Surface Effect on the Body-Centered-Cubic Phase of Diblock

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Macromolecules 2004, 37, 9646-9653

Surface Effect on the Body-Centered-Cubic Phase of Diblock Copolymers Hongge Tan and Dadong Yan* State Key Laboratory of Polymer Physics and Chemistry, Joint Laboratory of Polymer Science and Materials, Institute of Chemistry, Chinese Academy of Sciences, Beijing 100080, China

An-Chang Shi Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada Received April 8, 2004; Revised Manuscript Received September 29, 2004 ABSTRACT: In the weak segregation limit, we investigate the surface-induced structures in the bodycentered-cubic phase of diblock copolymers. Employing the Landau-Brazovskii mean field theory, we obtain the extent of surface-induced region as a function of temperature and surface field strength. At strong surface field strengthes, the lamellar and cylindrical characters are much more profound, and the surface-induced structure persists to a larger extent. A “phase diagram” of surface reconstructions is constructed, which describes the surface-induced structure. These results demonstrate that the coupling between the local average excess polymer concentration and the surface leads to lamella-like and cylinderlike structures close to the surface. Some of our theoretical results are in agreement with relevant experimental results. Our predictions about these interesting surface-induced structures should be observable in experiments under suitable conditions.

1. Introduction Block copolymers are composed of two or more chemically different subchains. Because of chain connectivity and incompatibility between the blocks, block copolymers can form a variety of ordered microstructures.1-7 The simplest block copolymer is the diblock copolymer, which is composed of two subchains. For bulk diblock copolymers and within mean field approximation, two parameters control the equilibrium phase behavior.6,7 The first parameter is χN, the product of the dimensionless Flory-Huggins parameter χ, describing the enthalpic interactions between statistical segments of type A and type B, and N, the total number of statistical segments on each diblock molecule. The second parameter is the molecular weight ratio f, the fraction of segments on each chain that are of type A. Periodic structures of lamellae, hexagonal arrays of cylinders, and body-centered-cubic (bcc) arrays of spheres, as well as a bicontinuous gyroid phase, appear in turn by changing χN or f. Extensive experimental and theoretical studies on diblock copolymers have been carried out,1-6 and a good understanding of the bulk phase behavior has been obtained. However, in many experiments, surface effects are usually present. The microstructure near the substrate may be different from the bulk structure because of the interaction between the diblock copolymers and the surface. In many cases surface effects may play a significant role, especially in microstructures of thin films.8-41 Surface-induced ordering of symmetric block copolymers, which form lamellar structures in the bulk, has been extensively studied in recent years. 11-22 Typically, one of the blocks is attracted to the substrate to minimize the surface energy, and lamellae are formed parallel to the substrate.21,22 The orientation of these lamellae persists over at least several lamellar periods. For the lamellar structure, a fairly simple physical picture can be used due to the one-dimensional nature of the structure. Theoretical predictions are available * Corresponding author. E-mail: [email protected].

for the lamellar structure.13-19 For asymmetric block copolymers with cylindrical bulk morphology, a number of studies on surface-induced ordering have also been carried out experimentally.23-37 Some of these studies focused on surface-induced orientational transition.23-29 In principle, cylinder domains often prefer to align parallel to the substrate to reduce the number of cylinder ends. Both parallel orientation and perpendicular orientation were observed due to the effects of surface field and confinement.25-28 The rest of the studies focused on structure reconstructions.25,26,30-37 A variety of structures near surfaces and in thin films have been observed, including a wetting layer,30 lamella,25 spherical microdomains,33 a perforated lamella,33 cylinders with necks,34 and more complicated structures.35,36 Theoretically, Turner and co-workers have studied the surface-induced lamellar ordering in a cylindrical bulk phase.32 Magerle and co-workers have studied the phase diagram of surface reconstructions both experimentally and theoretically,37 revealing that the surface-induced structures in thin films were determined by an interplay between surface field and confinement effects. In contrast to lamellar and cylindrical bulk phases, surface effects on the spherical bulk phase have received much less attention.38-41 Theoretically, Chen and Fredrickson applied self-consistent-field theory to study the thin film ordering in triblock copolymers.40 Pereira predicted the possibility for a bcc morphology to convert to a cylindrical morphology for bcc forming diblock copolymers confined in thin films.41 These valuable theoretical work focused on confinement effects or the interplay between surface field and confinement effects in thin films. To study the effects of the surface field, thick films were used to minimize the confinement effects. Experimentally, Kramer and co-workers studied the effects of surface field on laying of a spherical phase utilizing thick films.38,39 To understand the surface effects, theoretical studies are desirable. In particular, theory of surface effects on diblock copolymers with a bcc stable state in the bulk has been lacking to the best of our knowledge.

10.1021/ma049310i CCC: $27.50 © 2004 American Chemical Society Published on Web 11/12/2004

Macromolecules, Vol. 37, No. 25, 2004

Body-Centered-Cubic Phase of Diblock Copolymers 9647

This paper is devoted to the study of surface-induced ordering in bcc forming asymmetric diblock copolymers. Our study is carried out by employing the LandauBrazovskii model,42,43 which can be derived from the many-chain Edwards Hamiltonian for diblock copolymers in the weak segregation limit.1,2 Since we focus on the surface-induced behavior in the bcc phase close to the ordering transition, the weak segregation theory is appropriate. In section 2, we apply the single mode approximation, accurate at weak segregation, to the Landau-Brazovskii model, leading to an amplitude model studied previously in different contents.15,32,43 In section 3, we study the variation of the surface structure in bcc phase as a function of surface field strength and temperature. The results are summarized in a “phase diagram” of surface-induced reconstructions. Corresponding discussions are presented. Section 4 is devoted to the conclusions.

c, are defined by

1 (χN)s ) F(x*, fA) 2 c)

|

2 1 ∂ F(x,fA) 2 ∂x2

r1 ) q0r0

(9)

V ) q03V0

(10)

f)

F0 f0[φ(r0)] ≡ ) nkBT

{

}

2

ξ0 τ0 2 γ0 3 λ0 4 1 2 2 2 φ - φ + φ dr0 [(∇ + q )φ] + 0 V0 2 3! 4! 8q02 (1)

∫

Here, f0 is the Landau-Brazovskii free energy per block copolymer, ξ0 is the bare correlation length, q0 is the critical wavevector, τ0 is the reduced temperature, and γ0 and λ0 are expansion coefficients. For stability, λ0 > 0. Since the Landau-Brazovskii free energy can be derived from the many-chain Edwards Hamiltonian for diblock copolymers following the method of Leibler1 and Ohta and Kawasaki,2 we can express these parameters in terms of χN, fA, and Rg, which appeared in the Edwards model. Specifically, we have1-3,43 2

q0

x* ) 2 Rg

ξ02 ) 4x*cRg2

(4)

γ0 ) -NΓ3

(5)

λ0 ) NΓ4(0,0)

(6)

Here, Rg is the radius of gyration of the chains and x* is the position of the minimum of the function F(x,fA) appearing in the scattering function in ref 1. The disorder-to-order spinodal point, (χN)s, and the quantity,

(11)

(q0ξ0)2 ξ ) 4λ0

(12)

τ)

τ0 λ0

(13)

γ)

γ0 λ0

(14)

the Landau-Brazovskii free energy becomes

f [φ(r1)] )

{

}

ξ2 2 τ γ 1 1 [(∇ + 1)φ]2 + φ2 - φ3 + φ4 dr1 V 2 2 3! 4!

∫

(15)

For a bcc structure in the weak segregation regime, it is sufficient to restrict ourselves to the first mode in reciprocal space, whose reciprocal lattice vectors satisfy |Gi| ) q0 ) 2π/(b/x2) (b is the lattice parameter and |Gi| ) 1 in scaled units), instead of using the complete set of vectors Gi. For a bcc phase, the 12 shortest vectors Gi (in scaled units) may be chosen as

G1 ) G3 )

G7 )

1 1 (xˆ 1 + zˆ 1), G6 ) (xˆ 1 - zˆ 1) x2 x2

1 1 (- xˆ 1 + zˆ 1), G8 ) (- xˆ 1 - zˆ 1) x2 x2

G9 ) G11 )

1 1 (xˆ 1 + yˆ 1), G2 ) (xˆ 1 - yˆ 1) x2 x2

1 1 (- xˆ 1 - yˆ 1), G4 ) (- xˆ 1 + yˆ 1) x2 x2

(2) (3)

f0 λ0

2

G5 )

τ0 ) 2[(χN)s - χN]

(8)

x)x*

Finally, the vertex functions Γ3 and Γ4(0,0), which are functions of fA, are computed in ref 1. It is convenient to rescale eq 1 by expressing lengths in units of q0-1 and the free energy is in units of λ0. Under the rescalings

2. The Model We consider an incompressible melt of n AB diblock copolymers in a volume V0 at a temperature T. The total degree of polymerization of the diblock copolymer is N. The monomer density is F ) nN/V0. The degree of polymerization of the A block is fAN, where 0 e fA e 1. The order parameter of diblock copolymers is defined as φ(r0) ≡ φA(r0) - fA, the deviation of the local A monomer concentration from its average value. Because we are studying the surface-induced behavior near the ordering transition, which is in the weak segregation regime, the Landau-Brazovskii model can be used. Specifically, the Landau-Brazovskii free energy F0 is given by42,43

(7)

1 1 (zˆ 1 + yˆ 1), G10 ) (zˆ 1 - yˆ 1) x2 x2

1 1 (- zˆ 1 + yˆ 1), G12 ) (- zˆ 1 - yˆ 1) x2 x2

To study the surface-induced structures in the bcc phase explicitly, we choose the surface plane as the z ) 0 plane. We utilize particular epitaxial relations between the lamellar, hexagonal cylindrical, and bbc

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spherical structures and choose new orthonormal coordinates

xˆ )

1 (xˆ 1 - yˆ 1 + 2zˆ 1) x6

yˆ )

fbulk ) τ(a12 + 2a22 + 3a32) - 2γ(a1a22 + a1a32 + 1 2a2a32) + (a14 + 6a24 + 16a1a2a32 + 32a32a22 + 4 15a34 + 8a12a22 + 12a12a32) (18)

1 (xˆ 1 - yˆ 1 - zˆ 1) x3

zˆ )

For the equilibrium state of the bulk diblock copolymers, a1, a2, and a3 are position-independent, leading to a bulk free energy density

When a1 ) a2 ) a3 ) abcc, the structure is bcc spheres. The free energy corresponding to the bcc arrays of spheres is

1 (xˆ 1 + yˆ 1) x2

In the single mode approximation, the order parameter can be written as

˜ 1‚r) + 2a2(r)[cos(G ˜ 2‚r) + φ(r) ) 2a1(r) cos(G ˜ 4‚r) + cos(G ˜ 5‚r) + cos(G ˜ 3‚r)] +2a3(r)[cos(G cos(G ˜ 6‚r)] (16) with

fbcc ) 6τabcc2 - 8γabcc3 + with

abcc )

1 G ˜ 2 ) (x3xˆ + zˆ ) 2

(20)

15 a 4 4 hex

(21)

with

ahex )

1 (xˆ + x2yˆ ) G ˜4) x3 1 G ˜5) (xˆ - 2x2yˆ + x3zˆ ) 2x3 1 (-xˆ + 2x2yˆ + x3zˆ ) 2x3

where a1, a2, and a3 are space-dependent amplitude functions describing the variation of microstructure. Since the new coordinate is orthonormal, the form of the Landau-Brazovskii free energy is invariant. From eq 16, the structures of the lamellar component and the cylindrical components are parallel to the surface, which are chosen according to the well-established epitaxial relations.21,22,25 If we assume that the amplitudes a1, a2, and a3 are slowly varying functions on the scale of Ds, which is the period of the bcc phase, we can separate the length scale for variations in the amplitude from the length scale of the microstructure. Within this slowly varying amplitude approximation, the Landau-Brazovskii free energy can be written in terms of the amplitudes as

∫

2γ ( x4γ2 - 30τ 15

fhex ) 3τahex2 - 2γahex3 +

1 G ˜ 3 ) (-x3xˆ + zˆ ) 2

{

1 dr ξ2(∇2a1)2 + 2ξ2(∇2a2)2 + 3ξ2(∇2a3)2 + V ˜ 1·∇a1)2 + (G ˜ 2·∇a2)2 + (G ˜ 3·∇a2)2 + (G ˜ 4·∇a3)2 + 4ξ2[(G

f)

(19)

When a1 ) a2 ) ahex and a3 ) 0, the structure is the hexagonal arrays of cylinders. The free energy corresponding to the hexagonal arrays of cylinders is

G ˜ 1 ) zˆ

G ˜6)

45 a 4 2 bcc

˜ 6·∇a3)2] + τ(a12 + 2a22 + 3a32) (G ˜ 5·∇a3)2 + (G 1 2γ(a1a22 + a1a32 + 2a2a32) + (a14 + 6a24 + 4

}

16a1a2a32 + 32a32a22 + 15a34 + 8a12a22 + 12a12a32)

(17)

γ ( xγ2 - 10τ 5

(22)

For γ > 0, the solutions with positive roots in eqs 20 and 22 have lower free energies. The case for γ < 0 can be reconstructed from the γ > 0 case by recognizing that the free energies in eqs 19 and 21 are invariant under γ f -γ, abcc f -abcc, and ahex f -ahex. Define an effective temperature R ) τ/γ2. The effective transition temperature from the bcc arrays of spheres to the hexagonal arrays of cylinders is located by demanding fhex ) fbcc, leading to

R* ) -0.07345

(23)

When a diblock copolymer melt is put in contact with a surface, the translational symmetry is broken. To study the surface effect, we assumed that the surface is presented by a surface potential, leading to the free energy of the form

Ω)

∫

1 dr {[φ(r) U(r)] + f [φ(r)]} V

(24)

where U(r) is the reduced surface potential and f t[φ(r)] is the Landau-Brazovskii free energy shown in eq 15. We assume that the interaction of copolymers with the surface is short range. If surface lies in [x, y, 0] plane, U(r) can be written as U(r) ) -∆uδ(z), where ∆u is the differential affinity of surface with respect to polymer A.32 δ(z) is the delta function, which means the interaction is short range and only exists on the surface (z ) 0 plane). Moreover, amplitudes a1, a2, and a3 are assumed to be z-dependent. Because the amplitudes a1, a2, and a3 vary slowly on the scale of Ds as mentioned before, we only retain gradient terms up to quadratic order in the Landau-Brazovskii free energy. Under these assumptions, the free energy per unit surface can be



written as

Ω ) -2∆ua1(z)0) +

( )] ∂a3 ∂z

2

∫0∞dz

{ [( ) ( ) ξ2 4

∂a1 ∂z

2

+2

2

∂a2 ∂z

+2

+ τ(a12 + 2a22 + 3a32) - 2γ(a1a22 + a1a32 +

1 2a2a32) + (a14 + 6a24 + 16a1a2a32 + 32a32a22 + 4 15a34 + 8a12a22 + 12a12a32)

}

(25)

It is convenient to introduce the following new variables:

a1 ) γX, a2 ) γY a3 ) γZ, z )

2ξ t γ

˜ ∆u ) ξγ2σ, Ω ) 2ξγ3Ω Thus, X(t), Y(t), and Z(t) are related to the structures since they are proportional to a1(z), a2(z), and a3(z), respectively. Also, we introduce a reduced distance t and a reduced adsorption strength σ instead of z and ∆u, respectively. In terms of these new variables, the free energy density becomes

Ω ˜ ) -σX(0) +

∫0∞dt {(dX dt )

2

+

1 dY 2 1 dZ 2 + + 2 dt 2 dt

( )

( )

These equations for the variation of the surfaceinduced structure of diblock copolymers can be mapped to the motion of a classical particle of mass unity moving in the [X, Y, Z] space under the potential -f (X, Y, Z). The microstructure of the diblock copolymers corresponds to the position of the classical particle. We can draw an analogy between the solutions of the amplitude eqs 27-29 and the trajectory of a particle starting at a special point in the [X, Y, Z] space with a speed -σ parallel to the X-axis and terminating at [X(∞), Y(∞), Z(∞)] with zero speed. By introducing the effective surface field strength σ, which describes the effective interactions between polymers and surface, and the effective temperature R, which describes the bulk phase transition, the controlling parameters become σ and R instead of ∆u and T, respectively. In what follows, we will solve the above differential equations and discuss the results in the parameter space of (σ,R). 3. Surface-Induced Ordering In this section, we study how the surface-induced ordering is extended to the bulk structure. As a first step, we consider the case that the surface-induced structure does not deviate too far from the bulk phase; therefore, linearization of the Euler-Lagrange equations is valid. Within the linear approximation we obtain

∆X(t) ) X(t) - X(∞) ) σ[CX1(R)e-λ1(R)t +

}

f [X(t), Y(t), Z(t)] (26)

CX2(R)e-λ2(R)t + CX3(R)e-λ3(R)t] (34) ∆Y(t) ) Y(t) - Y(∞) ) σ[CY1(R)e-λ1(R)t +

where the function f (X,Y,Z) is given by

f (X,Y,Z) ) R(X2 + 2Y2 + 3Z2) - 2(XY2 + XZ2 + 1 2YZ2) + (X4 + 6Y4 + 16XYZ2 + 32Y2Z2 + 15Z4 + 4 8X2Y2 + 12X2Z2) Minimizing the new free energy per unit surface, we obtain three coupled Euler-Lagrange equations for the amplitudes X, Y, and Z

1 d2X ) (R + 2Y2 + 3Z2)X - Y2 - Z2 + 2YZ2 + X3 2 2 dt (27) d2Y ) dt2 (4R + 4X2 - 4X + 16Z2)Y - 4Z2 + 4XZ2 + 6Y3 (28) d2Z ) (6R - 4X - 8Y + 8XY + 16Y2 + 6X2)Z + 15Z3 dt2 (29)

CY2(R)e-λ2(R)t + CY3(R)e-λ3(R)t] (35) ∆Z(t) ) Z(t) - Z(∞) ) σ[CZ1(R)e-λ1(R)t + CZ2(R)e-λ2(R)t + CZ3(R)e-λ3(R)t] (36) We consider two different cases as follows. In the first case, we choose the effective temperature R ) -0.02 (far from the effective transition temperature R* ) -0.07345 in eq 23). In this case, we obtain λ1 ) 1.7046, λ2 ) 1.1034, and λ3 ) 0.5997. The corresponding coefficients are CX1 ) 0.00278, CX2 ) 0.1104, CX3 ) 1.456, CY1 ) 0.03268, CY2 ) 0.3496, CY3 ) 0.5503, CZ1 ) 0.04671, CZ2 ) 0.2314, and CZ3 ) 0.5585. These asymptotic solutions can be used as initial conditions to obtain exact numerical solutions of eqs 27-29, and we obtain amplitudes X(t), Y(t), and Z(t) varying with t for a given σ. We choose σ ) 0.9 and σ ) 2 to describe different surface field

which must be solved with the boundary conditions

|

dX ) -σ dt t)0 dY )0 dt t)0 dZ )0 dt t)0

| |

X(∞) ) Y(∞) ) Z(∞) )

2 + x4 - 30R 15

(30) (31) (32) (33)

Figure 1. Reduced amplitudes X(t), Y(t), and Z(t) as functions of t for R ) -0.02, σ ) 0.9 (solid lines), and 2 (dash lines).

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Tan et al.


Figure 2. (a) Three-dimensional contour plot of the order parameter φ(x, y, z) in the surface region with amplitudes from the solid lines in Figure 1. z is taken as 4πt (in length unit of q0-1). (b) Two-dimensional plot of the order parameter φ(2, y, z) in the same surface region. (c) Two-dimensional plot of the order parameter φ(x, 1, z) in the same surface region.

strengths, and the results are shown in Figure 1 by solid lines and dash lines, respectively. In the second case, we choose R ) -0.07 (close to R* ) -0.07345). In this case, we obtain λ1 ) 1.881, λ2 ) 1.092, and λ3 ) 0.5855. The corresponding coefficients are CX1 ) 0.00467, CX2 ) 0.1469, CX3 ) 1.419, CY1 )

0.04054, CY2 ) 0.3989, CY3 ) 0.6136, CZ1 ) 0.05729, CZ2 ) 0.2583, and CZ3 ) 0.6658. Following the same process as above, we obtain amplitudes X(t), Y(t), and Z(t) varying with t. The curves are quite close to those shown in Figure 1. To save the space, we do not show the results here.



Figure 3. (a) Three-dimensional contour plot of the order parameter φ(x, y, z) in the surface region with amplitudes from the dash lines in Figure 1. z is taken as 4πt (in length unit of q0-1). (b) Two-dimensional plot of the order parameter φ(2, y, z) in the same surface region. (c) Two-dimensional plot of the order parameter φ(x, 1, z) in the same surface region.

These results mean that changing R does not lead to appreciable changes in X(t), Y(t), and Z(t). However, changing σ leads to large changes in X(t), Y(t), and Z(t). This illustrates that the effective surface field strength plays a primary role in the surface-induced structures,

whereas the effective temperature R has small effect on the surface-induced structures. By eq 16 and the relations between a1, a2, a3 and X, Y, Z, we find that when X ) Y ) Z * 0, the structure is spherical phase. When X ) Y * 0 and Z ) 0, the

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Figure 4. “Phase diagram”, i.e., reduced absorption strength σ vs reduced distance t, at two different effective temperatures. a(R,σ) and b(R,σ) are defined as the distances from the surface (t ) 0) to the points where amplitudes Y(t) and Z(t) reach 20% of their bulk amplitudes, respectively. The solid lines and dash lines are for R ) -0.02 and R ) -0.07, respectively.

structure is cylindrical phase. When X * 0 and Y ) Z ) 0, the structure is lamellar phase. To demonstrate the extent of the surface-induced structure explicitly, real space profiles of the system are presented in Figure 2a-c and Figure 3a-c for σ ) 0.9 and 2.0, respectively. From Figure 2a-c, we find that small value of σ leads to surface-induced structures varying from cylinder-like structures to distorted spherical structures and then to spherical structures. From Figure 3a-c, we can find that large value of σ leads to surface-induced structures varying from lamella-like structures to cylinder-like structures, then to distorted spherical structures, and finally to spherical structures. Comparing Figure 2a-c with Figure 3a-c, we find that for the larger σ, the lamellar and cylindrical characters are more profound in the surface-induced region. At the same time, the width of the surface-induced region increases as a function of σ, as clearly shown in Figure 1. To illustrate the structure variation with surface adsorption strength, we construct a “phase diagram”, as shown in Figure 4. This “phase diagram” is obtained by defining a(R,σ) and b(R,σ) as the distances from the surface (t ) 0) to the points where the amplitudes Y(t) and Z(t) reach 20% of their bulk values, respectively. Such a “phase diagram” qualitatively describes surfaceinduced reconstructions. From Figure 4, we observe that if the surface field is weak enough, the bcc phase extends throughout the space. For intermediate surface field strength, cylinder-like structures appear near the surface and then convert to the bcc spherical structure far from the surface. If the surface field is strong enough, the lamella-like and cylinder-like structures appear in sequence near the surface and then convert to the bcc spherical structure in the bulk. It should be noticed that the parameter R has less effect on the surface-induced structures. Therefore, the surfaceinduced reconstruction is primarily determined by the interaction between polymer and surface. The surface-induced reconstructions in the bcc spherical phase of diblock copolymers are analogous to the surface-induced structures in the cylindrical phase of block copolymers. In the present work, the surfaceinduced lamella-like and cylinder-like structures in the bcc spherical bulk phase correspond to the surfaceinduced lamellar ordering in the cylindrical bulk phase, as reported theoretically and experimentally.25,32,37 So far, a systematic experimental study of surface effects on surface reconstructions in the bcc spherical phase of diblock copolymers is lacking. We expect that the


surface-induced structures predicted here should be observable in an appropriate experiment. Furthermore, there are lamellar and cylindrical components in the bcc spherical phase, and we assumed that the lamellar and cylindrical components parallel to the surface according to the experimental and theoretical evidences. This epitaxial relation leads to that the (110) plane of a bcc structure is parallel to the surface. If the surface field is weak enough, the surface-induced zone becomes insignificant (see Figure 4), leading to a direct substrate to bcc structure with the (110) plane parallel to the surface. This structural relation is consistent with the experimental observations obtained by Kramer and co-workers.38,39 4. Conclusions In summary, the extent of the surface-induced structure is studied based on the Landau-Brazovskii theory. In the weak segregation limit, the extent of the surface region is determined by the variation of the amplitudes along the direction perpendicular to the surface. For a weak surface field, the surface-induced structures vary from cylinder-like structures to distorted spherical structures and then to spherical structures. For a stronger surface field, the surface-induced structures vary from lamella-like structures to cylinder-like structures, then to distorted spherical structures, and finally to spherical structures. In other words, the lamellar and cylindrical characters are much more profound, and the surface-induced structure persists to a larger extent for the stronger surface field. Moreover, the surface-induced reconstructions are primarily determined by the surface field. Besides the prediction about the width of the surface-induced region, details of the real space structures are calculated. The surface-induced structures follow these specific deformation modes. These predictions can be compared directly with experiments. These results are summarized in a “phase diagram”, which is constructed from the distances at which the lamellar (a(R,σ)) and cylindrical (b(R,σ)) features become appreciable. We hope that this “phase diagram” will provide guidance to future experiments. Acknowledgment. We thank Prof. Shesheng Zhang and Prof. Charles C. Han for many helpful discussions. We acknowledge the support from National Natural Science Foundation of China (NSFC) 20340420327, 90103037, and 20490220. A.-C.S. acknowledges the support from Natural Science and Engineering Research Council (NSERC) of Canada. References and Notes (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

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Surface Effect on the Body-Centered-Cubic Phase of Diblock

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