NANO LETTERS
Simulation of Grafted Polymers on Nanopatterned Surfaces
2006 Vol. 6, No. 1 133-137
Michael Patra* and Per Linse Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund UniVersity, P.O. Box 124, SE-221 00 Lund, Sweden Received August 15, 2005
ABSTRACT Structural properties of polymer brushes on nanopatterned surfaces in good solvent have been determined by computer simulations. Scaling relations for the brush height and brush width are proposed. The properties of the central part of the patterned brush remain constant as long as the pattern is wider than a few times the brush height. The results agree qualitatively with recent AFM experiments, but some quantitative differences call for a reassessment of experimental procedures.
Polymers end-grafted onto surfaces form at sufficiently high grafting density so-called polymer brushes. Stimuli-responsive systems can be prepared by immersing polymer brushes in solvent. For example, significant structural changes can be triggered by changing temperature, pH, and solvent quality or by adding target molecules.1 After recent advances in the nanostructuring of surfaces,2,3 patterned polymer brushes can now be prepared on the nano and meso scale.4,5 For efficient fabrication and for many applications of such systems, the size of the polymeric structures needs to be minimized. For example, if the pattern size becomes comparable to the wavelength of light, diffraction measurements become possible and, hence, such nanopattered brushes can be used conveniently with optical sensors.6 The fabrication process is one of two constraints that limit the minimum system size that can be achieved. Apart from this technological limitation, which can eventually be overcome, there is one more fundamental limit: if the structures become too small, then the polymers do not form conventional proper brushes any more. At even smaller length scales, the properties of patterned surfaces are currently not well understood, making a reliable technical application problematic. The purpose of this letter is twofold: to examine, by simulation methods, how small nanopatterned polymer brushes can be while retaining properties comparable to conventional homogeneously grafted brushes and to present a basic characterization of the properties of smaller brushes. Currently, these questions are difficult to answer from experiments because the recent progress in the fabrication of patterned polymer structures has unfortunately been only partially matched by advances in the experimental methods * Corresponding author. Phone: +46 46 222 4812. Fax: +46 46 222 4413. E-mail:
[email protected]. 10.1021/nl051611y CCC: $33.50 Published on Web 10/08/2005
© 2006 American Chemical Society
used to study the properties of such systems.7 There are a number of established experimental techniques such as optical methods, X-ray scattering, and neutron scattering, used to study the properties of homogeneously grafted polymer brushes. Common to them is that the measured property is averaged over an area of meso or macro size. Hence, these methods are not applicable for nanopatterned polymer brushes. Instead, such brushes have almost exclusively been investigated by atomic-force microscopy (AFM). This is a powerful technique providing images with high spatial resolution, but measurements are done in contact and tapping modes that involve a physical contact between the AFM tip and the polymer brush. Thus, an AFM measurement distorts the systems under investigation.8 This distortion is significant; in the Supporting Information, strict mathematical bounds for the shape of a nanopatterned brush are derived, and these bounds are violated in most of the reported measurements. An additional experimental problem arises because the brushes are usually obtained by the “grafting-from” approach in which the covalently attached polymers are grown by surface-initiated polymerization from the substrate. Differences in the reaction conditions at the center and at the edge of the pattern introduce perpendicular heterogeneity in the polymer length, in addition to the conventional polydispersity encountered in homogeneously grafted brushes. Although the polymer length as well as the grafting density can be controlled qualitatively, the actual quantitative values usually remain unknown. Thus, the limitations encountered in experimental studies are twofold: there is only partial knowledge of the system parameters, and distortions are introduced by the measurement process. Computer simulations enable us to bypass these two types of experimental difficulties. Despite the
potential power of computer simulations in supplementing experimental studies, to the best of our knowledge nanopatterned polymer brushes have not yet been subjected to such studies. The only two existing theoretical studies9,10 on related systems are, however, only on a mean-field level and focus on the effects of the solvent quality. In this letter, we report on the structure of nanostructured polymer brushes as obtained by computer simulations, thereby partially filling this gap. Here, we restrict ourselves to polymers attached to an infinitely long stripe of a planar surface. This is also by far the most frequently used experimental geometry, and typical widths, ∆, of the stripe range from a few micrometers to a few tenths of micrometers but can be as low as 90 nm. In the limit ∆ f ∞, the properties of a traditional homogeneously grafted brush are recovered, which have been investigated thoroughly with different theoretical approaches.11-18 We have adopted a coarse-grained model system for studying the generic properties of patterned brushes. A schematic illustration of our model system is shown in Figure 1. Polymers are randomly attached onto an infinite long stripe with finite width ∆. The polymers are described as freely jointed chains composed of N spherical subunits connected by bonds. Each subunit (bead) corresponds to a small part of a real polymer, approximately one Kuhn length,19 and is described by a soft DPD-style potential (see the Supporting Information). Reduced units are used throughout this study. The bead diameter has been set to 1, which sets the length scale. The 2D polymer grafting density is denoted by σ. Further details about the model and simulation aspects can be found in the Supporting Information. We have performed simulations with different polymer contour lengths, N, from 25 through 200, grafting densities, σ, from (1/5)2 through (1/2)2 (σ ) 0.04, 0.063, 0.11, and 0.25), and stripe widths, ∆, from 1 through 200, giving a total of about 200 different systems. Our simulations cover the majority of the experimentally relevant parameter space in N, σ, and ∆. Typical heights of experimentally studied nanopatterned brushes are reported to be in the 20-100 nm range. For flexible polymers, the Kuhn length is about 0.5-1 nm so that the polymer length is of the order N ≈ 25-300. (This is shorter than typical values for homogeneously grafted brushes.) A homogeneously grafted polymer brush possesses a density profile, F(z), that depends only on the distance from
Figure 1. Snapshot illustrating the model system composed of a planar surface on which polymers are grafted onto a stripe with width ∆. The orientation of the coordinate system is also displayed. 134
the grafting surface. This problem has been studied for over thirty years both theoretically11-18 and experimentally,20-22 and its behavior is well understood. The stripe patterning of the surface creates an additional dependency, and the number density profile, F(x, z), now also depends on the lateral coordinate, x (see Figure 1). Figure 2 shows the density profile, F(x, z), from our simulations with N ) 100 and σ ) 0.11 for selected values of ∆ using logarithmically separated contour curves. The result for the homogeneously grafted brush (∆ ) ∞) is shown as well. As ∆ is increasing, we find (i) an increase in the brush height, (ii) an expansion of the region with high density close to the grafting surfaces, and (iii) an appearance of a considerable overshot of polymers outside the grafting region. As seen from the figure, the polymer density falls off rapidly when the perimeter of the brush is reached. Hence, the precise criterion used for defining the perimeter has only a minor effect, and we will be using the threshold F ) 0.05 throughout this study. We will now examine in more detail how the brush height depends on N, σ, and ∆. We will restrict ourselves to a discussion of the height, h, in the center of the pattern. Figure 3 displays the brush height, h, as a function of the stripe width, ∆, at various polymer lengths, N, and grafting densities, σ. Functional dependences are best seen by employing rescaled axes ∆/N and hσ-1/3/N. Within statistical error, all data points collapse onto a common curve. (The raw data is presented in the Supporting Information.) This implies that the dependence of the brush height h(∆, N, σ) factorizes as h(∆, N, σ) ) Nσ1/3h˜
(∆N)
(1)
The height of a homogeneously grafted brush scales as Nσ1/3,11-13 and, somewhat surprisingly, we have found the same scaling relation for finite values of ∆. The sole dependence on ∆ appears as h˜ (∆/N), where h˜ (∆/N) is a universal function of its argument ∆/N with the limit 0 for ∆/N ) 0 and a finite value of order one for ∆/N f ∞. Moreover, (i) half of the height of the homogeneously grafted brush is achieved when the width of the stripe is 10% of the polymer contour length, (ii) 90% of the full height is obtained when the stripe is equal to the polymer contour length, and (iii) the full height is practically recovered when the stripe is J4 times the polymer contour length, N. This thus marks the minimum pattern width, ∆, such that the nanopatterned brush, or at least its central part, has the same properties as a homogeneously grafted brush. Another interesting region of the patterned brush is the immediate vicinity of the edge of the pattern. Figure 4 shows how the characteristic shape of the brush develops as the grafting pattern becomes wider. Generally, the brush extends beyond the grafting region, as the osmotic pressure inside the brush can be released partly by expanding the brush laterally. Beyond the grafting region, the polymer density is low very close to the surface because of the loss in conformational entropy for chains located close to the surface. Nano Lett., Vol. 6, No. 1, 2006
Figure 2. Number density contour curves for polymer length N ) 100 and grafting density σ ) 0.11 at stripe width ∆ ) 15, 50, 200, and ∞ (from left to right). The contour curves mark the densities F ) 0.01, 0.02, 0.05, 0.1, and 0.2. The locations of the stripes are indicated by the filled black areas. Note that the scale of the abscissas differs among the panels.
Figure 3. Rescaled height as a function of reduced stripe width for all combinations of N and σ, indicating a universal behavior and suggesting the scaling relation h(∆, N, σ) ) Nσ1/3h˜ (∆/N).
Figure 5. Rescaled representation of w(∆) for all combinations of N and σ, indicating a universal behavior and suggesting the scaling relation (∆, N, σ) ) Nσ1/2w˜ (∆/N).
the edge, whereas h is determined by the larger number of polymers grafted in the central region of the stripe. The excess width, w, of all of the studied systems again collapses, within statistical error, onto a single curve after proper rescaling. The best agreement was found for the following scaling relation: w(∆, N, σ) ) Nσ1/2w˜ (∆/N) Figure 4. Perimeter of the polymer brush near the edge of the stripe for polymer contour length N ) 100 and grafting density σ ) 0.11 at indicated stripe widths ∆. The location of the stripe is shown by the filled black area. The origin of the x axis is placed at the right edge of the stripe.
We define the amount by which the brush extends over the edge of the stripe as the excess brush width, w, and the total width of the brush hence becomes ∆ + 2w. Frequently, for example, in applications such as sensors, multiple patterned brushes are needed in order to supply sufficiently large signals, and knowledge of w allows one to minimize the size of the multibrush structure. Figure 5 shows how the excess width, w, depends on the system parameters. As expected, w increases with grafting density and stripe width. Again, a limiting value appears to be reached at large ∆. The statistical error in the excess width, w, is larger than that in the height, h, because w is determined only by the polymers that are grafted close to Nano Lett., Vol. 6, No. 1, 2006
(2)
Hence, the dependence of the width, w(∆, N, σ), also factorizes and here isolates the dependence on the stripe width, ∆, into the universal function w˜ (∆/N) with the prefactor Nσ1/2. The maximal width, w, is achieved when ∆/N J 1. There thus seems to be a distinctly different scaling for the height of the polymer in the center of the pattern, h ∝ σ1/3, and for the amount by which the brush extends over the edge of the pattern, w ∝ σ1/2. Please note that although the brush height, h, is larger than the excess width, w, the excess width, w, increases faster with σ than does the height, h. The height, h, in the center basically quantifies the largest extension of each polymer between its grafting point and any point along its contour. This is also confirmed by a direct calculation of the radius-of-gyration tensor of the polymers (data not shown), and, consequently, height, h, polymer extension, and radius-of-gyration scale identically as a 135
Figure 6. Typical shapes of a nanopatterned brush as computed by simulation (solid line) and as measured by AFM (broken line). Horizontal and vertical length scales have been indicated in the figure.
function of N, σ, and ∆. This is different for the excess brush width, w, because it depends not only on the shape of each polymer but also on its inclination. The additional increase of w with increasing σ is due to the increase in tilting angle of polymers at the edge of the pattern as was compared by an explicit computation of this angle (data not shown). Unless the stripe is very small (∆ j N), the behavior of a polymer grafted at some position, x, does not depend on the strip width, ∆, explicitly but rather only on the distances d and ∆ - d to the two edges of the stripe. The closer edge is much more important because it sets the distance beyond which there is empty space that can be utilized to reduce the osmotic pressure. The local environment for a polymer that is grafted a distance d away from the nearest edge is thus similar to the environment seen by a polymer that is grafted in the center of a stripe with width ∆ ) 2d. The local height of the brush a distance d away from the edge thus is approximately the same as h(2d), that is, the height in the center of a brush with ∆ ) 2d. This is indeed observed. For example, the perimeter of the brush from the density profile (Figure 2) is similar to the shape of the h(∆) curve in Figure 3. (The two figures might look rather different at first but that is because the latter figure has a logarithmic abscissa.) The saturation of h(∆) for ∆ J 4N (cf. Figure 3) thus goes along with the formation of a plateau in the density profile, starting a distance ∼2N away from the edge of the stripe (cf. Figure 2). This is indicated schematically in Figure 6. The scaling relations for the brush height h(∆, N, σ) (eq 1) and excess brush width w(∆, N, σ) (eq 2) also give valuable information for a single polymer brush, that is, for constant ∆, N, and σ. For a brush width ∆ ) 10N, the plateau forms 60% of the entire brush, and the properties of the nanopatterned brush thus will be very similar to the corresponding homogeneously grafted brush. This limit for ∆ is surprisingly low, and significantly smaller than structures that have been studied experimentally so far. As mentioned previously, a number of experimental AFM investigations on the height profile, h(x), are available, but because of the experimental uncertainty in the parameters N and σ, a direct comparison of the experimental numbers to our results for h(∆) or w(∆) is not meaningful. Rather, we want to compare the shape of our computed density profiles against experimental AFM height profiles. The typical shapes of brush profiles from our simulations and AFM measurements are indicated in Figure 6. In contrast to the simulated profile, the experimental one displays no clear plateau in the center of the stripe even though the brush 136
is 10-100 times as wide as it is high. Furthermore, in the Supporting Information, we present a strict derivation for the maximal decrease of the brush height outside of the center. The line of the argument is straightforward: the overshot of polymers outside of the grafting region is the reason for the decrease of the brush height, h(x), outside of the center, and because the overshot is limited (because w < N), the decrease of h(x) is also limited. Most experimental brush profiles from AFM measurements violate this strict bound. We propose that an AFM measurement does not record the unperturbed brush surface, even in the tapping mode. Because a polymer brush is a soft material, it yields under the force of the AFM tip. This has been demonstrated by simulations of homogeneously grafted polymer brushes.23 Similar simulations are also possible for nanopatterned brushes, but they need a potential with a harder core than employed in this study to allow for accurate force estimation when a polymer bead collides with the AFM tip. Hence, a systematic underestimate of the brush height is expected. In addition, near the edges of the pattern the yield upon contact with the tip is anticipated to be even stronger, and the underestimation of the height is likely to be even larger as compared to the center of the pattern. Thus, the yield of the brush under the force of the AFM tip would (i) underestimate the brush height and (ii) underestimate the width of the plateau region, the latter in agreement with the concluded discrepancy between AFM results and our predictions. Finally, the AFM cone angle sets an upper limit of the slope, h(x), that can be measured directly, making it impossible study the very vicinity of the widest point of the brush. To conclude, our results provide two important scaling relations that we expect to be of general applicability to patterned brushes. We would like to encourage more detailed experimental investigations in the domain ∆ ≈ N to 5N because our results suggest that this domain contains the transition from narrow polymer brushes to the regime of wide brushes. Moreover, an improved understanding of this transition calls for a reassessment of the experimental conditions of the AFT technique as discussed above. An even more exiting task would be to experimentally investigate preferred orientation and conformational changes of chains near an edge by using, for example, labeled polymers. Many applications of nanopatterned polymer brushes would benefit from such a deeper understanding of the molecular arrangement in narrow polymer brushes. Acknowledgment. We thank Stefan Zauscher for valuable discussions and acknowledge financial support from the European Union (contract no. MRTN-CT-2004-512331) and the Swedish Research Council (VR). Supporting Information Available: Unscaled results for brush height, h, and excess brush width, w, derivation of strict mathematical bounds for the length over which the brush deforms near the edge of the pattern, and a description of the details of the simulation. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Nath, N.; Chilkoti, A. AdV. Mater. 2002, 14, 1243-1247. (2) Crooks, R. M. ChemPhysChem 2001, 2, 644-654.
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