HAL Id: jpa-00247542
https://hal.archives-ouvertes.fr/jpa-00247542
Submitted on 1 Jan 1991
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.
Scattering from random-sequenced-copolymers: RPA and replicas
T. Vilgis
To cite this version:
T. Vilgis. Scattering from random-sequenced-copolymers: RPA and replicas. Journal de Physique II, EDP Sciences, 1991, 1 (6), pp.585-591. �10.1051/jp2:1991191�. �jpa-00247542�
Classification
PlqsicsAbsmct£
61,4o-36.20-05.40
Shon Communication
Scattering from random-sequenced-copolymers: ItPA and
replicas (*)
TA~ Vflgis
Max-Planck Institut fur Po1ynlerforschung, PO Box 31 48, W~SW Mainz, Gernwny (Received 31 Janua~y I ml, nosed 9A phi 1991, accepted 11 Apnl I ml)
Abstract. This paper discusses the scattering from a melt of randomly sequenced copolymers.
Since the distribution of random sequences are quenched new collective variables have to be intro~
duced which carry two replica indices. The disorder has a strong influence as the resulting scattering
function. Moreover a mlcrophase separation is speculated,
t Intrtduction.
The problem of randomly sequenced copolymers has been discussed recently in the context of
biological polymers [1,2], where mean field models for protein folding have been developed. The
relationship to spinglass theories have been pointed out. These studies dhcuss the behaviour ofa
single random copolymer, I,e, a chain where the monomers consist of different chemical species.
Although these models are dealing with biopolymers the studies are also relevant for synthetical polymer which can be designed in a special way in order to find desired macroscopic properties.
An example has been discussed in [3] where a chain has been composed of sequences of different stiffness.
In this paper we are interested in randomly sequenced copolymers in dense systems, such as concentrated solutions and melts. We investigate the effect of disorder of the sequences, and we
find that the mean excluded volume parameter is renormalised by the disorder strength at higher disorder, the chains become collapsed and the scattering becomes infinite at a certain wave vector
~.
In this paper we use an oversimplified model: we assume that the excluded volume parameters
are distrtuted Gaussian, as it was used in [1,2]. More technically relevant problems such as an
A-B random copolymer would require different distributions and are related to the + J-model in spin glasses [4].
(*) Dedicated to Prof. H. Benoit in occasion of his 70th birthday.
586 JOURNAL DE PHYSIQUE II N°6
In order to develop a mean field theory for the melt we introduce new collective variables, which are extensions of the usual collective density variables appropriate for homopolymers. We show that in the case of heteropolymers new variables are required which carry replica indices
and two wave vectors. The introduction of replicas becomes necessary since the arrangements
of monomers along the chains are quenched. If we work in a Gaussian approximation, I.e. only
quadratic terms of the collective variables appear in the Hamiltonian, the model is exactly solvable and we do not have to break the replica symmetry.
2. The model.
We start from the Edwards Hamfltonian [~ which is also appropriate for block copolymers [fl and
other polymer systems [7j. In terms of chain variables Ra (s) it reads
H (lRo(S)I)
=
li f /~
dS ()) ~
+ L /~ dS/~dS'»op(S,S')b (R«(S) Rp(S')) +
+W f )~
dSf~ dS'/~ dS"b (Ro(S) RP(S'))b(Ro(S) R=(S")) (1)
where the first term is the usual Wiener measure, the second term the two body interaction. £ is the step length which we assume to be the same for all different monomers, N is the chain
length and w is the three body interaction parameter (~- £6) which will be ignored in most of our considerations. np is the number of polymers, a, p, and 7 label different chains, yap(s, s')
is a quenched random variable which accounts for the different excluded volume forces which
can be repulsive or attractive. For reasons of simplicity we assume a Gaussian distribution of the excluded volume parameters, I.e.
P (I»«p(S>S')I)
= N exP ~ ~ /~/~dSdS'(»op(S,S') »o)~
l(2)
Off ° °
where vo is the mean repulsion (or attraction with vo negative) and 6 is the standard variance which is a measure for the disorder. This assumption was also taken in [1,2].
Since the yap(s, s') are quenched random variables the replica trick has to be performed, For
example if the free energy F of the ensemble of heteropolymers is calculated one has to eval- uate F as a functional of the set (yap(s, s'))
,
I.e. F ((yap (s, s'))) and average afterwards over P ((yap(s, s') )) The average over the logarithm 1s done in the usual way
F " IF(ivoP(S>S')i) dP (ivoP(S>S')i) " ifi
= -jlog zj = Iim ) jz"j
[n=o 13)
n - 0 ~
where Z is the partition function
The replica trick has also to be employed if the scattering function of the melt of heteropolymers
1s of interest (which is the central point of this paper). The structure function [5] can be written
J
ijbR~is) £ j~ j~ dsds, e>q(«o(,)-R,(,>))~- ~~j~_j,~j~
sjq) = °=I off ° °
"P
(5)
f £ b Ra(s) e'H(lR"(s)I)
a=i
where is the disorder average as before. The random variables yapis, s') appear in the numer- ator and in the denominator and to perform the disorder average we can rewrite (5) into
"P n N N
S(q) = lim / fl fl b R$(s) £ / / dsds' e'~(Rll'J'~i("J)
x
n - o o=i a o~p ° °
'~jn H (IRS (~)l)
x
e
o=1 (6)
where a h the replica index (I < a < n) and H ((R$(s))) is the replicated Hamiltonhn. Equa-
tions (3-6) formulate the problem exactly, but the integration over the R$(s) variables are im-
posstle to carry out. Therefore the Hamiltonian of the dense system is transformed to couective coodinates, which turn out to be appropriate for dense systems (concentrated solutions and melts) 15-7l.
3. lkansformation to collective coordinates.
Averaging equation (6) gives immediately
S(q) = lim / fl fl bR$(s) £ /~/~ dsds' e'~((R[I')-Ri(")))
x
n - 0 ° ~ °>fl ~ ~
xexP
~ f f /~
°~j~~~
~
(vo 6) f f /~ /~
dSdS' b (IRS(s) Ri(s')))
~~~~~~ 0 ~~p~~~~~ 0
"P n N N
+ 6 L L f f dsds'b llRS(s) Ri(s')))
b (lRl(s) RI(s'))) (7)
o~p=I a#b ° °
and we see that we get a replica mkng term as given in references [1,2]. The last term in equation (7) gives rise to a definition of new collective coordinates (an order parameter field)Q([
Qab f /~
, ~~ ~
qk = ds e'l n(s)+,k R~(,)
o=1 0
(~)
which are an extension of the usual density variables p( = ~~£ /~ ds e+"l RI I'). These density
variables can be written as
n~ N
p( = Q(~o = £ /
ds e+"l ~Sl') (9)
'
~_~ 0
588 JOURNAL DE PHYSIQUE II N°6
Thus we can write the Hamiltonian as
~ N "P n ~~a 2 n n
H " ~ / d~ EL (#)
+ (~0 6) + ~ L Q(lQ [O 6 L L Q(lQ [-k (lo)
° o=la=I a=I q q,k a#b
Note that in equations (7) and (10) the mean excluded volume vo becomes renormalbed by the dhorder 6. If 6 is large enough we get an attractive pair interaction between the same and different replicas which lead to the frozen states (spin glass phase) in the heterogeneous globule li,21.
The next mathematical problem is to transform the entropy term (I.e. the Wiener measure).
The mathematical procedure for non-replicated collective coordination h outlined in [5-7] and we
find an effective entropy term
~ ((Q~~)) " ~°~/ ~ ~ bl~$(S)X
o=la=1
f f f~ ~l~$(S)
x fl fl b Q(~ £ /~ ds e'~ ~~~'~+'~ ~$~'~
eO=I a=1 ° ~~
(II)
qk a#b o °
The functional integral in equation (II can be evaluated in the so-called Gaussian approximation.
Therefore the b-function has to be represented by an auxiliary field in the usual exponential form which can be expanded. The functional integral over the R$(s) variables can then be carried out
term by term. If we truncate the series at the second term and reexponentiate the expression,
we have the effective entropy in the replicated collective coordinates Q(( This mathematical
procedure, if applied to ordinary homopolymers, h equivalent to the so-called Random Phase
Approximation RPA (see [6-8]), but is presented here in its replicated version.
Starting nom equation (I I) we have to average the b function over the replicated Wiener mea-
sure. This average will be denoted by lo We first introduce field variables tY([ as conjugated
fields to Q([ Thus we have
£ £ oab~yab ~yab £ ~ ~~ ~>q R[(s)+<k R$(s)
' qk -q-k -q'k
/ fl fl j
b~v([ e ~#~ 'lk ° (12)
q~k a#b
o
16 perform the R[(s) -integrals we expand the second term in the exponential in a series of the
auxiliary fields ~vf[_~ and integrate term by term, and we have expressions, such as
~~pab-q-k £ £ £ f d~ (e'q Rj(s)+>k R$(s)j j13)
~
o
a#b q,k O
from the first order, and
£ £ £ £ ~yab ~,cd ~~~~, £
(~>q Rj(,)+,k R$(,)+>P Rj(,>)+,j
Rj(,>)j j~~)
-q- -P-J
~
qk pj a#bc#d o,p 0
for second order. Higher orders will be (as usual in the RPA) neglected.
The first term would give only a contribution for a = b, but thin h not allowed and we have zero for this expression. The second term defines a higher order structure factor. In a pair approxima-
tion it gives rhe to two terms, I.e.
(~~) " £ £ ~~(~, k) ~~~~i ~q-k + ~~~~~~q-k (~~)
qk a#b
~~~~~
siiq,k)
= npj~ j~dsds'e'l~~+~~~"~"' l~~~
Under the assumption tY([ = tY([ we find the result
(l~) " ~ CC ~((~>k)(~$(~ (~~)
qk a#b
Reexponentiating this and carrying out the remaining tY(( -integration, we find for the entropy
S ((Q$)) " log / e~~~ ~°" ~~~~~ ~ fl fl bQ~~ (~~)
qk a#b
The effective Hamfltonian is now H ((Q([)) = S ((Q([)) + E ((Q([)) where the "energy"
E is given by equation (10) without the Wiener measure. Thus
H #
£ £ l~
6j (Q~(~ + £ £ (q 6) IQ((~ (19)
q~k a#b
~~~~~'~~ ~
q a
This is now the transformed Hamiltonian which enables us to calculate the scattering.
4. Scattering from a randomly sequenced copolymer melt.
We go now back to equation (6) where we use for the averaged weight the Hamiltonian calculated above;
s(q) = Em / fl fl bQ~[p~
p~ eH((Qll)) (20)
q q -q
n - 0 o=i a=i
We can write p( = Q(( as earlier. The calculation h simple and straightforward and we find
~~~~ ~~~~~~~°~ @~ V0 2f ~~~~
If we had taken into account the three-body interactions we would have obtained (in a RPA form)
1
2
()q,
0) ~ ~° ~~ ~ '°~° ~~~~
where po is the average density of the melt.
5W JOURNAL DE PHYSIQUE II N°6
Note that S((q,o)is proportional to the ordinary Debye function. Note further that the scattering
law is modified by a factor of1/2 1n the first term, and that the disorder counts twice. At a certain disorder 6 = )vo the second virial coefficient is cancelled and one can expect a e point (due to the disorder). If we neglect the third virial coefficient we find at large disorder 6 > vo/2 a wave
vector q* where the scattering becomes infinite, I.e. q*2 = ~p (2A vo)
,
indicating a dborder induced microphase separation.
5. Discussion.
We have presented a modified form of the random phase approxbnation which b appropriate for random copolymers. We defined new collective variables which are functions of two wave vectors and carry different replica indices. We could represent the ordinary density variables, relevant for the scattering on spechl cases of the new replicated collective coordinates. The result of the
quenched average was a modified formula for the inverse scattering function, which is substantially
different from the annealed average, I.e.
@ S°(q) ~ ~~ ~~ ~ ~~~
as one can expect [9,10].
The disorder hasa strong influence on the second virial coefficient which becomes reduced. More- over, for strong disorder we predict Ulat the scattering function becomes infinite at a certain q*
value which is also determined by the disorder. This microphase separation seems to be similar to that proposed in reference [iii. Note that we studied dense systems such as melts and concen-
trated solutions whereas in [I,iii the isolated chain behaviour has been studied.
Another point has to be made clear. It has recently been pointed out that there are deviations from the RPA in general [6,12] ifsingle chain properties in the melt are studied. For example in
di-block copolymer melts peculiar effects such as gradual stretching of the molecule have been found before the system undergoes the microphase separation. Such effects are beyond the RPA and have to be studied with different methods [6,13,14]. Nevertheless we find [14] that if one is interested in collective properties such as scattering functions, collective diffusion etc., the RPA results are sathfactory, whereas if single chain properties in the melt are studied the effective
Hamiltonian has to be calculated. From this all other properties of interest can be obtained [14].
We hope to extend this method developed in [14] for blends and di-block copolymer melts to random copolymer melts in a future publication.
References
II SHAKHNVOVICH E,I,, GUnN A~M,,J Phys.A Math Gem 22 (1989) 1647.
[2] GAREL I, ORLAND H., Eumpfiys, Len. 6(1988) 307.
[3] FREDRICKSON G,R, Macromolecules 22(1989) 2746,
[4] ME2ARD M., PARISI G., VIRUOROM.V, "Spin glass theory and beyond" (World Scienlific, Singapore)
1988.
[5j Doi M., EDWARDS S.E, "The theory ofpolymerdynamics" (OxfordUniversity Press, Oxford) 1986.
[fl ViLGts IA., BOFSAJJ R., Macmmolecules 23 (1990) 31#
[7j OLVERA DE LA CRtJz M,, SANCHES1,, Macromolecules19 (1986) 2051.
[8] ViLGis IA., BENMOUNA M., BENOrr H., subnutted to Macmmolecuks.
[9] SouRLAs N., Eumpfiys. Lent 3(1987~ 1007.
[10] CRISANU A~, et aL,I Phys. A23 (1990) 3083.
[III SHAKHNVOVICH E-I-, GUnN A.M., L Phys. Frmce 50 (1989) 1843.
[12] FRIED H., BINDER K., preprint.
[13] BRERErON M.G., VILGIS IA,, J Pfiys. France 5o (1989) 245-253.
[14] VILGIS IA~, BENMOUNA M., submtted to Phys. Rev LetL
