Ordered Phases of Diblock Copolymer Micelles in Selective Solvent

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by  Gregory M. Grason

Ordered Phases of Diblock Copolymers in Selective Solvent
Gregory M. Grason
Department of Physics and Astronomy, University of California at Los Angeles, Los Angeles, CA 90024, USA
(Dated: February 2, 2008)
We propose a mean-field model to explore the equilibrium coupling between micelle aggregation
and lattice choice in neutral copolymer and selective solvent mixtures. We find both thermotropic
7 and lyotropic transitions from face-centered cubic to body-centered cubic ordered phases of spher-
0 ical micelles, in agreement with experimental observations of these systems over a broad range of
0 conditions. Stabilityofthenon-closedpackedphasecanbe attributedtotwophysicalmechanisms:
2 thelarge entropy of lattice phononsnear crystal melting and thepreference of the inter-micelle re-
pulsionsforthebody-centeredcubicstructurewhenthelatticebecomessufficientlydenseathigher
n
solution concentrations. Both mechanisms are controlled by the decrease of micelle aggregation
a
and subsequent increase of lattice density as solvent selectivity is reduced. These results shed new
J
light on the relationship between micelle structure – “crewcut” or “hairy” – and long-range order
3
in micelle solutions.
2
PACSnumbers:

]
t
f
o I. INTRODUCTION scale – as in Yukawa systems – energetic considerations
s
alone show that the generic preference for close-packed
.
at structures at low particle densities gives way to a pref-
Well known for their ability to self-assemble into a
m erence for bcc at high densities which persists even at
seemingly endless array of nanoscale structures [1, 2],
zero temperature [14, 15, 16]. Indeed, the bcc lattice
- neutral block copolymer and solvent mixtures have re-
d is well-known to be the three-dimensional ground state
n centlyreceivedsignificantattentionasatestbedforphe- for Coulomb repulsions, when interactions exist over an
o nomena of a more classical nature: the crystallizationof
infinite range [17, 18].
c spherical particles [3, 4, 5, 6, 7, 8, 9, 10, 11]. In se-
In a series of systematic studies, Lodge and coworkers
[
lective solventcopolymer chains aggregateinto spherical
identified fcc-to-bcc transitions in copolymer micelle so-

2 micelles–amongotherstructuralmotifs–protectingthe lutionswhichareboththermotropic [5,8]–drivenbyin-
v insoluble polymer blocks within a core domain while al-
creasingtemperature,T –andlyotropic[6,7]–drivenby
9 lowing the soluble blocks to dissolve in a diffuse coronal
increasing polymer concentration, Φ. The former tran-
2 domain. At low concentrations when inter-micelle con-
sition scenario might suggest an entropic stabilization of
4
tact is rare, correlations are fluid-like. However,at suffi-
7 the bcc structures due to lattice fluctuations, while the
cientlyhighconcentrationstheswollencoronalbrushesof
0 latter scenario seems to implicate a density-driven tran-
neighboring micelles come into contact, and to minimize
5 sitioninwhichpositionalentropydoesnotplayacrucial
0 the repulsive effect of this contact, the micelles adopt role. In this article, we construct a mean-field model to
/ lattice configurations. Based on a series of experimental
t probe the relationship between the thermodynamics of
a studies of lattice-forming copolymer micelle solutions, a
micelle aggregationand the physical mechanisms under-
m basic thermodynamic picture has emerged: generically,
lyingstructuralphasetransitionsofmicellesincopolymer
- so-called“crewcut”micellescharacterizedbyshortcoro- solutions at fixed concentrationand temperature. While
d
nal brushes favor the face-centeredcubic (fcc) structure,
n simulations of ordered structures in star-polymer solu-
while“hairy”micelleswithlargercoronaefavorthebody-
o tions [19, 20] performed at fixed aggregation number, p,
centered cubic (bcc) lattice arrangement[3, 4, 6].
c and number density, n, shed some light onto the related
:
v At low temperatures, many repulsive particle systems problem of copolymer micelles in solutions, the thermo-
i favor close-packed lattice structures – such as fcc – pri- dynamicsofthepresentsystemhasthecomplicationthat
X
marilyduetothemoredistantnearestneighborsandsub- intra-micelleenergeticsmustbeinequilibriumwithinter-
r sequent weaker repulsions in these lattices. As tempera- micelle effects. That is, in copolymer solutions as Φ and
a
tureisraised,theentropiccontributiontothefreeenergy T arevaried,thesystemsimultaneouslyadjuststhemean
from lattice fluctuations eventually becomes comparable values of n and p as well as the structural organization
to repulsive contributions. Due to a relative abundance of micelles.
ofsoft-phononmodesofbcclattice,thisstructureispar- To address this issue, the present model combines
ticularlystableneartothecrystalmeltingtransition[12]. treatments of good and bad solvent polymer domains
Indeed, quite genericconsiderationsuggeststhatthe bcc with an effective theory for the free-energy cost – from
structure is always preferred near to the lattice melting micelleinteractionsandlatticeentropy–ofconfiningmi-
transition due to its large entropy [13]. Entropic consid- celleswithin alattice. Therefore,weaddress,ata mean-
erations,however,arenotnecessaryto drivea fcc-to-bcc field level, how the thermodynamics of the internal de-
structuraltransitioninsystemsofpurelyrepulsiveparti- grees of freedom of the constituent “particles” of this
cles. When repulsions are characterized by some length system, the micelles, drive the changes in inter-micelle
2
order observed on larger length scales. The principle able solvent contact only at the interface separating the
consequence of increasing temperature in these systems coreandcoronaldomains. Microscopicsolvent-monomer
is to decrease the solvent preference for coronal polymer interactionsgiverisetoasurfacetension,γ,forthe core-
domain leading to a decrease in the number of chains corona interface. We generically expect the importance
per aggregated micelle, p [5]. We argue below that the of molecular scale interactions to decrease with temper-
dissolution of micelles with increased T has two effects: ature, and hence, γ to be a decreasing function of T as
to reduce the strength of the effective micelle repulsions is typically the case for example, in immiscible polymer
and to increase the packing density of micelles. Both ef- blends [22]. In this modelthe effect ofsolventselectivity
fectstendtoenhancethestabilityofthebccstructureas will be encoded entirely in γ. The free energy of an iso-
solvent selectivity is decreased, although the two mech- latedmicelle configuration,F ,is the sumofthreeparts,
0
anisms are of qualitatively different origins. In the first
case,occurringatlowΦ,latticeentropystabilizesthebcc F0 =Fc+Fint+Fb , (1)
phase for sufficiently weak micelle interactions, while in
the secondcase,athigherconcentration,itis the micelle where Fc and Fb are the free energies of the core and
potentialitselfwhichprefersthebccstructureoverfccat coronal brush domains and Fint is the free energy cost
highmicelledensities. Varyingtherelativelengthsofthe of exposing the core domain to solvent along the inter-
coreandcoronalpolymerblockswithinthemodel,wein- domain surface.
vestigate the stability of orderedfcc and bcc phases as a Consideranisolatedmicellecomposedofptotalchains
function of the chain architecture. While the “crewcut” showninFigure1. The coredomainextends toaradius,
vs. “hairy”micelle distinction roughlydescribes ordered Rc, while the tips of the coronal brush reach to a radial
phase behavior, we find that the relative sizes of corona distance,R0. Becausesolventisabsentfromthecore,the
and core do not wholly determine lattice choice. Both polymerblocksthatconstitutethisdomainaredescribed
thermotropicandlyotropicfcc-to-bcctransitionsarepre- by the statistics of the molten polymer brush [23]. As
dicted over a broad range of polymer architectures. such, the free energy of this domain can be calculated in
This article is organized as follows. In the Sec. II we thestrong-stretchingapproximationforspherical,molten
construct a mean-field model for the free-energy of the polymer brushes [24],
internalstateofthemicelles,aswellasthecostoflattice
confinement. We present results for this model in Sec. Fc = 3π2p Rc2 , (2)
III and conclude with a brief discussion in Sec. IV. k T 80 N a2
B c
where a is the mean monomer-monomer distance.
Since excluded-volume interactions between chains are
II. FREE ENERGY MODEL FOR LATTICE
screened in the molten polymer state, the free energy
MICELLE PHASES
of this domain is purely due to the entropic (Gaussian)
costofextending the chains inthe brush. Giventhe sur-
In this section we construct an effective mean-field
facetensionbetweencoreandcoronalbrushdomains,we
modeltocomputethefreeenergyoflatticemicellephases
ofcopolymersolutionsatfixedtemperatureandpolymer
fraction, Φ. We first consider the energetics of isolated
micellesinsolutionbeforeturningtotheinter-micelleef-
fects. Note thatthe model forisolatedmicelle energetics
presentedherehasbeenrecentlyanalyzedindetailinthe
limitoflowconcentration(intheabsenceofinter-micelle
effects) by Zhulina et al. [21]. We present a complete
discussion here for the purpose of elucidating the results
ofthe full calculation,whichtakesintoaccounteffectsof
lattice confinement.
A. Internal Micelle Configurations
We consider a solution of diblock copolymers of N to-
tal monomers of volume fraction Φ. The lengths of the
insoluble core and soluble coronal blocks are Nc and Nb FIG. 1: A cartoon of an isolated micelle in solution. Solvent
respectively. Temperature dependence enters the model is excluded from a molten polymer core of radius, Rc. A
through the selectivity of the solvent for the coronal dissolve coronal brush extends from Rc to a radius of R0. In
brush domain over the core domain. Here, we assume the Daoud and Cotton model (see text) this brush can be
thatsolventisexcludedfromthecoredomainentirely,so divided into correlation “blobs” of size, ξ, determined bythe
thattheinsolublepolymerblocksareexposedtounfavor- distance between neighboring chains.
3
have,
F =4πγR2 . (3)
int c
Treatingthecoreblockashydrophobic,weexpectthatγ
is on the order of 10 mN/m [11], which in thermal units
corresponds to γa2/k T 0.5, assuming a segment size
B
of a=5˚A. Note that from≈the volume constraint on the
core we have,
4πR3ρ
p= c 0 , (4)
3N
c
where ρ is the monomer density.
0
The coronal domains of the micelles are under good-
solventconditions,leadingtoadramaticswellingofthese
blocks in comparison to the core domain. We treat the
FIG. 2: The solid line shows the crossover in the preference
free energy of this domainwithin the Daoudand Cotton
of U(r) for the bcc and fcc lattice structures as function of
modelof“star-like,”sphericalpolymerbrushes[25]. This micelle aggregation, p, and reduced density,η=(π/6)nσ3.
model has been studied extensively elsewhere [26, 27];
we present a detailed discussion here since the physics
of polymer corona has important implications for the
whereasfor“crew-cut”micelles,whenR R ,wehave
∗ c
micelle-micelle interactions. Under good-solvent condi- R (N p)1/3. ≪
0 c
tions,thepropertiesofasemi-dilutepolymersolutionare T∼he free energy of the good solvent domain is com-
governedlargely by a single length scale, ξ, the so-called
puted by counting the total number of these thermal
blob size, which is determined by the mean distance be- blobs, each of which represent a k T worth of energy.
B
tween neighboring chains [23]. At shorter length scales
This is most conveniently accomplished by integrating
thanξ the correlationsofthe chainsobey the Floryscal- the blob density, ξ−3(r), over the coronal domain vol-
ing, g3 ρ ξ5/a2, where g is the number of monomers
0 ume, from which it can be shown,
≈
per blob. For a spherical brush of p chains, the aver-
age lateraldistance between neighboring chains is deter- F p3/2 R
b 0
mined by the geometrical condition that 4πr2 = pξ2(r) kBT = √4π ln(cid:16)Rc(cid:17) . (9)
(see Figure 1) so that,
Intheabsenceofanyinter-micelleeffects,thescalingbe-
4π havior of the mean aggregationnumber can be obtained
ξ(r)= r . (5)
r p by considering the “hairy” and “crew-cut” limits of F0
[27]. When R R , the free energy of the coronal
0 c
The local concentration of monomer in the dissolved brush,F ,domina≫tes overthe corestretchingenergy,F ,
b c
brushisdeterminedfromφb(r)=g(r)/ξ3(r) p2/3/r4/3. so that p γ6/5N4/5. In the opposite crew-cut limit,
∝ c
Because the total number of monomers in the corona is ∼
F F , the mean aggregation grows as p γN . De-
b c c
fixed, integrating this concentration profile over the vol- ≪ ∼
creasing solvent selectivity (decreasing γ), decreases the
ume of the coronal brush allows us to solve for the size
meannumberofchainspermicelle,yieldingaroughlylin-
of the brush,
ear dependence of chain aggregation on the core-corona
surface tension.
R0
pN =4π dr r2φ (r) . (6)
b b
Z
Rc
B. Micelle Repulsion and Lattice Entropy
Inverting this condition we find that the size of the mi-
celle obeys,
At lowpolymer concentrations,the density ofmicelles
R5/3 =R5/3+R5/3 , (7) is sufficiently small that inter-micelle effects may be ig-
0 ∗ c nored [28]. As Φ is increased well beyond the critical
micelle concentration the coronal domains of neighbor-
with
ing micelles begin to overlap, leading to strong repulsive
125 pN3a2 effects. To minimize the free-energy cost of such repul-
R5 = b , (8)
∗ 108π ρ sions, micelle solutions adopt periodic arrangements, al-
0
ternately bcc or close-packed structures. We consider a
and R∗ is the size of a star polymer of p chains of length lattice of micelles of aggregation number, p, at number
Nb in the limit that Rc 0. Note that in the “hairy” density, n. Given the low solubility of single, unaggre-
→
micelle case, when R R , we have R N3/5p1/5, gated chains in solution [28, 29], we assume a negligible
∗ ≫ c 0 ∼ b
4
(zero) concentration of free chains. Therefore, a mean the bcc or fcc structures as a function of p and the re-
micelle aggregation number of p implies a micelle den- duced density, η (π/6)nσ3. At low densities, η < 0.7,
sity, micelles in either≡lattice interact via the long-dis∼tance
portion of the potential. In this regime, lattice geom-
ρ Φ
n= 0 . (10) etry dictates that the potential energy of fcc structure
pN is exponentially smaller than that of the bcc structure
due to its more distant nearest-neighbors. From studies
Notethatasmicellesdissolveandpdecreases,themicelle
of Yukawa systems [14, 15, 16], it is known that the bcc
densityincreasestomaintainafixedpolymerfraction,Φ.
structureisgenericallyfavoredathighdensities,andthis
When neighboring micelles begin to overlap at high
densities (i.e. n > (2R )−3), we must account for the fact accounts for the p-dependent crossover in preferred
0
structure shown in Figure 2 for low aggregationnumber
repulsive effects b∼etween coronal brushes. By a scaling
(p < 100). Near to η 0.7 nearest-neighboring micelles
analysis,Witten andPincus showedthat the pairpoten- rea∼chtheshort-distanc≈e,logarithmicregionsofU(r),and
tial for star-like,sphericalbrushes exhibits a logarithmic
the crossoverdensity becomes only weaklydependenton
interaction at short distances [26]. Physically this is jus-
p. Thecrossoverfrompreferredfcctobccstructureswith
tified by the demand that the free-energy cost of bring-
increaseddensityisarathergenericpropertyofrepulsive
ing two micelles ofp chains eachto a separationof order
systems, qualitatively insensitive to the precise form of
2R , should be roughly the same as the free energy of
c
the inter-micelle potential.
creating a single micelle of 2p chains (see eq. (9)). On
We compute the lattice entropy of competing struc-
these grounds, Likos and coworkersproposed the follow-
turesbyconsideringtheharmonicfree-energycostofdis-
ingformofthepairpotentialforstarpolymersofpchains
placingmicellesby∆rα fromtheirequilibriumpositions,
[30],
r ,
α
ln r + 1 r σ
UkB(rT) = 158p3/2 − σ/(cid:0)rσ(cid:1) e−1(+r−σσ/ξ)/mξamxax r >≤σ . (11) Hfluct = 12αXβ;ij∆riαDiαjβ∆rjβ , (14)
1+σ/ξmax
 wheretheiandjlabelCartesiandirections,αandβlabel
Atshorterseparationsthanasoft-coresize,σ,the repul-
lattice sites and Dαβ is the dynamical matrix computed
sion has the Witten and Pincus form. Since the micelles ij
from,
have a finite size, at separations beyond σ, U(r) has a
Yukawa form, which falls off exponentially on a length
∂2
scalesetbythe largestblobsize, ξ (seeFig. 1). Note Dαβ = U(r r ) , (15)
thatboththepotentialandforcearmeacxontinuousatr =σ. ij ∂riα∂rjβ γX6=δ | γ − δ|
Thecrossoverinthepotentialisdeterminedbytheradial
separation at which the outermost blobs of neighboring with ∂/∂rα the derivative with respect to the i compo-
i
micelles fully overlap. From this we have, nent of the position of the αth lattice site. This matrix
canbebediagonalizedinFourierspacetofindtheeigen-
σ = 2R0 , (12) values, λi(k), associated with the three phonon polar-
1+ π/p izations at wavevector k. In the harmonic approxima-
p tion [31] the equilibrium mean-square displacement can
and be computed by
π
ξmax =rpσ . (13) ∆r2 = kBT 3 1 , (16)
h| | i N λ (k)
This pair potential has been shown to agree well with m Xi=1k∈X{G} i
scattering experiments in dilute solutions of copolymer where G is the set of reciprocal lattice vectors corre-
micelles [10, 11] and star polymers [30]. { }
sponding to a lattice of N = nV micelles. The free-
m
To compute the free-energy cost of coronal overlap
energy contribution (per micelle) from positional fluctu-
for micelles in the various structures, we perform lat-
ations in the harmonic approximation is computed by,
tice sums of the potential in eq. (11) numerically. Be-
causeU(r)fallsoffexponentiallywithdistance,thesums F 1 3 λ (k)
are truncated to include a finite number of lattice neigh- fluct = ln i . (17)
k T 2N k T
bors, yielding a numerical precision of better than one B m Xi=1k∈X{G} (cid:16) B (cid:17)
part in 108. Unlike the case of pure Yukawa systems
which are characterized by a single length scale, the Although the potential in eq. (11) and its first deriva-
inter-micellepotentialdependsexplicitlyontwoseparate tivearecontinuousoverthefullrange,thesecondderiva-
length scales, namely, σ and ξ . To analyze the influ- tive is discontinuous at r = σ, which would enter the
max
ence of the inter-micelle potential on equilibrium struc- computed phonon spectrum through eq. (15). Hence,
ture,inFigure2weplotthepreferenceofU(r)foreither the form of U(r) proposed in ref. [30], would lead to
5
certain anomalies in the dependence of the fluctuation
contribution to the free energy on micelle density, n. To
circumvent this issue while maintaining some of the de-
tailed dependence of the vibrational spectrum on the
inter-micelle potential, we use only the long-distance,
Yukawa form of U(r) – even down to r σ – when
≤
computing the dynamical matrix. We expect that this
becomes a poor approximationwell above the density at
which the nearest-neighborseparationreachesσ. Hence,
inthismodelthevibrationalpropertiesofthemicellelat-
tices will be identical to those characterizedby Robbins,
Kremer and Grest for lattice phases of repulsive Yukawa
systems [15, 16].
To calculate the dependence ∆r2 and F on p
fluct
h| | i
and n, we compute Dαβ using eq. (15) and the long-
ij
distance form of U(r), numerically performing lattice
sums. The eigenvalues λ (k) are computed, and then FIG.3: Equilibriumphasebehaviorforsolutionofsymmetric
i
the Fourier sums in eqs. (16) and (17) are carried out copolymers Nc =Nb =50. The regions of stable fcc and bcc
by summing overafinite setofreciprocallattice vectors. phases are indicated, as well as the region where the crystal
structures are unstable to the micelle fluid phase as deter-
More than 300,000 reciprocal lattice points are used to
achieve a numerical convergenceof ∆r2 and F to minedbytheLindemanncriterion (seetext). Allphasetran-
better than one part in 104. h| | i fluct sitionsarefirstorder. Portionsofthefcc-bccphaseboundary
are labeled according to thephysical mechanism which dom-
When micelle interactions are sufficiently weak, lat-
inates thethermodynamics as discussed in thetext.
tice fluctuations will lead to a melting of the ordered
phases. To assess this instability to the fluid micelle
phase,weapplytheLindemanncriterionforcrystalmelt- lution conditions, the intensive free energy per chain,
ing, which defines the melting of lattice order to occur F /p, is minimized over micelle aggregation number,
tot
when ∆r2 1/2/rnn αm, where Rnn is the nearest- p. Mean-fieldfreeenergycalculationsfororderedfccand
h| | i ≥
neighbor lattice spacing and αm is a numerical constant bcc micelle structures are compared to construct phase
less than 1 [32]. Here, we use a Lindemann parame- diagrams. We beginwith a detaileddiscussionofmicelle
ter of α = 0.19, which was found to yield good agree- thermodynamics for solutions of symmetric copolymer
mentwiththe Hansen-Verletcriterionformelting [33]in chains.
numerical studies of repulsive Yukawa systems [16]. As
wasfoundinthese studies,the soft-phononmodesofthe
bcc crystalin the harmonic model are completely unsta- A. Symmetric Chains
ble to thermal fluctuations for sufficiently low densities
(i.e. nξ3 <0.002), and therefore, the model discussed
max In Figure 3 we show the predicted phase behavior
above canno∼t be used to compute the vibrational prop-
of copolymer solutions for symmetric molecules with
erties of the bcc structures at these densities. In this
N = N = 50. At sufficiently large core-corona surface
c b
regime, we use an extrapolation of Ffluct from the sta- tension, γa2/k T > 0.08, and polymer volume fraction,
B
ble bcc phase at higher densities to model the entropic Φ>0.05,themicel∼lesolutionordersintoeitherthefccor
contributionstothe freeenergyofthelowerdensityfluid bc∼cstructures. Theformerstructureispreferredatlower
phase.
Φ, while the stable bcc structure is predicted at higher
polymer fractions and atlower γ, before crystalmelting.
The topology of the mean-field phase diagram for the
III. MODEL RESULTS AND MICELLE
ordered micelle phases is consistent with experimental
SOLUTION THERMODYNAMICS
observations of symmetric diblock solutions, excepting
here that the temperature dependence of solvent quality
In this section we discuss the mean-field results of the is encoded in γ so that direct comparison to experiment
model for copolymer micelle solutions introducedin Sec. is not possible at present. Nonetheless, the predicted
II. Calculationsareperformedforfixedpolymercontent, fcc bcc fluid micelle phase sequence with increasing
Φ, and polymer composition, Nc and Nb. Temperature tem→perat→ure is a generic feature observed for a variety
dependence of the solvent selectivity is encoded in the of neutral copolymer solutions [6, 7, 8]. Moreover, we
core-coronasurface tension, γ. For simplicity we assume note that many systems (though not all) exhibit the fcc-
−1/3
that a=ρ . The total free energy (per micelle), F bccstructuraltransitionwithincreasingconcentrationas
0 tot
is the sum of internal micelle free-energy contributions predicted by the mean-field model (see. e.g. ref. [7]).
– eqs. (2), (3) and (9) – along with the inter-micelle Toputthethermodynamicsoflatticechoiceinthecon-
free-energy contributions, U and F . For fixed so- text of equilibrium micelle structure we plot the depen-
fluct
6
FIG. 4: Mean-field values of aggregation number, p, (solid
line)andreduceddensity,η=(π/6)nσ3 forsolutionsofsym-
metricchains(Nc =Nb =50) ofpolymerfraction Φ=0.175.
The arrow at γa2/kBT = 0.222 indicates the location of the
bcc-fcc phase boundary. Hence, for values of surface tension
smaller(larger)thanthistheplottedquantitiescorrespondto
theequilibrium bcc (fcc) structure.
dence of mean aggregation and reduced density as func-
tionsofthecore-coronasurfacetensioninFigure4. Since
γ isassumedtobeadecreasingfunctionofT,thisphase-
space trajectory describes a thermotropic fcc-bcc transi-
tion. Here,weseethatphasaroughlylineardependence
on γ, not unlike the case where inter-micelle effects are
ignored [27]. In the range γa2/k T >0.12, η is a mono-
B
tonically decreasing function of γ. This stems from the
factthatmicelledensity,n,isinverselyproportionaltop,
whileatlargeaggregationnumbersσ3 R3growsslower
than linearly with increased p (for ex∼amp0le, R3 p3/5
0 ∼
for hairy micelles). Hence, we have the generic property
thatatlargeaggregationnumbersincreasingcore-corona
repulsionmakesthe systemlessdense onthe scaleofthe FIG. 5: Differences between bcc and fcc the total, ∆Ftot,
intermicellerepulsion,σ. Indeed,theriseindensitywith interaction ∆U, and fluctuation, ∆Ffluct free energies of so-
increased temperature is observed in scattering experi- lutions of symmetric chains (Nc = Nb = 50) as a functions
of core-corona surface tension. The solid line corresponds to
mentsfrommicelle latticesinsolution[8]. Fromeq. (12)
results at Φ = 0.175 while the dashed line corresponds to
we have σ R p1/2 for small p, thus η decreases to 0
∼ 0 Φ=0.125.
as p 0, whichaccounts for the decreaseof the reduced
mice→lle density as γa2/k T is decreased below 0.12. For
B
a broad range of conditions in this model, when p is re-
ducedbelowroughly20chainspermicelle,orderedstruc- der to discriminate between these two distinct physical
tures are unstable to the fluid phase, in agreement with mechanisms for the thermotropic fcc-to-bcc transition –
the order of magnitude predictions for the critical chain driven by either lattice potential or lattice entropy – we
number for ordering in solutions of star polymers [34]. plot the difference between F , U and F (per poly-
tot fluct
In principle, both the results that the system effec- mer chain) for the bcc and fcc structures as a function
tively becomes moredense andthe number ofchainsper of γ in Figure 5. As is the case for pure Yukawa systems
micelledecreaseswithdecreasingcore-coronasurfaceten- [16], the relatively large entropy of the fluctuations in
sionpointtothebccstructureasthestablemicellelattice the bcc structure always favors that structure over fcc,
at high temperatures. The former effect drives an inter- so that ∆F <0 for all conditions. However, this en-
fluct
micelle potential preference for bcc (see Fig. 2) while tropicpreferenceamountstoroughly10−3k T perchain,
B
the latter effect suggests that lattice fluctuations stabi- which should be compared to differences in lattice po-
lize this phase as micelles dissolve, since the strength of tential, which tend to vary on the scale of 10−2k T per
B
effective micelle repulsions is proportionalto p3/2. In or- chain.
7
reduced density, η∗ 0.7, above which the long range of
≃
inter-micelle repulsion leads to a stabilization of the bcc
structure. Since n is linearly proportional to the poly-
mer fraction,increasingΦ allowsthe systemto passinto
the bcc-preferred region, η > 0.7 at larger aggregation
number, or alternatively, larg∼er γ. This accounts for the
Φ-dependence of the fcc-bcc boundary. For Φ 0.160,
≤
the weakening of the micelle potential with the loss of
chain aggregation supersedes the density-driven transi-
tion,leadingtotheentropicstabilizationofthebccphase
near to the melting point. Therefore, we find that the
interaction-drivenfcc-to-bcc transition tends to occur at
higher Φ, with the entropically-driven structural transi-
tion occurring at lower polymer concentrations.
Having analyzedthe thermotropic transitionin detail,
FIG. 6: Dependence mean aggregation on Φ for symmetric we now briefly consider the case of fixed γ. Clearly, the
copolymersolutionsatfixedγa2/kBT andfixedbccstructure. discussion above implies the low-Φ stability of the fcc
structure and the subsequent stability of bcc at higher
concentrations due to the micelle interactions. However,
Consider first the lower polymer fraction case, where the predicted fluid bcc fcc bcc phase sequence with
Φ = 0.125. Over the range of γ plotted in Figure 5, increasingΦ predic→ted fo→r the→range0.105<γa2/kBT <
∆U is always positive indicating that interactions fa- 0.265,implies anunusual thermodynamic dependence of
vor the fcc structure for this polymer fraction. Below micelle aggregation on Φ (shown in Figure 6). While at
γa2/k T = 0.105, however, the bcc structure is ther- low concentrations p will be insensitive to Φ, when mi-
B
modynamically stable over the close-packed lattice until celles begin to interact strongly the system adjusts the
melting for γa2/k T 0.067. In this regime of weak micelle structure in order to reduce the free energy cost
B
≤
core-corona repulsion, mean-aggregation of micelles is ofmicelleoverlap. Intheconcentrationregimewheremi-
sufficientlysmall(p<20)thatdifferencesinF which celles interactvia the Yukawa portion of U(r), the mean
fluct
favor the bcc lattice∼become comparable to the prefer- aggregation increases roughly linearly with Φ in order
ence of micelle interactions favoring the fcc structure. tomaintainadensitywherenearest-neighbormicellesdo
Hence, we view this thermotropic fcc-to-bcc transition not overlap strongly. Once the system reaches a den-
to be of entropic origin,resulting from the abundance of sity where nearest neighbors begin to interact via the
soft-phonon modes in the bcc crystal. short-distance, logarithmic portion of U(r), this trend
We now turn to the thermotropic fcc-to-bcc transition reverses and aggregation then decreases with increased
at Φ = 0.175 shown also in Figure 5. In contrast to Φ. This effect not only increases the density of the lat-
the lower concentration case we find a high temperature ticebutalsoreducesthestrengthofrepulsions. Thenon-
region, 0.057 γa2/k T 0.179, wherein micelle inter- monotonicdependenceofponΦshowninFig. 6explains
B
actions prefer≤the bcc str≤ucture over fcc. Indeed, from there-entrantbccphaseathighconcentration,sincethis
Figure 4, we find that the system becomes more dense structureis genericallypreferredathighdensity andlow
as γ is lowered, so that η > 0.7 in this region. While p. This unusual result that p first increases with Φ and
thegenericpreferenceofF ∼ forthebccstructuremay then decreases, is a fundamental consequence of the fact
fluct
further stabilize this lattice over the fcc structure, the that U(r) is finite-ranged and “ultra-soft” meaning that
plot of ∆U for this concentration demonstrates that the it should remain finite down to very small separations.
differences in micelle interactions dominate the entropic To prevent significant overlap, the system first slightly
contributionto the freeenergyovermuchofthe rangeof increases p, reducing density. Once strong overlap be-
the stable bcc lattice (note that the bcc lattice melts be- comes unavoidable, it becomes preferable to decrease p,
low γa2/k T =0.077 for this polymer fraction). There- since the free-energy cost of fusing two micelles cannot
B
fore, in this higher concentration regime the structural besignificantlylargerthancreatingasinglemicelleof2p
transitionis more appropriatelyattributedto micelle in- chains.
teractions which favor the bcc structure as the system
becomes effectively more dense.
Micelle aggregation is a decreasing function of tem-
perature,whichinturn weakensmicelle interactionsand
B. Copolymer Composition: “Hairy” vs.
increases the number density of micelles. In Figure 3,
“Crew-Cut” Micelles
crossover from entropy-driven to interaction-driven fcc-
to-bccstructuraltransitionsismarkedbyacuspinphase
boundary at Φ = 0.160 and γa2/k T = 0.120. For mi- We now consider the influence of copolymer composi-
B
celles with p>100, there is effectively a critical value of tiononthepredictionsofthemean-fieldmodel. Defining
∼
8
FIG. 7: Mean-field phase diagrams for asymmetric N = 100 copolymer micelle solutions. The phase are labeled as in Figure
3. Qualitatively, the f = 0.3 corresponds to a typical phase diagram for “hairy” micelles, while the f = 0.7 represents the
“crew-cut” micelle case.
f as the fraction of chain which is insoluble, theclose-packedstructure,whilethecorona-richmicelles
favorbcc. Thisphasediagramrevealsthatthere-entrant
N
f = c , (18) bcc fcc bcc lyotropic phase sequence is a feature
Nc+Nb of co→polym→er solutions over a broad range of chemical
compositions. Although the “hairy” vs. “crew-cut” dis-
we show the mean-field phase behavior for micelle solu-
tinction broadly describes ordered-phase thermodynam-
tions of corona-rich, f = 0.3, and core-rich, f = 0.7,
icsofmicellesolutions,wefindthatequilibriumstructure
copolymers in Figure 7. Qualitatively, these two cases
cannot be strictly determined by f (or even H /R ). In
correspondtothe“hairy”and“crew-cut”descriptionsof b c
fact, basic considerations of micelle interactions dictate
micelle structure, respectively. More quantitatively, we
that ordered-phase thermodynamics of micelles must be
can characterizethe micelle structure by the ratio of the
insensitivetothesizeofthecoredomain,R . Micellein-
height,H =R R ,ofthecoronalblocktothecoresize. c
b 0 c
At Φ=0.2 and−γa2/k T =0.2, we have H /R =2.734 teractionsgovernbothrepulsiveandentropicfree-energy
B b c
costs of lattice formation, and from eq. (11) we see that
for f = 0.3 and H /R = 0.768 for f = 0.7. Another
b c
U(r) depends only the number of chains per micelle and
significant difference is the mean aggregation since p is
the effective micelle size, σ. Certainly, this must be the
an increasing function of N : p = 28.5 and p = 188,
c
case, given that the self-similar statistics governing the
forthe“hairy”and“crew-cut”cases,respectively. These
monomer density in the coronal brush domain are inde-
theoreticalphasesdiagramsrecapitulatetheobservations
pendent of the length of coronal chains, N [25]. There-
of Lodge and coworkers that structural order of “hairy” b
fore, since neighboring micelles interact through overlap
(“crewcut”) micelles is principally of bcc (fcc) type [6].
of coronal domains where the monomer distribution is
It is the latter distinction in micelle aggregation for
governed purely by p and r, such interactions cannot
these two cases which accounts for the broad differences
encode information about the relative sizes of the core
in the mean-field phase diagrams in Figure 7. Scaling
domain. Or put another way, at the tips of their coro-
analysis of F suggests a roughly linear dependence of
0
nal brushes micelle interactions cannot “know” how far
mean-aggregrationnumber on the length N , which jus-
c
away the core domain is from the outermost edge of the
tifies the large differences in p for the two cases. Since
a smaller micelle aggregation implies a larger micelle micelle. According to a mean-field description of micelle
solutions, differences in ordered-phase thermodynamics
density, for given solution conditions “hairy” micelles
betweenthe“hairy”and“crew-cut”casesaremoreprop-
are characterized by highly overlapping coronal brushes
erlyattributedtodifferencesinsolvationpropertiesofthe
whilethe“crew-cut”micellestendtoformlessdenselat-
tices. In the low temperature regimes for which (p > 50 copolymersofdifferingcompositions,ratherthantherel-
this implies that micelle interactions tend to stab∼ilize ative size of core and coronal domains.
the bcc lattice in the former case and fcc in the latter.
Clearly,theeffectofmicellestructuredoesnotdrastically
alter the entropic stabilization of the bcc phase near to IV. DISCUSSION
lattice melting.
InFigure8weplotthe fixedγa2/k T =0.2phasebe- In the previous sections we have presented and ana-
B
haviorofN =100copolymersolutionsasafunctionofΦ lyzedamean-fieldmodelforthethermodynamicsofcon-
andf. Hereagain,thecore-richcopolymermicellesfavor centrated solutions of neutral block copolymers. This
9
analysis unifies a broad range of observed phenomena
regarding structural transitions in lattice phases of mi-
celles. In particular, the mean-field analysis elucidates
thephysicalmechanismsresponsibleforphasetransitions
between ordered states of micelles. In contrast to other
classical particle systems, such as charged-stabilizedcol-
loids [14] or even hard spheres [35], modeling the ther-
modynamics of micellar systems has the additionalcom-
plicationthattheinternalstructureofthemicellesthem-
selves establishes equilibrium with the inter-micellar de-
grees of freedom.
Thisfactunderliesthebasicphenomenologyofconcen-
trated solutions of copolymer micelles. As temperature
israisedandsolventselectivitydecreases,themeanvalue
of p decreases. Since the strength of inter-micelle repul-
sions grows as p3/2, the decrease in micelle aggregation
FIG.8: Mean-field phasebehaviorof N =100copolymer so-
leads to a rise in the relative importance of the entropic
lutionsasafunctionofchemicalcomposition,f,andpolymer
contributionstolatticefreeenergyresultingfromphonon
fluctuations. Asthe genericargumentsofAlexanderand content Φ, for fixedsolvent-core repulsion, γa2/kBT =0.2.
McTague dictate [13], this effect promotes the stability
of the bcc structure near to the micelle lattice melting.
However, we find that even far from the regimes where lattice melting [8]. While we have not included this ef-
entropic considerations are important – when solvent- fect in our mean-field treatment, we do not expect it to
selectivity is strong and p is large – micelle solutions change the qualitative view of ordered phase thermody-
exhibit broad ranges of bcc stability. In these regions namics presented here. Namely, our principle conclusion
of phase space, stability of the bcc structure over the that structural fcc-to-bcc transitions are driven by in-
close-packed fcc structure can be attributed to micellar creasesinmicelledensitywillremainunchanged. Indeed,
interactions which are of sufficiently long length scale. the swelling of the core domain will tend to counteract
the decrease in micelle size by loss of polymer chains as
While the presentmodeleffectivelydescribesthe ther-
temperature is raised. Thus, the basic prediction that
modynamics associatedwith the fcc-bcc structural tran-
η largely increases as temperature rises will hold in the
sitions in micelle solutions, the approach is limited in
case that solvent is permitted to swell the micelle core.
regardtootherequilibriumstructures. Inparticular,the
star-polymer potential of eq. (11) has been shown to A second type of fluctuation effect absent from the
stabilize non-cubic and even diamond lattice structures presentmodel are equilibrium fluctuations in micelle ag-
[19, 20]. Furthermore, studies of similar soft-molecular gregationnumber,p,leadingtoacertainamountofpoly-
systems suggest the stability of the relatively open A15 dispersityinmicellesize. Asinmicellarphasesofcopoly-
latticeofmicelles[36,37]. Duetotheabundanceofther- mer/homopolymerblends[38],fluctuationsinmicelleag-
modynamicstudiesofcopolymersolutionsexhibitingthe gregationabove the criticalmicelle concentrationcan be
structural fcc-bcc transition, we focus on this particu- estimatedtobeontheorderof10%orless. Whilethede-
lar phenomenon, and consequently, these lattice struc- gree of polydispersity in the size of micelles will be even
tures. Generally, we might expect the stability of other smaller than this, it is not clear how even this modest
micelle lattice phases at higher polymer concentrations. level of polydispersity should affect the relative stability
More likely, however, is the possibility that other self- of various ordered phases. Certainly, a broad distribu-
assembled morphologies intervene at higher concentra- tion in micelle size and aggregation number will frus-
tion, such as cylindrical, lamellar or network morpholo- trate periodic order, but given that copolymer solutions
gies. Sincethe inter-micellartermsinthefreeenergyare are known to form well-ordered structures over a broad
necessarily of the same order as the intra-micellar ones, rangeofequilibriumconditionsweexpectthiseffectmay
at high concentrations, the system achieves a more effi- not be so significant for this system.
cient packing by a rearrangementof aggregate topology. Finally,weconcludebycontrastingthethermodynam-
Such phase behavioris well-documentedin experimental ics of micellar phases to other spherically-aggregating
observations of more concentrated copolymer solutions molecular systems, such as ionic or non-ionic surfac-
[6, 7]. tants. Indeed,surfactantsystemsexhibitstructuraltran-
Other limitations of the model include the fluctuation sitions between competing ordered structures as concen-
effects of the micellar structure. One of these is the tration and solvent quality are varied [39]. However,
swelling of the core domain by solvent at high temper- unlike their polymeric counterparts, aggregates of low-
ature. Neutron scattering in dilute solutions of micelles molecular weight surfactants tend be characterized by
reveals that penetration of solvent into the core domain a fixed size, dictated by molecular-scale packing con-
may reach as high as 30% by volume near to micelle straints [40]. In contrast, micelles of flexible polymeric
10
amphiphiles adjust their aggregation by a factor 5-10 supported by NSF Grants DMR01-02459 and DMR04-
times over the accessible range of solution conditions. 04507,the Donors of the Petroleum ResearchFund, Ad-
This molecular flexibility is fundamentally linked to the ministered by the American Chemical Society and a gift
unusual feature that copolymer micelle lattices become from L.J. Bernstein.
moredenseastemperatureisraisedandmolecularforces
are weakened, ultimately driving transitions in the long-
ranged structural order of the system.
Acknowledgments
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