Quantification of Temperature-Dependent Order in Graphoepitaxially

Article pubs.acs.org/Macromolecules

Quantification of Temperature-Dependent Order in Graphoepitaxially Aligned Monolayer and Bilayer Films of Cylindrical Morphology Block Copolymer Vindhya Mishra†,§ and Edward J. Kramer*,†,‡ †

Department of Chemical Engineering and ‡Department of Materials, University of California, Santa Barbara, Santa Barbara, California 93106, United States S Supporting Information *

ABSTRACT: While the ability to direct order in block copolymer films using different techniques has been demonstrated by several researchers, comparison of the efficacy of these techniques has been limited by the lack of a robust method to quantify order in these systems. Measurement of the degree of order in block copolymer thin films can allow us to identify the conditions and guiding mechanisms that lead to an improvement in the quality of the block copolymer patterns as well as help us understand the underlying factors that lead to the destruction of order in these systems. In this work we have probed the temperaturedependent positional and orientational order in graphoepitaxially aligned1,2 monolayer and bilayer films of cylindrical morphology block copolymers. Previous works have drawn parallels between monolayers of cylindrical morphology block copolymers and 2D thermotropic smectics.3,4 Using grazing incidence small-angle X-ray scattering as a probe, we have studied the effect of film thickness and molecular weight on the temperature-dependent order in thin films. We have implemented a methodology to accurately measure order in thin films from the line shape of scattering peaks as an alternative to image analysis of micrographs. This technique allowed us to compare the effect of film thickness and molecular weight on the degree of order in thin films, with a very high degree of accuracy. We find that monolayer thick films of cylindrical morphology block copolymers when aligned in 2 μm wide channels display quasi-long-range translational order, in contrast to the short-range translational order characteristic of 2D smectics.5,6 We also confirmed that bilayer films exhibited better order than monolayer films of the same block copolymer at the same temperature and exhibited higher order−disorder temperatures than monolayers.7 We propose a model for the correlation function for a confined 2D smectic which confirms that the existence of quasi-long-range order arises due to suppression of long wavelength phonon modes responsible for destruction of order in unconfined systems.8,9

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INTRODUCTION Self-assembled block copolymer thin films are a potential candidate for a high-throughput, low-cost fabrication route for the next-generation, high-density lithographic devices.10−12 However, in absence of an external driving force, the patterns formed by self-assembly of block copolymers typically consist of randomly oriented grains with a large density of defects. A challenge for researchers over the past several decades has been to convert the random arrangement of block copolymer microdomains into a defect-free, single crystal spanning large areas. Several successful demonstrations of aligned microdomains have been reported in the literature, where an external field such as chemical13 or topographical patterning,1 electric field,14 or shear15 was employed as a directing force. However, typically the defect densities in the resulting patterns when measured over areas larger than a few micrometers are higher than the tolerance limits required for semiconductor device fabrication. Moreover, comparison between the efficacy of different directing techniques and processing conditions to © 2013 American Chemical Society

direct long-range order in block copolymer microdomains has been difficult due to the lack of an accurate technique to quantify the quality of large-area arrays. To date, quantification of block copolymer patterns has been performed by careful image analysis of scanning probe, scanning electron, or transmission electron micrographs, including in some cases by manual defect counting.3,4,16 Such parameters are not statistically representative and may be inaccurate as they are sensitive to the quality of the images and the accuracy of the image-processing techniques. Quantification of order in block copolymer thin films is also necessary to compare the nature of the order−disorder transition (ODT) in thin films to bulk systems. While accurate studies of the bulk order−disorder transition for block copolymers have been carried out using transmission SAXS Received: October 9, 2012 Revised: December 18, 2012 Published: January 15, 2013 977

dx.doi.org/10.1021/ma302111c | Macromolecules 2013, 46, 977−987

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and rheology,17−20 these techniques cannot be applied to thin films. The emergence of grazing incidence small-angle X-ray scattering (GISAXS)21−24 as a tool for structure characterization of thin films offers an alternative route to address these questions. However, the capabilities of this technique have not been fully utilized as its application to block copolymer systems has been limited primarily to morphology identification. While the position of the scattering peaks in GISAXS is sensitive to the symmetry and lattice dimensions of the unit cell, the shape of the scattering peaks can yield information regarding the rate of decay of order, grain size, and defect densities in the microdomain arrangement. Studies on epitaxially grown metal islands25 and monolayers of spherical morphology block copolymers26,27 have demonstrated that GISAXS can reveal fundamental information regarding the nature of order in confined materials. Behavior of monolayers of block copolymers can be different from the bulk since dimensionality of a system can critically affect its order due to the enhanced effect of thermal fluctuations and the lower cost of defect formation.7−9,29 While thermal fluctuations cause all molecules to be displaced from their mean position, the displacement in 3D solids is small compared to their lattice positions. Smectic liquid crystals, which are layered materials characterized by a density modulation in one direction and liquid-like arrangement in the other directions, belong to a unique class of materials where thermal fluctuations destroy translational order even in three dimensions.6 Theories regarding the nature of ordering in smectic phases have been proposed5,6,30 and confirmed experimentally.29,31−33 The absence of true long-range translational order in 3D smectics arises from fluctuations in layer displacements which are logarithmically divergent for infinitely large samples.8,34 Furthermore, upon increasing temperature, thermotropic smectics in 3D transform to a nematic and eventually an isotropic phase. The nematic phase is characterized by quasi-long-range orientational order and short-range positional order, while the isotropic phase has short-range positional and orientational order. Monolayers of cylindrical morphology block copolymers are similar to 2D thermotropic smectic A liquid crystals3,4 where the cylindrical microdomains can be considered equivalent to the smectic layers. Theory suggests that in 2D the smectic-to-nematic transition occurs at T = 0 K; i.e., translational order in 2D smectics is destroyed at any finite temperature.6 Moreover, the transformation from 2D smectic at 0 K to isotropic phase at any temperature occurs continuously via an intermediate nematic phase; i.e., loss of translational order is followed by loss of orientational order. Studies by Hammond and co-workers showed that the disordering transition in monolayers of cylindrical morphology block copolymers is similar to the smectic−nematic−isotropic transition.4 On the other hand, cylindrical morphology block copolymers in bulk are structurally similar to hexagonal columnar liquid crystal phases,35 which are regular two-dimensional lattices of onedimensional columns with liquid-like correlations along the columns. Since columnar liquid crystals display higher degree of order than smectic liquid crystals, it is possible that as film thickness is increased the degree of order in the films will increase. Bilayer thick films of cylinder forming block copolymers form an intriguing system that cannot be classified as either true smectic or columnar structures. While the physics of block copolymers can be similar to smectic and columnar liquid crystals, we cannot expect complete equivalence. For

example, one point of difference between the two is that liquid crystals are one-component systems while ordered block copolymers are characterized by local concentration variations.36 Moreover, all existing theories for 2D behavior were developed for infinitely large systems, whereas in graphoepitaxially directed assembly, the monolayers are restricted in their lateral extent. Thus, there is a need to quantify the nature of the order in these 2D and 3D block copolymer systems to uncover the physics governing their behavior.

Figure 1. Schematic of the smectic, nematic, and isotropic phases and the analogy with monolayers of cylindrical morphology block copolymer ordered and disordered phases.28

In this work, we have developed a methodology to quantify the temperature-dependent order in monolayer and bilayer films of cylindrical morphology block copolymers using GISAXS. This approach was motivated by previous smallangle X-ray scattering studies by researchers on bulk smectic layered systems.31−33,37 We used graphoepitaxy to direct the alignment of the block copolymer microdomains at low temperatures and studied the loss of the order as the annealing temperature was increased. The methodology developed for structural characterization allowed us to compare the effect of film thickness and molecular weight on the degree of order and the ODT for thin films with a very high degree of accuracy. If monolayers of cylindrical morphology block copolymers follow the same physics as 2D liquid crystals, it will be impossible to achieve long-range order in unconfined systems at any finite temperature. However, graphoepitaxy (which involves physical confinement of the block copolymer within topographic patterns) can potentially cut off the long wavelength phonon modes responsible for the destruction of order. We find that monolayer thick films aligned in 2 μm wide channels displayed quasi-long-range translational order, in contrast to the exponential decay characteristic of 2D smectics. Calculation of the correlation function by modeling the graphoepitaxially aligned monolayer as a confined 2D smectic confirms that the patterned substrate effectively suppresses destruction of order. In our previous work, we showed that monolayers of block copolymers have lower ODT temperatures than bulk, and the difference in the ODT temperatures was a function of molecular weight.7 The ODT for bilayer thick films were found to be higher than monolayers, so bilayers exhibited better order than monolayer films at the same temperature. In this work, we have used GISAXS to analyze the same system and find that it is an excellent probe for the quality of large-area arrays, revealing details of lateral film structure that were inaccessible using other microstructure characterization techniques such as microscopy and reflectivity. Furthermore, it also revealed information regarding the fundamental physics of laterally confined 2D and multilayer block copolymer systems. 978


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Small-angle X-ray scattering is an effective technique to determine the nature of the order in block copolymer systems since the measured structure factor (S(q)) is the Fourier transform of the pair correlation function G(r). G(r) was extracted by fitting the shape of the diffraction peaks with an assumed model since loss of phase information prohibits reverse transforming the measured intensity. Grazing incidence smallangle X-ray scattering was carried out at Sector 8-IDE at the Advanced Photon Source at Argonne National Laboratory. A schematic of the GISAXS setup is shown in Figure 2. The sample was irradiated with an

EXPERIMENTAL TECHNIQUES

The three block copolymers used for this study were poly(styrene-b-2vinylpyridine) with the following compositions: PS−PVP0.22 (Mn ∼ 21 kg/mol, minority block volume fraction f PVP = 0.217, PDI = 1.085), PS−PVP0.23 (Mn ∼ 21 kg/mol, f PVP = 0.232, PDI = 1.09), and PS− PVP0.26 (Mn ∼ 26 kg/mol, f PVP = 0.258, PDI = 1.075). All the three block copolymers were synthesized using anionic polymerization. The bulk order−disorder transition temperatures were characterized by variable temperature small-angle X-ray scattering (SAXS). Bulk samples were prepared by packing the block copolymer in a washer and sealed using PMDA-ODA polyimide (Kapton) and a vacuum compatible epoxy adhesive. The bulk samples were preordered by heating up to 250 °C, slowly cooled down to 140 °C, and held for 1.5 days under a vacuum of 10−6 bar. These preannealed samples were heated up to progressively higher temperatures in an INSTEC heat stage under continuous nitrogen flow and allowed to anneal at each temperature for 2 h before the SAXS data were collected. For thin film studies, patterned substrates for graphoepitaxy were prepared using optical lithography in the UCSB nanofabrication facility. Silicon oxide of desired thickness was deposited on cleaned silicon wafers using plasma enhanced chemical vapor deposition. The wafers were cleaned by sonicating in acetone, isopropyl alcohol, and deionized water for 3 min each and baked in a convection oven at 120 °C for 1 h to remove any adsorbed water. The substrate was primed with hexamethyldisilazane (HMDS) to promote photoresist attachment and coated with AZ4110 photoresist. The wafer was then subjected to a 1 min pre-exposure bake on a hot plate at 95 °C to remove all the volatile components of the photoresist and was subsequently exposed using a Karl Suss MJB3 contact aligner and developed in a 1:4 AZ400 K:DI water solution. A 2 cm by 2 cm array of 2 μm wide channels separated by 2 μm wide mesas was etched into the silicon oxide layer with CHF3 gas using a Panasonic inductive coupled-plasma etcher. The photoresist was removed by soaking overnight in a stripper solution held at 80 °C, followed by a 1 min oxygen plasma etch. Polymer films with thickness designed to produce a monolayer and bilayer of cylinders were deposited on these photolithographically patterned substrates by spin-coating from a well-mixed solution of the polymer in toluene. The oxide thickness (and therefore the depth of the channels) was adjusted to be as thick as the monolayer and bilayer, respectively. The height of the grooves were varied in accordance with the molecular weight of the block copolymer and were set to approximately the microdomain spacing in the film times the number of layers. These films were then subjected to thermal annealing under high-vacuum conditions by heating above the bulk ODT, slow cooling at a rate less than 0.05 °C/min to the annealing temperature and held there for more than 1.5 days. Because of capillarity, the block copolymer over the mesas of the pattern is transported to the channels, leaving only a brush layer covering the mesas.4,38 The thickness of the spun-cast films were controlled so that after annealing complete monolayer or bilayer films were formed with minimum islands and holes, and the height of the grooves plus polymer brush was at the same level as the polymer layer within the grooves (i.e., the film was flat overall). The films were then cooled at a rate faster than 0.5 °C/min to room temperature before removing from vacuum. Quenching of the samples from high temperatures was not performed because of the tendency of the films to dewet from the substrate when exposed to air at high temperatures. Because of the different surface energies of the two blocks, the PVP forms a brush layer next to the native SiOx on the silicon substrate, while the PS forms the polymer− air wetting layer. A Physical Electronics 6650 dynamic secondary ion mass spectrometry (SIMS) apparatus was used to etch through the film with a focused 2 keV, 40 nA O2+ oxygen ion beam that was rastered over a 300 μm by 300 μm area with a 0.6 eV electron flood gun providing charge compensation. By tracking the CN− signal, which is unique to P2VP, the number of microdomain layers parallel to the substrate could be determined. SIMS was also used to etch through the PS brush to expose the buried PVP cylinder for imaging using atomic force microscopy (AFM).

Figure 2. Schematic of the experimental setup for grazing incidence Xray scattering. X-ray beam of wavelength 0.16868 nm focused to a beam spot size of 100 μm by 50 μm. The scattered intensity was recorded either by a Pilatus 1MF detector or by a MAR-2 CCD area detector and stored as a 2048 × 2048 16-bit tiff image. A lead beam stop was used to block the specular beam. Data from both sides of the beam stop were collected and were corrected for possible drift of the beam center. Each sample was first aligned with the channel walls parallel to the direction of the incident X-ray beam by rotating the sample in the direction ϕ about the sample normal. The lateral periodic structure of the gratings patterned on the substrate leads to several orders of diffraction. Intersection of the grating truncation rods with the Ewald sphere leads to a series of diffraction spots located on a semicircle when the gratings are parallel to the incoming beam.39 A slight misalignment shows up clearly as asymmetry in this semicircle about the primary beamstop. This allowed us to determine the aligned position ϕ = 0° with an accuracy of 0.05°. A secondary beamstop was used to block the strong reflections arising from the periodic channel pattern on the substrate. In this nominal ϕ = 0° position, data were collected at an angle of incidence αi = 0.18° to probe the structure within the film. Orientational order can be characterized by plotting the intensity of the first-order peak as a function of azimuthal angle ϕ and measuring the width of the resulting peak. To collect data at different azimuthal angles, the sample was rotated up to 90° in its plane about this nominal direction with a step size of 1° for ϕ < 10° and 10° thereafter. At least two 2D diffraction patterns were recorded at each position, and the exposure time was varied to maximize the recorded intensity without detector saturation. In order to measure the structure factor of block copolymer thin films as a function of temperature, the 2D GISAXS maps I(2θ,αf) for samples annealed at different temperatures were converted to profiles of intensity I versus the in-plane wave vector q∥ by line integrations at constant values of αf. The background of the line profiles were fit with a polynomial baseline and subtracted from the I(q) data prior to lineshape analysis. The instrumental resolution, which is the shape of the direct beam and is shown in Figure 3, was recorded using a short exposure under heavy attenuation. This measured beam shape could be fit by a Gaussian with exponentially decaying tails. Since the tails 979


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S(0, 0, qz) ∼ |qz − qn|−2 + ηn S(qx , 0, qn) ∼ qx −4 + 2ηn where ηn =

⎡ (q*)2 k T B G(r) ∝ exp⎢ − B ⎢⎣ ⎤ |z | ⎥ 4πλ ⎥⎦

decay more steeply than 1/q2, the instrumental resolution was deemed sufficient for line-shape analysis. The measured line shape of the scattering peaks were fit using a convolution of the structure factor with the instrumental resolution function. Further details regarding the forms of the structure factor will be provided in the next section.

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MEASUREMENT OF ORDER USING X-RAY SCATTERING The positional order in a crystal is quantified by the pair correlation function G(r) of the density order parameter ρ(r) = exp[iq*u(r)]40 (where q* is the dominant wave vector and u(r) is the layer displacement field at position r) (1)

S(q) ∝

kBT 8π BK

1 ξ 2(q − q*)2 + 1

ψ (r) = ei2ϕ(r)

(7)

(8)

where ϕ is the angle between the symmetry axis of the molecule and the preferred direction of orientation of the molecules represented by the director n (as shown in Figure 1). The degree of orientational order is quantified by the orientational order parameter S.

(3)

where r2 = ς2 + z2 and ς2 = x2 + y2 (the axes are shown in Figure 1), λ is the penetration depth which is related to the layer compressional modulus B, and the elastic constant for splay K as λ = (K/B)1/2. For thermotropic liquid crystals, η is a temperature-dependent parameter given by

η = (q*)2

(6)

While translational order in these systems is a measure of correlations in layer displacement, orientational order depends on the alignment of the local directors. The correlation function quantifying orientational order gψ(r) can be defined in terms of the complex orientational order parameter ψ(r) which is

(2)

|ς| ≪ (λz)1/2

ς −2η |ς| ≫ (λz)1/2

|x| ≪ (λz)1/2

This implies that the shape of the scattering peaks assume a Lorentzian form, where the width of the peak is inversely proportional to the square of the correlation length ξ

The Fourier transform of G(r) is the structure factor measured in X-ray scattering. For a system with true long-range order, such as in perfect crystals, G(r) is a constant and X-ray scattering is characterized by true Bragg peaks. For a system with quasi-long-range order the pair correlation function G(r) decays algebraically in terms of the dimensionless index η30,31,37 G(r) ∝ z −η

(5)

⎡ (q*)2 k T ⎤ B |x|⎥ |x| ≫ (λz)1/2 G(r) ∝ exp⎢ − 4Bλ ⎢⎣ ⎥⎦

In the harmonic approximation ⎤ ⎡ (q*)2 G(r) ∼ exp⎢ − ⟨(u(r) − u(0))2 ⟩⎥ 2 ⎦ ⎣

8π BK

for ηn < 2, where qz and qx are components of the wave vector along the x and z axes as shown in Figure 1 and qn is the location of the nth-order scattering peak. Hence a power-law decay of the tails of the diffraction peaks is a signature of quasilong-range translational order for 3D systems. The analytical form of the structure factor for algebraic decay of order in 2D smectic systems has not been documented as it is rarely observed, and hence we numerically evaluated an approximate expression, as detailed in the Supporting Information. We find that the algebraic decay of order in 2D is also characterized by a power-law decay of tails, but with the exponent q−1.5+η. In 2D smectics of infinite extent4,6 and nematics, translational order is expected to decay exponentially at all finite temperatures, as shown in eq 6.

Figure 3. Direct beam profile, which is the instrumental resolution function, fit with the sum of a Gaussian and an exponential decay.

G(r) = ⟨ρ(r)ρ(0)⟩−⟨ρ(r)⟩⟨ρ(0)⟩

n2(q*)2 kBT

S=

3⟨cos2 ϕ⟩ − 1 2

(9)

We now detail the procedure used for extracting measures of positional and orientational order in our systems. Positional order correlations for the different samples were extracted by analyzing the line shape of the first-order Bragg peak (n = 1) after background subtraction, in the nominal ϕ = 0° position. The line shape was defined by the shape of the central portion of the peak as well as the rate of decay of the tails. The best fit for each line shape was determined by comparing the residuals

(4)

The conventional delta function nth-order Bragg peaks characteristic of long-range order are replaced by power law singularities30 of the form 980


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for fits for a Gaussian form, a Gaussian with power-law decaying tails or a Lorentzian. For a system with true longrange order, the shape of the peak should be resolution limited, but in practice the peak is broadened due to finite size effects. For quasi-long-range order, the decay of the tails of the diffraction peak depends on the rate of decay of translational order correlations and were fit by using an assumed form of the structure factor. For a system with short-range order, the structure factor transforms to a Lorentzian form according to eq 7, but since the instrumental resolution function is a Gaussian, the experimental line shape of the peaks thus assumes a Voigt form, which is the convolution of a Gaussian with a Lorentzian form. The relative Gaussian and Lorentzian contributions in the Voigt fit were quantified by the shape factorwhich is the ratio of the Lorentzian width to the Gaussian width. Thus, a shape factor of zero signifies a pure Gaussian, a shape factor of infinity signifies a pure Lorentzian, and for a shape factor of 1 the contributions are equal. The shape factor for the line shapes in this study were calculated using the Multipeak Fitting package in IgorPro. The shape factor can thus be thought of as a measure of the percentage of disordered regions in the films. This is relevant for our studies as the presence of the side walls suppress disorder in the microdomains next to the walls, even when the regions in the center of the channels become disordered. For estimating orientational order, the intensity of the background subtracted first-order diffraction peak I(ϕ) was recorded as a function of rotation angle ϕ from the nominal position ϕ = 0. The measured intensity is proportional to the number of microdomains aligned in the direction ϕ which enables calculation of the thermal average ⟨cos2 ϕ⟩ as

Figure 4. Voigt fits to the central part of the peaks for the PS−PVP0.22 bilayer.

Table 1. Shape Factor of the Voigt Line Shape for the PS− PVP0.22 Block Copolymer Bilayer as a Function of Temperature temp (°C) 140 150 170 175

∫ I(ϕ)cos ϕ dϕ ∫ I(ϕ) dϕ

temp (°C)

shape factor

± ± ± ±

185 190 210

2.08 ± 0.75 1.95 ± 0.55 2.7 ± 0.70

0 0.69 1.06 1.29

0.00 0.02 0.05 0.29

At 140 °C the shape factor was zero; i.e., the central portion of the peak shape is purely Gaussian, which implies that the system is very well ordered. Note that the tails of the peak are not captured by the Gaussian form, implying that true longrange order does not exist in the system. At 150 °C a nonzero value of the shape factor was obtained, implying presence of disordered areas in the system. As the temperature was increased, the shape factor increased, suggesting a growth of the disordered regions in the sampled area. The change in the shape of the first-order Bragg peak from Gaussian to Lorentzian form is demonstrated in Figure 5 where line shapes for a bilayer of the PS−PVP0.22 block copolymer annealed at 140 and 210 °C were fit to both Gaussian and Lorentzian forms. We can see that the peak shape at 140 °C is fit better by a Gaussian while the peak shape at 210 °C is a Lorentzian. The tails of the line shape at 140 °C in Figure 4 are not captured by the Gaussian form, which implies that the translational order correlation function is not a constant but instead decays to zero. The line profile of the first-order peak I(q) is shown on a log−log scale in Figure 6 to exaggerate the shape of the tails. The line shape is linear on a log−log scale, which suggests a power-law decay of the structure factor S(q) ∼ (q − q*)−ϑ. A power law fit shows that S(q) decreases as ∼(q − q*)−3.8 over several decades of intensity. This rate of decay of the structure factor is much faster than the (q − q*)−2 limit of Caillé’s line shape for 3D smectics with quasi-long-range order.30 This implies that the rate of decay of the real-space correlation function G(r) is slower than the z−η form shown in eq 3. This suggests that graphoepitaxy can stabilize the arrangement of the block copolymer domains against thermal fluctuations, resulting in improved order. The columnar-like nature of the bilayer may also contribute in suppressing destruction of order. A similar

2

⟨cos2 ϕ⟩ =

shape factor

(10)

The orientational order parameter S was then evaluated according to eq 9. This order parameter is an indicator of the fraction of microdomains that are aligned with the channel walls.

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RESULTS The bulk order−disorder temperatures (ODT) for PS−PVP0.22, PS−PVP0.23, and PS−PVP0.26 measured using SAXS were found to be 195 ± 10, 210 ± 5, and 260 ± 10 °C, respectively.7 Measurements of the order in the thin film systems were performed using GISAXS. We will first present results for bilayer thick films followed by comparison with the behavior of monolayers. Figure 4 shows the line shapes for the first-order peaks for bilayer films of PS−PVP0.22 annealed at different temperatures. The line shapes have been normalized to a maximum first-order peak intensity value of 1, and the intensity is plotted as a function of (q − q*) (where q* is the position of the first-order peak). In all cases the peaks are broader than the direct beam profile, indicating that the measurements are not resolution limited, and broadening occurs due to finite size effects or disorder in the system. The peak width was always broader than the instrumental resolution due to the finite size of the grains, which is limited by the channel width to a maximum of 2 μm. The central portion of the peaks excluding the tails were fit with a Voigt function form to assign the shape of the peaks as either Gaussian or Lorentzian. The fits are shown in Figure 4, and the shape factor is tabulated in Table 1. 981


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Figure 5. Change in line shape from Gaussian to Lorentzian form for the PS−PVP0.22 bilayer as the temperature is increased.

Figure 6. Line shape of the first-order diffraction peak for a PS−PVP0.22 bilayer thick film annealed at 140 °C. The unfilled diamond markers are the direct beam shape which defines the instrumental resolution function. The measured rate of decay of the structure factor is ∼(q − q*)−3.8. For comparison the (q − q*)−2 limit of Caillé’s line shape for 2D smectics is also plotted.30

effect resulting in a structure factor decay faster than (q − q*)−2 has been reported in the literature for lightly cross-linked smectic elastomers where the cross-links were responsible for suppressing fluctuations in the layered structure.41 As mentioned earlier, the best fits for the line shapes were determined by comparing the residuals for a pure Gaussian shape, a Gaussian with power law decaying tails, or Lorentzian. The power law fits for the tails of the first-order peaks for samples annealed at different temperatures are tabulated in Table 2. The slope of the tails decrease continuously upon increasing temperature, which indicates that the rate of decay of correlations increases with temperature. The slope crosses the (q − q*)−2 dependence at ∼185 °C. This analysis suggests that even at temperatures where the systems shows overall quasilong-range order as evidenced by the power law fit when averaged over large areas (between 150 and 175 °C), there

exists local regions of disorder as evidenced by the finite Lorentzian contribution to the shape factor which grow as temperature is increased (Table1). Thus, for complete order quantification both the shape of the peaks and the rate of decay of the tails should be considered. We will now compare these results with the order in a monolayer thick film to investigate the effect of reducing dimensionality. Translational order in an infinite layered smectic system in 2D has been predicted to decay exponentially (as shown in eq 6) as a result of the enhanced effect of fluctuations compared to 3D systems. 6 Using similar techniques, we found that a monolayer of the PS−PVP0.22 block copolymer is completely disordered at all temperatures. The reasons for this behavior are explained in our previous publication.7 Since the ODT for the PS−PVP0.22 was lower than 130 °C and the glass transition temperatures for PS and PVP are ∼100 °C, the change in the line shape as the system undergoes the order-to-disorder transition could not be captured. Instead, the behavior of a monolayer of the PS− PVP0.23 block copolymer is presented here in detail. The shape of the first-order scattering peak as a function of temperature for the PS−PVP0.23 monolayer is shown in Figure 7. Comparison between the residuals for Gaussian and Lorentzian fits to the peak at 135 °C revealed that the shape of peak is a Gaussian. This implies that positional order does not decay exponentially in this finite 2D-like system. Fits to the tails of the

Table 2. Fits to S(q) ∼ (q − q*)−ϑ for the PS−PVP0.22 Block Copolymer Bilayer as a Function of Temperature ϑ

temp (°C) 140 150 170 175

3.80 2.41 2.40 2.10

± ± ± ±

0.00 0.02 0.00 0.01

temp (°C)

ϑ

185 190 210

2.00 ± 0.05 1.95 ± 0.05 1.78 ± 0.06

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Figure 8. Comparison between the line shapes of a PS−PVP0.23 monolayer and bilayer at temperatures above and below the smectic/ columnar-to-nematic transition temperature.

Figure 7. Line shape of the first-order diffraction peaks for PS−PVP0.23 monolayers as a function of temperature. The solid lines are a guide to the eye.

Table 4. Comparison between Line Shapes of a PS−PVP0.23 Monolayer and Bilayer at the Same Temperature

Gaussian peaks at 130 and 175 °C are shown in Table 3 and follow a power-law decay, which is consistent with an algebraic

T (°C) 170

Table 3. Shape Factor for PS−PVP0.23 Monolayer as a Function of Temperaturea temp (°C) 135 170 185 190

shape factor

ϑ

± ± ± ±

2.45 ± 0.06 2.34 ± 0.14

0.1 0.97 77.2 734

0.12 0.24 2.0 6.0

185

monolayer

bilayer

Gaussian with tails that decay as (q − q*)−2.34±0.14 Lorentzian

Gaussian with tails that decay as (q − q*)−2.99±0.03 Gaussian with tails that decay as (q − q*)−2.47±0.08

In order to confirm our results that smectic-like quasi-longrange order can exist even in monolayer films, the monolayer behavior for PS−PVP0.26 polymer is also shown here. The line shapes at three different temperatures are shown in Figure 9

a Fits to S(q) ∼ (q − q*)−ϑ are also shown for temperatures where the peaks are Gaussian in shape.

decay of positional order. Moreover, S(q) decays faster than the (q − q*)−1.5 limit, which implies that in real space the decay of the positional correlation function g(r) is slower than r−1.5 in the direction normal to the microdomains. The Voigt fits show that destruction of translational order does not take place until about 185 °C. Thus, unlike the case of 2D smectics of infinite extent, quasi-long-range translational order can persist in block copolymers confined in 2 μm channels even in 2D-like systems at finite temperatures. Comparison between the line shape of a monolayer of PS− PVP0.23 with that of a bilayer of the same block copolymer is shown in Figure 8. Fit parameters for monolayer and bilayer films annealed at the same temperatures are shown in Table 4. The faster the rate of decay of the tails of S(q) (or the larger the power-law exponent), the slower the rate of decay of correlations in real space. So comparison of the power law exponent for S(q) for monolayers and bilayers of PS−PVP0.23 at the same temperature as shown in Table 4 confirms that the bilayers were better ordered than monolayers. The change in the line shape from Gaussian to Lorentzian also takes place at higher temperatures for the bilayer. Thus, at the same temperature the bilayer possesses better order than the monolayer and the smectic-to-nematic transition takes place at a lower temperature for the monolayer.

Figure 9. Line shapes for PS−PVP0.26 monolayer annealed at different temperatures.

while the shape factors are tabulated in Table 5. Fits to the line shape reveal that at the lowest temperature the peak shape is nearly pure Gaussian with the tails decaying as (q − q*)−3.10, and the Lorentzian contribution increases as the temperature is increased. We can also extract an effective correlation length from the GISAXS data. The full width at half-maximum (fwhm) of the 983


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Table 5. Shape Factor and Decay Rate of Tails for PS− PVP0.26 Monolayer as a Function of Temperature T (°C)

shape factor

ϑ

160 180 200

0.0 ± 0.0 0.72 ± 0.34 2.42 ± 0.74

3.10 ± 0.01 2.9 ± 0.04 2.7 ± 0.03

diffraction peaks is proportional to the size of the smectic domains for Gaussian peak shapes and the correlation length for Lorentzian peak shapes. Thus, an effective correlation length ξ can be estimated as 2π/fwhm. Figure 10 shows the

Figure 11. Left: decrease in intensity of the first-order peak for the PSPVP22 bilayer as the sample is rotated in ϕ. Right: the rate of change of I(ϕ) vs ϕ with temperature.

Figure 12. Variation of the orientational order parameter S with temperature for (A) PS−PVP0.22 bilayer and (B) PS−PVP0.23 monolayer.

close to the temperature at which the power law exponent for decay of translational order ϑ becomes nearly equal to 2. After the abrupt decrease, the rate of decay of S with temperature is much faster.

Figure 10. Effective correlation lengths ξ for PS−PVP0.23 monolayer (top) and PS−PVP0.22 bilayer (below) as a function of temperature.

variation of ξ with temperature for PS−PVP0.23 monolayer and PS−PVP0.22 bilayer. At low temperatures, the grain size for the PS−PVP0.23 monolayer is ∼45 cylinder spacings, while at 190 °C a correlation length of about 10 cylinder spacings is measured. As has been mentioned earlier, decay of order in layered systems is a two-step processthe system first loses its translational and then its orientational order. Quantification of orientational order of the arrangement of microdomains was performed using grazing incidence diffraction by rotating the sample in-plane and recording the variation of intensity of the first-order diffraction peak I(ϕ) as a function of rotation from the nominal position ϕ = 0, as shown in Figure 11 for a bilayer of the PS−PVP0.22 block copolymer. The variation of orientational order parameter with temperature for bilayer thick films of PS−PVP0.22 and monolayer thick films of PS− PVP0.23 copolymer as calculated using eq 10 is shown in Figure 12. At low temperatures, S is nearly equal to one, indicating that nearly all the microdomains are oriented along the channel walls. The order parameter remains almost constant with temperature and then undergoes a sharp drop at 185 °C for the PS−PVP0.22 bilayer and at 195 °C for the PS−PVP 0.23 monolayer. The abrupt decrease in S occurs at a temperature

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COMPARISON WITH REAL SPACE ANALYSIS For visualization of the structural transformation as a function of temperature in these systems, we used atomic force microscopy to image the order in the films annealed at different temperatures. As an example, AFM micrographs for monolayer films of PS−PVP0.23 are shown in Figure 13. The atomic force micrographs clearly show the gradual destruction of translational order by increased density of thermally generated defects. Note that at 185 °C the structure factor decay was found using GISAXS to be (q − q*)−2, and the peak shape was primarily Lorentzian. At 190 °C the orientational order parameter S showed a sudden drop. From the atomic force micrographs, an increased density of dislocation-type defects are seen at 185 °C while at 190 °C the defects are primarily disclinations. AFM micrographs for bilayer thick films of PS−PVP0.22 are shown in the Supporting Information. Previous studies of quantification of cylindrical morphology block copolymer order by Harrison et al.3 and Hammond et al.4 used image analysis techniques to extract order parameters from atomic force micrographs. To compare the results obtained from X-ray scattering with those obtained using 984


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channel walls, the order in the center of the channels is poorer than near the edges.4 GISAXS inherently averages over the entire channel area and thus includes the enhanced order near the walls. In most of the line-shape fits, we could quantify a separate Gaussian and Lorentzian contribution, which we think corresponds to scattering from the ordered microdomains near the edges and scattering from the more defective region near the center, respectively. Hammond’s results from the regions localized to the center of the channels are more representative of an unconfined block copolymer system, since even at the lowest temperatures the effect of the walls extends only to about 10 microdomain spacings, while the contribution of the microdomains next to the channel walls are significant in our analysis. Also note that the g(r) calculated by Hammond et al. were azimuthally averaged, so they incorporate correlations both along the cylinder axis and perpendicular to it. As can be seen from eqs 3 and 6, the decay of the correlations are more rapid along the cylinder axis (the “x” direction in Figure 1) than perpendicular to it (along the z-axis). The GISAXS line profiles, on the other hand, are more sensitive to the z direction, as the qx component of the measured wave vector is nearly zero. As is evident here, both AFM analysis and GISAXS are powerful techniques and should ideally be used in conjunction; however, when statistically accurate and detailed measurements are required, GISAXS measurements have an advantage. AFM measurements are representative of much smaller sample areas, and extracting order correlations from AFM data relies on image analysis techniques and artifacts due to AFM piezo hysteresis must be carefully controlled. GISAXS measurements inherently average over a large sample area, and the technique is essentially nondestructive, whereas for block copolymer systems where the microdomains are covered by a brush layer (such as the one used in this study) the microdomains cannot be imaged using AFM without etching through the brush layer and thus destroying the sample. Moreover, by analyzing the shape and width of higher order peaks, we can distinguish between the presence of long-range order, quasi-long-range order with defects, and short-range liquid-like order. The dislocation density in the samples can be extracted from the width of higher-order peaks in GISAXS, details of which are provided in the Supporting Information. This allows us a route to extract the defect density averaged over an area of 106 μm2. Obtaining similar information by AFM image analysis would mean averaging over roughly 105 images.

Figure 13. Atomic force micrographs of a monolayer of PS−PVP0.23 cylindrical morphology block copolymer as a function of temperature.

image analysis of atomic force micrographs, we reproduce translational order calculations performed by Hammond et al.4 for the PS−PVP0.23 monolayer. Figure 14 shows the decay of

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CALCULATION OF POSITIONAL ORDER CORRELATION FOR CONFINED 2D SMECTICS Our experimental studies reveal that graphoepitaxially aligned monolayers of cylinder forming block copolymers showed a power-law decay of the structure factor implying algebraic decay of positional order, in contrast to the exponential decay expected for infinite 2D smectics. In order to understand the driving force behind this behavior, we calculated the correlation function for a confined 2D smectic. The free energy density for an infinitely large 2D smectic is given by 1 F(q) = (Bqz 2 + Kqx 4)u 2(q) (11) 2 where K is the splay elastic constant and B is the compressional modulus normal to the layers. The correlation function g(x,z) for such a system has been found to have an exponentially decaying form,42 as is expected for a 2D smectic. However, the presence of the channel walls will have the effect of directing

Figure 14. Translational order correlation functions calculated by Hammond et al. for a monolayer of PS−PVP0.23 cylindrical morphology block copolymer as a function of temperature. Exponential decay in g(r) is observed in all cases with progressively shorter correlation lengths upon increasing temperature [from Macromolecules 2005, 38 (15), 6575−6585].

positional order g(r) calculated by Hammond et al.4 using AFM image analysis, which can be fit with an exponential decay at all temperatures. This is in contrast to our results from X-ray line shape analysis which predicts an exponential decay only at temperatures close to 185 °C. Furthermore, the correlation lengths are reported to be about 10 cylinder spacings, which again is smaller than our estimate for the same temperature. However, the order correlations extracted from the atomic force micrographs were calculated for a 1 μm wide and 1 μm high area from the center of the channels. As edge effects decay from the channel walls and defects are repelled from the 985


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standing questions regarding the nature of the melting transition in other 2D and 3D systems.

the alignment of the microdomains parallel to the channels and suppress fluctuations along the x-axis and hence result in another “line tension”-like contribution to the free energy 1 F(q) = (Bqz 2 + Kqx 4 + Tqx 2)u 2(q) 2

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S Supporting Information *

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Supporting Information Available: Effect of fluctuations in 3D and 2D, Evaluation of pair-correlation function for 2D smectics, Calculated line shape for 2D and 3D systems with algebraic and exponential decay of translational order, Extracting information from higher order peaks, AFM micrographs for PS−PVP0.22 bilayers. This material is available free of charge via the Internet at http://pubs.acs.org.

where T is a proportionality constant. We assume that T ≫ K such that the surface tension term is the dominant term compared to the splay elastic energy. The form of the pair correlation function is obtained by evaluating G(r) ∼ ⟨exp[iq·(u(r) − u(0))]⟩. In the harmonic approximation

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⎤ ⎡ (q*)2 ⟨exp[iq. (u(r) − u(0))]⟩∼exp⎢ − ⟨|u(r) − u(0)|2 ⟩⎥ 2 ⎥⎦ ⎢⎣

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected].

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Present Address

The average is calculated by integrating over all wave vectors

§

DuPont Central Research, Wilmington, DE.

2 ∞ ∞ (q*) ⟨ u(r) − u(0)|2 ⟩ = η dqx dqz −∞ −∞ 2 1 − cos[qxx + qzz] = −ln x 2η + ln(x 2 + λT 2z 2)η qz 2 + λT 2qx 2

∫

ASSOCIATED CONTENT

Notes

∫

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge funding from the NSF DMR-0704539 as well as the CSP Technologies Fellowship for partial support of V.M. Use of the MRL Facilities was supported by the MRSEC Program of the National Science Foundation under Contract DMR-11-21053. V.M. thanks Professor Gila Stein for her input, Dr. Matt Hammond for helpful discussions and synthesis of two polymers, Prof. Axel Müller and Dr. Felix Schacher for synthesis of the third polymer used in this work, and the beamline scientists at Argonne National Laboratory Dr. Joseph Strzalka and Dr. Zhang Jiang for assistance during the GISAXS experiments. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract DE-AC02-06CH11357. A portion of this work was carried out in the UCSB Nanofabrication Facility, part of the NSF-funded National Nanofabrication Infrastructure Network. A special thanks to Professor Cyrus Safinya for many helpful suggestions as well as his careful reading of the manuscript.

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where λT ≡ (T/B) . The details of the evaluation of the integral is provided in the Supporting Information. Thus, 2 2 G(r) ∼ e−(q* /2)⟨|u(r)−u(0)| ⟩ ∼ x2η + (x2 + λT2z2)−η. Hence, G(x→0,z) ∼ z−2η decays algebraically unlike the exponential decay expected for an infinite 2D smectic. This confirms analytically that the effect of the channel walls is to suppress loss of order by fluctuations and explains the powerlaw decay observed for monolayer thick films. 1/2

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CONCLUSION While GISAXS has been used in several block copolymer studies in the past, the extent of information that can be gleaned from this technique has not yet been fully exploited. The sensitivity of GISAXS to minor lattice variations and the inherent averaging over large areas can reveal information that is inaccessible by other techniques. We have developed a methodology to measure temperature-dependent order in thin films of cylindrical morphology block copolymers by line-shape analysis of X-ray scattering peaks. Our studies reveal similarities between the temperature dependent order in laterally confined monolayers and bilayers of cylindrical morphology block copolymers and thermotropic liquid crystals. However, lower critical dimensionality for finite block copolymers systems is 1D compared to 3D for smectics as even monolayers of block copolymers show effective quasi-long-range order under suitable conditions. We have shown, both through experiments and analytically, that physical confinement using graphoepitaxy can successfully suppress the exponential decay of positional order characteristic of block copolymer arrays, resulting in a well-ordered arrangement with a very slow decay of positional correlations. However, 2D arrays of block copolymers (monolayers) are more susceptible to disorder than bilayer or thicker films as they lose their order at lower temperatures. Here we have emphasized the power of GISAXS for quantification and detection of true long-range order in block copolymer systems. This methodology, particularly when coupled with in-situ thermal annealing, can help answer long-

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Quantification of Temperature-Dependent Order in Graphoepitaxially

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