Table Of ContentMulti-scale coarse-graining of diblock copolymer self-assembly: from monomers to
ordered micelles
Carlo Pierleoni,1 Chris Addison,2 Jean-Pierre Hansen,2 and Vincent Krakoviack3
1INFM CRS-SOFT, and Department of Physics,
6 U. of L’Aquila, Via Vetoio, I-67010 L’Aquila, Italy
0 2Dept. of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom
0 3Laboratoire de Chimie, Ecole Normale Sup´erieure de Lyon, 69364 Lyon Cedex 07, France
2
Starting from a microscopic lattice model, we investigate clustering, micellization and micelle
n
ordering in semi-dilute solutions of AB diblock copolymers in a selective solvent. To bridge the
a
J gapinlengthscales,frommonomerstoorderedmicellarstructures,weimplementatwo-stepcoarse
grainingstrategy,wherebytheABcopolymersaremappedonto“ultrasoft”dumbellswithmonomer-
8
averaged effective interactions between the centres of mass of the blocks. Monte Carlo simulations
1
of this coarse-grained model yield clear-cut evidence for self-assembly into micelles with a mean
] aggregation number n ≃ 100 beyond a critical concentration. At a slightly higher concentration
t themicelles spontaneously undergo a disorder-order transition to a cubicphase. Wedetermine the
f
o effectivepotential between these micelles from first principles.
s
.
t
a Block copolymers in solution show a remarkable ten- to be mutually avoiding(S). The resulting model will be
m
dency to self-assemble into a bewildering range of dis- referredtoasISS.Themodelisclearlyathermal,sothat
- ordered (liquid-like) and ordered structures, depending temperature will be irrelevant, and the polymer density
d
on the macromolecular composition, the relative sizes of ρ=N/L3 is the only thermodynamic variable (L3 being
n
o theblocks,solventselectivity,polymerconcentrationand the totalnumber ofsites ofthe cubic lattice, the spacing
c temperature [1, 2]. Complex phase diagrams of styrene- aofwhichwillbetakenastheunitoflength). Withtyp-
[ isoprenediblockcopolymersinvarioussolventshavebeen ical polymer sizes M ≃103, and expected micelle aggre-
1 mappedoutexperimentallyingreatdetailbyvaryingthe gation numbers n ≃ 102, very large system sizes would
v above physical parameters, using a combination of light be needed to generate the tens of micelles required to
7 scattering,small-angleneutronandX-rayscatteringand study their ordered and disordered structures.
1
4 rheologicalstudies[3,4]. Afrequent,partialscenariosees The first stage of our coarse-graining procedure, de-
1 the copolymersaggregateinto polydispersesphericalmi- signed to overcome the above bottleneck, is inspired
0 cellesatfairlylow polymerconcentration;uponlowering by earlier work on homopolymers [8, 9]. AB copoly-
6 the temperature or increasingthe concentration,the mi- mers are mapped onto “ultrasoft” asymmetric dumbells
0 celles undergo a disorder-order transition onto a cubic with effective interactions v (R) between the centres of
/ αβ
t lattice. A theoretical understanding of copolymer phase mass (CM) of A and B blocks on different copolymers,
a
m behaviourgenerallyreliesonphenomenologicalconsider- and an intramolecular “tethering” potential φAB(R) be-
ations[5],oronself-consistentfieldtheorysimilartothat tween the CM’s of the two blocks of the same copoly-
-
d appliedearlierto copolymermelts [6]. Amoremolecular mer [10]. Since the original model is athermal, these
n approach is obviously very difficult, in view of the wide effective interactions are purely entropic and are deter-
o range of length scales involved, from the monomer level minedby takingstatisticalaveragesovercopolymercon-
c
: to the mesoscopic scales characterising ordered micellar formations for fixed A-A, A-B and B-B CM distances
v structures. In particular, monomer-level Monte Carlo R. In practice this is achieved by inverting the CM-
i
X (MC) simulations have so far been restricted to short CM intermolecular and intramolecular pair distribution
r copolymer chains [7]. functions gAA(R),gAB(R),gBB(R) and sAB(R). The in-
a
In this letter we propose and implement a two-stage version was carried out in the low density limit, using
coarse graining strategy to investigate the micellization pair distribution functions generated in monomer-level
and disorder-order transition of a simple microscopic MC simulations of two diblock copolymers with 2M =
model of diblock copolymers. The AB copolymers are 500 monomers each, and an exact relation between the
madeupoftwoblocksofequalnumbersM =M =M g (R),s (R) and the corresponding v (R),φ (R)
A B αβ AB αβ AB
of monomers on a cubic lattice. N such chains occupy a [10]. The resulting pair potentials are shown in figure 1,
fraction of the lattice sites. The remaining sites are oc- where they are compared to the corresponding pair po-
cupiedbyanimplicitsolventchosentobeselective(good tentialsforanequimolarbinarymixtureofuntetheredA
solvent) for the B blocks, while it is a θ-solvent for the andBhomopolymers;inthelattercasetheinversionpro-
A blocks. This situation is crudely represented by mod- cedure is trivialin the zerodensity limit wherethe effec-
ellingtheAblocksasidealchains(I)andtheBblocksas tive potentials v (R) and φ (R) reduce to the poten-
αβ AB
self and mutually avoiding walks (S). MoreoverA and B tials of mean force, e.g. v (R)= −k Tlng (R). The
αβ B αβ
blocks of the same or different copolymers are assumed two sets of effective pair potentials v (R) and v (R)
AB BB
2
2.5 0.2 cellization sets in between ρ/ρ∗ = 4 and ρ/ρ∗ = 5, with
2 0.15 the ideal A-blocks forming the dense core, as one might
(R)B1.5 utenttheethreedred 0.1 (R)A ewxhpielectthbeecBa-ubsloecokfstfohremlotwheocrorvoannais,hwinhgereentehregiyr epneenragletyt-,
A 1 0.05 A
v v ically unfavourable overlap is minimized. An unambigu-
0.5 0
ous definition of a micelle remains somewhat arbitrary
0 -0.05 because of the size polydispersity, but we used the op-
2 erational convention that all A-blocks whose centres are
6
situatedwithinadistancer ofanyotherA-blockbelong
R) 1.5 R) c
(B 4 (B to the same cluster. The resulting distributions P(n) of
B 1 A
v φ
0.5 2 0.008
0 0
0 1 2 3 4 0 1 2 3 4 ρ/ρ∗=3
R/Rg R/Rg ) ρ/ρ∗=5
(n0.004 ρ/ρ∗=6
P
FIG. 1: (Colour on line) Effective pair potentials
a)
vAA(R),vAB(R),vBB(R) and φAB(R) for tethered and un-
tethered block copolymers in thezero density limit.
0
0 50 100 150 200
n
are seen to be surprisingly close. As in the case of ho-
3
mopolymers [8, 9] they are roughly gaussian in shape
withanamplitudeofafewk T,andarangeoftheorder
B )c 2
ofthenaturallengthscale,namelytheradiusofgyration R
(
R ofthecopolymer. Inlatticeunits,theMCsimulations P
g
1 b)
yield the following radii for an isolated copolymer with
2M = 500: R = 6.964,R = 11.274,R = 13.889,
g,A g,B g
0
whiletherootmeansquaredistancebetweentheCM’sof 0.5 1 1.5 2
the two blocks is R =20.661. The results for v (R) R /R
AB AB c g
and v (R) in figure 1 suggestthat tethering of the two
BB
blocks does not significantly modify their effective inter- FIG. 2: (Colour on line) Panel a): probability of a cluster of
actions. This is no longertrue for v (R), since this po- nmonomers. Panelb): probabilitydistributionofthecluster
AA
tentialisstrictlyzerointheuntethered(binarymixture) radius Rc.
case,while itis about10%ofv (R)for the copolymer;
BB
thisresidualeffectiveinteractionbetweentheidealblocks aggregationnumbersnareshowninpanelaoffigure2for
∗
is induced by the tethering to non-ideal B-blocks. ρ/ρ =3,5and6,basedonacut-offdistancerc =0.8Rg.
The key assumption made in this work is that these P(n) turns out to be very sensitive to the precise value
∗
zero density pair potentials are transferrable to finite of rc for ρ/ρ ≤ 4 but nearly independent of the cut-off
copolymer densities, i.e. we neglect the density depen- at higher densities, suggesting that well defined micelles
∗
dence of the effective potentials. Earlier experience with appear only beyond an estimated cmc of ρ/ρ ∼ 5. At
homopolymers shows that this is not an unreasonable that concentration P(n) exhibits a broad peak around
approximationwell into the semi-dilute regime ρ/ρ∗ <∼5 n≃70, which narrows and shifts to a mean aggregation
where ρ∗ =(4πR3/3)−1 is the overlap density [9]. number n≃100 for ρ/ρ∗ >∼6. Panel b of figure 2 shows
g
the correspondingdistribution P(R ) ofthe cluster radii
We havecarriedoutextensiveMC simulationsofsam- c
plesof5×103and104 ABcopolymersusingtheeffective (Rc) defined as
∗
pair potentials of figure 1, in the range 3 ≤ ρ/ρ ≤ 10.
1 n 2
The simulations using the coarse-grained representation Rc2 = nX(cid:12)(cid:12)R~iA−R~CM(cid:12)(cid:12) (1)
are at least two order of magnitude faster than simu- i=1(cid:12) (cid:12)
lations of the original all-monomer lattice model. This
allowed us to make detailed investigations, using both whereR~A isthepositionofthei-thA-blockbelongingto
i
sets of potentials in figure 1, in order to ascertain the an n-cluster and R~ the position of the instantaneous
CM
∗
sensitivityofthemicellizationbehaviourtodetailsofthe CM of that cluster. At ρ/ρ = 3, P(R ) is seen to peak
c
effective pair potentials. Both sets of potentials yield re- at a low value of R corresponding to small clusters; the
c
∗
sults in semi-quantitative agreement,and only those ob- distributionbecomesbimodalatρ/ρ =5,signallingthe
∗
tainedwiththeuntetheredpotentialswillbeshownhere. proximity of the cmc, while at ρ/ρ =6, P(R ) shows a
c
Inspection of snapshots of configurations shows that mi- sharp peak around R ≃ R , corresponding to micelles
c g
3
of well defined core size. The peak sharpens and moves
∗
to lower values of R at still larger values of ρ/ρ , in-
c
dicative ofacontractionofthe corewith copolymercon-
centration. The internal structure of the micelles is best 4 8
characterizedbythedensityprofilesρ (R)andρ (R)of
A B
the centres of A and B-blocks around the micelle CM.
∗
The profiles shown in figure 3 for ρ/ρ =7 are averaged 3 6
30 ) R)
R
25 ρA(R) ρB(R) 2 (mm2 4v(mm
g !
1.6
20 1 2
R) 1.2R)
( 15 (
A B
ρ ρ
0 0
0.8
10
2 4 6 8 10
5 0.4 R/Rg
0 0
0 1 2 3 4 FIG. 4: (Color on line) Pair distribution function gmm(R)
R/R ∗
g of the CM’s of the micelles at ρ/ρ = 5 (closed circles), 5.5
(open squares) and 6 (closed squares). The HNC potential
∗
FIG. 3: ρ/ρ∗ = 7. Density profiles of A-blocks (open sym- vm−m(R) at ρ/ρ =5is shown by a continuousline (in red).
Also the corresponding pair correlation is shown as a thick
bols) and B-blocks (closed symbols) with respect to the CM
of each cluster averaged over all clusters with n ≥ 40. Note curve in very good agreement with gmm(R) at ρ/ρ∗ = 5.
thedifferent scales for ρA (left axis) and ρB (right axis). Notethatthepotentialscaleinontherightaxis. Finally,the
snapshots in the upper part of the figure show the x-y plane
projectionofthepositionofthecentreofmassofthemicelles
over all micelles formed during the MC simulations, i.e. ∗
in a typical configuration at ρ/ρ =5 (left-disordered) and 6
over size polydispersity. As expected, ρA(R) peaks at (right-ordered).
the origin and drops rapidly to zero for R > R , while
g
the profile ρ (R) practically vanishes for r < R and
B g
peaks at R≃2Rg, thus defining the corona radius. The tation. Note that at ρ/ρ∗ = 6, an initially disordered
sharp distinction between micelle core and corona evi- micelle configurationevolvesspontaneously to the defec-
∗
dent from profiles at ρ/ρ = 7 gradually blurs as the tive orderedstructure shownin figure 4 after few tens of
copolymer concentration decreases thus signalling that thousands MC cycles.
the micelles dissolve into a quasi-homogeneous solution From the MC generated pair distribution function
of copolymers. g (R)atρ/ρ∗ =5 onecanextractaneffectivemicelle-
mm
A periodic sample of 5000 AB copolymers generated micellepairpotentialusingtheHNCintegralequationof
about 50 micelles, and the second step of our coarse- liquid state theory [11], according to
graining procedure is to consider the structure of the
system of micelles, regarded as well defined spherical v (R)=k T {g (R)−1−c (R)−ln[g (R)]}
mm B mm mm mm
colloids. Figure 4 shows the pair distribution function (2)
g (R) of the CM’s of the micelles, extracted from the where the direct correlation function c (R) is related
mm mm
∗
MC simulations for copolymers concentrations ρ/ρ = tog (R)bytheOrnstein-Zernikerelation[11]. There-
mm
5,5.5and6. Allg (R)gorapidlyto zerofor R<3R , sulting effective pair potential shown in figure 4 exhibits
mm g
indicative of a strong effective short-range repulsion be- a relatively soft repulsion, which culminates at a finite
∗
tween micelles. The structure at ρ/ρ = 5 is typical of value at r = 0, followed by a shallow attractive well. A
that of a dense colloidal fluid, but it is considerably en- precise value for the potential at the origin is difficult to
∗
hancedatρ/ρ =6,withtheamplitudeofthemainpeak obtainbecauseofthenoiseing atshortdistance. As-
mm
∗
jumping from 1.7 (at ρ/ρ =5) to over 3 while the split sumingagaussian-likecore,afittothedataintherange
secondpeak ing (R)is characteristicofa cubic struc- 0.5 ≤ r ≤ 3.5 provides v (0) ≃ 12k T. The accuracy
mm mm B
ture. The sudden change in pair structure is suggestive of the inversion procedure is vindicated by a direct MC
of a disorder/order transition, and snapshots of micelle simulationofmicelles(representedaspointparticles)in-
configurations, shown in figure 4 confirm this interpre- teractingviathisv (R): theresultingpairdistribution
mm
4
functions,alsoshowninfigure4,isindistinguishablefrom sight into clustering and micellization, as a consequence
the g (R) generated in the original simulation of the ofthecompetitionbetweenentropyandlowenergyover-
mm
underlying copolymer system, signalling the success of lap penalties [15, 16, 17].
the second step in our hierarchical coarse-graining pro-
C. Addison wishes to thank the EPSRC for a stu-
cedure. Potentials like v (R) shown in figure 4, which
mm dentship. Part of this work was carried out while C.
remain finite as R → 0, can give rise to a freezing tran-
Pierleoni was a Schlumberger Visiting Fellow at Cam-
sition at sufficiently high densities, as shown, e.g., in the
bridge, and he wishes to thank SCR for their support.
case of the gaussian core potential [12, 13]. The over-
The authors are grateful to A.A. Louis for useful discus-
all scenario described above is also observed when the
sions.
low density effective potentials v (R) for the tethered
αβ
AB copolymer are used (rather than the untethered bi-
nary mixture potential, cfg. figure 1), except that the
cmc and disorder-order transition are shifted to higher
∗
densities (ρ/ρ ≃ 6 and 7 respectively). Moreover the [1] T.P. Lodge, J. Bang, Z. Li, M.H. Hillinger and Y. Tal-
A-block density profiles(similar to those shownin figure mon, Faraday Discussions 128, 1 (2005), as well as sub-
3), now develop a minimum at R = 0 at higher densi- sequent papers published in those Proceedings
[2] Ausefulcompilation ofexperimentalandtheoreticalpa-
ties, so that the micelles tend to become “hollow” and
pers on block copolymer self-assembly is contained in:
increase in size. This is easily understoodfrom the weak
I.W. Hamley, “Block Copolymers in Solution” (Wiley,
short-range repulsion in vAA(R) shown in figure 1. Chichester, 2005).
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[5] for a recent example see: E.B. Zhulina, M. Adam, J. La
chainsaremappedontoultrasoftdumbells,andtheinter
Rue,S.S.SheikoandM.Rubinstein,Macromolecules38,
and intramolecular potentials are determined by invert-
5330 (2005).
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Macromolecules34,1881(2001);Y.Termonia,J.Polym.
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Science B 40, 890 (2002).
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(1994).
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quantitativeinvestigationofmicelleordering. Ourcoarse J. Chem. Phys. 114, 4296 (2001).
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transferability of the low-density effective potentials to Louis, Molec. Phys. 103, 3045 (2005).
[11] seee.g.J.P.HansenandI.R.McDonald,“TheoryofSim-
semi-dilute copolymer solutions, which we are presently pleLiquids”2ndedition(AcademicPress,London,1986).
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(1997).
tureeffectsmaybeeasilyincludedbyintroducinginterac- [13] A. Lang, C.N. Likos, M. Watzlawek and H. L¨owen, J.
tionsbetweenadjacentmonomersonthelattice[14]. The Phys. Condens. Matter 24, 5087 (2000).
relatedsymmetricISImodelinvestigatedin[10],whereA [14] Forarecentapplicationtohomopolymersolutionsee: V.
andBareideal, butmutually avoiding,exhibits a lamel- Krakoviack, J.P. Hansen and A.A. Louis, Phys. Rev. E
67, 041801 (2003).
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[15] C.N.Likos,A.Lang,M.WatzlawekandH.L¨owen,Phys.
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Finally we believe that coarse-graining provides new in- C.N. Likos, to appear in Phys.Rev.Lett. (2006).
