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Coarse-grained modeling of mesophase

dynamics in block copolymers

Zhi-Feng Huang and Jorge Vinals

McGill Institute for Advanced Materials and Department of Physics, McGillUniversity, Montreal, QC H3A 2T8, Canada

1 Introduction

Block copolymers not only are well suited for studying self-assembly andnanostructure formation in soft matter, but also provide a wide range of newapplications in fields such as microelectronics, biomedicine, etc. Viewed as asimplified picture, a block copolymer is composed of two or more segments(blocks or sequences of monomers) that are chemically different and mutuallyincompatible, joined by covalent bonds (see Fig. 1a for schematic of diblockand linear triblock). Of most interest in block copolymers is the emergence ofmesophases below the order-disorder transition temperature TODT, showingas spatially ordered compositional patterns of different types of symmetries,such as lamella (see Fig. 1b), cylinder, sphere, gyroid, etc. [1, 2]

Fig. 1. (a) Schematic of block copolymer molecules: AB diblock and ABC tri-block. (b) Polycrystalline, defected configuration for a diblock copolymer system ina lamellar phase. Also indicated are three types of topological defects: grain bound-ary, dislocation, and disclination.

2 Zhi-Feng Huang and Jorge Vinals

However, when a block copolymer melt sample is processed by tempera-ture quench or solvent casting from a high temperature isotropic, disorderedstate, the resulting configuration is usually polycrystalline with randomly ori-ented domains or grains, instead of a uniform, completely ordered structureas desired in most applications. The characteristic of such mesoscopic, poly-crystalline state is the presence of large amount of topological defects, suchas grain boundaries, dislocations, and disclinations, as shown in Fig. 1b for atwo-dimensional configuration of a lamellar phase. Consequently, there is nolong-range orientational order in the whole sample. These topological defectsdetermine the longest relaxation times in the system which is outside of ther-modynamic equilibrium, and hence their structure, interaction, and motioncontrol the coarsening and evolution towards the equilibrium ordered state,as well as the response of the copolymer microstructure to external forces.

Widespread applications of block copolymer materials, especially in nan-otechnology [3] such as nanolithography [4, 5, 6, 7], photonic crystals [8, 9],high density storage systems [10], etc., require well-ordered nanostructuresand therefore free of topological defects that are usually detrimental for de-vice performance. However, full development of such ordered structures takesplace over very long times. Furthermore, the essential mechanisms behind suchevolution are still poorly understood. Therefore, increased emphasis has beenput on the microstructural control of the copolymer configuration, throughe.g., external forces or confinement, particularly for defect removal in orderto achieve long range order.

In this article we survey recent research on a theoretical description ofthe mechanisms controlling the dynamics of mesophases in extended blockcopolymer systems, including the motion of topological defects, coarsening ofdomains in partially ordered, nonequilibrium configurations, and the responseof mesophases to external shear flows for the control of long range order. Ourfocus here is on a mesoscopic modeling of block copolymer dynamics, basedon the fact that the characteristic spatial and temporal scales of structuralevolution in nanostructured copolymer materials are far beyond the individualmolecular dimension, and hence a coarse-grained, reduced description can beadopted to account for the complex behavior of the system. The mesoscopicmodel equations as well as the associated amplitude/envelope equations de-rived from a multiple scale analysis will be introduced below in Sect. 2. Theapplications to various issues in block copolymer mesophase dynamics aregiven in Sect. 3 and Sect. 4.

2 Mesoscopic Modeling

2.1 Ginzburg-Landau Model Equations

At a mesoscopic level over length scales much larger than the microscopicmonomer scale and time scales long enough compared with the polymer chain

Coarse-grained modeling of mesophase dynamics in block copolymers 3

relaxation time, the evolution of a block copolymer melt can be described byan order parameter field ψ(r, t) which represents the local density differenceof the constituent monomers, and a local velocity field v(r, t). Given thatrelaxation of the concentration field ψ is driven by local dissipation due to freeenergy minimization, we can adopt the following time-dependent Ginzburg-Landau equation

∂ψ/∂t+ v · ∇ψ = −ΛδF/δψ, (1)

where Λ is an Onsager kinetic operator (with Λ = −M∇2 reflecting conserveddynamics for the order parameter). For a copolymer system under imposedshear flow, the advection term (v · ∇ψ) in Eq. (1) cannot be neglected, anda hydrodynamic equation governing the velocity field has to be incorporatedinto the description of copolymer dynamics. In Eq. (1), the coarse-grained freeenergy functional F is given by (the Ohta-Kawasaki energy [11])

F [ψ] =

∫

dr[

−τ2ψ2 +

u

4ψ4 +

κ

2|∇ψ|2

]

+B

2

∫ ∫

drdr′(ψ(r) − ψ)G(r − r′)(ψ(r′) − ψ), (2)

with ψ the spatial average of ψ. Here the first part of the r.h.s. is identical tothe Ginzburg-Landau free energy used for the description of phase separationin a binary mixture, and the second part (with the kernel G(r − r

′) definedby ∇2G(r − r

′) = −δ(r − r′)) represents the long range interactions arising

from the connectivity of different blocks in a copolymer.In the weak segregation limit (i.e., near the order-disorder transition

point), a block copolymer melt can be described by a simpler coarse-grainedfree energy (the Brazovskii or Leibler energy) [12, 13]

F [ψ] =

∫

dr

{

−τ2ψ2 − g

3ψ3 +

u

4ψ4 +

ζ

2

[(

∇2 + q∗02)

ψ]2

}

, (3)

where the reduced temperature variable τ measures the distance from theorder-disorder transition, with τ > 0 for T < TODT, and q∗0 is the wavenum-ber of the periodic structure. Note that here the order parameter has beenreplaced by ψ → ψ− ψ, the local deviation of the concentration field from itsspatial average. Substituting Eq. (3) into the Ginzburg-Landau equation (1),approximating Λ by Mq∗0

2 which is valid near TODT with negligible long rangediffusion [14], and rescaling all quantities to be dimensionless, we obtain theso-called Brazovskii or Swift-Hohenberg model equation [15]

∂ψ/∂t+ v · ∇ψ =[

ǫψ −(

∇2 + q20)2

]

ψ + gψ2 − ψ3, (4)

where ǫ = τ/ζq∗04 with 0 < ǫ ≪ 1 corresponding to the weak segregation

limit, and q0 = 1 after rescaling although we retain the symbol q0 in whatfollows for clarity of presentation. For lamellar phases which correspond to


symmetric block copolymers (ψ = 0) and are our major focus in this article,we have g = 0 in Eq. (4).

Note that the free energy expressions given above are mainly applied todiblock copolymers, and that the generalization to multiblocks is much morecomplicated. Recent results of more general free energy expressions can befound in Ref. [16].

2.2 Multiple Scale Approach and Amplitude Equation Formalism

In the limit of weak segregation (ǫ≪ 1), a multiple scale approach [15, 17] canbe used to separate fast spatial/temporal scales of a base periodic or modu-lated structure and slow scales of evolution of its envelope (see Fig. 2), andthen be used to derive the amplitude/envelope equations for slowly varyingamplitudes of the base pattern. Within this amplitude equation formalismproperties and motion of a single topological defect in ordered phases havebeen well studied [15]. Here we present an example for a tilt grain boundaryconfiguration of a lamellar phase [18, 19] as shown in Fig. 2. Results for othertypes of defect configuration can be obtained similarly (some of them will begiven in Sect. 3.2 and Sect. 4.1).

Fig. 2. Envelope description of a tilt grain boundary configuration of a lamellarphase in multiple scale approach. The grain boundary is located at x = xgb, withvelocity vgb and boundary thickness ξ ∼ λ0ǫ

−1/2 (λ0 = 2π/q0).

For a three-dimensional (3D) 90◦ configuration comprising two lamellardomains of mutually perpendicular orientations (x orientation for domain Aand z (the vertical direction in Fig. 2) for domain B), we can expand theorder parameter field as the superposition of two base modes exp(iq0x) andexp(iq0z):

ψ =1√3

[A exp(iq0x) +B exp(iq0z) + c.c.] , (5)

with complex amplitudes A and B slowly varying in space and time:


A = A(X = ǫ1/2x, Y = ǫ1/4y, Z = ǫ1/4z, ǫt),

B = B(X = ǫ1/4x, Y = ǫ1/4y, Z = ǫ1/2z, ǫt).

Following a standard multiple scale expansion procedure (see e.g., Refs. [15,18]), from the model equation (4) (with g = 0 for lamellar phases) we obtainthe following amplitude equations (at O(ǫ3/2)) [19]

∂tA =[

ǫ− (2iq0∂x + ∂2y + ∂2

z )2]

A− |A|2A− 2|B|2A, (6)

∂tB =[

ǫ− (∂2x + ∂2

y + 2iq0∂z)2]

B − |B|2B − 2|A|2B. (7)

The corresponding equations for a 90◦ grain boundary in two dimensions[18] are the same as Eqs. (6) and (7) except for the absence of the termsproportional to ∂2

y simply due to the lack of the y direction in two-dimensional(2D) space.

3 Defected Structure and Dynamics

The Ginzburg-Landau model equation introduced above has been widely ap-plied to the study of mesophase dynamics in block copolymers, includingdomain evolution and the dynamics of topological defects. The discussion inthis section is restricted to some recent research on 2D domain coarsening inlamellar and hexagonal phases controlled by grain boundary motion, as wellas on grain boundary configurations and stability in 3D lamellar patterns.

3.1 Domain Coarsening and Pinning (2D)

Lamellar/Stripe Phase

Coarsening of defected, multidomain configuration in lamellar phases (orstripe phases in 2D space) is a long-lasting problem but still remains not wellunderstood. The existence of statistical self-similarity in domain coarseninghas been addressed both theoretically [20, 21, 22, 23, 24] and experimentally[4, 25], with the characteristic length scale R(t) of a domain expected to scaleas R(t) ∼ tα. However, no consensus has been reached on the value of thecoarsening exponent α, which ranges from 1/5 to 1/2 in different studies andappears to depend on the distance to TODT (i.e., the value of ǫ), thermal noise,and the selected linear scale for analysis.

A typical configuration in the weak segregation regime obtained by numer-ical integration of the model equation (4) is shown in Fig. 1b. Large amountsof 90◦ grain boundaries exist, separating differently oriented domains withcurved lamellae, as well as pointlike defects such as disclinations and dislo-cations. In the case of ǫ ≪ 1, the evolution is dominated by the motion ofgrain boundaries since bulk curved lamellae remain largely immobile as a re-sult of the topological constraints set by the disclinations. Grain boundaries


move over large distances driven by lamellar curvature and distortion withindomains [26, 23]. Given the characteristic curvature κ of the lamellae aheadof the grain boundary, the average boundary velocity can be obtained fromthe analysis of the 2D amplitude equations [26] as

vgb ∼ ǫ−1/2κ2 ∝ R−2 (8)

due to the fact that the distance between disclinations, which is proportionalto the characteristic domain size R, determines the inverse curvature of lamel-lae constrained by these disclinations. Therefore, a coarsening law R(t) ∼ t1/3

is implied from dimensional analysis of Eq. (8). This coarsening behavior hasbeen verified by the direct numerical solution of Eq. (4), which yields the sameexponent 1/3 from the probability distribution of lamellar curvatures, a finitesize analysis of the total grain boundary length, and the structure factor ofthe order parameter ψ [23].

However, when a system is far enough from the order-disorder threshold(i.e., with larger, finite value of ǫ), the above coarsening results no longer hold.In this case, the coupling between fast scales of base lamellar patterns andslow scales of the associated amplitudes/envelopes (as described in Sect. 2.2and Fig. 2), referred as the nonadiabatic effects, has to be taken into account[24]. This results in an extra contribution to the grain boundary velocity(8), which is proportional to the amplitude of an effective periodic potentialinduced by the underlying periodicity of the base pattern. Grain boundariesbecome pinned in this potential, with domains growing up to a pinning lengthscale given by

Rg ∼ λ0ǫ−1/2 exp

(

|α0|ǫ−1/2/2)

, (9)

with the lamellar period λ0 = 2π/q0 and α0 a constant of order unity. Forsmall enough ǫ (in the weak segregation limit) so that Rg is much largerthan the system size, pinning effects are negligible and domain coarseningobeys the t1/3 scaling law as discussed above. With increasing ǫ and hencedecreasing Rg, defect pinning becomes more pronounced, leading to smallereffective coarsening exponents (α < 1/3) in an intermediate time regime aswell as a saturation of domain size and the appearance of glassy configurationsat long times [23, 24]. Defect pinning effects are believed to account for therange of coarsening exponents in previous studies.

Hexagonal Phase

For nonsymmetric block copolymer melts, mesophases with hexagonal sym-metry have been observed and extensively studied in experiments, and canalso be well modeled by the coarse-grained equations in Sect. 2. Similar tolamellar patterns, macroscopic samples do not completely order; instead, poly-crystalline configurations with the presence of topological defects are found,with a typical example given in Fig. 3 which shows three hexagonal domains


Fig. 3. A defected, multidomain configuration in a hexagonal phase obtained bynumerical solution of the model equation (4).

separated by grain boundaries and is obtained by numerically integrating themodel equation (4).

The amplitude equation formalism has been used to analyze defect dynam-ics in hexagonal patterns. By considering nonadiabatic effects arising from thecoupling of slowly varying amplitudes and the phase of the defects, one candetermine the grain boundary velocity to be [27]

vgb = −phex sin [2q0xgb sin(θ/2)] /D, (10)

where xgb(t) is the grain boundary position, θ is the relative misorientation an-gle between the two uniform, hexagonal domains on either side of the bound-ary, and D is a friction coefficient. The pinning force (or Peierls’ force) phex

in Eq. (10) decays exponentially with the grain boundary thickness ξ, that is,

phex ∼ exp [−2a∗q0 sin(θ/2)ξ] , (11)

where a∗ is a dimensionless constant of order unity. In contrast to the caseof lamellar phases, the pinning force here cannot be neglected in the weaksegregation limit ǫ → 0 due to the finite value of ξ(ǫ = 0) ≃ 15λ0/(8

√6πg)

(with g given in Eq. (4)). For finite values of ǫ, phex is much larger thanthe pinning force in a lamellar phase (which scales as exp(−ǫ−1/2) due toξlamellar ∼ ǫ−1/2; see also Eq. (9)). Therefore, defect pinning is unavoidable inthe hexagonal phase, and the coarsening behavior in multidomain, partiallyordered samples, as observed in both block copolymer experiments [28] andnumerical studies of Ginzburg-Landau equations (1) and (2) [29] (with aneffective coarsening exponent around 1/4), likely occurs in an intermediatecrossover regime before the system becomes pinned.


3.2 Grain Boundaries (3D)

Due to the experimental observation [30, 31] that grain boundary defects pre-dominate in block copolymer bulk samples, in the following we focus on grainboundary configurations in lamellar systems, which in 3D can be classifiedinto two types: tilt and twist boundaries, as discussed respectively.

Tilt Boundaries

For a tilt grain boundary separating two differently oriented domains of lamel-lar symmetry, the structure can be viewed as the tilting of two grains withrespect to an axis lying in the grain boundary plane, and the plane formed bylamellar normals of the two grains is perpendicular to the boundary plane. 90◦

tilt boundaries are commonly observed in 2D systems [17], and a stationarysolution for the corresponding boundary configuration (see Fig. 4a) is knownto exist, with the same, unique value of wavenumber (q0) of the lamellae inboth domains [18]. However, this is not the case in 3D patterns. In experi-ments of 3D lamellar block copolymers, 90◦ tilt boundaries are rarely observed[32], which we believe can be attributed to the finite wavenumber instabilityshown below.

Fig. 4. 90◦ tilt grain boundary configurations in (a) two-dimensional and (b) three-dimensional lamellar phases, as calculated from either the order parameter modelequation (4) or the amplitude equations (6) and (7). Note that a finite wavenumberinstability develops along the grain boundary plane in the 3D configuration (b).

The amplitude equations governing the evolution of a 90◦ tilt boundaryare already given by Eqs. (6) and (7). It is convenient to expand the complexamplitudes A and B in Fourier series (note that the grain boundary is on they-z plane):

A = A0(x) +∑

qy,qz

A(qy, qz, x, t)ei(qyy+qzz),

B = B0(x) +∑

qy ,qz

B(qy, qz , x, t)ei(qyy+qzz), (12)


with A0(x) and B0(x) the base state solution corresponding to a planar andstationary grain boundary structure [18] and only dependent on the coordi-nate x that is normal to the boundary plane. Substituting the above expansioninto Eqs. (6) and (7) and retaining terms up to first order of the perturbationamplitudes, we can obtain the time evolution equations for A and B whichgovern the stability of grain boundary configuration. It can be found from thenumerical study of these perturbed amplitude equations [19] that any initialperturbations of A and B will be amplified around the grain boundary plane,indicating an instability characterized by a positive perturbation growth rateσ. The maximum growth rate occurs at a finite wavevector (q∗y , q

∗

z), and thecorresponding wavelengths of maximum instability along the y and z direc-tions (λ∗y,z = 2π/q∗y,z) both decrease with the increase of ǫ, as shown in Fig.5a. Fig. 5a also indicates that wavelengths along both two directions of thegrain boundary plane are larger than the wavelength of the base lamellar pat-tern, and the mode of this finite wavelength instability is anisotropic on theboundary plane. This can be seen in Fig. 4b which presents a typical 3D grainboundary configuration with finite wavenumber undulations of the boundaryand is obtained from numerical solution of the amplitude equations (6) and(7) in a 2563 system for ǫ = 0.08. A similar instability (e.g., anisotropic in-stability wavelengths given in Fig. 5a and configuration in Fig. 4b) can bederived from direct numerical solution of the order parameter model equation(4).

0.01 0.1ε

10

100

λ y,z*

(a)

λz*

λy*

0 100 200 300 400 500 600 700 800 900 1000t

-6×10-8

-4×10-8

-2×10-8

0

2×10-8

4×10-8

∆F/V

3D

2D

(b)

Fig. 5. (a) Log-log plot of most unstable wavelengths λ∗

y and λ∗

z along two directionsy and z of the grain boundary plane as a function of ǫ. (b) Effective free energydensity difference ∆F/V as a function of time, for both 2D and 3D grain boundaryconfigurations. See also Ref. [19].

Further insight into this instability can be gained from an effective freeenergy analysis of the amplitude equations. By rewriting Eqs. (6) and (7) ina potential form

∂tA = −δF/δA∗, ∂tB = −δF/δB∗, (13)

with ∗ denoting complex conjugation and F the effective free energy given by


F =

∫

dr

[

−ǫ(

|A|2 + |B|2)

+1

2

(

|A|4 + |B|4)

+ 2|A|2|B|2

+∣

∣(2iq0∂x + ∂2y + ∂2

z )A∣

∣

2+

∣

∣(∂2x + ∂2

y + 2iq0∂z)B∣

∣

2]

, (14)

we can examine the instability by the net energy change∆F = F−F0, with F0

the effective energy of the base state. Time evolution of ∆F per unit volumeV is plotted in Fig. 5b, for both 2D and 3D grain boundary structures. Asexpected, a stable 2D configuration yields ∆F > 0, due to the energy penaltyof any modulation on the stationary state. In contrast, we have ∆F < 0 for a3D configuration, indicating the instability as found in the stability analysis ofthe amplitude equations. Substituting the expansion (12) into Eq. (14), we findthat the negative contribution to ∆F arises from the coupling between A andB lamellar modes which is nonnegligible only in the region around the grainboundary plane. This cross mode coupling dominates in three dimensions, butis not strong enough compared to other positive, stabilizing contributions inthe free energy of a 2D system. Further details can be found in Ref. [19].

Twist Boundaries

A twist grain boundary can be constructed by rotating two lamellar domainswith respect to an axis perpendicular to the grain boundary plane, with lamel-lar normals of the two joint domains staying in a plane that is parallel to theboundary plane. An example for a 90◦ twist boundary is shown in Fig. 6,with the boundary plane exhibiting a doubly periodic morphology. Despitethe importance of twist boundaries in block copolymer samples [30], the re-lated theoretical analyses are very limited [33, 34, 35]. Here we present acoarse-grained modeling of the twist boundary structure, based on the orderparameter model equation and the amplitude description given above.

Fig. 6. A 90◦ twist grain boundary configuration of lamellar phase, which is stable,in contrast to the unstable 3D tilt structure shown in Fig. 4b.

We first apply the amplitude equation formalism to the description oftwist boundaries. Similar to the tilt boundary case, we separate slowly varying


amplitudes and fast spatial and temporal variables of the base lamellar patternby expanding the order parameter field

ψ =1√3

[A exp(iq0y) +B exp(iq0z) + c.c.] (15)

for a single boundary of the configuration of Fig. 6, with the coordinate xnormal to the y-z boundary plane. The amplitudes A and B scale as A =A(ǫ1/4x, ǫ1/2y, ǫ1/4z, ǫt) and B = B(ǫ1/4x, ǫ1/4y, ǫ1/2z, ǫt) respectively, giventwo different orientations (y and z) of constituent lamellar domains and thecorresponding anisotropic spatial scalings. Thus, a multiple scale approach tothe model equation (4) yields

∂tA =[

ǫ− (∂2x + 2iq0∂y + ∂2

z )2]

A− |A|2A− 2|B|2A, (16)

∂tB =[

ǫ− (∂2x + ∂2

y + 2iq0∂z)2]

B − |B|2B − 2|A|2B. (17)

Note the difference between the above amplitude equations and those for tiltgrain boundaries given in Eqs. (6) and (7), which is due to different relation-ship between domain orientations and the boundary plane in this two typesof grain boundaries. We perform the stability analysis on Eqs. (16) and (17)by using the procedure described above for tilt boundary, and obtain the per-turbation growth rate σ ≤ 0 for all wavenumbers. This indicates stability ofthe twist configuration, a result qualitatively different from the tilt structureand consistent with the experimental findings in block copolymer melts.

A simple dimensional analysis of Eqs. (16) and (17) along the x direction(the boundary normal) yields a ǫ−1/4 dependence of the boundary thickness,in contrast to the ǫ−1/2 result of tilt boundaries, in agreement with both anumerical solution of the amplitude equations and of the model equation (4).Thus, in the limit of ǫ → 0, i.e., close to the order-disorder threshold, thewidth of twist boundary region is much smaller than that of tilt boundary.All the above results hold for twist boundaries of different angles, as shownin Ref. [36].

4 Control: Shear Alignment

As discussed in Sect. 1, polycrystalline copolymer samples are highly unde-sirable for most applications. Hence, external forces are usually imposed ondefected samples to control microstructural evolution and achieve long rangeorder. In this case, we must address not only the competition between dif-ferently oriented domains, but also the response of phases of diverse orienta-tions to the external field. Significant effects arise from the coupling betweenmesophase dynamics and hydrodynamic flows in response to imposed fields.Here we are interested in the particular case of oscillatory shears, a widespreadmethod to promote global order in bulk samples of block copolymers. The fo-cus of the discussion that follows is on recent theoretical analyses within theframework of the coarse-grained modeling introduced in Sect. 2.


A topic of special concern regarding shear alignment of lamellar blockcopolymers is the problem of orientation selection, that is, the selection of aparticular lamellar orientation relative to the shear as a function of the pa-rameters of the block copolymer and of the imposed shear. It is convenientto distinguish among three possible orientations of a lamellar phase: parallel(with lamellar layers parallel to the shearing surface), perpendicular (withlamellae normal parallel to the vorticity of the shear flow), and transverse(with lamellae normal along the shear direction), as shown in Fig. 7. Themechanisms responsible for the response of the copolymer mesostructure andthe orientation selection by shear, especially between parallel and perpen-dicular, still remain unknown despite intensive studies in recent years, bothexperimental (see Ref. [2] for a recent review, and Ref. [37] for recent progressin multiblocks) and theoretical [38, 39, 40, 41, 42, 43, 44].

Fig. 7. Shear alignment of a polycrystalline, defected configuration, with three pos-sible uniaxial orientations of lamellar phase: parallel, perpendicular, and transverse.

The coarse-grained model, Eqs. (1) and (2), combined with the equationgoverning the local velocity field, has been used to study secondary instabil-ities of the lamellar phase under oscillatory shears, in both 2D [45] and 3D[46]. It has been found that subjected to long wavelength perturbations, uni-form lamellar configurations of all three orientations (parallel, perpendicular,and transverse) could be linearly stable under specific shearing conditions;however, the stability region turns out to be wider near the perpendicularorientation compared to the parallel one, with a much smaller projection onthe transverse direction. Importantly, the viscosity contrast between the mi-crophases of a copolymer is shown to play a negligible role in the stability ofwell-aligned lamellar structures, although this effect has been proven crucialnear the order-disorder transition point [39].

These stability results determine the parameter range over which specificorientations in steady state can be in principle observed experimentally. How-ever, the problem of orientation selection is of a dynamical nature, while thestability analysis does not provide for a selection mechanism among simul-taneously stable states below the order-disorder transition. Attention below


focuses on recent theoretical efforts to model the competition between coex-isting lamellar phases of different orientations under oscillatory shear flowsof low enough shear amplitude γ and frequency ω (with ω < ωc, where ωc

is the characteristic relaxation frequency of polymer chains). In the follow-ing we separately discuss parallel/transverse and parallel/perpendicular grainboundaries.

4.1 Parallel/Transverse Grain Boundary

We consider a 90◦ grain boundary in a 2D x-z space (see Fig. 8), separatingtwo lamellar domains A and B with the same wavenumber q0 but mutuallyperpendicular orientations. In the absence of shear this configuration is knownto be stationary, whereas boundary motion is expected upon the impositionof shear even though both bulk domains are linearly stable. As shown in Fig.8, lamellae in domain A are transverse to the shear, hence the individualcopolymers are compressed, leading to elastic response to the shear, whileparallel lamellae in domain B respond like a fluid to the shear. Consequently,extra elastic energy is stored in the compressed domain A, which can berelieved only through the motion of the grain boundary from the parallel (B)to the transverse (A) region.

Fig. 8. Schematic representation of a 2D grain boundary configuration separatingparallel and transverse lamellar domains subjected to an oscillatory shear. Alsoindicated are different advection effects of the shear on the two domains. Note thatboth domains are linearly stable in the corresponding bulk state.

Details of such grain boundary motion can be described at a mesoscopiclevel by the model equation (4), with the velocity field v simply approximatedby v0, the imposed shear flow, i.e.,

v ≃ v0 =da

dtzx = γω(cosωt)zx, (18)

with γ the shear strain amplitude, ω the shear frequency, and the shear straina(t) = γ sinωt. The shear flow will advect the lamellae through the termv · ∇ψ in Eq. (4), but with different effects on the two domains (see Fig.


8): no advection occurs in the parallel domain B because v0 · ∇ψ = 0, buttransverse lamellae are distorted as a result of a nonzero advection term. Theresulting motion of the grain boundary can then be obtained by numericalsolution of Eq. (4) [47], with the introduction of a sheared frame of referencewith coordinates

x′ = x− a(t)z and z′ = z, (19)

corresponding to an nonorthogonal basis set

x′ = x and z′ = a(t)x+ z.

Typical results for the grain boundary location x′gb and velocity vgb (bothin the sheared frame) as a function of time are given in Fig. 9a. Note theoscillatory motion of grain boundary, both forward (with positive vgb) andbackward (with negative vgb). Boundary motion occurs through diffusive re-laxation of the order parameter field around the grain boundary, in additionto the rigid distortion of the transverse lamellae that accounts for the freeenergy imbalance between the two domains. This diffusive relaxation leadsto the break-up and reconnection of the transverse lamellae at the bound-ary, and plays an important role in enabling dissipation of the excess energystored in the transverse bulk region. Note also that the net motion of the grainboundary is still driven by free energy reduction, with positive value of timeaveraged boundary velocity 〈vgb〉t (over a shearing period T0) as shown in Fig.9b. This net average velocity monotonously increases with shear frequency ωand strain amplitude γ [47].

Further information can be obtained from the analysis of the correspond-ing amplitude equations, based on the fact that in the vicinity of the order-disorder transition, the characteristic length and time scales of defect motionare governed by slowly evolving amplitudes of the base patterns. Expressedin the sheared frame (x′, z′), up to O(ǫ3/2) the amplitudes A and B obey theequations [48]

∂tA = {ǫ− [2iq0(∂x′ − a∂z′) + ∂2z′ − q20a

2]2}A− |A|2A− 2|B|2A, (20)

∂tB = {ǫ− [2iq0(∂z′ − a∂x′) + ∂2x′ ]2}B − |B|2B − 2|A|2B. (21)

Note that the term proportional to q20a2 contributes to the equation for A in

the entire transverse region, and represents the compression effect describedabove. On the other hand, terms proportional to a∂x′ and q20a

2∂x′ are nonzeroonly around the grain boundary and incorporate local diffusive relaxation ofthe order parameter (or of the copolymer monomer concentration). Resultscalculated from Eqs. (20) and (21) agree with those from the model equation(4), especially for small frequencies, as shown in Fig. 9.

Diffusive relaxation of the monomer concentration field incorporates the re-orientation of copolymer molecules at the grain boundary, leading to diffusivemonomer redistribution around the boundary so that the extent of the phaseof transverse lamellae of higher energy can be reduced. We observe additional


0 1 2 3 4t/T

0

-1

0

1

2

x’gb

/ λ 0

1 1.5 2 2.5 3

-0.05

0

0.05

v gb

(a)

0 0.02 0.04 0.06 0.08 0.1ω

0

0.002

0.004

0.006

0.008

0.01

<v gb

>t

model equationamplitude equationsanalytic approximation

(b)

Fig. 9. (a) Time evolution of the grain boundary location in the sheared frame forǫ = 0.04, γ = 0.3, and ω = 0.01. Here T0 = 2π/ω is the shear period. Inset: boundaryvelocity as a function of time; the results calculated from the model equation (4) areplotted as the solid curve, while those from the amplitude equations (20) and (21)are indicated as the open circles. (b) Plots of temporal average of grain boundaryvelocity vs. shear frequency ω, for ǫ = 0.04 and γ = 0.3. Solid line, results obtainedfrom numerical solution of the model equation (4). Symbols, from the amplitudeequations (20) and (21). Dashed line, from the analytic approximation (23). Formore details see Refs. [48] and [47].

phase diffusion in the transverse phase corresponding to local wavenumberreadjustment. It develops near the boundary and slowly propagates into thebulk, showing as a linear spatial dependence of the phase ϕA of the complexamplitude A, i.e.,

ϕA ∝ −δq · x′, (22)

with δq > 0, and hence a wavenumber decrease qx′ → q0 − δq. We haveobtained ϕA from a one-dimensional (1D) approximation to Eqs. (20) and(21) by assuming a planar grain boundary with only spatial dependence onx′. The results are illustrated in Fig. 10, with the phase shift δq agreeing withthat directly calculated from the model equation (4).

At intermediate frequencies for which diffusive effects are less importantthan elastic distortion effects, we can derive an analytic expression for theboundary velocity [48],

vgb =F (x′ → +∞) − F (x′ → −∞)

2∫ d

0dz′

∫ +∞

−∞dx′(|∂x′A′|2 + |∂x′B|2)/d

, (23)

with the effective driving force

F (x′ → +∞) − F (x′ → −∞)

=1

2(−2q0δq + q20γ

2 sin2 ωt)2[2ǫ− (−2q0δq + q20γ2 sin2 ωt)2] (24)

arising from the free energy increase in the transverse region. The denomi-nator of Eq. (23) represents an inverse mobility coefficient that depends on


3000 3500 4000ix’

-1

0

1

2

3

4

5

6

ϕ A

t=50

50 100

100

(b)

(γ=0.4)

(γ=0.3)

Fig. 10. Spatial dependence of phase ϕA of the complex amplitude A in terms ofthe grid index i′x along the x′ direction, given by the 1D amplitude equations withLx′ = 4096, ǫ = 0.04, ω = 0.04, and γ = 0.3 (solid lines) and 0.4 (dashed lines),and at times t = 50T0 and 100T0. Also shown for comparison are dotted lines withslopes −δq∆x′ = −3∆x′/128 (for γ = 0.3) and −5∆x′/128 (for γ = 0.4), with ∆x′

the grid spacing. Values of δq originate from the long time solution of the modelequation (4).

the grain boundary profile (A′, B), with A′ defined by A = A′ exp(−iδq · x′)as a result of wavenumber compression. Time average of Eq. (23) is plottedin Fig. 9b as a function of ω, agreeing well with the direct numerical inte-gration of the amplitude equations at high enough frequencies. However, asfrequency decreases the deviation between this analytic approximation andthe numerical result increases, and Eq. (23) fails at very low frequency, sincediffusive effects become more pronounced and lead to a smaller amplitude ofthe boundary velocity as compared to that given by the approximation (23).

Note that the above results and discussion are not confined to mesophasedynamics of block copolymers; instead, they should hold in other systems ofthe same symmetry in which solid-like or elastic response occurs on one sideof a grain boundary, resulting in a structural distortion by the shear, andfluid-like response is expected on the other side.

4.2 Parallel/Perpendicular Configuration

As shown above, we already have a fairly complete picture of the competitionbetween coexisting parallel and transverse orientations, including the drivingforce for boundary motion that originates from the different response to theshear flow, as well as local phase diffusion effects. On the other hand, fluid-like response to the shear is expected for both parallel and perpendicularorientations; thus, the treatment given above for the parallel/transverse casecannot apply here.


An alternative approach regarding mesoscopic hydrodynamics in blockcopolymers has been pursued [49, 50]. The emphasis is put on the local velocityfield v = (vx, vy, vz) for a block copolymer, which satisfies the equation

ρ (∂v/∂t+ v · ∇v) = −∇p+ ∇ · σD, (25)

as well as incompressibility ∇ · v = 0, with ρ the copolymer density, p thepressure field, and σ

D the dissipative stress tensor that still needs to be spec-ified. Instead of addressing the velocity field at spatial scales of the lamellarspacing, we adopt a long wavelength description of the dissipative part of thestress tensor σ

D in Eq. (25) [51]

σDij = ηDij + α1ninj nknlDkl + α56(ninkDjk + nj nkDik) (26)

with i, j, k = x, y, z, which is the most general form appropriate for a phase ofuniaxial symmetry, such as nematic liquid crystals [52, 53]. In Eq. (26), η is theNewtonian viscosity, α1 and α56 are two independent viscosity coefficients foruniaxial phases, Dij = ∂ivj + ∂jvi, and n = (nx, ny, nz) represents the slowlyvarying normal to the lamellar planes. Note that the last two terms of Eq.(26) reflect the lowest order deviation from Newtonian behavior in the limitsof low frequency and large scale flow that is compatible with the lamellarphase symmetry. The assumption (26) leads to different viscous response inparallel and perpendicular regions: the dynamic viscosity η′ (= G′′/ω, withG′′ the loss modulus) for a fully-ordered lamellar structure is a function of itsorientation,

η′parallel = η + α56 and η′perpendicular = η, (27)

consistent with experimental evidence [54, 55].This proposed deviation from Newtonian response leads to a dynamical

mechanism that favors one orientation over the other, even in regions in whichboth are equally stable in bulk. We consider the simple configuration shownin Fig. 11 to address the competition between coexisting parallel and per-pendicular domains under oscillatory shears. Both domains are linearly stablein the corresponding bulk state. To study the dynamics of competing lamel-lar domains, in principle the coupled equations (1), (25), and (26) should besolved, but this is complicated due to the ψ dependence of the dissipativestress tensor σ

D, and nonzero fluid inertia. Here we adopt the approximationthat the dependence of σ

D on ψ perturbations can be neglected. Therefore,we first obtain the velocity field from Eqs. (25) and (26) without consideringorder parameter diffusion in Eq. (1), and then discuss the resulting advectioneffects of the flow field obtained on the order parameter.

The base state for the configuration shown in Fig. 11 (with a planar inter-face at z = dA) is the same as that of two superposed Newtonian fluids withviscosities µA = η and µB = η+α56. We examine the hydrodynamic stabilityof this configuration and the effects of any instability on domain competition.Note that the configuration here is analogous to that of two-layer Newtonian


Fig. 11. Schematic representation of a parallel/perpendicular configuration underoscillatory shear. A hydrodynamic instability occurs at the interface, with the re-sulting secondary flows (of ux = 0 and uy, uz 6= 0 corresponding to the critical modeof instability) also indicated.

fluid with viscosity stratification which is known to be unstable under steady[56] and oscillatory [57] shears. We extend the analysis to the current systemcharacterized by the additional viscosities given in Eq. (26).

Expanding both velocity and pressure fields of A and B domains as

vA,Bi = VA,Bδiy +

∑

qx,qy

uA,Bi (qx, qy, z, t) exp [i(qxx+ qyy)] ,

pA,B = p0 + p′A,B, (28)

with VA,B (periodic in time t) and p0 the base state solutions, and substitutinginto Eqs. (25) and (26), we obtain the following linear stability equationsgoverning the perturbed velocity fields uA,B

x and uA,Bz :

Re[

(∂t + iqyVA)(∂2z − q2)uA

z − iqy(∂2zVA)uA

z

]

= (∂2z − q2)2uA

z

−α56q2x(∂2

z − q2)uAz + iqx[2α1q

2x − α56(∂

2z − q2)]∂z u

Ax , (29)

Re[

∂t(∂2z − q2)uA

x + iqy(∂2z − q2)(VAu

Ax ) + 2qxqy(∂zVA)uA

z

]

= (1 + α56)(∂2z − q2)2uA

x − 2α1q2x(∂2

z − q2y)uAx , (30)

for the perpendicular domain A (0 ≤ z ≤ dA), and

Re[

(∂t + iqyVB)(∂2z − q2)uB

z − iqy(∂2zVB)uB

z

]

= (1 + α56)(∂2z − q2)2uB

z − 2α1q2∂2

z uBz , (31)

Re[

∂t(∂2z − q2)uB

x + iqy(∂2z − q2)(VB u

Bx ) + 2qxqy(∂zVB)uB

z

]

= (∂2z − q2)2uB

x + α56(∂2z − q2)

(

∂2z u

Bx − iqx∂zu

Bz

)

− 2iqxα1∂3z u

Bz , (32)


for the parallel domain B (dA ≤ z ≤ 1). Here Re = ρωd2/η (with d thedistance between shearing planes) is the Reynolds number, q2 = q2x + q2y, andthe above equations have been made dimensionless by introducing a lengthscale d, a time scale ω−1, and by rescaling all viscosities by η (so that µA = 1,µB = 1 + α56). Boundary conditions include the continuity of velocity andstress and the kinematic condition at the interface z = dA + ζ(x, y, t), aswell as rigid boundary conditions on the planes z = 0 and z = d (= 1 afterrescaling) [50].

It is important to first address the magnitude of two dimensionless pa-rameters: the Reynolds number Re and the rescaled interfacial tension Γ ′ =Γ/(ηωd) (with Γ the interfacial tension, as appearing in the boundary condi-tion for the normal stress). For a typical block copolymer system of ρ ∼ 1 gcm−3, η ∼ 104 − 106 P, Γ ∼ 1 dyn/cm, and d ∼ 1 cm, we have

Re/ω ∼ 10−4 − 10−6 s and Γ ′ω ∼ 10−4 − 10−6 s−1. (33)

Hence, Re ≪ 1 for the frequency range of interest, and the above stabilityequations (29)–(32) combined with boundary conditions can be solved ana-lytically by an expansion of velocity and interfacial perturbations in ordersof Re. Detailed solutions are given in Refs. [50] and [49]; in the following weonly discuss major results.

The solutions of the perturbed velocity fields uA,Bz,x and interfacial profile

ζ (the Fourier transform of ζ) can be written as

uA,Bz,x (qx, qy, z, t) = eσtφA,B

z,x (qx, qy, z, t),

ζ(qx, qy, t) = eσth(qx, qy, t), (34)

where σ is a Floquet exponent, representing the perturbation growth rate, andfunctions φA,B

z,x and h are periodic in time when qy 6= 0 (according to Floquet’stheorem) or time independent for qy = 0. In the limit of Re → 0 with finitesurface tension Γ ′ = O(1), which might correspond to very low frequency ωaccording to Eq. (33), we find σ ≤ 0 for all wavenumbers qx and qy due tothe stabilizing effect of surface tension, leading to a stable configuration, orto coexistence of parallel and perpendicular domains under shear.

However, different results can be obtained for larger frequencies in therange ω ∼ 1 s−1, as in most experiments, yielding Re ≪ 1, Γ ′ ≪ 1, andΓ ′/Re = O(1). In this case, the perturbation growth rate σ is given by

σ/Re = fBz0(qx, qy)Γ ′/Re+

1

2δ2γ2fB

z1(qx, qy), (35)

=1

2δ2f1γ

2q2 − (θf0ω−2 − 1

2δ2f2γ

2)q4 + O(q6), (36)

where δ = (µA/µB)/(dA + dBµA/µB) with dB = 1 − dA, θ = Γ/(ρd3), andfB

z0, fBz1, f0, f1, and f2 are complicated but known functions of α1, α56, and

dA, but do not depend on the shear parameters γ and ω. Equation (36) is


obtained from the long wave expansion of Eq. (35) along the y direction only,with qx = 0 and qy = q, based on the result that the maximum instabilitygiven by the numerical evaluation of Eq. (35) occurs at qx = 0. Equation (35)presents the stabilizing effect of surface tension (given by the first term ofthe r.h.s., with fB

z0 ≤ 0 for all wavenumbers), and the destabilizing effect ofthe imposed shear (given by the last term proportional to γ2, with maximumvalue of fB

z1 being positive for certain range of parameters α1, α56, and dA).A typical profile of the perturbed velocity fields is given in Fig. 12. At the

most unstable wavevector (qx = 0, qy 6= 0), the associated secondary flow isgiven by ux = 0 and uz = φz exp(σt), with the amplitude φz plotted in Fig.12 as a function of spatial position z at t = T0. We find that flow fluctuationsare directed only along the z and y directions (due to the incompressibilitycondition), develop around the interface, and relax respectively into the bulkregions of the perpendicular (A) and parallel (B) domains.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z

-0.1

-0.05

0

0.05

0.1

φ z (t=

T0)

Real(φz)<<1

Imag(φz), α56=−0.9

φx = 0

Imag(φz), α56=9

(Perpendicular A) (Parallel B)

Fig. 12. Spatial dependence of the velocity amplitude φz for the most unstablemode, at time t = T0 (= 2π here) and with parameters α56 = −0.9 or 9 (i.e.,µB = 1/10 or 10), α1 = 1, γ = 1, dA = 1/2, and Re = 5 × 10−4.

The resulting stability diagram as a function of the effective viscosity con-trast µB and the domain thickness ratio dA/dB, as determined by Eq. (36), isshown in Fig. 13 (with σ > 0 indicating instability). The diagram is symmetricwith respect to µB → µ−1

B and dA/dB → (dA/dB)−1, and shows a thin-layereffect: instability occurs when the thinner domain has larger effective viscos-ity. This thin-layer effect is analogous to the case of viscosity stratificationin Newtonian fluids [58], but of different origin: here the viscosity contrastis caused by the orientation dependence of the viscous response in uniaxiallamellar phases.

Given the above interfacial instability and related perturbed flows, we caninfer the response of the lamellar domains, and argue about orientation se-lection. Since at instability ux = 0 and uz 6= 0, as shown schematically inFig. 11, parallel lamellae will be compressed and expanded by the z-direction


0.02 0.1 1 10 50d

A / d

B

10-4

10-3

10-2

10-1

100

101

102

103

104

µ B (

=1+

α 56)

Perpendicular

Perpendicular

CoexistenceParallel / Perpendicular

Parallel / Perpendicular

Coexistence

(σ > 0)

(σ > 0)

Fig. 13. Stability diagram of viscosity contrast µB vs. domain thickness ratiodA/dB , for phases of uniaxial symmetry. The selection between parallel and per-pendicular orientations is determined by hydrodynamic stability, as addressed inRefs. [49] and [50].

secondary flow due to a nonzero advection term u · ∇ψ in Eq. (1), while nosuch effect will occur for perpendicular lamellae. This situation is similar tothe case discussed in Sect. 4.1 in which the flow induced free energy imbalanceleads to interface motion towards the distorted region (which is the paralleldomain in the present case). Therefore, we argue that the hydrodynamic in-stability results in the growth of the perpendicular lamellae at the expenseof the parallel lamellae and hence the selection of the perpendicular orienta-tion, whereas stability indicates the coexistence of parallel and perpendicularphases, as shown in the diagram of Fig. 13.

Equation (36) can also be used to examine the dependence of the maximuminstability growth rate σm on shear parameters. The line of constant σm isgiven by

γω3/4 =2(θf0ησm)1/4

ρ1/4δ(f1d)1/2, (37)

which is equivalent to γω3/4 = const. for a given block copolymer system.Since σm is proportional to Re ≪ 1, it is a very small quantity, perhapsof the order of the inverse observation time for typical experiments. In thiscase, this line can determine an effective stability boundary. In this regard wenote that the experimentally determined boundary separating perpendicularand parallel orientations in poly(styrene)-poly(isoprene) (PS-PI) copolymers[59, 37] is approximately given by γω = const.

Finally, it is important to note that a polycrystalline configuration alwaysinvolves a variety of domain orientations and sizes. We would expect thatlamellar domains with a component of the orientation along the transversedirection with respect to the shear will be eliminated quickly, according toSect. 4.1, leaving only parallel and perpendicular domains. Their competition


has been addressed in Sect. 4.2. Further efforts beyond the stability analysispresented here are still necessary for a complete understanding of orientationselection in polycrystalline samples.

5 Perspectives

One of the major challenges for widespread applications of nanostructured softmaterials such as block copolymers is achieving precise microstructural con-trol, and hence the study of the mechanisms underlying structure, nonlineardynamics, and response of nanoscale phases needs to be undertaken. All threeaspects are strongly coupled to each other. In this article we have presentedsome recent research regarding these aspects in block copolymer mesophases,including structure and stability of topological defects, mesoscopic dynamicsof domain evolution, as well as the effects of external shears on response andorientation selection of mesophases. Although our primary focus of attentionis on block copolymers, the methods and results are applicable to other typesof mesophases of the same symmetry.

However, our understanding in this field is still far from complete, andfuture research would focus on further clarifying the interplay between struc-ture, dynamics, and response. Examples include defect dynamics and domaincoarsening, including hydrodynamic effects that follow from the mesoscopicviscoelasticity of copolymers. Of particular interest are the nonlinear responseof the copolymer system to external forces (e.g., shears, electric fields, etc.),defect removal and anisotropic domain coarsening processes, and the dynamicsof block copolymer thin films under surface confining which serve as an im-portant system for self-directed self-assembly of nanostructured copolymers.

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