring

the energy ^cost of self-intersections

entirely.

The

self-consistent vector field is _a strange

object,

_as it

depends

not

only

_on the

polymer

position

r(s),

but also the

point along

the

polymer

_s. This is necessary from _a

geometric point

of view,

and _suppresses tangent vector correlations between two _monomers far apart

along

the

polymer

sequence. In

(2)

_{we can}

imagine integrating

out the

velocity

field _u. The details of this _are in

1668 JOURNAL DE PHYSIQUE I N°8

appendix B,

and the

resulting

free _energy looks similar to that for _a

self-avoiding walk,

with

an energy ^cost for self-intersections.

Reproducing

the

Flory

exponent

then, suggests

that the

Flory theory

_may be _more

robust,

and that the model studied here is in the _same

universality

class. The

quality

of the

Flory prediction

_may lie in the fact that the self-consistent

analysis

here

incorporated

_an exact

result,

within the

epsilon expansion.

Acknowledgements.

It is _a

pleasure

to

acknowledge stimulating

discussions with L.

Balents,

M-E-

Fisher,

M. Gou-

lian,

P. Le

Doussal,

T.

Lubensky

and D. Nelson _as well _{as a} critical review

by

P.

Fendley.

This work _was

supported

in part

by

the National Science

Foundation, through

Grant No. PHY92-

45317,

and

through

the Ambrose Monell Foundation.

Appendix

A. Exact Value of _z.

In the above discussion _we used the result that _z

= 2 _-e at the fixed point, to all orders in _e ~12].

This result is similar to that in

~9] where Galilean invariance assured that the

scaling

field of the interaction would

only

rescale

by

its naive dimension. Our argument will be

perturbative.

The first

requirement

is that _u

only

rescales

by

its naive dimension. Since the

quenched

and annealed

problems

_are the _{same, we can}

regard

_{u as a}

quenched

random field. While it is

typical

to absorb the

coupling

I into _{u, we} will not do _so

here,

and hence it will _not be _non-

trivially

renormalized. We also note that

(3)

is linear in

~b ^so any nontrivial

rescaling

of _~b will not affect I.

We consider _a

general graph

with _an

incoming

_~b

line, carrying

momentum p, and

outgoing

~b*

line, carrying

momentum

p',

and _an

outgoing

_u

line, carrying

momentum p

p'.

The

incoming

_~b ^line must first meet _a vertex with _u. At this vertex, ^let u carry away ^momentum

ki

The contribution _to the

graph

from this vertex is then

proportional

to:

p~

((ki)~d"P k(k[)

= p~

((ki)~d"P k(k[) (A.1)

Thus the

graph

will be

explicitly proportional

to pH.

Similarly,

the

outgoing

_~b* line will emerge from such _a vertex. If the _u line carries momentum k2> then it will contribute _a term

proportional

to:

(l"

~2)~ ((~2)~~~~ ~~~i)

"

(l")~ _((~2)~d~~ k~~i) (J~.~)

due to the _transverse nature of the _u propagator. Thus the

graph

will generate a term with

at least two powers of the external momentum. This will _not then renormalize I since it is the coefficient of _a term with

only

_{one power} of momentum.

Indeed,

this

graph

will generate

terms, which

by

_power

counting,

_are irrelevant operators. Hence, the recursion relation for I will

simply

be

([~~

=

>(-i

+

z) (A.3)

~~ ~i orders.

(A.4) Now,

dB(t)

B(d

2A ₊

z)

dt

and if D has the recursion relation:

~

~/~

=

D(

2 + _z +

f(g)) (A.5)

where

f(g)

represents the

perturbation expansion renormalizing D,

then _{g must} have the recursion relation:

~~~~~

= g

~2(-1

+

z) (d

2A +

z) (-2

+ z +

f(g))]

~~

=

g~2A

^d

fig)] ^(~.~)

"

9k fig)]

since the dimension of _g

+~

l~/BD

will include _a contribution from B and D.

Thus,

if there is

a fixed

point,

then

fig)

_{= e} to all orders _{in g.}

Hence,

_{we see} that at the fixed point z = 2 _e to all orders.

Appendix

B. Ewective self-avoidance interaction.

Starting

with

(2),

the

coupling

of the

polymer

to the random field _can be written

~int

_"

~

/d~Zd8~~lrl8) ^{Z)VIZ, 81'} ~j)~ ^lB.I)

Upon integrating

out u, we find _to order l~

(note

that this will not include the

llJ(l~)

term in

(1))

~

/

^d~k _,

~~~

^~

(2~r)d ^~~~~

~

e~4~~"'~~~"

j~2

^{~ ~ ~} ~,k.j~(~>-~(~>jj

dr,18) dry(8') ^~~

^~~

16D/1kA+2

^{" ' ~} ds ds'

Because of

phase oscillations,

the

integral

_over k is small if

r(s) -r(s')

is

large.

If

r(s) -r(s')

= 0

we can

replace k~d;j by k,kjd

in the k

integration, through

rotational invariance.

Making

this substitution then is _a

good approximation,

and the corrections _are

suppressed

due to the oscillations. We then have

l~(d-I) /

^d~k _{~ ~,}

^e~4~~~'~

^{~" d d}

~,k.y(s)-~(s')I Fint

"

~~

~/@ (2~rjd

^{~ ~ kA+2} ds ds'

l~(d I) /

^d~k _{~ ~~,}

e'~'l~(4~~(~"l

_{d d}

~-@@k"j~-~'j

"

i~

~fi

_(2~r)d ^~ kA+2 ds ds'

(B.3)

>2(d i)W j

^ddk _{~ ~ ,}

~a-~~,~,j~~~,-~~~,,j~-wm~Aj~-sip

16D/@

_(2~r)d ^{~ ~}

+ constant

Within the self-consistent

approximation

we found A

=

(d

₊

2)/3,

_so A 2

= d 2A

= -e.

We _now

expand

in powers of

is s'i,

since

by

self-avoidance

large

values of is

s'[

will

typically

1670 JOURNAL DE PHYSIQUE I N°8

be

accompanied by large

values of

r(s) r(s')

which will suppress the

integral.

^In addition _we

can

expand

in _powers of _e

assuming

that _away from the critical dimension d

= 4 _we

only

have

a

slightly

modified

potential, corresponding

to a renormalized interaction. The first term in _a double

expansion

in _powers of _e and

is s'i

is

Fi~t

+~

l~ dsds'd~(r(s) r(s'))~ (B.4)

looking

_very similar indeed to a self-avoidance term. We _can also _see that the

repulsion

is

proportional

to

l~,

which itself is

proportional

to the

expansion

parameter of the model

discussed in this _paper.

References

[1] Flory P., Principles of Polymer Chemistry, Chap. XII

(Cornell

University Press, Ithaca,

1971).

[2] ^{For another} way of interpreting the Flory exponent, see Bouchaud J-P-, Georges A., Phys. Rep.

195

(1990)

_127~ and references therein.

[3] ^Nienhuis B., Phys. Rev. Lett. 49

(1982)

1062.

[4] ^{de Gennes P-G-} Phys. Lett. A 44

(1973)

271;

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[5] Grest G.~ private communication.

[6] Bouchaud J-P-, Cates M-E-, Phys. Rev. E 47

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1455;

Doherty J., Moore M-A-, Bray A.

(unpublished).

[7] Feynman R-P-, Rev. Mod. Phys. 20

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[8] We would like to thank Le Doussal P. for pointing this out. See also Le Doussal P., Kamien R-D-,

in preparation

(1993).

[9] ^Forster D., Nelson D-R-, Stephen M-J-, Phys. Rev. A16

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732.

[10] Kamien R-D-, Le Doussal P.~ Nelson D-R-, Phys. Rev. A 45

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8727.

[11] Kardar M., Zhang Y-C-, Phys. Rev. Lett. 58

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2087.

[12] ^{After submission of this} manuscript, J. Bouchaud informed

us of

a simian result in Honkonen J., Karjalainen E., Phys. Lett. A 129

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333.