Nanopattern of Diblock Copolymers Selectively Adsorbed on a Plane

Sep 18, 1999 - Aggregation behavior of a polystyrene–b-poly(phenylsilsesquioxane) H-type copolymer at the air/water interface. Yongsok Seo , Sang Gi...
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Nanopattern of Diblock Copolymers Selectively Adsorbed on a Plane Surface I. I. Potemkin,*,† E. Yu. Kramarenko,† A. R. Khokhlov,† R. G. Winkler,‡ P. Reineker,‡ P. Eibeck,§ J. P. Spatz,§ and M. Mo¨ller§ Physics Department, Moscow State University, Moscow, 117234 Russia, Department of Theoretical Physics, University of Ulm, 89069 Ulm, Germany, and Laboratory of Organic Chemistry/Macromolecular Chemistry, University of Ulm, 89069 Ulm, Germany Received January 25, 1999. In Final Form: April 28, 1999 Surface interaction controlled microphase separation leading to the formation of chemically heterogeneous surface nanopatterns in dry ultrathin films of A-B diblock copolymers is studied experimentally and theoretically in the strong segregation limit. On a planar surface one of the blocks (A block) is strongly adsorbed, forming a tightly bound monomolecular layer (two-dimensional melt). The nonadsorbed B blocks can aggregate due to incompatibility with the A-block layer on the substrate and with the air. As a result, a chemically heterogeneous surface pattern can emerge. Depending on the block length ratio and the interaction parameters, the dewetting B blocks can assemble to either globular surface micelles or wormlike surface aggregates, that is, a point or a striped surface pattern. The region of stability of these morphologies and the main parameters, such as aggregation number, size, and periodicity, of the surface micelles and wormlike surface aggregates have been determined as functions of the lengths of the blocks and the interaction energies of the blocks with each other and with the air. The phase diagram is compared with experimental data for poly(styrene)-block-poly(4-vinylpyridine) on mica.

Introduction Generation of a chemically heterogeneous surface pattern by the controlled adsorption and self-assembly of organic molecules and polymers represents a topical challenge for functional substrates and templates in nanotechnology, molecular biology, biomineralization, colloid science, and supramolecular chemistry. Present concepts include soft lithography, controlled dewetting, and microdomain formation in thin block copolymer films.1-11 The symmetry of the structures formed by diblock copolymers in the bulk depends on the chain architecture and the relative length of the blocks NA and NB, respectively. For instance, in the case of strongly asymmetrical diblock copolymers, two-dimensional hexagonal and three-dimensional bcc micellar structures can emerge while the stability of a one-dimensional lamellar * To whom correspondence should be addressed. † Moscow State University. ‡ Department of Theoretical Physics, University of Ulm. § Laboratory of Organic Chemistry/Macromolecular Chemistry, University of Ulm. (1) Chou, S. Y.; Krauss, P. R.; Renstrom, P. J. J. Vac. Sci. Technol., B 1996, 14 (6), 4129. (2) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Fery, A.; Ibn-Elhaj, M.; Schlagowski, S. Science 1998, 282, 916. (3) Spatz, J. P.; Roescher, A.; Sheiko, S.; Krausch, G.; Mo¨ller, M. Adv. Mater. 1995, 7, 731. (4) Jackman, R. J.; Wilbur, J.; Whitesides, G. M. Science 1995, 269, 664. (5) Harrison, C.; Park, M.; Chaikin, P. Polymer 1998, 39, 2733. (6) Park, M.; Harrison, C.; Chaikin, P.; Register, R.; Adamson, D. Science 1997, 276, 1401. (7) Liu, Y.; Zhao, W.; Zheng, X.; King, A.; Rafailovich, M. H.; Sokolov, J.; Dai, K. H.; Kramer, E. J.; Schwarz, S. A.; Gebizlioglu, O.; Sinha, S. K. Macromolecules 1994, 27, 4000. (8) Ge, S.; Takahara, A.; Kajiyama, T. Langmuir 1995, 11, 1341. Wilbur, J. L.; Biebuyck, H. A.; MacDonald, J. C.; Whitesides, G. M. Langmuir 1995, 11, 825. (9) Krausch, G. Mater. Res. Rep. 1995, 14, 1. (10) Anastasiadis, S.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. Phys. Rev. Lett. 1989, 62, 1852. (11) Bassereau, P.; Brodbreck, D.; Russel, T. P.; Brown, H. R.; Shull, K. R. Phys. Rev. Lett. 1993, 71, 1716.

structure is observed for diblock copolymers with symmetrical composition. The thermodynamical variable which describes the phase state of A-B diblock copolymers in the bulk is the product χABN of the Flory-Huggins interaction parameter between monomer units of type A and B and the degree of polymerization of the copolymer N.12-15 For symmetrical copolymers the transition between the ordered and disordered states occurs when χABN ∼ 10.13 In the vicinity of the phase transition the A-B interactions are weak, individual chain statistics is unperturbed Gaussian, and the ordered composition profile is approximately sinusoidal (weak segregation regime). Far from the transition point, that is, at χABN . 10, nearly pure A and B microdomains are formed. The interfacial region becomes very narrow, and the chains adopt extended conformations. The characteristic size of microstructures is determined by the balance between energetically favorable separation of A and B blocks and entropically unfavorable stretching of the individual blocks.12,14 An additional interaction of copolymers with a surface can lead to essential changes in structure morphology or even to its disappearance. Any slight surface preference of A or B monomer units induces compositional order with the formation of lamellae oriented parallel to the surface even above the bulk order-disorder transition.10,15,16 For thin films this leads to the formation of surface-induced, highly ordered multilayers of A-B microdomains. In the case of an equilibrated supported film, its thickness is equal to an integral value of the bulk lamella periodicity (eventually plus one-half). Mass balance is achieved by the formation of holes in the top layer. One obtains a (12) Helfand, E.; Wasserman, Z. R. Macromolecules 1976, 9, 879. (13) Leibler, L. Macromolecules 1980, 13, 1602. (14) Semenov, A. N. Sov. Phys. JETP 1985, 61, 733. (15) Bates, F. S.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990, 41, 525. (16) Radzilowski, L.; Carvalho, B.; Thomas, E. J. Polym. Sci. B 1996, 34 (17), 3081.

10.1021/la9900730 CCC: $18.00 © 1999 American Chemical Society Published on Web 09/18/1999

Nanopattern of Diblock Copolymers

chemically homogeneous surface of the lower surface energy block without any lateral order. Recently, it has been shown that laterally ordered microdomain structures can appear on the surface if the film thickness is much smaller than the thickness of a complete lamella.17,18 Such structures appear when one of the blocks is strongly adsorbed and forms a tightly bound monomolecular layer on the surface while monomer units of the other blocks are incompatible with the surface. These ordered surface patterns were first observed experimentally in thin films formed by adsorption of symmetrical polystyrene-block-poly(2-vinyl-pyridine) copolymers from a dilute nonselective solvent onto a mica substrate.17,18 It has been shown by scanning force microscopy that the observed structures consist of a thin layer of P2VP blocks covering the substrate and surface micelles of PS blocks arranged on top of this layer in hexagonal near range order.17,18 Steric crowding of the chains in the course of micelle formation is found to lead to the stretching of P2VP blocks on the substrate. In a previous paper19 we proposed a simple model to describe the parameters of surface micellar nanopatterns and reveal the main physical factors leading to microphase separation in adsorbed block copolymers. The results of our calculations seem to be in good agreement with the experimental data. We have shown that the equilibrium structure of micelles is determined by the interplay of the surface energy which favors the aggregation process and the energy of stretching of the adsorbed blocks. The parameters of surface micelles were shown to differ significantly from those for micelles in the bulk. In ref 19 we considered only the case when the length of nonadsorbed blocks forming micelles is much smaller or comparable to the length of adsorbed blocks. It is in this case that microphase separation leads to the formation of globular micelles on the surface. However, one could expect that in some situations other morphologies (wormlike and brush structures) can arise on the surface. In this paper we analyze this possibility; that is, our aim is the construction of the full phase diagram of block copolymer films. The paper is organized as follows. In the next two sections we describe the model and the method of calculation of the free energy of surface micelles used in the previous paper19 and recall briefly the main results on micellar structures. In the following section we apply this method to evaluate the free energy of the wormlike surface aggregates (stripes) and define the main characteristics of these patterns. Then we discuss the conditions of stability of micelles and stripes and the transition between these phases. The theoretical results are compared with experimental data. Model Let us consider a dry film of A-B diblock copolymers which is formed by adsorption of one of the blocks (e.g., A blocks) on a flat surface. We suppose that the blocks are flexible and are characterized by statistical segments of equal length a. According to the experimental observations, the adsorbed blocks strongly flatten on the surface so that they form practically a monomer thick layer with very low probability of loop formation.17.18 Thus, as in the previous (17) Spatz, J. P.; Roescher, A.; Sheiko, S.; Krausch, G.; Mo¨ller, M. Adv. Mater. 1995, 8, 731. (18) Spatz, J. P.; Mo¨ller, M.; Noeske, M.; Behm, R. J.; Pietralla, M. Macromolecules 1997, 30, 3874. (19) Kramarenko, E. Yu.; Potemkin, I. I.; Khokhlov, A. R.; Winkler, R. G.; Reineker, P. Macromolecules 1999, 32, 3495.

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Figure 1. Schematic representation of model surface micelles (a) and surface stripes (b) formed by A-B diblock copolymers with A blocks strongly adsorbed on a planar surface.

paper,19 we consider the adsorbed blocks as two-dimensional and we assume that the density of A monomer units on the surface corresponds to the case of a two-dimensional melt. We suppose that the contacts of B blocks with A monomeric units and with the surface are unfavorable and that B chains form three-dimensional coils or other structures over the surface. Thus, the adsorbed block copolymers form a kind of a brush with the grafting density depending on the length of the adsorbed blocks. If the compatibility of B monomer units with the air is poor, individual B chains would collapse and at NA , NB collapsed B blocks would form a uniform layer completely covering the surface. However, when the value of NB is not so large, the aggregation of B chains due to the incompatibility with the air can lead to lateral phase separation, resulting in the formation of either surface globular micelles of B blocks or a striped structure (cylindrical rods laying on the surface) (see Figure 1). The type of the surface structure depends on the relative ratio between the length of the blocks and the interaction parameters. To define the parameters of the aggregates (their size, aggregation number, periodicity) and the conditions of stability of micelles and stripes on the surface, we write down the free energy of the system under consideration using the strong segregation approximation.14 Globular Micelles The free energy of a micelle in the strong segregation limit can be written as a sum of three terms:19

Fmic ) Fsurf + FAel + FBel

(1)

The first term Fsurf is the surface free energy, which is the energy of the interfaces. Its effect is to minimize the number of unfavorable contacts of monomer units of different blocks with each other and with the air:

Fsurf ) γ1S1 + γ2S2 + γ3S3

(2)

The first two terms in eq 2 are the energies of the interfaces between the air and blocks B and A, respectively. The third term accounts for the energy of the block A/block B interface. In this expression γi and Si, i ) 1, 2, 3, are the surface tension coefficients and the surface areas of the corresponding interfaces (see Figure 2).

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πD2a NA ) V NB

Figure 2. Schematic representation of a surface micelle (a surface stripe) formed by diblock copolymers on the adsorbing surface Z ) 0. Adsorbed A chains form a 2D melt on the surface. The B block micelle (stripe) is described by a part of a sphere (a cylinder) of radius RS. H and R are the height and the radius (half-width) of the micelle (stripe) on the plane Z ) 0, respectively. γ1, γ2, and γ3 are the surface tension coefficients of the air/polymer B, the air/polymer A, and polymer A/polymer B interfaces, respectively.

We like to stress that in our model of a two-dimensional melt the contribution from the energy of interaction of adsorbed blocks with the surface to the total energy of the system is constant for any type of the structures formed by B blocks on the surface. Thus, in the following we shall not take into account this contribution to obtain the equilibrium characteristics of the surface structure and the conditions at which the transition between micellar and cylindrical phases takes place. According to the simplest symmetry considerations, we shall describe the form of the micelle by a spherical segment of radius RS (see Figure 2). Contrary to the case of the micelles formed in the bulk, which can be characterized by one length scale due to symmetry arguments and the conditions of dense packing, in the system under consideration the surface micelles should be characterized by two independent length parameters, for example, by the radius R and the height H (see Figure 2). We assume that monomeric units of B blocks are strongly incompatible with the air and that their volume fraction in aggregates is equal to unity. The condition of dense packing of monomer units in the surface micelle can be written as follows:

π H(3R2 + H2) ) QNBa3 6

z(r,NB) ) xRS2 - r2 + H - RS

(4)

S3 ) πR2 where SA is the surface area per one micelle in the system. We suppose that the micelles are ordered on the adsorbing surface with the symmetry of a hexagonal lattice, and hence the surface per one micelle has the form of a regular hexagon. With high accuracy one can approximate it by a circle of radius D ) (SA/π)1/2.14 According to the conditions of dense packing, the value of D is connected with the length of A blocks NA and with the height H and the radius R of the surface micelle through

(6)

RS ) (R2 + H2)/2H Thus, for the distribution of the junction points σm(r) we obtain the following expression:

σm(r) )

2πr z(r, NB) NBa3

(7)

The total number of blocks in one micelle is connected with σm(r) simply via Q ) ∫R 0 dr σm(r). The local elongation of B chains starting at the distance r from the micellar center is

dz(r, nB) ) z(r,NB)/NB dnB

(8)

where nB counts the B monomers along the chain (see Figure 4). The free energy of stretching of B blocks in the micelles FBel can be found by summation of the local elongations along each chain and then over all chains in the micelle:

FBel ) kBT

S1 ) π(R2 + H2) S2 ) SA - πR2

where V is the volume of the micelle, V ) (π/6)H(3R2 + H2). The terms FAel and FBel in eq 1 are the free energies of stretching of A and B blocks. To calculate these contributions, we use again symmetry considerations and we suppose that the mean contours of B chains are just the straight lines perpendicular to the adsorbing surface and that the contours of A blocks possess a radial symmetry. In analogy with the Alexander-de Gennes approximation for polymer brushes,20,21 we suppose that free ends of B blocks are located on the surface of the micelle. To find the distribution of the A-B junction points on the adsorbing plane under the micelle σm(r), we use again the condition of the dense packing of B monomeric units in the micelle. The number of chains whose junction points are located in the thin layer of width dr at the distance r from the center of the micelle is equal to σm(r) dr. They occupy the cylindrical layer (see Figures 3a and 4) of the volume 2πrz(r,NB) dr, where z(r,NB) is the mean end-toend distance of B blocks in this layer:

(3)

where Q is the aggregation number, that is, the mean number of chains in one micelle, and a3 is the volume per monomer unit. The surface areas of the boundaries between different phases have the following forms:

(5)

∫0Rdr σm(r)∫0N

B

( ) 1 dz a dnB

2

dnB

(9)

where T and kB are the temperature and the Boltzmann constant, respectively. Substituting here the expressions for σm(r) and dz(r,nB)/dnB (eqs 7 and 8) and integrating, we obtain

FBel π H3 3 ) R2 + H2 kBT 4 a5N2 5 B

(

)

(10)

To calculate the free energy of stretching of the adsorbed A blocks, it is convenient to express it as a sum of two terms (20) Alexander, S. J. Phys. 1977, 38, 983. (21) de Gennes, P.-G. J. Phys. 1976, 37, 1443.

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of the A segmets which are exposed to the air can be found from the dense packing arguments, dr/dnA ) Qa2/(2πr), and the corresponding contribution to the free energy assumes the following simple form:

FAout Q ) kBT a2

2

dr Q D ln( ) ) ∫RDdr dn 2π R A

(12)

To calculate FAin, we also use the condition of the dense packing of monomer units applied to every circular layer. In the ring of radius r and of width dr there are dnA∫r0 σ(r′) dr′ monomer units, where dnA is the average number of monomer units per chain. Hence, the average elongation of the chains in the ring is defined by the following expression

a2 dr ) dnA 2πr

∫0rdr′ σm(r′)

(13)

and the elastic energy of this ring is

( )∫

dFAin dr dr ) kBT a2 dnA

r

0

dr′ σm(r′)

(14)

Integrating this expression over the radius r, one obtains the following result for the elastic energy of the A block segments laying under the micelle (in the “core” region)

Figure 3. Schematic representation of the adsorbed A blocks in the surface micelle (a) and in the surface stripe (b) (top view). We denoted as a “core” region the area of size R where the junction points of the blocks are located and a “corona” region at R < r < D where A blocks are in contact with the air.

FAin ) kBT )

r dr [∫0 dr′ σm(r′)]2 ∫0R2πr

3 H3 Q 3 Km(x), x ) R/H 64 a N

(15)

B

The function Km(x) is calculated in the Appendix. It is easy to show that for all values of the parameter x (the asymptotics of the function Km(x) are presented in the Appendix) FAin ∼ FAout if the length of the A blocks is comparable with the length of the B blocks. On the other hand, comparing the stretching free energies of adsorbed A blocks and B blocks forming the surface micelle, one can see that Figure 4. Schematic representation of the assumed contours of B blocks in the surface micelle (stripe).

FAel ) FAin + FAout

(11)

where FAin and FAout are the free energies of stretching of segments of A blocks which lay under the surface micelle (“core” region, see Figure 3a) and which are in contact with the air (“corona” region, see Figure 3a), respectively. The number of the chains in the “core” region that start within the circle of radius r is equal to q(r) ) ∫r0 σm(r′) dr′. The number of chains in the “corona” region is constant and equal to the aggregation number of the micelle Q (it is not favorable for a whole chain to lay in the “core” region, since in that case other chains would be extra stretched). The free ends of the adsorbed blocks are also somehow distributed at the periphery of the “corona” region. However, the contribution to the stretching free energy from the end segments of the blocks is small, and with high accuracy14 one can use an approximation according to which all the block ends are “grafted” to the boundary r ) D of the Wigner-Seitz cell. Thus, the local stretching

FBel FAel

∼

{

a/H, x∼1 2 a/(Hx ), x . 1

(16)

Since a/H , 1 it is clear that FBel , FAel and one can neglect the contribution from the stretching of B chains, that is, FBel. Thus, the total free energy of the surface micelle consists of two main contributionssthe surface free energy and the energy of stretching of the adsorbed A blocks. To find the equilibrium values of the micelle dimensions, one has to minimize the free energy per chain with respect to the corresponding parameters. As variables, it is convenient to choose the normalized height of the micelle

h)

H 1/4 aN1/2 B γ1

(17)

where γ1 is the dimensionless surface tension coefficient, γ1 ) γ1a2/(kBT), and x the ratio of the micelle radius R to the micelle height H, x ) R/H. The total free energy as a function of these two variables has the following form:

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2 6 1 + x (1 + δ) 3 ) + h3Km(x) + 2 1/2 3/4 h 64 kBTQNB γ1 1 + 3x 3 γ2 t1/4 h 1 (1 + 3x2) ln h 1 + 2 + t1/4 (18) 24 2 γ1 3x

corresponding to the minimum of the free energy in the case of δ ) 0. The last term accounts for the energy of translational motion of chains in micelles. Comparing the free energies FI and F (eqs 19 and 20), we obtain that the critical value of the length of A blocks above which there is no micelle formation is

where δ is a combination of surface tension coefficients, δ ) (γ3 - γ2)/γ1. The dependence on the degree of polymerization of the blocks A and B is included in the expression (18) through the dimensionless parameter t ) γ1N4A/N2B. The detailed analysis of the free energy of surface micelles has been carried out in our previous paper,19 and the dependences of micelle parameters on the variables δ and t have been calculated. It has been shown that the value of δ influences the form of the surface micelles. According to the experimental situation, the value of δ is negative (for the PS-P2VP block copolymers used in the experiments the incompatibility of the P2VP blocks with air is higher than that with PS blocks forming micelles). When δ is negative, the radius of the micelles is larger than their height. It has been shown that with the increase of δ the values of R and H tend to become equal to each other and at δ ) 0 the form of the micelle is practically semispherical.19 One of the main results of ref 19 is the conclusion that, apart from logarithmic factors, the radius R and the height H of the micelles, as well as their aggregation number Q, scale with the length of nonadsorbed B blocks as N1/2 B . The decrease of the dimensions of the micelles on the surface in comparison with micelles in the bulk (for which R ∼ N2/3 B ) is connected with the strong stretching of the adsorbed chains. In fact, the radius D of the WignerSeitz cell and (approximately) the size of the A blocks in 1/4 the system of consideration scale as RA ∼ N1/2 A NB while for a spherical micelle in the bulk (in the case NA . NB) 1/3 and, hence, the stretching for this radius Rsp A ∼ (NANB) the latter case is smaller. Thus, to compensate the gain in the surface free energy of the micelle on the surface, one needs a smaller number of chains Q than that for the micelle in the bulk and therefore the size of the surface micelle R ∼ (QNB)1/3 will also be smaller. One should mention that when the length of the adsorbed blocks is much larger than the one of nonadsorbed blocks (NA . NB), B chains do not aggregate but form single globules on the surface.19 To estimate the critical value of the ratio NA/NB, one has to compare the free energy of individual chains with the free energy of chains in the micellar aggregate. When B chains do not aggregate, the free energy per one chain has a simple form:

1/4 2/3 (NA)c ∼ N1/2 B /γ1 exp(γ1NB )

Fmic

[ (

)]

2 FI/N ∼ kBT ln(1/NAa2) + N2/3 B γ1a

F 1/4 3/4 1/2 2 ∼ N1/2 + ln(γ1/4 B γ1 (ln t) 1 /NB a ) kTQ

that is, it grows exponentially with the length of B blocks and with the increase of incompatibility of B monomer units with the air. Let us consider now the other limiting case of large values of the ratio NB/NA. Wormlike (Stripe) Structure With the increase of the length of B blocks, the surface of the micelles increases and at some value of NB micellar structures become unfavorable and the chains begin to form a quasi-one-dimensional stripe structure: alternating “ridges” of B polymers in the “sea” of adsorbed A polymers. As for the case of micelles, the free energy per unit length of the stripe can be written in the form of eq 1, where one can also neglect the contribution from the stretching of B blocks FBel in comparison with that of A blocks FAel. We shall describe the form of a stripe by a cylindrical segment of radius RS (we do not consider here defects of the one-dimensional structure on large length scales), see Figures 1b and 2, and introduce the same notations as in the case of surface micelles; that is, H is the height of the stripe and R and D are the half-widths of the stripe and the Wigner-Seitz cell dimension, respectively. The expressions for the corresponding surface areas of the interfaces between A and B blocks and the air per unit length of the stripe assume the following form

h 1(x) S1 ) HS S2 ) 2(D - R) S3 ) 2R

(20)

The first term in this expression is written down with the use of eq 18, in which we put h ∼ 1/(ln t)1/4,

( )

S h 1(x) ) (1 + x2) arccos

x2 - 1 x2 + 1

(22)

and the condition of dense packing of monomer units in the stripe and in the adsorbed layer can be expressed as follows

H2V h (x) ) NBQa3 NA 2Da ) 2 N HV h (x) B

(19)

Here the first term is the energy of the independent translational motion of the chains on the surface and the second term is the surface free energy of the B block globule. To estimate this latter contribution, we assumed for simplicity that γ2 ) γ3 and considered the globules to be semispheres of radius N1/3 B a. The free energy of a chain in micelles in the case of δ ) 0 has the following form:

(21)

V h (x) )

( )

(1 + x2)2 x x2 - 1 arccos 2 + (1 - x2) (23) 4 2 x +1

where x ) R/H. The density distribution σl(r) of the junction points between A and B blocks under the stripe is determined also from the condition of the dense packing of B monomer units in the stripe

σl(r) )

2 z(r,NB) a NB 3

(24)

where the distance z(r,NB) from the adsorbing surface to

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Figure 5. Phase diagram of the layer of adsorbed diblock copolymers in the coordinates t ) γ1N4A/N2B and δ ) (γ3 - γ2)/γ1. The solid line separates the regions of stability of micellar and stripe surface structures.

the polymer B/air interface as a function of the coordinate r is defined by eq 6. We suppose that all the A block ends are “grafted” to the boundary r ) D. Then the local stretching of the A blocks dr/dnA ) (D - r)/NA depends only on the location of the A-B junction points under the stripe. Taking into account the variation of the number of chains under the stripe, q(r) ) ∫r0 σl(r′) dr′ (q(R) ) Q), we obtain the following expression for the corresponding free energy

FAel ) kBT

(D - r)2

∫0 dr σl(r) R

a2NA

(25)

The integral in eq 25 is calculated in the Appendix. To minimize the free energy, we choose again as relevant variables the normalized height of the stripe

h)

1 H a N1/2γ1/4 B

(26)

1

and the ratio x ) R/H. Then the expression for the total free energy takes the following form

Fstr 3/4 kBTQN1/2 B γ1

)

h 1(x) + 2xδ 1S + t1/4h4G1(x) h V h (x) h3G2(x) +

γ2 h2 G (x) + t1/4 (27) 1/4 3 γ1 t

The functions Gi(x), i ) 1, 2, 3, are presented in the Appendix. As for the case of surface micelles, the dependence of the free energy on the surface tension coefficients and the lengths of A and B blocks appear through the parameters δ and t. Before analyzing the equilibrium size of the stripe and the period of the formed structure, let us find the stability region of this structure, that is, the range of δ and t values for which the free energy of the stripe (eq 27) becomes smaller than the free energy of the micellar structure (eq 18). The corresponding phase diagram is presented in Figure 5 in the variables δ and t. One can see that the stability of the stripe structure is realized at small values of t ) γ1N4A/N2B, that is, at large length of the 2 blocks forming the stripe, NB ∼ γ1/2 1 NA. With the decrease of δ, which corresponds to an increase in incompatibility

of the adsorbed A blocks with the air in comparison with that of the B blocks, the region of stability of the stripe phase increases and the transition to the micellar structure takes place at smaller values of NB. This behavior is due to an increase of asymmetry of the micelle with a decrease of δ. The critical value of the surface area of the micelle at which the transition into a stripe structure occurs is reached for more asymmetrical micelles at smaller values of their volume, that is, at smaller values of NB. Near the transition shown in Figure 5, the emerging 1/2 stripes have Gaussian size, R, H ∼ γ1/4 1 NB . Simultaneously, the adsorbed blocks become very stretched, RA ∼ γ1/2 1 NA, and the distance between neighboring stripes becomes of the order of their width, D/R ∼ 1. For instance, for δ ) 0 the ratio D/R ) 1.62 and for δ ) -0.2 we have D/R ) 1.56. We like to point out that at very large values of NB/NA and/or negative values of δ, when the width of the stripes becomes much larger than their height and the neighboring stripes can touch each other, the wormlike structure becomes unfavorable. This situation is close to the case of the planar brush of homopolymers which are grafted to the surface by short blocks. To estimate an upper boundary of the region of stability of the wormlike structure, one has to compare the free energies of the chains in stripes and in the planar brush. The free energy per chain of a polymer brush with a fixed grafting density, F ) 1/(NAa2) (the number of grafted chains per unit area), has the following form:20,21

( )

Fbr NB N2A 1 + ln 1/4 ) (γ1 + γ3)NA + 2 + kBTN NA NB t

(28)

Here the first term incorporates the surface energies of B/air and AB interfaces, respectively. The second and the third terms in eq 28 are elastic contributions of stretching (∼h2/a2NB) of the B chains and their contraction (∼a2NB/ h2), where h ) aNB/NA is the height of the brush. The last term in eq 28 is the difference of the energies of translational motion of the chains on the surface in the brush and a stripe. Comparing the free energies Fbr, Fmic, and Fstr (eqs 18 and 27) and also taking into consideration the condition (eq 21) for the micelle formation, we construct schematic phase diagrams of the system under consideration in the variables (NA - NB) for two different values of the tension coefficient γ1. At a small enough value of γ1, γ1 ) 0.1 (see Figure 6a), there is a wide interval of NA values where the wormlike phase is stable. With the increase of γ1 (i.e., increase of the incompatibility of the B monomer units with the air), the region of stability of the wormlike phase becomes more narrow and the direct transition from the micellar phase to the brush structure is possible at certain values of NA and NB (see Figure 6b). This means that, for those “short” B chains that cannot form a brush structure at low γ1, it is more favorable to form a brush being compressed at high γ1 to minimize the surface area. Discussion and Comparison with Experimental Data In the present paper the theory of strong segregation has been developed to describe the microstructures in the system of diblock copolymers when one of the blocks is adsorbed on the surface and the other blocks can aggregate in structures of various types due to incompatibility with the air.

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Figure 6. Phase diagrams of the dry film of A-B diblock copolymers with A blocks strongly adsorbed on a planar surface for δ ) (γ3 - γ2)/γ1 ) 0 and γ1 ) 0.1 (a) and γ1 ) 0.75 (b).

The aggregation process starts at NA less than the critical value (NA)c, eq 21, and in a wide range of values of NA the micellar structure appears to be thermodynamically the most stable one. The form of the micelles is determined by the relationship between the surface tension coefficients γ2 and γ3 of the adsorbed A blocks with the air and with B blocks, respectively. If γ2 ∼ γ3, the form of the micelles is close to a hemisphere. If γ2 > γ3, the height of the micelles is less than their radius. It is this relationship between micellar dimensions that is observed experimentally in refs 17 and 18. Contrary to the case of three-dimensional diblock copolymer micelles in the melt, which are characterized by one length scale (their radius Rbulk ∼ N2/3 B ), the surface micelles have 1/2 Gaussian size and R, H ∼ γ1/4 1 NB . This difference is connected with the stronger stretching of the adsorbed blocks in comparison with the stretching of the chains forming “the corona” of three-dimensional micelles. This leads to a decrease of the aggregation number Q of the 1/2 surface micelles, Q ∼ γ3/4 1 NB , and hence their size decreases, too. Simultaneously, the period D of the micellar structure on the surface more strongly depends on the length NA of the adsorbed blocks than on the length NB 1/4 of the aggregated blocks, D ∼ N1/2 A NB , while the micellar

Potemkin et al.

structure in the bulk exhibits the same dependence on both NA and NB, Dbulk ∼ (NA NB)1/3. With the increase of the length of B blocks up to the 2 value NB ∼ γ1/2 1 NA, the micellar structure becomes unstable, and a quasi-one-dimensional wormlike structure (stripes laying parallel to the surface) can be formed on the surface. In the transition region the micelles and stripes exhibit the same scaling dependence of their characteristic size, R, H ∼ γ1/4N1/2 B ; however, for larger NB the period of these structures becomes of the order of their size D ∼ R. With a further decrease of the ratio NA/NB neighboring stripes start to touch each other and merge into a uniform layer of B chains completely covering the surface. We like to mention that the stability of the stripe phase depends essentially on the value of γ1. At high γ1 the decrease of the ratio NA/NB can induce the transformation of the micellar structure right into a homogeneous B chain planar brush without an intermediate stripe phase. Our theoretical results are in good agreement with the experimental observations. Recently, structures of different morphologies have been observed in thin films of polystyrene-block-poly(4-vinylpyridine) (PS-b-P4VP) diblock copolymers. PS-b-P4VP molecules were dissolved in chloroform, which is a nonselective solvent for both blocks. The polymer concentration varied from 0.01 to 0.1 g/L. Freshly cleaved plates of mica were dipped in these solutions and pulled out at a constant rate (typically 10 mm/min). After drying, the structure of the films was investigated by scanning force microscopy (SFM). The images of structures formed by adsorbed PS-b-P4VP copolymers with different degrees of polymerization of the blocks are presented in Figure 7. They are a topography of a AFM micrograph having a size of 5 × 5 µm2. The height information is encoded in the brightness of the structure: the bright color corresponds to the clusters of PS blocks with a maximum height of 10 nm while dark regions depict the P4VP layer. One can see that the type of the structure formed on the substrate depends essentially on the relative length of the blocks. When the length of PS blocks (molecular weight MPS) is smaller than the length of P4VP blocks (molecular weight MP4VP), the island-like structure (micelles) is more favorable (Figure 7a). With the increase of the length of PS blocks, this structure transforms into stripelike structures (quasi-one-dimensional structures on the small length scales) (Figure 7b,c). Comparing the images of the stripe structures, we may conclude that with the increase of both the length of PS and P4VP blocks the width of the bright regions (PS blocks) remains practically the same whereas the width of the dark regions (adsorbed P4VP blocks) increases significantly. It means that within the strong segregation limit the characteristic distance (period) between the stripes more strongly depends on the length of the adsorbed blocks, in agreement with our theory. In Figure 8 we plot a phase diagram of the system under consideration with the parameters γ1 and δ, which ensure the best approximation of the experimental data. The experimental data obtained for thin films of PS-blockP4VP copolymers with various lengths of PS and P4VP blocks are shown on the diagram; the filled circles correspond to observed micellar structures while filled squares show the systems in which stripelike patterns emerge on the adsorbing substrate. One can see that the experimental data fit the calculated phase diagram very well. The detailed quantitive comparison of theoretical

Nanopattern of Diblock Copolymers

Langmuir, Vol. 15, No. 21, 1999 7297

Figure 8. Phase diagram of the dry film of A-B diblock copolymers with A blocks strongly adsorbed on a planar surface for δ ) -0.5 and γ1 ) 0.1. The experimental data obtained for thin films of PS-block-P4VP copolymers with various lengths of PS and P4VP blocks are shown on the diagram; the filled circles correspond to observed micellar structures while filled squares show the systems in which stripe patterns emerge on the adsorbing substrate.

It is worth noting that structures of multiple morphologies have been observed also in the systems of diblock copolymers with one hydrophobic and one polyelectrolyte or polar block at the water/air interface.22-24 In particular, with the increase of the ratio between the lengths of hydrophobic and hydrophilic blocks, surface micelles, stripes, and planar brush-type structures have been found. These observations seem to be in good qualitative aggrement with our theoretical predictions. Acknowledgment. The support of the Deutscher Akademischer Austauschdienst (DAAD) is gratefully acknowledged (I.I.P. and E.Yu.K.). This work has been carried out in the framework of the Sonderforschungsbereich 239 and the Graduiertenkolleg 328 of the Deutsche Forschungsgemeinschaft. I.I.P., E.Yu.K., and A.R.Kh. are also grateful to Russian Foundation for Basic Research for financial support. Appendix The function Km(x) in eq 15 is calculated by substitution of σm(r) (eqs 6 and 7) taking into account the incompressibility condition (eq 3)

Km(x) )

(1 + x2)6 2x (I(y) - I(0)), y ) 1 + 3x2 1 + x2

(29)

where the function I(y) has the following form

2 2 y4 8 ln(1 + x1 - y2) - y2 + - y6 9 3 3 27 8 y4 2 4 8 (1 - y2)3/2 - x1 - y2 + l l + y2 + (1 - y2)5/2 27 9 4 3 15

I(y) )

Figure 7. Scanning force micrographs of ultrathin PS-b-P4VP films (5 × 5 µm2) for different molecular weights of PS and P4VP blocks: MPS ) 380, MP4VP ) 470 (a); MPS ) 330, MP4VP ) 130 (b); MPS ) 480, MP4VP ) 210 (c).

results with experimental data will be given in a subsequent publication.

(

)

(22) Zhu, J.; Lennox, R. B.; Eisenberg, A. J. Phys. Chem. 1992, 96, 4727. (23) Zhu, J.; Eisenberg, A.; Lennox, R. B. Macromolecules 1992, 25, 6547. (24) Li, S.; Hanley, S.; Khan, I.; Varshney, S. K.; Eisenberg, A.; Lennox, R. B. Langmuir 1993, 9, 2243.

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l)

Potemkin et al.

functions Gi(x), i ) 1, 2, 3 of eq 27:

1 - x2 1 + x2

G1(x) )

The function Km(x) has the following asymptotic behavior:

{

22 2 x , x.1 Km(x) ) 9 16x4, x , 1

}

(30)

In the case of the stripe surface structure using eq 24 for σl(r), one obtains the following expressions for the

G2(x) ) G3(x) )

[

]

x2V(x) arcsin(y) + x1 - y2 + 2l 4y y 2x2 3 1 - (1 - y2)3/2 + y2l 2 3y3

[

x4

]

8 arcsin(y) - yx1 - y (1 - 2y ) + y l] [ 3 4y V(x) 4

where V(x) is defined in eq 23. LA9900730

2

2

3