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Exploiting Photoinduced Reactions in Polymer Blends to Create Hierarchically Ordered, Defect-Free Materials Rui D. M. Travasso, Olga Kuksenok, and Anna C. Balazs* Chemical Engineering Department, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261 ReceiVed December 11, 2005 Computer simulations reveal how photoinduced chemical reactions can be exploited to create long-range order in binary and ternary polymeric materials. The process is initiated by shining a spatially uniform light over a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and phase separation. We then introduce a well-collimated, higher intensity light source. Rastering this secondary light over the sample locally increases the reaction rate and causes formation of defect-free, spatially periodic structures. These binary structures resemble either the lamellar or hexagonal phases of microphase-separated diblock copolymers. We measure the regularity of the ordered structures as a function of the relative reaction rates for different values of the rastering speed and determine the optimal conditions for creating defect-free structures in the binary systems. We then add a nonreactive homopolymer C, which is immiscible with both A and B. We show that this component migrates to regions that are illuminated by the secondary, higher intensity light, allowing us to effectively write a pattern of C onto the AB film. Rastering over the ternary blend with this collimated light now leads to hierarchically ordered patterns of A, B, and C. The findings point to a facile, nonintrusive process for manufacturing high quality polymeric devices in a low-cost, efficient manner.
Introduction One of the current challenges in polymer science is developing processes to create hierarchical structures that exhibit controlled ordering at different length scales.1 The creation of defect-free polymeric structures, which simultaneously encompass features from the submicron to macroscopic scales, would be of significant importance in a number of technological applications, including the fabrication of optoelectronic devices and electromagnetic storage media. Previous approaches to fabricating such hierarchically ordered materials involved the self-assembly of various novel block copolymers, which encompassed the chemical linkage of soft and rigid segments1 or linear and branched chains.2 The creation of these unique copolymer architectures typically required specialized synthetic chemistry.1 Furthermore, the resulting materials exhibited the structural defects that are common in self-assembled systems.2b In a recent paper,3 we used a computational model to demonstrate a facile method for creating hierarchically ordered polymeric materials that exhibit periodicity on different length scales and are macroscopically defect-free. In this paper, we extend the study by pinpointing conditions that yield the optimal structures and demonstrating a broader range of structures that can be created through this approach, which harnesses both photoinduced, reversible chemical reactions and the phase separation of homopolymer blends. Both of the latter features are appealing for the following reasons. First, there exist a number of photosensitive chemical species that can be utilized to carry out the reaction.4-6 Second, the approach involves homopolymers, rather than complicated copolymers; consequently, it not only * To whom correspondence should be addressed. E-mail: balazs1@ engr.pitt.edu. (1) Muthukumar, M.; Ober, C. K.; Thomas, E. L. Science 1997, 277, 1225 and references therein. (2) (a) Ruokolainen, J.; et al. Science 1998, 280, 557. (b) Ruokolainen, J.; Saariaho, M.; Ikkala, O.; ten Brinke, G.; Thomas, E. L.; Torkkeli, M.; Serimaa, R. Macromolecules 1999, 32, 1152. (3) Travasso, R. D. M.; Kuksenok, O.; Balazs, A. C. Langmuir 2005, 21, 10912. (4) Tran-Cong, Q.; Kawai, J.; Endoh, K. Chaos 1999, 9, 298.
opens up the type of components that can be used in the process but also provides a new means of creating periodic structures from systems that would not normally form such ordered morphologies. To explain this concept more fully, we start our discussion by pointing to previous experimental studies where reversible photochemical reactions induce phase separation in a binary polymer mixture. For example, Tron-Cong and co-workers4-6 have worked extensively with blends of trans-stilbene labeled polystryene and poly(vinyl methyl ether) (PSS/PVME). Upon irradiation, the stilbene moieties on the PSS chains undergo a reversible trans-cis photoisomerization. Once the reaction reaches a certain threshold, the mixture undergoes phase separation since the cis-labeled polystyrene and PVME are immiscible.4 Thus, phase separation and a reversible chemical reaction are taking place simultaneously within this binary blend. Other photosensitive pendent groups can also be used to yield a similar behavior.4-6 The next important point is that both experimental and theoretical studies have demonstrated that the morphology of polymer blends encompassing both phase separation and reversible chemical reactions resembles the structure formed by microphase-separated diblock copolymers.4,7-12 By tuning the forward and reverse rates of the chemical reaction, hexagonal, cylindrical, or lamellar structures can be obtained.7 Another important point is that the reaction rates themselves can be tuned by varying the light intensity.4-6,13,14 At higher (5) Nishioka, H.; Kida, K.; Yano, O.; Tran-Cong, Q. Macromolecules 2000, 33, 4301. (6) (a) Tran-Cong-Miyata, Q.; Nishigami, S.; Ito, T.; Komatsu, S.; Norisuye, T. Nat. Mater. 2004, 3, 448. (b) Nakanishi, H.; Satoh, M.; Norisuye, T.; TranCong-Miyata, Q. Macromolecules 2004, 37, 8495 and references therein. (7) Bahiana, M.; Oono, Y. Phys. ReV. A 1990, 41, 6763. (8) Oono, Y.; Bahiana, M. Phys. ReV. Lett. 1988, 61, 1109. (9) Glotzer, S.; Di Marzio, E.; Muthukumar, M. Phys. ReV. Lett. 1995, 74, 2034. (10) Glotzer, S. C.; Stauffer, D.; Jan, N. Phys. ReV. Lett. 1994, 72, 4109. (11) Liu, F.; Goldenfeld, N. Phys. ReV. A 1989, 39, 4805. (12) Christensen, J. J.; Elder, K.; Fogedby, H. C. Phys. ReV. E 1996, 54, 3, R2212.
10.1021/la053350d CCC: $33.50 © 2006 American Chemical Society Published on Web 02/11/2006
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sections, we articulate the method for using the light to comb out defects within the materials and present the novel morphologies that can be created in the ternary mixtures.
Methodolgy The above process requires an immiscible ternary A/B/C mixture where the phase separation among all of the components occurs simultaneously with a reversible chemical reaction between components A and B, as indicated below Γ+
Figure 1. Schematic representation of the combing process. A region with reaction rate Γ2 * Γ1 is rastered over the sample. The size of the simulation box is 198 × 198 lattice sites. The width of the rastering stripe is 25 lattice sites.
light intensity, the rate of reaction is increased; consequently, the domains become more intermixed and the characteristic domain size is smaller.5,7 In what follows, we build on this knowledge to devise a means of creating defect-free, periodic structures in binary blends and hierarchically ordered structures in ternary mixtures. We start by irradiating an AB binary blend with a uniform background light, which is used to initiate a reversible chemical reaction and subsequent phase separation between A and B. Then, we introduce a higher intensity, spatially localized beam that locally increases the rate of reaction. By rastering over the sample with this secondary light (see schematic in Figure 1), we effectively “comb” out any defects in the material and create highly regular structures. Through the studies presented herein, we isolate the optimal range of rastering speeds for creating both ordered lamellar and hexagonal patterns. Note that the structural elements in these ordered materials can be as small as nanoscale. By adding a third component, C, we can “write” a well-defined pattern onto the binary system. In particular, C is driven to migrate to the region illuminated by the higher intensity light; in this manner, the free energy of the entire system is minimized. By exploiting this concept, we can create a variety of patterned structures by focusing the light in the desired spatial motif. Once the collimated light is rastered over the ternary system, the defects are again “combed” out of the sample and the defect-free material displays periodicity over two distinct lengths scales. Visually, the combing process described herein appears analogous to the zone refinement or directional quenching processes,15,16 which harness heat to induce a phase transition. However, the process described herein is distinct from the latter methods in a number of ways. First, as we describe in the Results and Discussion section, the factors that control the ordering in our process and in the heat-driven methods are quite different. Second, the use of light to promote a transformation does not require a physical coupling to the system. This is particularly important in nanostructured thin films where the lateral dissipation of heat would be detrimental. Finally, the process involved in creating spatially ordered ternary materials is distinct from zone refinement since the light is used to effectively “imprint” a pattern of the C component onto the AB system. Below, we present the governing equations for the ternary blend. We then show how the equations can be used to model both the two and three component mixtures. In the following (13) Daniels, F.; Alberty, R. A. Physical Chemistry, 3rd ed., Wiley: New York, 1966; p 622. (14) Colvin, V. L.; Larson, R. G.; Harris, A. L.; Schilling, M. L. J. Appl. Phys. 1997, 81, 5913. (15) (a) Furukawa, H. Physica A 1992, 180, 128. (b) Zhang, H.; Zhang, J.; Yang, Y.; Zhou, X. J. Chem. Phys. 1997, 106, 784 and references therein. (16) Hashimoto, T. Bull. Chem. Soc. Jpn. 2005, 78, 1.
ASB Γ-
(1)
The parameters Γ+ and Γ- represent the forward and reverse reaction rate coefficients, respectively. In our ternary system, C is nonreactive; it simply phase separates from the A and B components. In the case of a homogeneous quench, the structural evolution of a similar ternary mixture has been studied previously.17 The incompressible A/B/C ternary mixture is characterized by two order parameters, φ ) FA - FB and ψ ) FC, where Fi is the volume fraction of the ith component. With the reaction in eq 1, the structural evolution of this mixture is now given by18
∂φ ) Mφ∇2µφ - (Γ + + Γ-)φ + (Γ- - Γ+) ∂t
(2)
∂ψ ) M ψ ∇ 2µ ψ ∂t
(3)
where the constants Mφ and Mψ are the mobilities of the respective order parameters φ and ψ. The chemical potentials, µφ and µψ, are defined through the free energy functional F(φ,ψ) as
µφ )
δF(φ,ψ) δF(φ,ψ) , µψ ) δφ δψ
(4)
A suitable form of F(φ,ψ) is
F(φ,ψ) )
∫ dr[f(φ,ψ) + κφ(∇φ)2 + κψ(∇ψ)2]
(5)
where the local free energy f(φ,ψ) is taken to be19
f(φ,ψ) ) -aφ2 + bφ4 + cψ2 - dψ3 + eψ4 + gφ2ψ2 (6) The two gradient terms in eq 5 are related to the interfacial tensions between the components. The values of the constant coefficients in eq 6 are obtained by constraining f(φ,ψ) to have equal minima at φ ) (1; ψ ) 0 (pure A and B), and at φ ) 0; ψ ) 1 (pure C); this yields a ) 0.02, b ) 0.01, c ) 0.06, d ) 0.16, e ) 0.09, and g ) 0.06.20 We note that this is not a necessary constraint; we have carried out studies where the depths of the different minima are not equal (altering the values of the coefficients in eq 6) and observed similar behavior as reported below. (17) (a) Ohta, T.; Ito, A. Phys. ReV. E 1995, 52, 5250. (b) Tong, C.; Yang, Y. J. Chem. Phys. 2002, 116, 1519. (18) Equations 2 and 3 describe phase-separation in ternary mixtures of polymers with relatively small molecular weights or shallow quench depths, and all hydrodynamic effects have been neglected. For mixtures of polymers with high molecular weights, the above system of equations should be modified. In particular, the free energy equations should include an explicit dependence on chain length and the Onsager coefficients should reflect a dependence on the polymer concentration and chain length. (19) Good, K.; Kuksenok, O.; Buxton, G. A.; Ginzburg V. V.; Balazs, A. C. J. Chem. Phys. 2004, 121, 6052. (20) Travasso, R.; Buxton, G. A.; Kuksenok, O.; Good, K.; Balazs, A. C. J. Chem. Phys. 2005, 122, 194906.
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For the majority of the calculations presented here, we set κφ ) κψ/3 ) 0.015 and Mφ ) 3Mψ ) 0.15; consequently, the interfacial tensions between the different components are equal, and the mobilities of all of the phases are the same. However, the latter constraints are also not necessary conditions for the observed phenomena. To illustrate this point, we considered a number of cases where the interfacial tensions between the species are not the same and the mobilities are not equal. Examples of these cases are provided in the ensuing figures. Note that for Γ( ) 0, eqs 2 and 3 together describe phase-separation in a ternary mixture.19-21 When Γ( * 0 and C is absent, eq 2 describes a phase-separating binary blend where the reaction in eq 1 suppresses long wavelength fluctuations and arrests the domain growth at a characteristic wavelength.7-12 Consequently, the system’s morphology resembles the structures formed by microphase-separated diblock copolymers,4,7-11 forming lamellar-like structures when Γ+ ) Γ- ) Γ, and a hexagonal morphology when Γ+ * Γ-. In addition, we note that eq 2 for the case of equal reaction rate coefficients (Γ+ ) Γ- ) Γ) can be rewritten in the form
δF ˜ (φ,ψ) ∂φ ) Mφ∇2 ∂t δφ
Figure 2. Examples of steady-state morphologies for (a) Γ+ ) Γ- ) 0.0003 and (b) Γ+ ) 0.0007 and Γ- ) 0.0003.
(7)
where
F ˜ (φ,ψ) ) F(φ,ψ) +
Γ Mφ
∫ dr dr′ G(r - r′)φ(r)φ(r′)
(8)
and the Green’s function G(r - r′) can be found from the Poisson equation ∇2G(r - r′) ) - δ(r - r′) with the appropriate boundary conditions.11,12 Herein, we consider reaction rates that vary both dynamically and spatially; hence Γ((t,r) is a function of both time and space. In this manner, we can describe a light source that is moved or rastered over the sample. Below, we first describe how we exploit this feature in a binary blend and then discuss the consequences of rastering the light over a ternary mixture. To carry out the ensuing investigations, we solve eqs 2 and 3 numerically via an efficient computational scheme. We use the lattice Boltzmann technique detailed in ref 19; however, we neglect hydrodynamic effects in this study (by setting the velocity field to zero). We note that other schemes could also be adopted for the numerical integration (such as the cell dynamical systems approach7,8).
Results and Discussion Binary Blend. The binary system is described by eq 2 alone. In these studies, we turn on a spatially uniform, “background” light, initiating the reaction in eq 1. With Γ+ ) Γ- ≡ Γ the system forms the lamellar-like structure7-12 in Figure 2a, which resembles the morphology of microphase-separated, symmetric diblock copolymers. For Γ+ * Γ-, the system can form a hexagonal pattern,7 as seen in Figure 2b. In both these examples, the presence of the reversible chemical reaction arrests the coarsening process, so that the domains assume a characteristic size. The actual equilibrium domain size arises from a competition between the mixing due to the reversible chemical reaction and the demixing due to the phase separation.9,11,12 Focusing on the case where Γ+ ) Γ- ≡ Γ, we plot the domain size, λ, as a function of Γ in Figure 3. The domain size is seen to decrease with increases in Γ as λ ∼ Γ(-1/4). This value is consistent with previous results (see, for example, ref 7). (21) (a) Huang, C.; Olvera, de la Cruz, M. Phys. ReV. E 1996, 53, 812. (b) Huang, C.; Olvera de la Cruz, M.; Swift, B. W. Macromolecules 1995, 28, 7996.
Figure 3. Stripe width λ (measured after combing) as a function of Γ. The line has a slope of (-1/4).
The value of Γ also dictates the equilibrium value for the order parameter, φ0, for the A-rich phase (and -φ0 for the B-rich phase). In Figure 4a, we plot φ0 versus Γ and find that the value of φ0 systematically decreases with increases in Γ; these observations agree with earlier findings.7,17 In other words, the domains are more intermixed for larger values of Γ. To illustrate this point, we plot the variations in the order parameter along the x direction (for a fixed value of y) for two values of Γ. As can be seen in Figure 4b, the absolute value of φ0 is smaller for the sample with the larger Γ, which is also consistent with results obtained earlier.12 To this uniformly irradiated system, we introduce a spatially localized, secondary light source, which is represented by the dark stripe in Figure 1. To facilitate the discussion, we will refer to the reaction rate coefficient of the background light as Γ1. We first focus on the case where the intensity of the secondary light is higher than that of the background light; we briefly consider the inverse case at the end of this section. The higher intensity, secondary light source locally increases the rate coefficients to Γ+ ) Γ- ≡ Γ2 > Γ1. Here, we numerically solve eq 2 in two dimensions with the reaction rate coefficients equal to Γ2 within the dark stripe and to Γ1 elsewhere. From the discussion above, it is clear that, if Γ2 > Γ1, the domains in the Γ2 region will be smaller and more intermixed than in the Γ1 region (see Figures 3 and 4). Hence, the Γ2 region is effectively more homogeneous or “neutral” relative to the Γ1 domains. The A/B interface will form a 90° contact angle with a neutral boundary;22 that is, the domains orient perpendicular to a neutral boundary. The fact that the lamellar domains within the Γ1 region are oriented perpendicular to the Γ1/Γ2 boundaries (22) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827.
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Figure 6. System combed at a rastering speed V ) 0.1, with Γ2 ) 0.003 and Γ1 ) 0.0003. Simulation times are t ) 2 × 102 and 1.5 × 103 for a and b, respectively. The arrows point to the location of the moving Γ2 stripe.
Figure 4. (a) Average value of the order parameter in the bulk, φ0, as a function of Γ. (b) Order parameter profile along the x direction for Γ+ ) 0.0015 (dashed line) and Γ+ ) 0.0001 (solid line).
Figure 5. Morphologies obtained for a system with Γ1 ) 0.0003 and Γ2 ) 0.003 for the case of a stationary stripe in a and a moving stripe with velocity V ) 0.001 in b-d. The simulation times are t ) 7 × 104, 1.2 × 105, and 3 × 105 for b-d, respectively.
is clearly illustrated in Figure 5a, where the Γ2 region is stationary. Since eq 2 can also be used to describe the dynamic behavior of diblock copolymers,7,11 we can draw an analogy between the system in Figure 5 and the behavior of diblocks confined between
neutral or nonselective walls. In the latter system, it is known that the lamellar domains of symmetric diblocks align perpendicular to neutral walls since this configuration lowers the free energy of the system.23 The above discussion of the system’s behavior in the case of a stationary secondary light helps explain the observed behavior for the case when the secondary light source is moving over the sample. Figure 5b-d shows snapshots of the structural evolution of the film as this Γ2 region is moved from the bottom to top of the system. (In other words, eq 2 is now solved for both spatially and temporally varying values of the reaction rate coefficients.) If we move the Γ2 region sufficiently slowly (the estimates for the effective speed are given below), at each moment of time, the lamellar domains within the Γ1 region orient perpendicular to the Γ2 boundary. In this manner, as the Γ2 region is moving over the sample, it effectively combs out the defects within the Γ1 region. In effect, the lamellar domains are aligning perpendicular to the moving neutral interface. The effectiveness of this procedure depends on both the speed of rastering, V, and the value of Γ2. If Γ2 is too low (i.e., its value is close to Γ1), the structure within the stripe is not sufficiently different from that in the surrounding Γ1 regions and the procedure is ineffective at any rastering velocity. For relatively high Γ2, the combing can be effective if the rastering velocity is sufficiently slow that the domains in the Γ2 region reach their equilibrium size. It is only in the latter case that the neighboring Γ1 regions experience a neutral interface. We can estimate the characteristic time for the morphology to reach steady-state within the Γ2 region as (1/2Γ2); thus, the rastering speed must be lower than Vmax ≈ 2Γ2w, where w is the width of the stripe (see Figure 1). If V is on the order of or greater than Vmax, then the secondary light is being rastered too quickly for the lamellar domains within the Γ2 stripe to reach both their equilibrium size and bulk order parameter (i.e., for the domains to become intermixed). This case is shown in Figure 6, and it can clearly be seen that rastering at this speed does not lead to defect-free structures. To quantify our observations on the efficiency of the “combing” process, we measure β, the regularity of the domains in the combing direction, as a function of Γ2 for Γ1 ) 0.0003. We define β ) 〈〈φ〉y2/〈φ2〉y〉x, where 〈‚‚‚〉i represents the average along the i direction (this definition is similar to that described in ref 16). The case β ) 1 corresponds to the completely ordered phase (along the combed direction). In Figure 7, each value of β represents an average over six independent runs. In each case, we calculate the value of β from the morphology that is created after the second passage of the light over the film. The reasons we choose the second passage are the following. On one hand, after the first passage, the alignment of the “combed” lower part of the sample is influenced by the “uncombed” upper part due (23) Kellog, G. J.; et al. Phys. ReV. Lett. 1996, 76, 2503.
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Figure 7. Dependence of the regularity of the structure, β, on Γ2. Each point represents an average over six independent runs after a second passage of the combing stripe. Data are presented for three different rastering speeds.
to the periodic boundary conditions at the top and bottom of the sample. Therefore, it is not sufficient to use the morphology obtained after a single passage to quantify the efficiency of the process. On the other hand, we did not observe any improvement on the third or fourth passages,3 so that each following passage is equivalent to the previous one. We note that even though in simulations a second passage is necessary (due to the periodic boundary conditions) in experiments a single passage over the sample would potentially be sufficient to “comb” the structure. The data in Figure 7 are for three different values of V, the rastering velocity. The plot clearly reveals that optimal performance is achieved for the higher values of Γ2 for all chosen velocities. It is in this high Γ2 region where our “neutral moving boundary” argument is most applicable. For the intermediate values of Γ2 (approximately 0.001 eΓ2 e0.003), the mixing within the Γ2 region is not sufficiently great to create a truly neutral interface. For this intermediate Γ2 range, the regularity of the patterns can depend significantly on the rastering velocity. For example, for Γ2 ) 0.0012, the slower the rastering speed, the better the ordering in the sample (as can be seen in Figure 8a-c); however, even for the lowest velocities, the morphologies still exhibit some defects (see Figure 8c). The improvement of the ordering at the lowest velocities can be understood from the following arguments. Even though the Γ2 stripe is moved sufficiently slowly (V , Vmax) that the domains inside the stripe reach their equilibrium size, the system still might not reach the lowest free energy state. To achieve the latter configuration, the system needs sufficient time for the lamellae both within and outside Γ2 to reorient and effectively match-up. In particular, if we move the Γ2 stripe slowly enough, the system evolves to minimize its free energy by connecting the appropriate domains across the Γ1/Γ2 boundary. The alignment of these domains introduces the ordering in the system (see Figure 8c). (For intermediate values of Γ2, continuous domains that cross the Γ1/Γ2 boundary can form since the mismatch in the thickness of the lamellae in the Γ1 and Γ2 regions are not drastically dissimilar.) If we move the stripe faster, there is not enough time for these domains to rearrange and coincide; therefore, the ordering in such system is lower (see Figure 8a,b). At the upper end of the intermediate Γ2 range, we observe relatively high ordering within the system and no longer observe such a strong dependence on V. Examples of morphologies for three different velocities at Γ2 ) 0.0024 are shown in Figure
Figure 8. Examples of morphologies obtained at the end of the second passage of the combing process for different values of Γ2 and different rastering speeds. Γ2 ) 0.0012 for a-c and Γ2 ) 0.0024 for d-f. The rastering speed is V ) 0.002 for a and d, V ) 0.001 for b and e, and V ) 0.0005 for c and f. Here, Γ1 ) 0.0003.
8d-f. Here, the ordering is primarily due to the lower values of the order parameter within Γ2 and, hence, the relative neutrality of the interface. In other words, there is a strong tendency for lamellar domains in the Γ1 region to orient perpendicular to the Γ2 domain, yielding high values of β. We also note that the ordering within Γ1 is affected by the actual pattern within the moving Γ2 stripe (and vice versa) and even by the width of the stripe. The latter fact explains the wide error bars for the cases of relatively good but not perfect ordering for the different random initial seeds. As Γ2 increases, the contribution from the neutrality of the interface becomes stronger, whereas the contribution from the actual pattern within the Γ2 stripe becomes negligibly small; that is, the order does not depend on the initial fluctuations, and therefore, for all of the cases of complete ordering, the error bars are very small. Additional studies show that if we decrease the rastering velocity even further (V < 0.0005), the dependence of β on Γ2 remains almost identical to the case of V ) 0.0005 (for example, the curve for the case V ) 0.00033 lies on the top of the curve for V ) 0.0005). In other words, at the lowest velocity for high Γ2 in Figure 7, the system is in the quasi-stationary state at each time t, and thus, no further changes in the system’s evolution can be made by moving the Γ2 stripe even slower. We now briefly consider the case where the intensity of the light within the moving region is less than that of the background light, i.e., Γ2 < Γ1. In this case, not only is the intensity of the secondary light less than that of the background light but also the background light is not applied within the Γ2 region (i.e., this region is shielded from the Γ1 source). Although this scenario
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Figure 9. Examples of morphology obtained when Γ1 > Γ2. Here, Γ2 ) 0.0003 and Γ1 ) 0.003. The rastering velocity is V ) 0.001 and the sampling times are t ) 2 × 104, 7 × 104, 1.9 × 105, and 2.6 × 105 for a-d, respectively.
Figure 10. Morphologies at the end of the second passage of light with rastering speed V ) 0.001 for Γ1+ ) 0.0007 and Γ1- ) 0.0003. For panel a, Γ2 ) 0.004. For panels b-d, the ratio Γ2+/Γ2- is 7/3 and Γ1- ) 0.004, 0.002, and 0.0012, respectively.
may be somewhat experimentally challenging to achieve, it is nonetheless worth noting the interesting behavior that is observed in this case. Even in this case, the secondary light can be rastered over the sample to create defect-free materials. Figure 9 shows snapshots of this system as a function of time, and again, we see that the secondary light source effectively combs out structural imperfections. However, the reasons for this ordering are somewhat different. Here, the more intermixed Γ1 regions provide the neutral interfaces for the Γ2 region and cause the lamellar domains in the latter area to assume a perpendicular orientation. As the Γ2 domain is shifted upward, the material trailing behind is exposed to Γ1 radiation, which leads to thinner lamellae. However, recall that the thinner lamellae is formed at the place originally occupied by the thicker stripes, i.e., effectively, each thicker stripe “divides” into the thinner stripes, keeping the “memory” of it’s previous orientation. This process happens at each consecutive step of the Γ2 stripe moving upward and eventually leads to the perfectly regular structure seen in Figure 9d. Due to the periodic boundary conditions in the simulation, two passes are needed to create this highly ordered pattern. We also note that even though the case of Γ2 < Γ1 can be used to create defect-free structures in the binary system it is not efficient for the ternary systems (see below), so in the case of the ternary fluids, we will study the systems with Γ2 > Γ1. Up to this point, we have concentrated on the lamellar morphology, which is produced by setting Γ+ ) Γ- in eq 1. However, eq 2 can also yield a hexagonal pattern when Γ+ * Γ- (see, for example, Figure 2b). As we illustrate in Figure 10, the combing technique can be applied to this hexagonal morphology to produce a spatially regular structure. In Figure 10a-d, the spatially localized secondary light has a higher intensity than the background light; the actual values of Γ1+/and Γ2+/- for each case are given in the figure captions. The example in Figure 10a is distinct from the others because, within the Γ2 region, Γ2+ ) Γ2-, so that the combing light and the background light produce structures with different symmetry. In the other cases, we fixed the ratio Γ1+/Γ1- ) Γ2+/Γ2- to 7/3. As can be seen, rastering over the sample with this light source yields a defect-free film for the examples in Figure 10a-c.
To understand this phenomenon, it is worth reiterating the fundamental processes that are occurring in this system. In particular, eq 1 can be expressed in words as “A produces B” (and visa versa). Additionally, since A and B are immiscible, B diffuses away from A (and visa versa). Overall, the chemical reaction establishes a characteristic length scale for the domains. Returning to the image in Figure 10a, we see that the bottom of the Γ2 region is occupied by a relatively uniform red (A-like) lamellar layer. This A-rich layer is driven to produce a B-rich (blue) material. Due to the immiscibility of A and B, the B forms a narrow stripe underneath the A domain and within the Γ1 region. Now this B is driven to form A, but the reaction in the Γ1 area is asymmetric with respect to the forward and reverse rates. Consequently, the A (red) that is formed is the minority component, and as dictated by the choice of Γ+ and Γ-, the system forms a hexagonal pattern. If the rastering of the secondary light is carried out sufficiently slowly; that is, V < λ1(Γ1+ + Γ1-), the system has time to reach the equilibrium domain size for the hexagonal morphology. One possible reason the approach works so effectively is that the structures that develop along the Γ2 interface form a propagating, ordering front as the stripe is moved. To understand this concept more fully, let us consider the scenario where the Γ2 stripe starts near the bottom of the sample. A thin layer of hexagonal film orders at the Γ2/Γ1 boundary. Unlike a random coarsening event, where the different domains are nucleated and grow at different times, in this case, the pattern in this small region is essentially created at one time. When the stripe is moved, the hexagonally ordered layer close to the Γ2 surface templates the structure for the neighboring layers. The above logic also explains the production of the defectfree pattern in Figure 10b,c. Again, the intensity of the secondary light source is higher than the background light. But now, however, Γ+ * Γ- within the Γ2 domain; in particular, the rate constants in both the stripe and background region are tailored to produce a 30:70 mixture of A-to-B. Because the reaction rates are higher in the Γ2 region, this domain appears more uniformly “blue” relative to the remainder of the system. At the bottom of the Γ2 boundary, the blue material is driven to produce the red
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Figure 12. Evolution of a ternary system with 33% of C and Γ ) 0.0015. Panels a and b correspond to t ) 3 × 104 and 1 × 106, respectively.
Figure 11. Dependence of the regularity of the structure βH on Γ2+. Each point represents an average over six independent runs after a second passage of the combing stripe for V ) 0.001.
component. However, the red is the minority phase, and as can be seen a few sites away from the boundary, these A’s form distinct droplets in the Γ1 region (where the rate constants dictate larger domains than in Γ2). Again, if the rastering is carried out sufficiently slowly, the system forms a spatially regular hexagonal pattern. In contrast to these examples, Figure 10d shows a case where the reaction rates (forward and reverse) in the Γ2 region are not sufficiently greater than that in the Γ1 domains to promote the creation of a spatially regular film. In particular, the system forms a clearly disordered hexagonal pattern in the moving stripe, a pattern that cannot induce order within the bulk of the material. As in the lamellar case, we define a parameter βH to characterize the regularity of the domains in the combing direction. In a defect-free structure and along this y direction, the circular domains of the minority phase are aligned and equally separated by a distance Λ; hence, the order parameter φ is a periodic function of y. We calculate the period Λ by computing the l that minimizes the expression Σx,y (φ(x,y) - φ(x,y + l))2. We then define βH ) 〈〈φ〉(Λ)2/〈φ 2〉(Λ)〉all(Λ), where 〈‚‚‚〉(Λ) represents an average that involves only the set of points in the sample for which the elements are separated along the y axis by multiples of Λ. The average 〈‚‚‚〉all(Λ) is a weighted average (with a weighting function 〈φ 2〉(Λ)) performed over all of these sets of points. The plot of βH as a function of Γ2+ is shown in Figure 11; as in the examples shown in Figure 10b-d, the ratio Γ+/Γ- is equal to 7/3 both inside and outside the combing stripe in the simulations used to obtain the data in this plot. We obtain defect free structures for high Γ2+, whereas for low Γ2+, the combing stripe is not able to order the sample. We also observe that the transition region between these two regimes is quite narrow. Unlike the lamellar case for relatively high values of Γ2 at the slower rastering speeds, we do not observe large error bars for the data in the hexagonal system. This is due to the fact that in the hexagonal system, the domains can easily rearrange themselves locally in the case a defect is introduced by the moving stripe. In the lamellar case, the situation is different since the domains extend throughout all the system, and a small defect in a domain can be more easily enhanced. We note that the formation of regular binary structures can also be achieved through directional quenching of binary systems.15,16 In the latter process, heat is applied at the boundary of the film (so that the edge is in the disordered phase) and slowly moved over the sample. Thermal fluctuations in the moving
Figure 13. Schematic representation of the process used to create hierarchical structures. (a) Two stationary regions with reaction rate Γ2 > Γ1 are defined within the sample (dark stripes). (b) Another Γ2 stripe, which is rastered over the sample, is added to the system in panel a.
front allow the heated region to reach the lowest free energy. As the front propagates, the system becomes ordered. Thus, the underlying physics controlling the ordering in directional quenching is different from that in our “combing” process. Ternary Blends. The system becomes even more intriguing when we consider an A/B/C ternary blend. Here, the chemical reaction in eq 1 is occurring in the presence of the C. The latter component is nonreactive and is simply immiscible with both the A and B components. Since most polymer pairs are immiscible, this condition does not impose considerable constraints on our choice of C. To describe this ternary system, we now numerically solve the coupled eqs 2 and 3. The majority of this discussion will focus on the case where the interfacial tensions between the different components (controlled by the values of κφ and κψ in eq 5) are equal and the mobilities (dictated by Mφ and Mψ) are also identical. In other words, κφ ) κψ/3 and Mφ ) 3Mψ. However, we also illustrate the fact that these are not necessary conditions for the observed behavior by showing cases where the respective interfacial tensions and mobilities are not equal. Figure 12a,b shows the morphology for the ternary blend when the system is irradiated with a uniform background light where Γ+ ) Γ- ≡ Γ1. As can be seen, the A and B components form the characteristic lamellar-like domains and the C phase separates into distinct droplets. We now introduce the secondary light sources, as shown schematically in Figure 13a,b. Our desired “written” pattern involves two stripes of C, and thus, we utilize two stationary lights; for each of these lights, Γ+ ) Γ- ≡ Γ2 > Γ1. As discussed in the previous section, in the binary AB mixture, the higher intensity light produces thinner lamellae and more intermixing between the A and B species. The total free energy of such an AB binary mixture, F ˜ (φ,0) (as defined in eq 8), monotonically increases with increasing Γ (see, for example, ref 12). Therefore, the total free energy density of the binary AB mixture is higher
Exploiting Photoinduced Reactions in Polymer Blends
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Figure 15. Diffusion of C to the high Γ region for κφ ) κψ ) 0.015. The other parameters are the same as in Figure 14 except for the times, which are t ) 3 × 105 and 2 × 105 for panels a and b, respectively.
Figure 14. Diffusion of C to the high Γ region. Here Γ1 ) 0.0003, Γ2 ) 0.003 and the volume fraction of C is 0.25. The times sampled are t ) 1 × 104, 1 × 105, 1.7 × 105, and 3 × 105 for a-d, respectively.
in the Γ2 region than in the Γ1 region. In the ternary case, the A and B components diffuse to the lower free energy region (i.e., to the Γ1 region or in the other words, away from the Γ2 region); thus, the Γ2 region becomes more and more “depleted” of AB domains and “enriched” with the nonreactive C component. Therefore, in the system that consists of two regions, one with a higher and one with a lower reaction rate coefficient, the A and B components diffuse out of the region with the higher reaction rate coefficient, whereas the C component migrates into this region. At the late times, the C component will always be located in the region with the higher reaction rate coefficient, since, in this way, the total free energy within the system (see eq 8) is minimized. A detailed theoretical description of the migration of the C component to the high Γ2region will be the subject of a separate study. However, we note here that the actual time needed for the C component to occupy the Γ2 region depends on the Γ2/Γ1 ratio, on the concentration of the C component in the system, and on the values of interfacial tensions and mobilities. For example, for a higher Γ2/Γ1 ratio, the gain in the total free energy for “replacing” the AB domains with the C domains within the Γ2 region is larger than for a smaller Γ2/Γ1 value. Therefore, the higher the Γ2/Γ1 ratio, the faster C migrates to the Γ2 domains (we have, in fact, observed this behavior in our simulations). Figure 14 shows the morphology of the system in the presence of the secondary lights and reveals the diffusion of the C to the regions with the higher Γ2. As noted above, this behavior occurs even when the interfacial tensions between the components, γij, and mobilities are not identical. Differing values of these parameters alter the dynamics of the process, however, the localization of C to the Γ2 regions occurs even for values as high as γCA ) γCB ) 3γAB and as low as γCA ) γCB ) 0.3γAB. Comparable variations in the relative mobilities yielded no effect on the regularity of the patterns. Figure 15 illustrates this point for a selected example. Thus, by exploiting this effect, we can effectively “write” on the sample with the C component. The image, however, is not ordered. To create the hierarchically ordered A/B/C structures, we exploit the approach used in the binary blends. In particular, while keeping the two higher intensity lights fixed in position, we now introduce an additional Γ2 light source and raster this
Figure 16. Combing the pattern in Figure 14b with a stripe with Γ2 ) 0.003, moving with velocity V ) 0.001. Morphologies at the beginning and end of the combing are represented in panels a and b, respectively.
beam over the sample, as illustrated schematically in Figure 13b. As can be seen in Figure 16, by “combing” the sample with the light, one can obtain a defect-free structure where vertical stripes of A and B alternate with larger, horizontal stripes of C. The above approach can be utilized to create more complex, hierarchically ordered structures, such as the example shown in Figure 17. Here, we set the background radiation so that Γ1+ ) Γ1- ≡ Γ1 ) 0.0003. The two stationary Γ2 stripes (see schematic in Figure 13a) are characterized by Γ2+ ) Γ2- ≡ Γ2 ) 0.004. Distinct from the example in Figure 13, in the region between the two stationary Γ2 stripes, we introduce an additional stationary light source where Γ+/Γ- ) 7/3 and Γ- ) 0.0003. Figure 17a,b shows the evolution of the system during the first step of the process, where the C component migrates to the stationary stripes with high Γ2. The morphologies in Figs. 17 c-d correspond to the beginning and end of the combing process, respectively (where Γ2 ) 0.004 within the rastering stripe). The resulting ordered structure combines both the lamellar and hexagonal patterns of A and B with the horizontal stripes of C. In combing the ternary mixtures, the secondary light is moved relatively quickly, so that the C domain does not diffuse from its original position. Consider the case where the width of the C domain is L . λ1. For a rastering speed on the order of Γ1λ1, the influence of the moving stripe on the C domains is very small. For each passage, C can only diffuse by a maximum length ≈ λ1 (just as the A and B). Finally, we comment on the relationship between our dimensionless simulation parameters and actual, physical values. In particular, if we equate the distance between two lattice sites as ζ ≈ 10-8 m, then the choice of the dimensional diffusion constant D gives the time scale for one simulation time step as τ ≈ ζ2Ma/D ≈ ζ210-3/D (where we use the typical dimensionless value for the diffusion constant from our simulation, Ma ≈ 10-3). If we take the value of D within the range from D ) 10-14 to
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Figure 17. Creating hierarchically ordered structure that combines lamellae and hexagonal patterns. Within the two stationary stripes, Γ2+ ) Γ2- ≡ 0.004. In the region between the two stationary Γ2 stripes, Γ+ ) 0.0007 and Γ- ) 0.0003. In the rest of the sample (below the bottom Γ2 and above the top Γ2stripes), Γ1+ ) Γ1- ≡ 0.0003. The simulations times are t ) 2 × 104 and 3.3 × 105 for panels a and b, respectively. The frames in panels c and d show the beginning and end of the combing process, respectively.
10-12 m2/s, we find that the time scale is in the range from τ ≈ 10-5 to 10-7 s and the dimensional combing speed, Vζ/τ, is in the range from 10-6 to 10-4 m/s, respectively (where we use V ) 10-3 for the dimensionless value of the combing velocity). In the other words, in the above example, a sample 1 cm long can be “combed out” in 102-104 s with the smallest features being ≈ 80 nm. The size of the C domain is limited by the wavelength of light that is used to create the pattern; thus, for the case of visible light, the C domain is constrained to greater than roughly 700 nm in width. On the other hand, the width of the interface between the A and B domains establishes the smallest length scale in the A/B pattern. If the chains are sufficiently short, this length, and the A/B domains themselves, can be on the nanometer length scale. Thus, the overall A/B/C pattern can encompass significant variations in features sizes.
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component materials that show hierarchical ordering, with periodicity over two distinct length scales. Since our approach involves homopolymers, it significantly expands the range of materials that can be fashioned into a periodic pattern. The method could potentially provide advantages over current photolithographic processes from a number of vantage points. First, functionality or contrast can be selectively placed on a surface in just two processing steps, eliminating the need for conventional multistep processes. For photovoltaic, sensor, and media applications, the ability to select the lateral placement on materials is key. Second, the process is fully reversible, thereby producing “read-write” media in a straightforward way. Third, since light is used, the process is noninvasive, eliminating the need to physically contact the film. Fourth, the technique points to a novel means of writing patterns in three dimensions within polymeric materials. One area where this technique could potentiallyhave important consequences is in the fabrication of organic photovoltaic (PV) devices. The active material in these devices is usually a blend of two materials, one is good at transporting holes and the other is good at transporting electrons.24 The two materials should lie in close proximity to each other (i.e., on the order of 10 nm apart), and there should be a large interfacial area between these two domains. In addition, the morphology must provide a percolating path for both holes and electrons. The spatially regular morphology shown in Figure 5d could provide the optimum desired structure for the organic component of a PV cell. In particular, if the domains are on the order of 10 nm and the material is sandwiched between a cathode and anode, electron and hole transporting materials would form a continuous path, facilitating exciton splitting and carrier collection. Although microphase-separated diblock copolymers could also be harnessed for this application, it is relatively challenging and limiting to create diblocks that have the necessary electronic properties. By using blends, rather than diblocks, the designer has a greater range of materials that can be utilized to create the PV devices.
Acknowledgment. The authors gratefully acknowledge discussions with Prof. Mary Galvin and Prof. Tom Russell. This work was supported by DOE. Partial salary support for some of the work was provided by NSF to O.K. and ARO to R.D.M.T. LA053350D
Conclusions In summary, by sweeping films with a collimated light source, we establish a facile process for creating defect-free, multi-
(24) (a) For a special issue on organic based photovoltaics, see: MRS Bull. 2005, 30, 10-53. (b) For a review of the field of organic photovoltaic films, see: Nelson, J. Curr. Opin. Solid State Mater. Sci. 2002, 6, 87-95.
