Diamond-Forming Block Copolymers and Diamond-like Morphologies

Article pubs.acs.org/Macromolecules

Diamond-Forming Block Copolymers and Diamond-like Morphologies: A New Route toward Efficient Block Copolymer Membranes I. Erukhimovich,*,† Y. Kriksin,‡ and G. ten Brinke§ †

Nesmeyanov Institute of Organoelement Compounds, RAS, Moscow 119991, Russia Keldysh Institute of Applied Mathematics, RAS, Moscow 125047, Russia § Department of Polymer Chemistry and Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands ‡

S Supporting Information *

ABSTRACT: A new promising class of ordered block copolymer morphologies in thin films we call diamond-like morphologies (DLM) is theoretically described in detail. This class comprises the cubic morphologies possessing diamond symmetry Fd3̅m as well as its tetragonal and orthorhombic generalizations. The characteristic property of the DLM is that the corresponding percolation cluster contains both perpendicular and parallel (with respect to the substrate) intertwining channels, which is expected to improve noticeably the permeability efficiency of such quasi-isotropic morphologies as compared to the anisotropic lamellar and cylindrical ones. On the basis of both the weak segregation theory (WST) analytical consideration and self-consistent field theory (SCFT) numerical procedure, we find relationships between the microscopic parameters of the ternary ABC symmetric block copolymers of a specified composition and their morphology in thin films. The effects of confinement (film width) and the film walls selectivity are analyzed, and the corresponding phase diagrams are built. Our key theoretical prediction is that the way the selectivity is distributed over the film substrate makes a drastic difference in the selectivity influence on the various phases’ stability. The homogeneous selectivity distribution enhances the parallel lamellae stability. On the contrary, a simple 1D (lamellar) selectivity modulation could (depending on the modulation period) enhance the DLM stability. The explicit recommendations are made how to facilitate the DLM stabilizing in real block copolymers. We address also the issue of spectral and 3D visual ways to identify the SCFT equations solutions in thin films as the DLM.

1. INTRODUCTION

The typical morphologies under consideration in the existing literature are the one-dimensional (1D) lamellar and twodimensional (2D) cylindrical morphologies, the competition of the perpendicular and parallel lamellae (Figure 1) as well as perpendicular and parallel cylinders (Figure 2) being the main issue under study. Generally, an increase of incommensurability between the bulk value of the lamellar period L and the film width H favors the perpendicular morphologies whereas increase of the wall selectivity as to any of the block copolymer components does parallel ones. The corresponding phase diagrams were obtained both for microscopic and semiphenomenological models of diblock copolymers via theoretical calculations based on the weak segregation theory (WST) and self-consistent field theory (SCFT) as well as via computer simulations (see, e.g., refs 17−23 and references therein). Some richer situations including the

One of the promising routes to create isoporous membranes with desired transport properties is to search for the proper ordered (self-assembled) block copolymer nanostructures in thin films.1−5 Indeed, such thin films would relieve the corresponding spatial periodicity of their local permeability and, thus, mimic naturally the desired isoporousity (for the basic facts on the order−disorder transition in block copolymers in both bulk and thin films, see, e.g., refs 6−16 and references therein). Usually, the channels providing the transport of the desired species through the block copolymer membranes are formed by (or on the base of) the regions filled by the minority component. It could be done, e.g., via etching and subsequent removing all blocks of a specified sort. Alternatively, if a species has a thermodynamic affinity to the repeated units forming blocks of certain sort, then the species accumulates and moves basically within the domains formed by the units of this sort. In any case, the perpendicular morphologies, where the channels span across the whole film, are obviously much more permeable than the parallel one. © XXXX American Chemical Society

Received: July 9, 2015 Revised: October 2, 2015

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DOI: 10.1021/acs.macromol.5b01513 Macromolecules XXXX, XXX, XXX−XXX

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conducting channels characteristic of bicontinual 3D morphologies. To specify the set of the desired morphologies, we make use of the following remark by I. Lifshitz:26,27 “For a complex morphology to exist it should possess both (i) thermodynamic stability and (ii) kinetic attainability”. Stated originally for the protein folding problem, as applied to our case it means that the desired ordered bicontinual 3D morphology should both (i) evolve continuously (i.e., via the second-order phase transition) from the disordered state and (ii) be more thermodynamic stable than the lamellar and cylindrical morphologies. An additional requirement would be that the wall selectivity as to one of the blocks should not affect considerably the thermodynamic equilibrium between the quasi-isotropic (cubic) and anisotropic morphologies in favor of the latter. The purpose of our paper is to illustrate this new route with the diamond (D) and diamond-like morphologies (DLM) (theoretically predicted to exist first for a phenomenological model28 and next in melts of the linear ternary ABC-like block copolymers29−33 with the middle block effectively nonselective with respect to the end blocks). A general visual idea of the DLM is presented in Figure 4, and the quantitative description of the DLM in various films is given throughout the paper.

Figure 1. Lamellar morphologies with different orientation of the layers with respect to film walls: the perpendicular (left) and parallel (right) ones. H is the film width; L and L⊥ (see definition of L⊥ in section 3) are the lamellar periods in the bulk and for the parallel lamella, respectively. The bold solid lines correspond to the film walls.

Figure 2. Cylindrical morphologies with different orientation of the layers with respect to film walls: the perpendicular (left) and parallel (right) ones. Reproduced with permission from ref 20. Copyright 2000 AIP Publishing.

formation of nonbulk-like morphologies have been also studied (see, e.g., refs 24 and 25 and references therein). However, the problem cannot be reduced just to comparison of the free energies for these two types of morphologies only. In fact, the competition between the tendencies to fit both the wall confinement and wall selectivity could result, too, in formation of mixed (two-phase) morphologies shown in Figure 3. Indeed, a

Figure 4. Two types of the diamond morphology visualization: (a, b) show the order parameter profile over the walls of a properly selected cell; cross sections with channels laying in the lateral (a) and normal (b) planes are clearly seen; (c, d) are the intermaterial dividing surfaces (i.e., the surfaces that correspond to the zero level of the order parameter) seen from different directions. See more explanations in the text.

Figure 3. Mixed (two-phase) lamellar morphologies. Left: a cartoon for one selective and one neutral wall. Reproduced with permission from ref 17. Copyright 1997 AIP Publishing. Right: a snapshot from Monte Carlo simulation. Reproduced with permission from ref 19. Copyright 2000 AIP Publishing.

The further presentation is organized as follows. In section 2 we remind the reader some basics on the D phase and its thermodynamic stability in the bulk. Next, we present our results on the DLM in thin films with both nonselective and uniformly selective walls in sections 3 and 4, respectively. The key point of our discussion is section 5, where the idea of a completely new effect, which is the DLM stability enhancement due to a lamellarlike patterning the substrate, and some related preliminary results are presented. The final discussion of the results is given in section 6.

parallel lamellar of a self-adjusted width formed along the selective film wall would provide an energetic win whereas the perpendicular lamellar formed in the rest of the film would minimize the loss due to confinement. It is important to realize that as soon as a mixed state including some parallel layers adsorbed to the selective wall is formed in a thin f ilm, its transport ability degrades considerably. Thus, due to instability with respect to formation “parasitic” parallel layers, the strongly anisotropic (1D and 2D) perpendicular block copolymer morphologies are not really suitable to form the membranes with good transport properties in thin films with even weakly selective walls. Instead, a natural new route to design efficient block copolymer membranes would be to look for morphologies that possess an isotropic (at least, quasi-isotropic, which means cubic symmetry) structure of

2. DIAMOND MORPHOLOGY IN THE BULK To begin with, let us remember that the specific free energies of the perpendicular lamellar and cylinder morphologies in thin films are the same as those in the bulk. Thus, a quasi-isotropic (cubic) phase can only be preferable as compared to lamellar or B


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Macromolecules ⎧ ⎛ 2πx ⎞ ⎛ 2π (z + y) ⎞ Ψ(r) = 2A ⎨cos⎜ ⎟ ⎟ cos⎜ LD ⎠ ⎩ ⎝ LD ⎠ ⎝

cylinder morphologies in thin films if it is dominant in the bulk. So, before considering the DLM in thin films, we remind the reader some facts on the D phase in the bulk. Diamond Phase Morphology in the Bulk. The block copolymer morphology possessing the diamond symmetry Fd3m ̅ (space group no. 227) has been first (even though implicitly) discussed by Ludwik Leibler in his seminal paper on microphase separation in AB diblock copolymer melts.34 Nowadays, it is convenient to summarize our understanding as follows. Among other weak segregated morphologies, the order parameter profile for the face centered cubic family morphologies can be written as follows:

⎪

⎪

⎛ 2πx ⎞ ⎛ 2π (z − y) ⎞⎫ − sin⎜ ⎟ sin⎜ ⎟⎬ LD ⎝ LD ⎠ ⎝ ⎠⎭

⎪

⎪

(two equivalent expressions can be also obtained by permutations of the coordinates x, y, z). Here LD =

k=1

(1)

where Ψ(r) = (ϕA (r) − ϕB(r))/2

(1a)

(ϕi(r), i = A, B are the local volume fractions of the repeated units of the ith sort at the point r). The four vectors qk that make (with four inverse vectors −qk) the first coordination sphere of the reciprocal lattice read

q k = q nk *

(2)

where q* is the wavenumber of the critical fluctuations destroying the disordered state stability34,35 and the four unit vectors ni point out to the nodes of the first coordination sphere of the reciprocal lattice conjugated to the face-centered cubic Bravais lattice and form the tetrahedral angles (ninj = −1/3). Say, for definiteness, ( −1, 1, −1) (1, −1, −1) , , n2 = 3 3 (1, 1, 1) ( −1, −1, 1) , n4 = n3 = 3 3

n1 =

(3)

At last, the Ωk are the phase shifts of the corresponding plane composition (order parameter) waves. Two different choices of these phase shifts Ω1 = Ω 2 = Ω3 = Ω4 = 0

(4a)

Ω1 = Ω 2 = Ω3 = 0,

(4b)

Ω4 = π

3L

(6)

is the diamond period along of any axes (remember that L = 2π/ q* is the lamellar period in the bulk) . The relationship (6) is crucial for the effect we discuss in section 5. Diamond Phase Stability in the Bulk. Within the weak segregation realm, the diamond (D) morphology generated by the choice (4b) is always more stable (corresponds to the lower free energy) than the FCC morphology generated by the choice (4a). But for molten diblock (and various other) copolymers even the D morphology has higher free energy than the lamellar, hexagonal, and body-centered cubic ones; i.e., it is always only metastable. However, some architectures were shown,29−33 indeed, to reveal well permeable nonconventional bulk morphologies. To such architectures belongs, e.g., the class of symmetric ternary linear ABC triblock copolymer melts with the χ-parameters satisfying the conditions χAB = χBC = χAC/4 that correspond to the well-known Hildebrand approximation (see, e.g., discussion in ref 29). Obviously, the middle block in such copolymers is nonselective with respect to the side ones. This class comprises well studied poly(isoprene-b-styrene-b-2-vinylpyridine)36−38 and poly(isoprene-b-styrene-b-ethylene oxide)39,40 copolymers. Naturally enough, a similar phase behavior has been also found 30,31,33 in two-scale multiblock AB copolymers AmN/2(BN/2AN/2)nBmN/2 that mimic the ABC ternary copolymers with the proper composition. For the sake of simplicity, we focus in the present paper on the ternary linear ABC triblock copolymer melts with a middle block nonselective as to the side ones. For these copolymers the order parameter definition (1a) is slightly modified:

k=4

Ψ(r) = A ∑ cos(q kr + Ωk)

(5)

Ψ(r) = (ϕA (r) − ϕC(r))/2

(1b)

where ϕi(r), i = A, B, C, are the local volume fractions of the repeated units of the ith sort at the point r, with A and C corresponding to the side blocks. The corresponding phase diagram calculated via the SCFT32 for symmetric ABC block copolymers with χAB = χBC = χAC/4 is presented in Figure 5. In what follows we study, for definiteness, the system whose compositions and χAC lays inside the region DA in Figure 5:

generate different extinction rules33 or, in other words, different structures of the reciprocal lattices and therefore the morphologies possessing different symmetries. Namely, the choice (4a) corresponds to the face-centered cubic (FCC) symmetry Fm3̅m (space group no. 225) whereas (4b) does to the diamond Fd3̅m (space group no. 227) one. Therewith, the structures of the first coordination spheres of the reciprocal lattices for the FCC and D morphologies are identical, which prompted Leibler34 and one of us28,29 not to differentiate (somewhat misleadingly) these morphologies within the weak segregation approach. The distinction has been recognized explicitly only recently.32,33 It follows from eqs 1−3 that the order parameter profile for the weakly segregated D phase reads

fA = fC = 0.15,

fB = 0.7,

χAC N = 66

(7)

To avoid any ambiguity, we specify that the diamond (D), alternating gyroid (GA), and simple cubic (SC) phases correspond to the symmetries Fd3m ̅ (space group no. 227), I4132 (space group no. 214), and Pm3m (space group no. 221), respectively. It is worth to notice that as we argued earlier,33 the permeability of the single (alternating) gyroid GA and the D phases is much higher than that of the SC one. However, it is only the D phase that is compatible with the boundary conditions on the nonselective walls (see below), and accordingly, it is this fact that determines the special role of the diamond morphology for thin films. C


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the bulk that possess the mirror planes, the film walls and width corresponding respectively to any pair of the parallel mirror planes and the distance between the latter. In other words, the quantity q should obey the reflecting boundary conditions on the film boundaries (walls): n ∂q/∂r|z = 0, D = 0

where n is the unit vector parallel to the z-axis (and, thus, normal to the film walls). It is worth to make a comment here. Strictly speaking, the account of the film surfaces via the reflecting boundary condition 9 for the diffusion eq 8 is an approximation only. Indeed, the density derived from the exact solution of this diffusion equation is expected to drop smoothly to zero,17,18 and Wu et al. suggest that some mixed boundary conditions are more appropriate.47 On the other hand, the very diffusion eq 8 is also an approximation only, which does not allow for discrete nature of polymer chains. A more correct (and sophisticated) way to describe both the connectivity and discreteness of polymer chains is based on the Lifshitz integral equation,26 which enables ones to see27,48 that sometimes the density profile could undergo a discrete jump instead of a smooth dropping to zero. Summarizing, the combination of the diffusion eq 8 and reflecting boundary condition 9 is both reasonable and simple enough to be used for the further SCFT calculations in the present paper, even though, in general, the rigorous way to allow for the film confinement still stays an open issue. Now, one can see easily (e.g., via straightforward calculation) that the phase GA, which is described in detail in ref 29, does not possess any mirror planes. Therefore, the single gyroid morphology is incompatible with the boundary conditions on the film walls. D Morphology in Thin Films (Commensurability Case). On the contrary, the D morphology characterized by the order parameter (1) does possess six families of the mirror planes:

Figure 5. SCFT phase diagram of the melts of symmetric ternary linear ABC block copolymers with a nonselective middle block in the plane (composition of the middle block f B; effective Flory−Huggins parameter χ̃AC) at χAB = χBC = χAC/4. The stability regions for the lamellar, diamond, and alternating gyroid, tetragonal planar, and simple cubic phases are labeled as L, D, GA, CA, and SA, respectively. The dashed line corresponds to the transition to the disordered phase. The coordinates of the points of intersection between the dashed and bold lines correspond to the weak segregation theory29 predictions. Reproduced with with permission from ref 32.

3. DIAMOND AND DIAMOND-LIKE MORPHOLOGIES IN THIN FILMS WITH NONSELECTIVE HOMOGENEOUS WALLS Theory and Boundary Conditions in Thin Films. To calculate the equilibrium composition profiles and the corresponding free energies of the relevant phases needed to build the phase diagrams, we make use of the current standard technique, which is the weak segregation theory (WST), first elaborated by Leibler for diblock copolymers34 and later modified by one of us for ternary triblock copolymers,29 as well as self-consistent field theory (SCFT) elaborated by Matsen.41,42 More specifically, for the SCFT numerical procedure we are using the pseudospectral version43,44 appended by the Ng iterative method.45 Besides, to speed up the calculations, we choose the initial guesses for the iterative procedure based on the information on the plausible morphologies provided by the weak segregation theory.30,31,33 For brevity, we omit here a more detailed description of the SCFT procedure, which is already given, e.g., in refs 13, 31, 33, and 41−44. However, to describe some peculiarities in the way we make use of the SCFT in thin films more clearly, it makes sense to write out here the corresponding master equation (modified diffusion equation): ∂q(r, s)/∂s = ∇2 q(r, s) − wα(r)q(r, s)

(9)

y − z = m1LD /2,

y + z = (2n1 + 1)LD /4

(10a)

x − y = m2LD /2,

x + y = (2n2 + 1)LD /4

(10b)

z − x = m3LD /2,

z + x = (2n3 + 1)LD /4

(10c)

(mi, nj are integers, i, j = 1, 2, 3) and, thus, can be embedded into a thin film. With due regard for the film symmetry it is convenient to replace the coordinates via the rotation x1 = x ,

y1 = (y + z)/ 2 ,

z1 = (y − z)/ 2

(11)

the z1-axes being directed normal to the walls. As the result, two of the vectors (2) become parallel to the walls, whereas the plane determined by two other vectors is normal to the walls:

(8)

q1± = q (c0 , ±s0 , 0), q±2 = q ( −c0 , 0, ±s0), * * c0 = cos θ0 , s0 = sin θ0

where the auxiliary function q(r,s), which satisfies the initial condition q(r,0) = 1, is involved in the calculation of both the chain partition function and local density ϕ(r,s) at the point r of all monomer units located in the position s (0 ≤ s ≤ 1) along the chain; α = α(s) and wα(r) is a self-consistent field applied to such monomers of the kind α = A, B, C at the point r.13,31,33,41−44 Equation 8 is to be appended by some boundary conditions. In the bulk we impose the periodic boundary conditions corresponding to the chosen Bravais lattice. Unlike that to study thin films in the present paper, we follow Angerman et al.,22 who adopted the Silverberg argument46 and suggested that block copolymer morphologies in films are just those morphologies in

(12)

where 2θ0 is the tetrahedral angle (cos(2θ0) = −1/3). Accordingly, the weak segregation (one-harmonic) order parameter in the new coordinates reads Ψ(r) = 2A(cos(2πx1/L ) cos(2πz1/L⊥) + sin(2πx1/L ) sin(2πy1 /L⊥))

(13)

where D


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Macromolecules L = L /c0 ,

L⊥ = L /s0

(14)

Taking into account definition (12) and that of the tetrahedral angle (c0 = 1/√3), one gets L∥ = √2L⊥ = LD. It is the function (13) (but the factor 2A) which is visualized in Figure 4 in the reduced coordinates x̃ = 2πx1/L∥, ỹ = 2πy1/L⊥, and z̃ = 2πz1/L⊥. A remarkable feature of the diamond morphology shown in Figure 4a,b is that the order parameter pattern over the cross sections normal to the x-axes is rotating on 90° every quarter of the period along the x-axes. In other words, the parallel and perpendicular channels are alternating along the axes. This fact clearly demonstrates quasi-isotropic nature of the D phase. We see from (13) that the planes z1 = 0 and z1 = mL⊥/2 satisfy the boundary conditions (9), and thus, the set of the film widths commensurable with the genuine diamond morphology is Lz = H0(m) = L⊥m/2

Figure 6. Dependence of the DLM periods of the film width H. The WST data calculated via eqs 17 and 18 and those calculated via the SCFT procedure are presented by lines and symbols, respectively. The solid curves (solid circles) and dashed lines (open circles) correspond to L∥ and L⊥, respectively. The curves are labeled by the number of the halfperiods m.

(15)

the positive integer m being the number of half-periods across the film. Note that unlike ref 22, we admit existence of both symmetric (m is even) and asymmetric (m is odd) morphologies.49 Then the planes Lz = 0 and Lz = L⊥m/2 coincide with the mirror planes of the diamond morphology. In this case the diamond morphology in the film is not distorted as compared to the bulk one, and we can choose the calculation cell as a rectangular parallelepiped with the dimensions L = LD ,

L⊥ = LD / 2 ,

Lz = H0(m)

reduced value of L we calculated at χ̃ = 66 (L = 3.4508) is somewhat bigger than the value (19) calculated at the spinodal (χ̃ = 56). These results have a simple physical meaning: when described within the WST, the diamond structure confined into an incommensurable film keeps its topological (permeability) properties but changes symmetry from a cubic to tetragonal one. Therewith, the length L⊥, which is just that of the double half-period along the normal to the walls, characterizes the periods along both z1- and y1-axes. The corresponding morphology we call the diamond-like morphology (DLM). Therewith, the morphologies with m half-layers across the film are designated Dm. When we calculate the periods within the SCFT approach (beyond the WST) we start with the initial (oneharmonic) description of the DLM via the order parameter (13) and next obtain the higher harmonics and equilibrium values of L∥ and L⊥ via the iteration solution of the SCFT equations. Incommensurability WST Case: The DLM Free Energy. As discussed in refs 29, 30, and 33, where the diamond phase was still mixed up with the FCC phase, with due regard for the D phase symmetry its excess (as compared to the disordered state) free energy reads

(16)

DLM in Thin Films. Incommensurability Weak Segregation Case: Tetragonal Structure. If a film width is an arbitrary positive value H that does not satisfy equality (15), then the cell dimensions Lx and Ly differ, generally, from the bulk values (16). Within the WST, these dimensions still satisfy eqs 14, but the subscript “0”, which is to be dropped. Instead, the angle θ is to be related to the film width via the straightforward generalization of the equalities (14) and (15): q smH = mπ → sm = sin θm = mL /(2H ) (17) * and, accordingly, the periods L⊥ and L∥ are to be found via the relationships L (m) = L /cos θm ,

L⊥(m) = L /sin θm = 2H /m

(18)

Remember, that for definiteness, we consider further in this paper the ternary symmetric ABC block copolymers that have the compositions (7) fA = fC = 0.15,

ΔFD = −τ 2/(4βD)

fB = 0.7

(20)

where

and satisfy the conditions χAB = χBC = χAC/4. Within the WST the dependences (18) hold at the spinodal, which for compositions (7) is located at χ̃ = χACN ≈ 56. They are shown in Figure 6. Besides, we present in Figure 6 the data calculated via the SCFT numerical procedure at χ̃ = χACN = 66. All lengths are measured in the units of the radius of gyration Rg = (Na2/6)1/2 for the whole ABC chain. Therewith, the reduced WST value of the lamellar period L in the bulk appearing in (18) reads

βD = {λ 0(0) + 6λ 0(4/3) − 2λ(4/3, 4/3)}/16

(21)

34

and the Leibler designations for the fourth vertices are used. The free energy of the DLM in thin films with m half-layers is also described by similar expression

ΔFDLM = −τ 2/(4βD)

(20a)

where the effective fourth vortex could be found similar to eq 20 of ref 29 (see Supporting Information for detailed calculation):

L = 2π /x = 3.2854 (19) * where x* = q*Rg is the reduced critical wavenumber we calculated for compositions (18) as described earlier.29 As seen from Figure 6, the WST and SCFT data for the DLM lattice scales fit each other rather well. A noticeable (and increasing with H increase) shift of the L∥ data to the right is simply related to the fact that the reduced SCFT value for

βDLM (m , H ) = {λ 0(0) + 2λ 0(4sm 2) + 4λ 0(2sm 2) − 2λ(2sm 2 , 2sm 2)}/16

(21a)

where the parameter sm is related to the film width H and the number m via eq 16. Taking into account that the excess free energy of the lamellar phase reads34 E


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Macromolecules ΔFL = −τ 2/(4βL),

βL = λ 0(0)/4

orthorhombic nature of the DLM formed in film is illustrated in Figure 8 representing the normal and lateral amplitudes An and Al in eq 13a as functions of the film width H.

(20b)

we find the conditions for the phase transitions between the lamellar perpendicular phase and the DLM structures with neighboring numbers m: βDLM (m , H ) = βDLM (m + 1, H ),

βDLM (m , H ) = βL (22)

In Figure 7 we present the plots of the reduced DLM free energy F ̃ = ΔFDLM /|ΔFL|

(23)

Figure 8. Dependence of the amplitudes An and Al (solid and dashed, respectively) in eq 13a and the ratio η = An/Al (short-dashed) on the film width H for diamond-like phases (Dm). The vertical dotted lines demarcate the regions of stability of the Dm phases.

Accordingly, the lateral and normal wavenumbers y

ql = 2π((L )−2 + (L⊥1)−2 )1/2 , Figure 7. Plots of the reduced free energy (23) for the DLM consisting of m half-layers calculated within the WST (solid lines) and SCFT (full circles) and labeled by the corresponding numbers m. The horizontal solid line F̃ = −1 corresponds to the perpendicular lamellar phase; the dashed lines are the guides for eyes.

qn = 2π((L )−2 + (L⊥z1)−2 )1/2 = 2π((L )−2 + (m/(2H ))2 )1/2 (24)

are presented as the functions of the film width H in Figure 9.

One can see from Figure 7 that the WST and SCFT data are in quite reasonable agreement (at least, at the semiquantitative level), even though the WST clearly overestimates stability of the DLM. A shift to the right of the SCFT data is due to the same reason as in Figure 6. As usual, the WST provides a deeper understanding of the physics underlying formation of the nonconventional ordered phases whereas the SCFT is more accurate. Incommensurability SCFT Case: Orthorhombic Structure. Remarkably, even though the SCFT data for L⊥ (open circles in Figure 6) deviate from those of the WST only slightly, this deviation is crucial. Indeed, a small but noticeable (rather than zero) curvature of the corresponding curves implies that the periods along the z1- and y1-axes are different. In other words, the DLM is in fact an orthorhombic rather than tetragonal structure when we treat it within the more precise SCFT procedure. It means that, in fact, even the initial try for the order parameter now loses its symmetry (13) between the lateral and normal modes. Instead, it should be written as follows: Ψ(r) = Al sin(2πx1/L )

Figure 9. Dependence of the wavenumbers qn and ql (solid and dashed, respectively) defined by eqs 24 and the ratio η = ql/qn (short-dashed) on the film width H for diamond-like phases (Dm). The vertical dotted lines demarcate the regions of stability of the Dm phases.

The nondistorted value of the ratio L∥/L⊥ = √2 (ql = qn, see Figure 9 and eq 16) and the equality Al = An of the amplitudes appearing in eq 13a take place simultaneously only if the film width H is multiple to one-half of the period observable in bulk for the D phase. One can check that the approximate equality Al ≈ An holds in the inner part of the regions of stability of the Dm phases. The DLM is mostly distorted (as compared to the genuine D morphology) for the thinner films (D1) mostly constrained as compared to the bulk conditions. In spite of the symmetry changes (cubic−tetragonal− orthorhombic) and other distortions the phases Dm keep the quasi-isotropic permeability and stay topologically equivalent to the D morphology in bulk. It is seen from Figure 10, where some

y sin(2πy1 /L⊥1)

+ A n cos(2πx1/L ) cos(2πz1/L⊥z1)

(13a)

Lz⊥1

where = 2H/m is the length of one period (i.e., that of two y half-periods) and L∥, L⊥1 are to be found via the free energy minimization. Our SCFT calculations are carried out at χ̃ = χACN = 66 for the ternary symmetric ABC block copolymers that have compositions (7) and satisfy the Hildebrand conditions χAC = 4χAB = 4χBC. All energetic values and the values of length in the following text are measured in the units of kBT and the radius of gyration Rg for the whole ABC chain, respectively. The F


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Figure 10. Isosurface ϕA(r) = 0.15 of diamond-like phase D2: (a) L∥ = 12.38, L⊥ = 3.59, H = 3.51; (b) L∥ = 5.98, L⊥ = 3.59, H = 4.23; (c) L∥ = 4.74, L⊥ = 4.97, H = 5.21. The case (b) corresponds to the D morphology in bulk.

direction in order to take into account the reflective boundary conditions (9). To relate the strengths of the surface fields εα appearing in eq 26 to the conventional surface tension, let us remember that the presence of an external surface field γα(r) and eq 26 result in an extra surface term in the free energy

isosurfaces are presented for the order parameter profiles calculated within the SCFT. (Note the difference in the ratios L∥/L⊥ indicated in the capture for Figure 10 for various H.)

4. DIAMOND-LIKE MORPHOLOGIES IN THIN FILMS WITH HOMOGENEOUS SELECTIVE WALLS We demonstrated in the preceding sections that for some block copolymer architectures the diamond-like morphologies are stable in a rather broad interval (actually, a set of intervals) of the film widths H and that transport properties (channel connectivity) of such (DLM) do not change substantially with H. However, our consideration has been based, until now, on the assumption that the film walls are nonselective with respect to adsorption the block copolymer components. Meanwhile, as noticed in the Introduction, the selective walls favor formation of the parallel morphologies (in our case parallel lamellar morphology) substantially. So, it could seem natural to expect that the DLM would be suppressed in thin films with selective walls. Fortunately, as we show in this and subsequent sections, the selectivity effect is not as straightforward. In fact, the DLM would be suppressed in the films with homogeneously selective walls only; in thin films with heterogeneous selective walls the DLM stability could be even enhanced for a specially designed selectivity pattern. To begin with we allow for the selectivity effects via introducing a surface field.51,52,54 To this end we represent the total external field wα(r) appearing in the modified diffusion eq 8 as the sum of a self-consistent (volume) field ωα(r) (α = A, B, C) and an external (surface) field γα(r) wa(r) = wa(r) + γa(r)

α = A,B,C

= 2S

∑

ϕα(S)εα(S) = σ(ψS)Stot

(27)

α = A,B,C

where σ(ψS) is the interfacial tension on the wall−block copolymer boundary, S and Stot = 2S being the areas of one and both walls of the film, respectively. Now, let, for simplicity, the surface fields strengths be represented as follows: εA = ε0 + ε(ψS),

εC = ε0 − ε(ψS),

εB = ε0

(28)

Such a choice implies that the walls attract the component A and repel the component C as compared to the component B, the absolute values of the corresponding interaction increments being the same. It follows from eqs 27 and 28 that in this case σ = ε0 + 2ψSε(ψS)

(29)

Thus, ε0 is just surface tension of the block copolymer in the disordered state when ψ = 0. It determines the global properties of the block copolymer melt (like contact angle) but does not influences ordering in the melt. The surface field ε

(25)

ε = (σ(ψS) − σ(0))/(2ψS)

Let us first consider the situation when the surface field is laterally homogeneous (i.e., it does not depend on the lateral coordinates). Then the surface field appearing in eq 23 can be defined as follows: γa(r) = 2εa(ψS)[δ(z) + δ(z − H )]

∑ ∫ dVϕα(r)γα(r)

ΔFS({γα(r)}) = −

(30)

whose influence on the ordering is studied in the paper could be measured as a rate of changing of the surface tension with increase of the order parameter. In what follows we assume, for simplicity, that ε does not depend on ψ. Thus, the selectivity is described by the only parameter ε. In further calculations in this subsection the values 0 ≤ ε ≤ 0.15 are explored. Similar to the case of fully nonselective boundaries we dealt with in section 3, the initial phases D as well as the perpendicular L⊥ and parallel Li lamellae are considered only (the subscript i denotes the number of the parallel lamellar layers). The results calculated within the SCFT procedure are shown in Figure 11, where the H dependences of the corresponding free energies are presented. It is seen from Figure 11 that the D phase is stable (has the minimal free energy) when the film width H is approximately multiple to one-half of the period observable in bulk for the D

(26)

where ψS is the value of the order parameter at the wall and the factor 2 in eq 26 is introduced for convenience. The deltafunction sources on the walls are incompatible with the Neumann boundary conditions (9). So, we interpret these conditions in a generalized sense, i.e., that the walls are the reflection planes for the desired solution. The numerical algorithm uses a rectangular uniform grid. The sources are only in the grid nodes belonging to the walls. The calculations are performed via the pseudospectral technique where the 3D Fourier expansion is applied for the cosine basis in the ZG


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in the limit H → ∞ the parallel lamellae can be observed in the limit ε → ∞ only. This observation could be also expressed somewhat differently: the bigger the selectivity ε is, the more narrow the film width interval to provide the DLM stability is. The small region of the perpendicular lamella L⊥ is observed only when the width H is small enough. Therewith, the morphology L⊥ is almost not distorted for the thinnest films (see Figure13α) whereas with increase of H it

Figure 11. H dependence of the specific (per one chain) free energies for symmetric linear ternary ABC triblock copolymer with fA = f B = 0.15 at the selectivity parameter ε = 0.1. The labels Dm and Li mark the DLM composed of m half-layers (solid lines) and parallel lamella composed of i layers (dashed lines), respectively. The dash-dot line marked by an arrow corresponds to the perpendicular lamella L⊥. Figure 13. Color visualization of the volume fraction ϕA(r) of the stable perpendicular lamellar phases L⊥; the selectivity parameter ε = 0.1. The pictures are labeled as consistent with the succession of the solid circles in Figure 12. The top and bottom horizontal planes correspond to the film walls z = 0 and z = H.

morphology. Similarly, the stable parallel lamellar phases Li appear when the film width H is approximately multiple to the lamellar half-period in bulk. The points, where the free energy plots for various morphologies intersect, define the selectivity dependence H(ε) of the film widths at which the morphologies transform into each other. It is worth to notice that there is no surprise that the free energies of the gyroid (G) and cylinder (C) phases are not shown in Figure 11, contrary to the conventional (for AB diblock copolymers) expectations saying that the G and C phases have a wider range of bulk stability. The case is that the physics of ABC copolymers differs substantially from that of AB ones. In particular, the symmetric nonselective ABC block copolymers were shown by one of us29 not to reveal the G and C phases at all. The resulting phase diagram is presented in Figure 12. (It is worth to notice that this phase diagram ignores the possibility of island/hole formation and refers to a rigidly confined rather than a supported film.) We see from Figure 12 that the Li stability regions shift to the higher selectivity values with increase of the film width H so that

becomes much less regular (see Figure 13β) and could be identified only by the dominant harmonic (we skip a more detailed harmonic analysis in this paper). When the film width H increases, the distortions of the perpendicular lamellar phase increase either until L⊥ becomes fully unstable. As far as the DLM phases Dm is concerned, their distortions due to both the confinement and selectivity of the walls do not affect much the topological structure of the conducting clusters (see Figure 14). Thus, the homogeneous wall selectivity affects equilibrium between the lamellar and DLM morphologies in thin films as expected: increase of the selectivity parameter ε results in a broadening of the regions where the parallel lamellar morphologies Li are stable at the cost of the corresponding squeezing of those where the DLM are stable. So, it could appear that the selectivity effect would destroy the quasi-isotropic DLM, which therefore would cast doubts on the value of our results for nonselective walls. Fortunately, as we demonstrate in the next section, the properly designed inhomogeneous boundary selectivity pattern affects the DLM stability in a quite opposite way.

5. DIAMOND-LIKE MORPHOLOGIES IN THIN FILMS; WALLS WITH LAMELLAR PATTERNED SELECTIVITY Patterned Boundary Selectivity as Confinement. When we speak of patterned substrate selectivity, we imply that the surface field introduced in eq 24 is redefined to act on the only bottom wall as follows: γα(r) = εα(ρ)δ(z),

εA (ρ) = − εC(ρ) = ε(ρ),

εB(ρ) = 0 (31)

where 2D vector ρ = (x, y) indicates the lateral location of the surface point the surface field is acting (the top boundary of the film being assumed neutral) and ε(ρ) is a 2D periodic function. There are many ways to arrange such a two-dimensional periodic selectivity profile. For example, one can use a surface of some properly chosen ordered diblock copolymer morphology as the substrate for the thin film ordering from the ABC-like block

Figure 12. Phase diagram in the plane (film width H; selectivity ε). The regions of the DLM composed of m half-layers and parallel lamellar morphologies composed of i layers are marked as Dm and Li, respectively. The areas of the perpendicular lamella L⊥ are dashed. The open and solid circles and their labels correspond to the DLM and perpendicular lamellae imaged in Figures 13 and 14, respectively. H


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Figure 14. Isosurfaces ϕA(r) = 0.15 of the stable diamond-like phases Di at the selectivity ε = 0.1 and various film widths. (a) i = 1, H = 2.21; (b) i = 2, H = 4.59; (c) i = 3, H = 5.86; (d) i = 4, H = 8.43; (e) i = 5, H = 9.61; (f) i = 5, H = 11.07. The pictures are labeled as consistent with the succession of the open circles in Figure 12.

Figure 15. Spinodal values of the effective Flory−Huggins parameter for L⊥ and D (cos θ = 1/√3) phases (bold and thin solid lines, respectively) as the functions of the squared pattern wavenumber, left, and their difference, right, as the function of the pattern period. The horizontal dashed and vertical dotted lines (left) are the guides for eyes to demonstrate the shift of stability for the pattern fitting the genuine D phase. Right: the full and open circles correspond to genuine D and L⊥ phases, respectively; the dashed lines are the guides for eyes to define the value of L* = 3.8544.

morphology LD/L = √3 as specified in eq 6). Thus, choosing the selectivity pattern period Lp to fit well to that of the D phase, we automatically make it poorly compatible with the intrinsic bulk period of the lamellar one. To evaluate this effect quantitatively let us make use of the general representation of the free energy of morphologies formed on the patterned substrate as the sum of the volume and surface contributions:50,54

copolymer under consideration. Thereafter, in this section we specify the selectivity profile (31) to read ε(ρ) = 2ε cos(qpx)

(32)

It could be realized by using as the substrate the top surface of an ordinary perpendicular lamella prepared from a diblock copolymer with the properly chosen degrees of polymerization and incompatibility of blocks, qp and Lp = 2π/qp being the wavenumber and period of the substrate lamellar pattern. It would seem natural to expect that the 1D substrate selectivity pattern (32) would induce formation of the 1D lamellar morphology L⊥ of period Lp within the film volume. Such an expectation is based, actually, on the hypothesis that the symmetry of an effect is determined by the impact symmetry. However, as noticed long ago by Birkhof,53 this hypothesis is not always true. In particular, for the diamond-forming block copolymers the proper choice of the lamellar pattern period Lp lets to induce the DLM (rather than LAM) formation. The physical reason for such a paradoxical behavior is that the intrinsic periods L and LD of the L⊥ and DLM phases, respectively, are essentially different (e.g., for the genuine D

F = min{Fvol + Fsurf }

(33)

Here Fsurf =

∫ Ψ(x , y , z)|z=0 ε(x , y) dx dy = 2εA p

(34)

where Ap = A(qp) and A(q) = ∫ Ψ(x,y,z)|z=0 cos (qx) dx dy is the 1D Fourier transform of the order parameter Ψ averaged over the lateral direction normal to that of the pattern. Therewith the equilibrium morphology is to be sought via minimization over the set {A} of all Fourier amplitudes A involved. It follows from eqs 33 and 34 that the free energy change due to modulated affinity to the substrate is I


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(35)

In other words, a selectivity pattern can diminish the ordered nanostructure free energy only if it induces an inhomogeneity with the pattern wavelength. Such inhomogeneities, in general, read Ψ(pk)(r) = A p cos(qpx) cos(kπz /H )

with k = 0, 1, 2, ... (the case k = 0 corresponds to the lamellar morphology). WST Treatment. Now we can evaluate the spinodal value of the effective Flory−Huggins parameter χ̃AC for the inhomogeneities induced by the pattern with the wavenumber qp. (Remember that at χ̃AC < χ̃sp AC(qp) the harmonic order parameter fluctuations with q = qp can appear in bulk only as some labile (heat) fluctuations whereas at χ̃AC > χ̃AspC (qp) they grow spontaneously until some finite value.) The corresponding calculation for the symmetric ternary ABC block copolymers considered in this paper is based on eq 87 of ref 29 (see Supporting Information for detailed calculation). The result reads χAC ̃ sp (qp) = τ(q2),

τ(q2) = g11 ̃ −1(q2) − g12 ̃ −1(q2)

Figure 16. Phase portrait of thin patterned film of the linear ternary ABC block copolymers (ε = 0.3, f B = 0.7, χ̃AC = 66). The region of parallel cylinders C∥ is restricted by the bold dashed line. The boundary between the DLM and L⊥ phases is marked as the solid line. The locations of the numbered circles, triangles, and stars define the Lp, H values for the accordingly numbered L⊥, C∥ and DLM images shown in Figures 17−19, respectively. The vertical thin dashed lines are the guides for eyes that correspond to the accordingly labeled Figure 22.

17−19 give a reasonable idea of the morphologies, which arise at various scales of the substrate pattern and film width, in terms of their topology. In particular, the images shown in Figure 19 clearly demonstrate the intertwined channels characteristic of the DLM. However, they could mislead the reader concerning the actual scale of the inhomogeneities formed. The case is that to build the images one has to involve that or another computer program relating the inhomogeneity amplitudes to colors (in our case it is the open-source “Paraview 3.12”). But any such a program involves intrinsically a nonlinear operation introducing an additional source of errors. In fact, the actual plots of the A, B, and C volume fractions presented in Figure 20 do not support the idea of well separated layers, which could be evoked by images shown in Figures 17 and 18. Just contrary, even though the variations of the volume fractions in max Figure 20 are quite considerable ( fA/ϕmin A ≈ 4, ϕA /fA ≈ 2), the corresponding profiles look almost harmonic, which implies the weak (or, at the most, intermediate) segregation regime. To advance further and get a more quantitative insight on segregation and thermodynamics of the thin film morphologies, we employ their spectral analysis. More precisely, we expand the order parameter (1b), which characterizes morphology of certain symmetry, into the Fourier series

(36)

where the matrix g−1 ̃ (q2) is calculated in ref 29 (see also Supporting Information) and54−56 q2 = qp 2 + (kπ /H )2

(37)

The functions χ̃sp AC(qp) for various values of the half-layer width H/k are plotted in Figure 15. As seen from Figure 15, for Lp > L* (the bigger values of the pattern period) the DLM is more stable than L⊥ and, vice versa, for Lp < L* (the smaller values of the pattern period) the DLM is less stable than L⊥. SCFT Phase Diagram. To verify the presented WST considerations, we made use of the SCFT numerical procedure to study how is the dominant morphology of the symmetric ternary ABC block copolymers considered in the paper influenced by the selectivity pattern period and film width (see Figure 16). First of all, unlike the situation with the homogeneous selectivity studied in section 4 we found no stability islands of the parallel lamellar morphology Ln within the region explored. Surprisingly, within the region explored to delineate the D phase stability island in Figure 16 the Ln turned out to be completely unstable (even not metastable!) phase. More precisely, starting our iteration procedure with the initial trial function ψ(r) = A cos(2πnz/H) + ξ(r), where n corresponds to the most commensurable parallel lamella and ξ is a small noise (|ξ/A| < 10−2), we found that other harmonics start to grow until the final order parameter profile converges to the proper DLM one. Note that the periodic boundary conditions imposed in the lateral plane to solve the SCFT equations are not compatible with the limit Lp → ∞, d = constant expected to generate the Ln phases (here d is the length of the film in the direction normal to the stripes). Instead, the morphology of parallel cylinders C∥ (Figure 17), the perpendicular lamella L⊥ (Figure 18), and the DLM phase (see Figure 19) are found to be dominant in the corresponding parts of the explored region (see Figure 16). Degree of Segregation, Spectral Analysis, and Thin Film Thermodynamics. The images presented in Figures

Ψ(r) =

∑ Aq exp(iqr) (38)

q

with the vectors q running all the nodes of the corresponding reciprocal lattice: q = 2π (l gx + mg y + n gz),

gα = iα /Lα

(39)

where α = x, y, z; Lx = L∥, Ly = L⊥, Lz = H/2; l, m, n = 0, ± 1, ± 2, ..., and iα is the unit vector along the α-axes. Further we specify the harmonics with the wave vector q by the triple (l, m, n). Generally, identification of the crystal structure via the spectral analysis is a formidable task.57 But for weakly and moderately segregated ordered block copolymers the task is reduced to determination of the leading (first) harmonics as well as a few of secondary harmonics and, thus, simplified tremendously. In particular, for morphologies under study in this paper the leading harmonics can be enumerated readily. (i) The perpendicular J


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Figure 17. Color visualization of the volume fraction ϕA(r) of the phase C∥ (parallel cylinders). The pictures are numbered as consistent with the succession of the open triangles in Figure 16. The top and bottom horizontal planes correspond to the film walls z = 0 and z = H.

Figure 18. Color visualization of the volume fraction ϕA(r) of the stable perpendicular lamellar phases L⊥. The pictures are numbered as consistent with the succession of the open circles in Figure 16. Note the difference between the images 1 and 2−4. The vertical asymmetry of all morphologies is due to the boundary conditions asymmetry (the selective boundary is the bottom one only). The profile 1 is somewhat rippled since it includes, apart from the dominant lamellar harmonic, also some other lateral ones (see Figure 21). The top and bottom horizontal planes correspond to the film walls z = 0 and z = H.

Figure 19. Visualization of the DML phases via the intermaterial dividing surfaces (ϕA(r) = ϕ̅ A = 0.15). The pictures are numbered as consistent with the succession of the stars in Figure 16.

by two first harmonics either: (1, 0, 1) and (−1, 0, 1), but their amplitudes could differ since the periods L⊥, Lz could be different. (v) At last, the DLM is characterized by four first harmonics: the lateral (1, 1, 0) and (1, −1, 0) and normal (1, 0, 1) and (−1, 0, 1) ones, the amplitudes being the same for each pair

lamella with the layers normal to the x-axes: (1, 0, 0). (ii) The parallel lamella with n half-layers: (0, 0, n). (iii) The tetragonal array of perpendicular cylinders is characterized by two first harmonics with identical amplitudes: (1, 1, 0) and (1, −1, 0). (iv) The parallel cylinders directed along the y-axes are characterized K


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values of H the dominant harmonic is (1, 0, 0) (the blue line corresponding to the L⊥ morphology), but at H ≈ 2.2 its amplitude drops and two new dominant harmonics (the black line (1, 0, 1) and the red one (1, 1, 0)) appear jumpwise. (Here and in what follows indications of the harmonics (−1, 0, 1) and (−1, 1, 0) are omitted.) As discussed above, these harmonics correspond to the parallel and perpendicular cylinders, respectively, and form together the diamond-like morphologies as demonstrated in Figure 19. Thus, the point H ≈ 2.2 belongs to the boundary between L⊥ and DLM phases (see Figure 16), the latter having here the orthorhombic symmetry (the amplitudes of the lateral and normal modes are different). Now, in the case Lp = 4.08 the thin film morphology undergoes three rather than one discrete symmetry changes: (i) for H < 1.7 the dominant harmonic and morphology are (1, 0, 0) and L⊥, respectively; (ii) for 1.7 < H < 2.95 the dominant harmonic and morphology are (1, 0, 1) and parallel cylinders, respectively; (iii) for H > 2.95 the DLM phase becomes dominant. Therewith within the interval 2.95 < H < 4.65 the harmonics (1, 0, 1) and (1, 1, 0) are dominant, which implies the one-half-layer diamond structure, whereas for H > 4.65 the first normal harmonic (1, 0, 1) is replaced by the second harmonic (1, 0, 2), which implies formation of the two half-layer diamond structure. With increase of the selectivity pattern period Lp the critical width, where the perpendicular lamellar is stable, shrinks until L⊥ is fully replaced by the DLM. For example, for Lp = 5.58 thin film undergoes (at H = 1.56) the transition D1−C∥ rather than L⊥− C∥. Next, at H ≈ 2.3, the discrete reentrant transition C∥−D1 occurs. The further increase of H results in a gradual decrease of the one-half-layer normal mode (1, 0, 1) and increase of the two half-layer normal mode (1, 0, 2), which corresponds to reconstruction D1−D2. It is worth to notice here an increase (for H > 4.5) of the three half-layer mode (1, 0, 3) (dashed red line), which results, with further increase of Lp (see Figure 21, Lp = 6.53), in a new discrete transition D2−D3 at H ≈ 5. Therewith morphology of parallel cylinders disappears at all, and the transition D1−D2 becomes gradual only. (Note that in Figure 16 we disregard the boundaries between the DLM phases Di with different values of i since all of them are equivalent in terms of permeability properties.) The presented way to study the phase behavior of block copolymers in thin films basing on its spectral properties can be readily verified via conventional comparison of the free energies for the competing morphologies.

Figure 20. Dependences of the volume fraction profiles ϕ on the reduced coordinate X = x/Lp for the lamellar morphology L⊥ of the A, B, and C units (labeled by the corresponding letters) corresponding to the open circle 3 in Figure 16 (Lp = 3.48, H = 3487) at the heights z = 0.05H and z = 0.50H (solid and dashed curves, respectively).

but, generally (for orthorhombic symmetry) different for lateral and normal harmonics. In other words, the DLM is nothing but a superposition of the parallel and perpendicular cylinders, which is in accordance with Figure 4. To show which information can be extracted from the spectral analysis, we plot, at various values of the pattern period Lp, the amplitudes Aq as functions of the film width H for the first and some secondary harmonics. The plots are to be compared with the phase portrait shown in Figure 16. Let us start with comparatively small pattern period (see Figure 21). It is seen from Figure 21 that here the leading

Figure 21. H dependences of the basic harmonics amplitudes for perpendicular lamellae corresponding to comparatively small pattern period (indicated on the graph). Explanations are given in the text.

6. CONCLUSION We can summarize the key points of our consideration as follows. 1. The conventional (typical) morphologies of molten block copolymers confined in thin films are strongly anisotropic (1D lamellar and 2D cylindrical). However, for some specially designed block copolymer architectures such morphologies can possess quasi-isotropic (cubic, tetragonal, orthorhombic) symmetry. In particular, the symmetric linear ternary ABC block copolymers AnBmCn with the middle block B nonselective with respect to the short side blocks A and C ( fA = f C ≈ 0.15) could form nanostructures, which possess the diamond symmetry Fd3̅m (space group no. 227) and its tetragonal and orthorhombic generalizations we call diamond-like morphologies. 2. The A and C domains, which form together a structure supporting the transport of low-molecular species through the DLM, look like intertwined channels (see Figure 4), which

harmonics are (1, 0, 0) and (2, 0, 0), which clearly indicates that the perpendicular lamella is formed. The negative sign of the second harmonics (2, 0, 0) means only that the first and second harmonics are in antiphase. The modulus of the second harmonics is considerably smaller than that of the first one, which confirms, in accordance with Figure 20, that the lamella is not strongly segregated. Besides, we see some other harmonics, which are present either but rather small. It means that the actual solution of the SCFT equations does not possess the rigorous 1D symmetry and corresponds to slightly rippled layers. At last, the H dependence of the amplitudes is rather weak, which is quite natural for the perpendicular lamella. With increase of the pattern period the system behavior is changed dramatically (see Figure 22). To begin with, let us consider the case Lp = 3.83. As seen from Figure 22a, for small L


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Figure 22. H dependences of the basic harmonics amplitudes at various values of the selectivity pattern period Lp indicated on each graph in accordance with the corresponding dashed lines in Figure 16. Explanations are given in the text.

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improves substantially both the permeability of the DLM and its stability with respect to all sorts of defects. 3. The characteristic scales between the parallel and perpendicular channels forming the DLM depend considerably on the film width but their topology does not. 4. The selectivity of the substrate with respect to adsorption of different copolymer blocks can both diminish and enhance the thermodynamic DLM stability. In particular, a spectacular revelation of the ABC-like block copolymer peculiarity could be observed under a 1D (lamellar) modulation of the substrate selectivity with the period Lp ≈ 1.7L. (Remember that it could be realized by using as the substrate the top surface of an ordinary perpendicular lamella prepared from a diblock copolymer with the properly chosen degrees of polymerization and incompatibility of blocks.) Surprisingly, it results in the DLM (rather than lamellar!) stability enhancement. It is worth to emphasize that the new striking fact here is of course, not that surface interaction can qualitatively change the block copolymer morphology. (For example, the lamella-forming copolymers were shown to form a sort of bicontinuous morphology when ordered on a pattern consisting of square array of spots.58) The real message is that a simple modulation of surface can induce formation of the morphology with a higher symmetry than the surface pattern. 5. The presented theoretical predictions seem to open a new route to finding new technological applications of the block copolymer physics. It makes sense to add that some further advances along this route could be related to consideration of the ABC block copolymer mixtures with other block copolymers and/or homopolymers, which would considerably diminish the volume fraction (and, thus, cost of producing) necessary to provide the DLM formation.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b01513. Some details on calculation of the higher vertices used to analyze the system within the framework of the weak segregation theory (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (I.E.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The initial stage of our work was supported by the Alexander von Humboldt Foundation (I.E.) as well as Ministry of education and science of Russian Federation through State Contract 02.740.11.0858 and by European Commission within the FP7 Project NMP3-SL-2009-228652 (SELFMEM) (I.E. and Y.K). SCFT calculations have been performed under assistance of the Center for Information Technology of the University of Groningen and the Research Computing Center of M.V. Lomonosov Moscow State University. I.E. thanks V. Abetz for helpful discussions.

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REFERENCES

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