Dynamical scaling exponent z for polymer chains in a
good solvent
D. Jasnow, M.A. Moore
To cite this version:
D. Jasnow, M.A. Moore.
Dynamical scaling exponent z for polymer chains
in a good solvent.
Journal de Physique Lettres, 1977, 38 (23), pp.467-471.
<10.1051/jphyslet:019770038023046700>. <jpa-00231422>
HAL Id: jpa-00231422
https://hal.archives-ouvertes.fr/jpa-00231422
Submitted on 1 Jan 1977
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LE JOURNAL DE PHYSIQUE
-
LETTRES
TOME
38,
ler DECEMBRE
1977,
I
L-467
’
Classification
Physics Abstracts
36.20
DYNAMICAL SCALING EXPONENT z FOR POLYMER CHAINS
IN A GOOD SOLVENT
D. JASNOW
and M. A. MOORE
(*)
Department of Theoretical Physics, The University, Manchester,
M13 9PL, U.K.
(Re~u le 12 aofit 1977, accepte le 3 novembre 1977)
Résumé.
L’exposant dynamique z d’une chaîne polymérique isolée dans un bon solvant est
4 - d pour le modèle de Rouse et Zimm. Dans la limite où les interactions
obtenu à l’ordre 03B5
hydrodynamiques sont absentes, on trouve z 4 - 03B5/4 + ... en accord avec le résultat z 2 + 1/03BD
du développement phénoménologique de de Gennes. L’accord cesse hors de cette limite, les valeurs
de z sont respectivement 4 - 5 03B5/4 + ... et d, comme le confirment de récents résultats expérimentaux
qui s’écartent des prédictions phénoménologiques.
2014
=
=
=
The dynamical exponent z for a single polymer chain in a good solvent is obtained
(= 4 - d) for the bead and spring (Rouse-Zimm) model. In the free-draining limit
2 + 1/03BD due
z
4 - 03B5/4 + ..., a result in agreement with a phenomenological prediction that z
4 - 5 03B5/4 + ..., which disagrees with de Gennes’
to de Gennes. In the non-free draining limit, z
prediction that z = d for this limit, but which is consistent with recent experimental work which finds
departures from the phenomenological predictions.
Abstract.
to order
2014
03B5
=
=
=
A central question in the study of polymer dynamics
is the determination of the spectrum of relaxation rates
for a single polymer chain in a good solvent ~1]. One
is particularly interested in the interplay between
hydrodynamic interactions, which are induced by flow
fields associated with the motion of a monomer unit,
and excluded volume interactions.
There have been considerable advances in elucidating the static (equilibrium) behaviour of such polymer
chains. It is believed [2] that universal properties are
described by the usual Landau-Ginzburg-Wilson
isotropic n-vector model [3] iny the limit ~ -~ 0. For
example, the mean square end-to-end distance of an N
link chain behaves as (Rl - RN)2 &#x3E; I"OW N 2v with,
4 - d expansion,
in an 8
=
The exponent v is universal and does not depend on,
say, the strength of the excluded volume forces, but
does depend crucially on the dimensionality.
Dynamical aspects of polymer chains in solution are
described in terms of the exponent z. For example, the
characteristic times t of the internal relaxation modes
behave with a power law dependence on N, viz.
z ~ NZV. The translational diffusion coefficient D of
the chain [4] is given by D - N . In this letter, we
present an 0(s) calculation of z. In the so-called freedraining limit, where the hydrodynamic interactions
are neglected, the excluded volume forces can be
handled by an E expansion and it is found that
z
4 - B/4 + ’". This result is consistent with the
conjecture of de Gennes [4] that in this limit z 2 + 1 Jv.
Unfortunately, the free draining limit is not realizable
in practice, for the hydrodynamic interactions always
dominate the dynamics. Even in the absence of
excluded volume effects the relaxation times r in the
non-free draining limit cannot be given in closed
form [1]. However, near four dimensions we are also
able to treat the hydrodynamics by means, of an E
expansion. Our final result for z in the presence of
excluded volume effects is z
4 - 5 s/4 + ’", which
disagrees with de Gennes’ conjecture [4] z = d
whether or not excluded volume forces are present.
The bead-spring (Rouse-Zimm) [5] model is described by the equations of motion
=
=
=
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019770038023046700
JOURNAL DE
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where
Rj(t)
is the
vector of the
position
PHYSIQUE -
jth bead
d labels the Cartesian
1, 2,
N) and x 1,
component. U { R } is the potential energy of a
configuration { Rj } and fj(t) represents the effects of
(j
=
=
...,
...,
LETTRES
If excluded volume and hydrodynamic interactions
neglected ( Vk 0 in (2) and T { R } 0) the
system of eq. (1) or (3) with A -+ A (0) is directly soluble
in terms of the normal (Rouse) coordinates
are
=
=
random collisions with solvent molecules. The distribution function of this random force is taken as
Gaussian. The tensor D is given by
where y is the friction coefficient [6] and T { R } is the
Oseen tensor, generalised to d-dimensions, whose form
in k-space is given in eq. (19), and which incorporates
the long-range hydrodynamic interactions. In the
free-draining limit T { R } 0. The potential energy
U { R } Uo { R } + U 1 { jR } is given by
=
=
In these coordinates A (0) becomes diagonal, i.e.
and the eigenfunctions
nomials in the argument
of_Aq° ~
are
Hermite
~c~q pqa with, for q &#x3E;
poly1,
The ground state, j(qa)
0 all q &#x3E;, 1, corresponds to
the equilibrium state. Note that the center-of-mass
motion separates out completely; we have
=
where the first term, Uo, represents harmonic forces
between neighbouring beads and the second, Ul, is the
excluded volume term. In the actual calculation we
shall take Vk = V
constant, which implies a delta
function self-interaction.
The time evolution of the polymer configuration
distribution function P[ { R } ; t] is governed by the
operator [7] (sums over repeated indices implied)
=
corresponds to motion of the whole chain. The
dynamics of the center-of-mass motion corresponds
to diffusion of a particle with diffusion constant
ykB TIN. For the free-draining limit (T { R ~ 0)
generally, this result is recovered for any
which
=
by summing (1)
Rouse
according to
where Z
=
d
means
{ R } exp(-
U
{ R }/kB T) is the ordi-
nary partition function. The characteristic relaxation
times of the system are the inverse of the eigenvalues Àm
as
any function of
according
configuration ~ {7~}
relaxes
to
where { Ro } symbolizes here the configuration
t = 0 and
at
over
all beads. Note that for
chain, ~,l~qa~ N (ykB T) q2 1[21b2 N2,
a
long
which
that the internal relaxation times
The aim of this letter is to discover the modifications
of the power law when excluded volume and hydrodynamic interactions are present. Our technique is to
do first order perturbation theory in the excluded
volume and hydrodynamic interactions in four dimensions, followed by the usual renormalization group
technique of exponentiating the logarithms. We
consider first the free-draining limit.
In this case we set T ~ ~ } = 0. Write A = A ~°~ + A ~ 1~
with
tim
/I,n
bation
=
An -
T
u/l,n~
expansion
r
wmi
u/!,n
in A ~ 1~ just
LV 111~L umci 111 ~:1
given by
pciLui-
DYNAMICAL SCALING EXPONENT
Here
(kB
T)-1 U° _ ~ c~q p~, corresponding
to the
qa
first term of (2) and u~ are the unperturbed eigenfunctions of eq. (7). We consider the typical state with
the (~oc) oscillator once excited, namely
A (1)
can
dinates {
be expressed in terms of the Rouse coor~ }, and evaluation of (9) directly yields
z
L-469
FOR POLYMER CHAINS
n-vector model. In the formalism adopted here,
equilibrium averages are ground state expectation
Perturbation theory to first
values
{R~
( Vo
=
1).
order in V for the end-to-end distance yields
where
U1 is the excluded volume
term
cumulant average in the
implies
finds
One
system.
a
which yields the special value 2 K4
Hence for the relaxation times we have
where
and
&#x3E;0,,
unperturbed
Vo b-4 = e/8.
and
implies via the definition Lq ’" NZV that
4 - 814 + O(~). This is one of our central
results. The 0(s) expression agrees with the phenomenological prediction of de Gennes [4]
which
z
=
BN(lj) come from the
small q region. Furthermore, for a long chain we can
assume ~7, ! ~ 2013./! ~ 1. Going over to the continuum
limit one finds BN(lj)
b2I I - j 1/2.
At this point in the analysis the dimensionality of If the
phenomenological argument is correct to all
space begins to play a role. When d &#x3E; 4 the integrals orders in B it means that no new dynamical exponent is
over k and then the sums over I and j in (10) can be
required in the free draining limit, that is, z is compleperformed without divergences occurring. One finds tely determined by equilibrium exponents. This situathat - N-2 so that the characteristic relaxation tion
usually arises when there are conservation laws
times remain "Cq ’" N2 in the presence of excluded
present (cf. Model B [8]). As noted above, there is a
The main contributions to
=
volume interactions. The power law is not modified,
although particular amplitudes will be affected. For
d
4, the power law is modified strongly (e.g.
6A - N 1/2 AO in d
3) and standard perturbation
theories are not justified. In the special dimensionality
4 the leading correction is logarithmic. Specid
fically, one finds for large N
=
=
(YkB ~) 1 6A I (,.)
=
kind of conservation law for the center-of-mass
motion in that the form it takes is independent
of U { R }. The decoupling of center-of-mass motions
from internal motions is removed when hydrodynamical interactions are included, as in the non-free
draining limit, to which we now turn.
The non-free draining limit is experimentally much
more important than the free-draining limit. Perturbation expansions in the Oseen tensor T { R } can also
be formally constructed. If we neglect excluded volume
interactions temporarily we have the perturbation
where the dots symbolise terms which are weaker
than logarithmic. In evaluating (10) we used
One therefore has a relaxation time spectrum
special choice Vo Vo(e) for obtaining expo4 - s dimensions can be found as
in
d
nents
follows. The equilibrium exponents are believed to be
given by the ~ -~ 0 limit of an ordinary isotropic
The
=
=
Diagonal matrix elements yield a shift in 6A.
unperturbed (Rouse) eigenvalue ~B where
in the
JOURNAL DE
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PHYSIQUE -
Zo is the partition function appropriate to Uo { R ~.
If we take the typical state with one excited mode,
LETTRES
dependence r~ ~ Nd12. It is easy to show
from the exact solution that there does indeed exist a
value of (yY~bd-2)-1
E/3 + 0(E2), for which exponentiation of the expansion about d
4 yields the
correct power law dependence for Tq. Notice that the
pre-averaged Oseen tensor (which is obtained from the
full Oseen tensor T { R } by assuming that the bead
positions { R~a } in it are Gaussian random variables
and averaging over them) and first-order perturbation
theory on the full Oseen tensor yield the same logarithmic corrections (in first order) to the eigenvalues
(but not the eigenvectors), thereby at least recovering
the result T, - N2 -E~2 to 0(E).
We are now in a position to put together the effects
of excluded volume and hydrodynamics. The excluded
volume interaction V is again supposed of0(s). Hence
in a first-order calculation of its effects on the non-free
power law
=
=
we
find by direct computation that
where we have used the form for the Oseen tensor
in k space
draining eigenvalues, we can, correct to 0(s), just use
the Rouse free-draining eigenfunctions of (7) rather
than the non-free draining ones, as the two sets of
which is valid for general dimensionality. In first order eigenfunctions differ only by terms of 0(s). In other
only 7~(k) enters the calculation. We find once again words, the perturbations from T { R } and U1 do not
4 is the special dimensionality. When d &#x3E; 4 mix to 0(s) and we can use the previously established
that d
the hydrodynamical interactions do not appear to special values Vo and (yr~b2) -1. Thus to 0(E) we have
modify the power law for relaxation times, while for the case of both excluded volume and hydro4 the first order correction (18) strongly dynamic interactions at the special values of the
for d
4 we constants,
dominates the unperturbed eigenvalue. In d
=
=
find
logarithms arise separately from the two
perturbations. On exponentiating 1:q ’" (N/q)2-~3ls~E.
This result implies 1: q ’" NZV with
where the
plus convergent terms. This means
typical relaxation time goes like
that for
large N a
As noted above, the value z = d has been suggested by
de Gennes [4] on the basis of phenomenological
Since it is believed [1, 4, 9] that in the absence of
excluded volume forces iq ~ Nd/2, one can deduce
that the special value of (y~b2) ^ 1 which allows such
exponentiation must be 8/3. (This value for (yqb 2)-l
could presumably be established directly by going to
next order in perturbation theory in T { R }).
The validity of this typical renormalization group
technique for what is essentially only a problem in
hydrodynamics was checked by applying it to a
similar problem which is, however, also exactly
soluble. The pre-averaged Oseen tensor [1, 9], for’
which
arguments.
A consequence of the failure of de Gennes’ prediction is that the hydrodynamic radius RH will not have
the same exponent describing its N dependence as the
end-to-end distance. This can be seen as follows. On
the basis of scaling [4] one expects a characteristic
wave-number dependent relaxation rate Fk of the form
rk r-1 f (kNv) where r - A~B Such a rate can be
determined by measurements of the polymer’s dynamic structure factor S(k, t) [10]. At long wavelengths
rk ’" Dk2 so that the translational diffusion coefficient
D - N . One refers to the quantity
=
as the hydrodynamic radius (D ~ kT/r~ RH- ~ according to Stokes law in d-dimensions). The de Gennes
prediction yields RH ~ ~ ~ No.6 (d 3), while we
=
is such that the eigenvalues of the A associated with it
can be calculated analytically for all d (in the absence
of excluded volume forces) and again they have the
find RH - N - N . While we do not take the
numerical estimate seriously, we note that it deviates
in the right direction from the phenomological
prediction as seen in the recent experiments of Adam
DYNAMICAL SCALING EXPONENT
and Delsanti [10] who quote RH ~ N’-" with an
uncertainty of ± 0.02 in the exponent. The phenomological statement that the equilibrium and hydrodynamical radii are proportional is appealing, but
evidently incorrect.
z
L-471
FOR POLYMER CHAINS
One of us (D. J.) gratefully
of the National Science Foundation and the Alfred P. Sloan Foundation. We also
wish to thank Dr. A. J. Bray and Professor B. U. Felderhof for helpful discussions.
Acknowledgments.
acknowledges support
-
References
For
general review see YAMAKAWA, H., Modern Theory of
Polymer Solutions (Harper and Row, New York) 1971.
[2] This was first noticed by DE GENNES, P. G., Phys. Lett. 38A
(1972) 339. See also EMERY, V. J., Phys. Rev. B 11 (1975)
229 and JASNOW, D. and FISHER, M. E., Phys. Rev. B 13
(1976) 1112.
[3] For a review of results pertaining to the n-vector model, see
[1]
a
WALLACE, D. J., in Phase Transitions and Critical Phenomena (ed. C. Domb and M. S. Green, Academic, N.Y.)
1976, Vol. VI, along with additional articles in the volume.
[4] DE GENNES, P. G., Macromolecules 9 (1976) 587.
P. E., J. Chem. Phys. 21 (1953) 1272;
ZIMM, B. H., J. Chem. Phys. 24 (1956) 269.
[6] In d 3, according to Stokes’ Law, 03B3 1/6 03C0~a, where
[5] ROUSE,
=
=
[7]
[8]
a
is the radius of a bead. In general d, 03B3 ~ (~ad-2)-1.
See, e.g., Section 34 of Ref. [1].
HOHENBERG, P. C. and HALPERIN, B. I., Rev. Mod. Phys. 49
(1977).
[9] DUBOIS-VIOLETTE,
E. and
DE
GENNES, P. G., Physics 3 (1967)
181.
[10] ADAM, M. and DELSANTI, M., Macromolecules, to be published.