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Nanostructured Ultrathin Films Obtained by the Spreading of Diblock Copolymers on a Surface Elena S. Patyukova† and Igor I. Potemkin*,†,‡ Department of Physics, Moscow State UniVersity, Moscow 119992, Russian Federation, and Department of Polymer Science, UniVersity of Ulm, 89069 Ulm, Germany ReceiVed July 5, 2007. In Final Form: August 29, 2007 We propose a theory of nanopattern formation in ultrathin films obtained by the spreading of A-B diblock copolymers on a surface. A blocks are strongly adsorbed (spread) on the substrate, and the strength of the adsorption can be varied. B blocks are incompatible with the substrate, but they can spread atop the layer of the A blocks. We predict disk-shaped micelles ordered with hexagonal symmetry and parallel stripe-shaped micelles and bilayers to be stable. The type of resulting structure and its geometrical parameters depend on the composition of the copolymer and interaction parameters. We interrelate these results with those obtained for the case of the spreading of both blocks on the substrate,19 and we construct a unified phase diagram.
1. Introduction The formation of nanostructures is an inherent property of block copolymer systems,1 so they are actively used in numerous applications. In most cases, exploitable systems are block copolymer films with chemically nanostructured surfaces.2,3 These films are used as templates for growing biological cells with controlled shape and size,4 for the fabrication of optoelectronic devices,5 for the self-assembling of proteins in regular arrays,6 and as templates for lithographic masks.7,8 An easy, reliable way to obtain block copolymer films with patterned surfaces is based on the preparation of ultrathin films7-13 whose thickness is smaller than the period of the equilibrium structure in the bulk. Significant progress has been achieved in the preparation of the ultrathin films via the adsorption of polystyrene-block-poly(2,4-vinylpyridine) (PS-PVP) from a dilute solution on mica.7-11 Here, PVP (A) blocks are strongly attracted to the surface and form a nearly monomer-thick layer on it. PS (B) blocks are incompatible with the substrate, the PVP blocks, and the air (dry films). To reduce the number of unfavorable contacts of PS blocks with the environment, the blocks aggregate into clusters (Figure 1a). Depending on the * To whom correspondence should be addressed. E-mail: igor@polly. phys.msu.ru. † Moscow State University. ‡ University of Ulm. (1) Bates, F. S.; Fredrickson, G. H. Annu. ReV. Phys. Chem. 1990, 41, 525. (2) Foester, S.; Matthias, K. J. Mater. Chem. 2003, 13, 2671. (3) Farmi, A. W.; Stamm, M. Langmuir 2005, 21, 1062. (4) Singhvi, R.; Kumar, A.; Lopez, G. P.; Stephanopoulos, G. N.; Wang, D. I.; Whitesides, G. M.; Ingber, D. E. Science 1994, 264, 696. (5) Morkved, T. L.; Wiltzius, P.; Jaeger, H. M.; Grier, D.; Witten, T. Appl. Phys. Lett. 1994, 64, 422. (6) Kumar, N.; Hahm, J. Langmuir 2005, 21, 6652. (7) Spatz, J. P.; Sheiko, S.; Mo¨ller, M. AdV. Mater. 1996, 8, 513. (8) Spatz, J. P.; Eibeck, P.; Mo¨ssmer, S.; Mo¨ller, M.; Herzog, T.; Ziemann, P. AdV. Mater. 1998, 10, 849. (9) Spatz, J. P.; Mo¨ller, M.; No¨ske, M.; Behm, R. J.; Pietralla, M. Macromolecules 1997, 30, 3874. (10) Potemkin, I. I.; Kramarenko, E. Y.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P.; Eibeck, P.; Spatz, J. P.; Mo¨ller, M. Langmuir 1999, 15, 7290. (11) Spatz, J. P.; Eibeck, P.; Mo¨ssmer, S.; Mo¨ller, M.; Kramarenko, E. Yu.; Khalatur, P. G.; Potemkin, I. I.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P. Macromolecules 2000, 33, 150. (12) Kramarenko, E. Yu.; Potemkin, I. I.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P. Macromolecules 1999, 32, 3495. (13) Eibeck, P.; Spatz, J. P.; Potemkin, I. I.; Kramarenko, E. Yu.; Khokhlov, A. R.; Mo¨ller, M. Polym. Prepr. (Am. Chem. Soc., DiV. Polym. Chem.). 1999, 40, 990.
composition of the copolymer (the relative length of the blocks), mushroomlike micelles ordered with the symmetry of a hexagonal lattice, stripelike micelles, and a double layer (planar brush) were observed10,13 (Figure 1a). Theoretical studies of such structures explained the physical reasons for the ordering and quantified the experimental data.10-13 Similar periodic surface micelle structures were observed in a system of diblock copolymers with one hydrophobic block and one polyelectrolyte block adsorbed at the air/water interface.14-16 The PVP-PS diblock on mica can be attributed to a copolymer with “sticky” (PVP) and “nonsticky” (PS) (toward the substrate) blocks. However, diblock copolymers with both blocks being sticky are known. Examples are poly(ethylene oxide)-blockpoly(2-vinylpyridine) (PEO-P2VP)17 and poly(butyl acrylate)block-poly(2-vinylpyridine) (PBA-P2VP) on mica. Furthermore, the stickiness of the blocks can be varied by external action (temperature, atmosphere, etc.). For instance, a cyclic spreading/ desorption of PBA comblike macromolecules on mica was realized by alternate treatment of the molecules with water and ethanol vapors.18 Theoretical studies of the adsorbed diblock copolymers with variable stickiness of one of the blocks were performed in ref 19. In this paper, both blocks were considered to be in contact with the substrate: one of the blocks is completely adsorbed (2D), and the conformation of the second block varies from 2D to 3D globular (a partially desorbed block) (Figure 1b). Various nanostructures including flat, prominent disks and stripes were calculated. The most important prediction of ref 19 is that the variation of the spreading parameter (degree of adsorption) of one of the blocks can result in morphological changes. For the case of a single molecule adsorbed on the patterned solid surface, periodic variation of the stickiness of one of the blocks can result in the directional motion (reptation) of the copolymer.20 (14) Zhu, J.; Eisenberg, A.; Lennox, R. B. Macromolecules 1992, 25, 6547. (15) Zhu, J.; Eisenberg, A.; Lennox, R. B. Macromolecules 1992, 25, 6556. (16) Zhu, J.; Eisenberg, A.; Lennox, R. B. Langmuir 1991, 7, 1579. (17) Albrecht, K., Ph.D. Thesis, Aachen Technical University, Aachen, Germany, 2007. (18) Gallyamov, M. O.; Tartsch, B.; Khokhlov, A. R.; Sheiko, S. S.; Borner, H. G.; Matyjaszewski, K.; Mo¨ller, M. Macromol. Rapid Commun. 2004, 25, 1703. (19) Potemkin, I. I.; Mo¨ller, M. Macromolecules 2005, 38, 2999. (20) Perelstein, O. E.; Ivanov, V. A.; Velichko, Yu. S.; Khalatur, P. G.; Khokhlov, A. R.; Potemkin, I. I. Macromol. Rapid Commun. 2007, 28, 977.
10.1021/la701989s CCC: $37.00 © 2007 American Chemical Society Published on Web 10/20/2007
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Figure 1. Schematic representation of three kinds of ultrathin diblock copolymer films: (A) A blocks are strongly adsorbed on the substrate, whereas B blocks are not compatible with either the substrate or the layer of A blocks. Structured films with spherical and cylindrical micelles or planar brushes were predicted in refs 10 and 13. (B) Both blocks are adsorbed on the substrate. Possible morphologies of the film are calculated in ref 19. (C) A blocks are adsorbed on the substrate (variable degree of adsorption). B blocks are incompatible with the substrate, but they can spread atop the layer of A blocks.
In this article, we generalize theories of refs 10, 12, 13, and 19 for the case in which both blocks can be partially desorbed and when one of them can spread atop the other block (Figure 1c). We construct a series of unified phase diagrams where various regimes of strong adsorption of the diblocks depicted in Figure 1b,c and possible microstructures are analyzed.
2. Model Let us consider a dry, ultrathin film of A-B diblock copolymers that is formed by adsorption of the blocks on a flat surface from a dilute solution. A typical preparation procedure for ultrathin films involves dipping the substrate in the solution and pulling it out. Therefore, we assume that the number of macromolecules on the surface is fixed and that the area of the surface is larger than the total area of the adsorbed segments. This assumption allows us to analyze all possible morphologies obtained via variation of the spreading parameter (i.e., the film area). We suppose that the blocks are flexible and have statistical segments of equal length a; NA and NB are the numbers of segments of A and B blocks, respectively. Strongly adsorbed A blocks form a thin layer whose thickness is much less than the radius of gyration of the block. We assume that the density of A-monomer units on the surface corresponds to the case of a 2D melt, ensuring a minimum number of unfavorable contacts of the units with the air. We assume that B blocks are strongly incompatible with the substrate and spread atop the A blocks. The conformation of each block can be varied between that of a completely adsorbed (2D) chain and that of a partially desorbed, “shrunken” chain.
Interactions of the A and B blocks with the surface and with each other control such conformational variations. Competition of the interfacial interactions with the entropic elasticity of the blocks can lead to lateral phase separation, resulting in the formation of various nanostructures. The type of surface structure depends on the relative length of the blocks and the interaction parameters. For the described system, we will examine the conditions for the stability of (i) hexagonally packed micelles with a disklike core formed by B blocks spread atop the A layer (substratephobic disks), (ii) parallel stripes (substratephobic stripes), and (iii) a bilayer (Figure 1c). The analysis will be done within the strong segregation approximation. 2.1. Disk-Shaped Structure. We model the disklike micelle as two coaxial disks: the bottom disk of radius R0 (the radius of the Wigner-Seitz cell) and of thickness h0 is formed by the A blocks whereas the upper disk of sizes R and h is formed by the B blocks (Figure 2). We will study the regime of strong spreading of both blocks when their end-to-end distances are much larger than the corresponding thicknesses, a eh0 , R0, a eh , R. In this regime, a smoothed profile of the edge of the upper disk (a cross section of the disk in the plane perpendicular to the substrate) can be approximated by a steplike one. In the strong segregation limit, the free energy of the micelle can be written as a sum of three terms:
Fdisk ) Fsurf + Fconf + Fel
(1)
The first term is the surface energy. In the equilibrium state, this term is responsible for the minimization of the number of
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Figure 2. Sketch of the structure of a disklike micelle. A blocks (blue) and B blocks (cyan) are radially stretched.
unfavorable contacts of monomer units of different blocks with each other and with the surrounding media:
Fsurf ) πR2γAB + (πR2 + 2πRh)γBa + (πR02 - πR2)γAa + πR02γAs + (σ0 - πR02)γ0 (2) The first term of eq 2 is the energy of the A-B interface. The second and the third terms are the energies of interactions of the B and A blocks with the air, respectively. The fourth term is the energy of interaction of the A blocks with the substrate, and the last term is the energy of contact of the air with the substrate. Here γi, i ) AB, Aa,..., are the corresponding surface tension coefficients, and σ0 is the area of the substrate per micelle. Each micelle comprises Q chains (Q is the aggregation number). The condition of the dense packing of monomer units in the micelle takes the form
πR2h ) QNBa3, πR02h0 ) QNAa3
(3)
Using these equations, we can write the surface energy per chain as follows:
F h surf )
SA2NAa SB1NBa 2γ j BaNBa Fsurf )+ + const (4) kTQ h0 h R
Here γ j Ba ) (a2/kT)γBa is a dimensionless surface tension coefficient; SA2 ) a2/kT(γ0 - γAs - γAa) and SB1 ) a2/kT(γAa - γBA - γBa) are dimensionless spreading parameters that describe the spreading of A blocks on the substrate and B blocks atop the A blocks, respectively. Notice that spreading is possible if the parameters are positive, SA2 > 0 and SB1 > 0. The term σ0γ0/kTQ is denoted as a constant because σ0/Q is the area of the substrate per chain, and the number of chains on the surface is fixed. This constant is the same for all kinds of structures, which will be analyzed, so we will omit it. The second term of eq 1 describes a loss of conformational entropy of the chains because of their adsorption (confinement
free energy). It comprises two parts: the free energies of A and B blocks, Fconf ) FAconf + FBconf. These free energies can be calculated as those of chains gripped in the slits of thicknesses h0 and h. Owing to the conditions R0 . h0 and R . h, we can use the so-called ground-state approximation21 where the free energy per monomer unit is the minimum eigenvalue λ of the differential equation:
|
2 ∂χ(x) a2 ∂ χ(x) + λχ(x) ) 0, 2 6 ∂x ∂x
)
x)(h/2
h h 0 (boundary conditions for B blocks), - e x e 2 2 ∂χ(x) ∂x
|
)
x)(h0/2
0 (boundary conditions for A blocks), -
h0 h0 exe (5) 2 2
The boundary conditions are taken to satisfy a requirement of constant density of monomer units inside the slits. The solution of eq 5 has to be symmetric with respect to the origin of coordinates (the middle of the slit), χ(x) ) const cos(x x6λ/a) and
F h Aconf )
FAconf 2π2 NAa2 ) kTQ 3 h2 0
F h Bconf )
FBconf 2π2 NBa2 ) kTQ 3 h2
(6)
The third term of eq 1 is the free energy of radial stretching of the blocks. It also consists of two terms: the stretching free energy of A blocks, FAel, and the stretching free energy of B (21) Grosberg, A. Yu.; Khokhlov, A. R. Statistical Physics of Macromolecules; AIP Press: New York, 1994.
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blocks, FBel. To calculate these terms, we make a number of approximations. First, let us assume that the free ends of the B blocks are located on the periphery of the upper disk and that the AB junction points are distributed as free ends of inner blocks of cylindrical micelles in the bulk22 (Figure 2). Then we can approximate the stretching free energy of the B blocks as the free energy of the core of the cylindrical micelle:22
F h Bel
FBel π2 R2 ) ) kTQ 24 N a2
The free ends of the A blocks are also supposed to be located on the boundary of the Wigner-Seitz cell. The stretching free energy of the A blocks is represented as a sum of two terms: the free energy of the inner part of the A disk that has a radius R, FA,in el , and the free energy of the outer part of the A disk (a ring of width R0 - R), FA,out . Let us assume that Nin el A monomer units of each A block are in the inner part of the micelle and Nout A out in are in the outer part of it, Nin A + NA ) NA. We can evaluate NA out A,in and NA using the dense packing condition. Thus, Fel can also be approximated by the free energy of the core of the cylindrical micelle 2 FA,in el π2 R2 π2 R0 π2 R2 h ) ) ) ) 2 2 in kTQ 24 N a 24 N a 24 N a2 h0 A
A
(8)
B
and the stretching free energy of the outer part of the A blocks takes the form
F h A,out el
FA,out el ) ) kTQ
∫R
R0
E(r) a2
dr
()
( )
R0 NAh Qa R2h ) ln ) ln 2 2πh0 R NBh0 4NBh0a
(
Fdisk SA2NAa SB1NBa 2γ j BaNBa 2π2 )+ + kTQ h0 h R 3
NAa2 h02
+
)
NBa2 h2
+
(
( ))
N Ah π2 R2 h 6h 1 + + 2 ln 2 24 N a h0 π h NBh0 B
0
(10)
)
SA2NAa SB1NBa 2π2 NAa2 NBa2 + + 2 + F h disk ) h0 h 3 h2 h 0
( ((
( ) N Ah NBh0
1/2
)
( )))
NAh h 2h π2 3 γ j Ba2NB 1+ + ln 2 3 h0 h0 NBh0 (22) Semenov, A. N. SoV. Phys. JETP 1985, 61, 733.
((
24γ j BaNB2
)
1/3
( ))
NAh h 6h π 1 + + 2 ln h0 π h NBh0 2
0
(11)
2 3 (SA2NAx + SB1NB) F h disk ) - 2 + 8π NAx2 + NB
( (
( )))
N Ax 3 π2 γ j Ba2NB (1 + x) + 2x ln 2 3 NB h0 ) a
NAx2 + NB 4π2 3 S N x2 + S N x A2 A B1 B
1/3
(12)
If spreading parameters SA2 and SB1 are not very small, then we can use perturbation theory for the minimization of eq 12 over x because the first term is ∼NB or ∼NA whereas the second term is ∼NB1/3:
F h disk ≈ -
x≈
3SA22NA
-
3SB12NB
+ 8π2 8π2 SA2 2SA2 NASA2 π2 3 γ j Ba2NB 1+ + ln 2 3 SB1 SB1 NBSB1
( ((
)
( )))
SA2 4π2 4π2 , h0 ≈ a ,h≈a , SB1 3SA2 3SB1
((
24γ j BaNB2 R≈a SA2 6SA2 NASA2 π2 1 + + 2 ln SB1 π S NBSB1
( ) NASA2 NBSB1
((
1/2
B1
)
1/3
( ))
)
24γ j BaNB2 SA2 6SA2 NASA2 π2 1 + + ln SB1 π2S NBSB1 B1
1/3
1/3
( ))
(13)
A quantitative criterion of the applicability of perturbation theory (validity of eq 13) reads
The equilibrium value of F h disk is calculated by minimization over parameters R, h, and h0. Minimization over R can be done explicitly:
(
( ))
,
Further explicit minimization is possible if we introduce a new variable x ) h/h0 and use h0 and x as independent minimization parameters. Minimization over h0 leads to
R0 ≈ a
The total free energy per chain of the disk-shaped micelle (eq 1) takes the form
F h disk )
R0 ) a
0
)
1/3
N Ah h 6h π 1 + + 2 ln h0 π h NBh0 2
(9)
where E(r) ) dr/dn is the local stretching of the block (a derivative of the radial coordinate r over the number of segments n) that can be calculated from the space-filling condition a3Q dn ) 2πrh0 dr, E(r) ) a3Q/2πrh0. Therefore
F h A,out el
((
24γ j BaNB2
(7)
B
F h A,in el
R)a
1/3
( (
SA22NA + SB12NB . γ j Ba2NB 1 +
SA2 const SB1
))
1/3
(14)
In a manner similar to the control of single-chain adsorption,23 the thicknesses of the A and B layers, h0 and h, are controlled by the balance between adsorption and confinement (i.e., they are determined by the corresponding spreading parameters). One of the assumptions of our model is the validity of inequalities R . h and R0 . h0. Notice that these inequalities and eq 14 are fulfilled at SA2, SB1 ≈ 1. In the examined regime, the heights of the A and B layers do not depend on the lengths of the blocks whereas the radii of the disks have a strong dependence on NA and NB: R ≈ NB2/3 and R0 ≈ NA1/2NB1/6. Keeping in mind that (23) Bouchaud, E.; Daoud, M. J. Phys. (Paris) 1987, 48, 1991. (24) Alexander, S. J. Phys. (Paris) 1977, 38, 983.
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parameters h0 and h cannot be smaller than the size of the segment a, eqs 13 are valid only at SB1 e 4π2/3 and SA2 e 4π2/3. Above these values, h0 ) h ) a and the equilibrium free energy of the disklike structure has the form (see eq 11 at h0 ) h ) a)
F h disk ) -SA2NA - SB1NB +
(
(
( )))
NA 2π2 3 π2 j Ba2NB (NA + NB) + 2γ + ln 3 2 3 NB
1/3
4π2 4π2 , SB1 g 3 3
{
SA2 g
(15)
In the intermediate case, when only one of the spreading parameters exceeds the 4π2/3 threshold value, the free energy takes the form
( ((
)
( )))
( ((
)
( )))
3SA22NA 3 3SA2 3SA2 2π2 π2 2 -SB1NB + + γ j N 1 + + ln NB Ba B 3 2 3 8π2 4π2 2π2 4π2 4π2 SA2 e , SB1 g 3 3 F h disk ) 2 3SB12NB 3 π2 2π 4π2 8π2 2 -SA2NA + NA + γ j N 1 + + ln Ba B 3 2 3 3SB1 3SB1 8π2 4π2 4π2 SA2 g , SB1 e 3 3
3SA2NA
1/3
4π2NB
4π2NA 3NBSB1
1/3
(16)
It is well known that if one of the blocks of a diblock copolymer is too short then microphase separation does not occur. In our case, this means that there should be the critical value NcB and disk-shaped micelles appear only if NB > NcB. To estimate NcB, let us write down an expression for the free energy of nonaggregated chain. We assume that each nonaggregated chain forms its own globule, which is modeled by two coaxial disks: a bottom disk of radius R0 and thickness h0 composed of the A block and an upper disk of radius R and thickness h composed of the B block. Neither block forming disklike globules is subjected to radial stretching, and the corresponding contribution to the total free energy can be omitted. The conditions of the dense packing of the blocks are πR02h0 ) NAa3 and πR2h ) NBa3. The resulting free energy of the nonaggregated diblock takes the form
{
-
3(SA22NA + SB12NB)
+ 4πγ j Ba
8π2
x
πNB 4π2 , SA2, SB1 e 3SB1 3
2π2 4π2 -SA2NA - -SB1NB + (NA + NB) + 2γ j Ba xπNB, SA2, SB1 g 3 3
F h homo )
(17)
x
3SB12NB πNB 2π2 4π2 4π2 -SA2NA + NA + 4πγ j Ba , SA2 g , SB1 e 2 3 3SB1 3 3 8π 2 2 2 2 3SA2 NA 2π 4π 4π - SB1NB + + 2γ j BaxπNB, SA2 e , SB1 g N 2 3 B 3 3 8π
A comparison of F h disk (eqs 13, 15, and 16) with F h homo allows the determination of NcB(NcA). 2.2. Stripe-Shaped Structure. If the length of the B block increases, then one can expect the stripe-shaped structure to be stable. The stripe-shaped micelle is modeled as two infinite bars composed of A blocks (bottom) and B blocks (upper) (Figure 3). The heights and hemiwidths of the bars are denoted by h0, h, R0, and R, R . h, R0 . h0. The total free energy of the stripe is calculated in a similar way as for the case of the disks
F h str ) -
) (
(
)
SA2NAa SB1NBa γ h2NA j BaNBa 2π2 NAa2 NBa2 R2 + + 1 + + + h0 h R 3 h2 h2 NBa2 h02NB 0
(18)
where the space-filling conditions are a3QNB ) 2LRh and a3QNA ) 2LR0h0. Here, L and Q are the length of the stripes and the aggregation number of the micelle, which are assumed to be infinite, L,Q f ∞. The first three terms of eq 18 are the interfacial energy, the next term is a confinement free energy of A and B blocks, and the last term is the elastic free energy of A and B blocks, which is calculated under the assumption of equal stretching of the blocks (Alexander approximation24). By minimizing eq 18 over R, we get
F h str ) -
(
) ( (
))
SA2NAa SB1NBa 2π2 NAa2 NBa2 h2NA 3 2 + + + 2γ j N 1 + h0 h 3 h2 2 Ba B h2 h02NB 0 R)a
( ( )) γ j BaNB2
2 1+
h2NA
h02NB
1/3
NA h , R0 ) a NBh0
( ( )) γ j BaNB2
2 1+
h2NA
h02NB
1/3
1/3
(19)
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Furthermore, we again use the perturbation theory for minimization over h and h0:
To calculate NB at which a disk-stripe transition occurs, we should solve equation F h str ) F h disk. 2.3. Bilayer. If the B blocks are long enough or the spreading of the B blocks is very strong, then the only possible structure is a planar bilayer. (The adsorption of B blocks on the substrate is unfavorable in the considered regime.) In this case, we can assume that the blocks are not stretched laterally and their spreading provides a loss of conformational entropy. Then the total free energy of the double layer takes the form Fbilayer ) Fconf + Fsurf, where Fconf is calculated via eq 6. The surface energy is
Fsurf ) Σ(γBa + γBA + γAs) + γ0(σ - Σ) ) -Σ(γAa - γBa - γBA) - Σ(γ0 - γAs - γAa) + γ0σ
(21)
where Σ and σ are the areas of the film and the substrate, respectively. The space-filling conditions of the layers, Σh ) Q ˜ NBa3 and Σh0 ) Q ˜ NAa3, give the dependence between the thicknesses of the layers, h0 ) hNA/NB. Here, Q ˜ is the number of chains on the substrate and
F h surf )
Fsurf kTQ ˜
)-
SA2NAa SB1NBa h0 h
(22)
where the constant γ0σ/kTQ ˜ , which is equal for all of the structures analyzed, is omitted. The equilibrium free energy of the bilayer is
The first expression of the free energy in eq 23 is obtained by minimization of the sum of eqs 6 and 22 over h0 (or h), taking into account the dependence between h0 and h. The next two expressions in eq 23 correspond to the free energies of the bilayer where one of the layers has a minimal thickness a.
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Figure 3. Sketch of the stripelike micelle.
Figure 5. SA2 - (NB/(NA + NB)) phase diagram of the film at SB1 ) 5, γ j Ba ) 9.5, NA + NB ) 104.
Figure 4. SB1 - (NB/(NA + NB)) phase diagram of the film at SA2 ) 5 (a) and 13 (b). γ j Ba ) 9.5 and NA + NB ) 104.
To determine the boundaries of the stability regions of various structures of the film, we should compare the corresponding free energies.
3. Results and Discussion There are three parameters (SA2, SB1, and f ) NB/(NA + NB)) that are responsible for the structures formed in the ultrathin films. Let us plot SB1 - f and SA2 - f phase diagrams at a fixed value of the third parameter. The boundaries between phases in the diagrams are solutions of equations obtained by a pairwise equating of the free energies of various structures (i.e., F h homo ) F h disk, F h str ) F h disk, etc.). The SB1 - f diagrams at SA2 ) 5 and SA2 ) 13 are shown in Figure 4. One can distinguish two kinds of behavior of the film: SB1-independent and SB1-dependent. If the spreading parameter of the B blocks is larger than 4π2/3 (SB1-
independent behavior), then the B blocks form monomer-thick structures atop the A blocks. With the increase in the length of the B blocks, they form hexagonally ordered disks, parallel stripes, and a full coverage layer (Figure 4). Note that nonaggregated diblocks are stable only at very small values of f and the corresponding stability region is too narrow to be shown in Figure 4. The decrease in the spreading parameter below 4π2/3 (SB1dependent behavior) results in the expansion of the stability region of the disks and a narrowing of the regions of the stripes and bilayer. It is seen in Figure 4 that the smaller the SB1, the larger the f of the disks-to-stripes transition. This effect is related to the thickness of the B layer that increases with the decrease in SB1. If we introduce the effective number of strongly adsorbed (2D) segments of the B block, Neff B ) NB(a/h), then this number definitely decreases with decreasing SB1. Keeping in mind that the 2D ordering in the film is controlled byNeff B , the decrease in Neff acts to stabilize the disklike structure. If the spreading B parameter SB1 is small enough, then a direct disk-bilayer transition is possible. The increase in the spreading parameter of the A block shifts the boundaries of the SB1 - f diagram toward larger values of f (Figure 4b). This effect is due to the formation of a monomer-thick layer by the A blocks. The SA2 - f phase diagram of the film is shown in Figure 5. It is seen that effect of SA2 on a sequence of transitions between the phases is opposite to the effect of SB1: the decrease in SA2 stimulates disk-stripe-bilayer transitions. Now let us unify the results of the present work with the results obtained earlier for ultrathin films where both blocks can spread on the substrate.19 In the general case, there are six different surface tension coefficients that characterize adsorption and the spreading of diblock copolymers on a substrate. They are substrate/air (γ j 0), B block/air (γ j Ba), A block/air (γ j Aa), A block/ substrate (γ j As), B block/substrate (γ j Bs), and A block/B block (γ j AB) coefficients. Using these coefficients, it is possible to compose four spreading parameters that characterize the spreading of A blocks on the film formed by B blocks (SA1), A blocks on the substrate (SA2), B blocks on the film formed by A blocks (SB1), and B blocks on the substrate (SB2):
SA1 ) γ j Ba - γ j Aa - γ j AB SA2 ) γ j0 - γ j As - γ j Aa SB1 ) γ j Aa - γ j Ba - γ j AB SB2 ) γ j0 - γ j Bs - γ j Ba
(24)
Nanostructured Ultrathin Films
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Figure 6. Spreading parameters determine the kind of the ultrathin film: |γ j Ba - γ j Aa| > γ j AB (a) and |γ j Ba - γ j Aa| < γ j AB (b).
If one of the blocks wets the substrate and the other one dewets both the substrate and the film composed of the first blocks, then we deal with the structures of the film shown schematically in Figure 1a. This case corresponds to SA2 > 0, SB2 < 0, SB1 < 0 (A blocks wet the substrate) or SB2 > 0, SA2 < 0, SA1 < 0 (B blocks wet the substrate). If both blocks wet the substrate, SA2 > 0, SB2 > 0, then we have the second type of the film (Figure 1b). In the case when one of the blocks wets the substrate and the other block dewets the substrate but wets the film composed of the blocks of first type, we deal with the system studied in the present work (Figure 1c): SA2 > 0, SB2 < 0, SB1 > 0 (A blocks wet the substrate) or SB2 > 0, SA2 < 0, SA1 > 0 (B blocks wet the substrate). If both blocks dewet the substrate, then the
ultrathin film cannot be formed. These regimes are depicted schematically in Figure 6. It is easy to see that if |γ j Ba - γ j Aa| < γ j AB then both SB1 and SA1 are negative and the structures depicted in Figure 1c are not realized at any values of SA2 and SB2. Otherwise, when |γ j Ba - γ j Aa| > γ j AB, the spreading parameters SB1 and SA1 have different signs and all regimes are realized (Figure 6a). Keeping in mind that in the present article and in ref 19 we used similar approaches of strong spreading of the blocks (R . h,R0 . h0), it is reasonable to unify only regimes depicted in Figure 1b,c. Examples of the resulting phase diagrams are shown in Figure 7. We suppose that γ j Aa > γ j Ba and |γ j Ba - γ j Aa| > γ j AB so that SB1 > 0. These SA2 - SB2 phase diagrams correspond to
12364 Langmuir, Vol. 23, No. 24, 2007
PatyukoVa and Potemkin
Figure 7. SA2 - SB2 phase diagram of the film at SB1 ) 1, f ) 0.9, γ j Aa ) 15, γ j Ba ) 13.5, γ j AB ) 0.5, NA + NB ) 104 (a); SB1 ) 1, f ) 0.7, γ j Aa ) 15, γ j Ba ) 10, γ j AB ) 4, NA + NB ) 104 (b); SB1 ) 5, f ) 0.5, γ j Aa ) 15, γ j Ba ) 9.5, γ j AB ) 0.5, NA + NB ) 104 (c); and SB1 ) 5, f ) 0.3, γ j Aa ) 15, γ j Ba ) 9.5, γ j AB ) 0.5, NA + NB ) 104 (d). The A and B blocks are depicted in blue and cyan, respectively.
the films with substratephobic (disks, stripes, and bilayer) and substratephilic (disks, stripes, and holes) structures. Films with substratephobic B block are definitely stable at small (or negative) values of SB2, whereas the substratephilic structures arise at SB2 ≈ SB1. For example, Figure 7a corresponds to the strongly asymmetric copolymer with f ) 0.9 (the B block is much longer than the A block) and with the relatively small parameter SB1, SB1 ) 1, which characterizes the spreading of the B blocks atop the A blocks. If parameter SB2 is negative or slightly positive (B-block contact with the substrate is unfavorable or these contacts are less favorable than contacts with the A blocks), then the bilayer structure of the film is the most stable. In this case, the thickness of the film (A and B layers) depends on the SA2 parameter that characterizes the spreading of the A blocks on the substrate: the larger SA2, the smaller the thickness of the A layer. The film has a minimum thickness at SA2 g 4π2/3. The increase in SB2 makes B-block contact with the substrate favorable. When the A blocks form a monolayer on the surface (SA2 g4π2/3), then the increase in SB2 leads to a series of transitions: bilayerprominent disks to prominent stripes to holes to flat disks. Such a transition sequence is determined by the composition of the copolymer. First (for small enough SB2), the number of strongly adsorbed (2D) segments of the B blocks (Neff B ) NB(a/h)) is smaller than NA, and the B blocks form the core of the disklike eff micelles. Increasing, SB2 gradually enlarges Neff B . Thus, when NB
is comparable to NA, the stripes becomes favorable. The inverse structures, holes or flat disks (the B blocks form coronas of the micelles), are formed at Neff B > N A. If the spreading of the A blocks on the substrate is not full (SA2 < 4π2/3), then other sequences of transitions are possible: bilayer to stripes to holes to prominent disks or bilayer to holes to prominent disks (Figure 7a). This peculiarity is related to the greater thickness of the A layer (i.e., it is related to the decrease in the number of adsorbed segments of the A blocks (Neff A )). The effect of the composition of the copolymer is demonstrated in Figure 7b. Decreasing NB (f ) 0.7 in Figure 7b) excludes the stability of the bilayer at large values of SA2, SA2 g4π2/3, and provides morphological variations with SA2 at negative values of SB2: the spreading of the A blocks on the substrate stimulates transition bilayer to substratephobic stripe to substratephobic disk transitions. The next two diagrams (Figures 7c,d) correspond to the films with higher “philicity” of the A and B blocks (SB1 ) 5) differing in the copolymer composition, f ) 0.5 (c) and 0.3 (d). The main difference of these diagrams from those in Figure 7a,b is that the transition from the substratephobic to the substratephilic structures shifts toward larger values of SB2. Besides, a longer A block of the copolymer (f ) 0.3) is responsible for the stability of the stripes under full spreading of the B blocks, SB2 g4π2/3 (Figure 7d).
Nanostructured Ultrathin Films
In conclusion, we have developed a theory of nanostructures in ultrathin diblock copolymer films in the case when one of the blocks (A) is attracted to the substrate and the other block (B) is repelled from the substrate but spreads atop the A layer. It has been shown that the morphology of the film is controlled by the composition of the diblock copolymer and by the strength of attraction of the A blocks to the substrate and the B blocks to the A layer. The structures of the film calculated include disklike, stripelike, and bilayer. Using the results of ref 19, where the
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spreading of both blocks on the substrate was analyzed, we have plotted a unified phase diagram including all diversity of the film structures. Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft within the SFB 569 and the Russian Foundation for Basic Research is gratefully acknowledged. LA701989S
