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Particle Assembly on Patterned Surfaces Bearing Circular (Dots) and Rectangular (Stripes) Surface Features Zbigniew Adamczyk,* Jakub Barbasz, and Małgorzata Nattich Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, 30-239 Krakow, ul. Niezapominajek 8, Poland ReceiVed August 29, 2007. In Final Form: October 15, 2007 Irreversible and localized adsorption of spherical particles on surface features of various shapes (collectors) was studied using the random sequential adsorption (RSA) model. Collectors in the form of dots and rectangles were considered, including the two limiting cases of squares and stripes. Numerical simulation of the Monte Carlo type enabled one to determine particle configurations, average coverage of particles, and the distribution for various collector length to particle size ratios L h ) L/d and collector width to particle size ratios bh ) b/d. It was predicted that particle coverage under the jamming state was highly nonuniform, exhibiting a maximum at the center and at the periphery of the collectors. The averaged number of particles 〈Np〉 adsorbed at the jamming state was also determined as a function of the L h and bh parameters, as well as the averaged number of particles per unit length in the case of stripes. It was revealed that 〈Np〉 was the highest for the circular and square collectors (for a fixed value of L h ). On the other hand, for L h > 5, our numerical results could be well approximated by the analytical expressions 〈Np〉 ) θ∞L h2 2 for circles, 〈Np〉 ) 4θ∞L h /π for squares, 〈Np〉 ) 4θ∞bhL h /π for rectangles, and 〈Np〉 ) 4θ∞bh/π for stripes (per unit length). It was demonstrated that the theoretical results are in agreement with experimental data obtained for latex particles adsorbing on patterned surfaces obtained by a polymer-on-polymer stamping technique of gold covered silicon and on photolitographically patterned silane layers on silica.
I. Introduction Deposition (irreversible adsorption) of colloids and bioparticles on solid/liquid interfaces is of major significance for predicting the efficiency and kinetics of many processes such as selfassembly, filtration, separation by affinity chromatography, immobilization at interfaces, removal of pathological cells, immunological assays, biofouling of transplants and artificial organs, and so forth. Accordingly, particle deposition on homogeneous surfaces has been extensively studied both theoretically1-6 and experimentally.7-15 From a practical point of view, however, especially interesting is the problem of particle deposition at heterogeneous surfaces, covered by adsorption sites such as polyelectrolyte chains,16-19 * To whom correspondence should be addressed. E-mail: ncadamcz@ cyf-kr.edu.pl. Telephone: (+48 12) 6395104. Fax: (+48 12) 4251923. (1) Adamczyk, Z.; Siwek, B.; Zembala, M.; Warszyn´ski, P. J. Colloid Interface Sci. 1990, 140, 123-137. (2) Viot, P.; Tarjus, G.; Ricci, S. M.; Talbot, J. J. Chem. Phys. 1992, 97, 5212-5218. (3) Adamczyk, Z.; Weron´ski, P. J. Chem. Phys. 1996, 105, 5562-5573. (4) Oberholzer, M. R.; Stankovich, J. M.; Carnie, S. L.; Chan, D. Y. C.; Lenhoff, A. M. J. Colloid Interface Sci. 1997, 194, 138-153. (5) Weron´ski, P. AdV. Colloid Interface Sci. 2005, 118, 1-24. (6) Adamczyk, Z.; Senger, B.; Voegel, J. C.; Schaaf, P. J. Chem. Phys. 1999, 110, 3118-3138. (7) Harley, S.; Thompson, D. W.; Vincent, B. Colloids Surf. 1992, 62, 163176. (8) Johnson, C. A.; Lenhoff, A. M. J. Colloid Interface Sci. 1996, 179, 587599. (9) Bohmer, M. R.; van der Zeeuw, E. A.; Koper, G. J. M. J. Colloid Interface Sci. 1998, 197, 242-250. (10) Adamczyk, Z.; Siwek, B.; Zembala, M. J. Colloid Interface Sci. 1998, 198, 183-185. (11) Semmler, M.; Mann, E. K.; Ricka, J.; Borkovec, M. Langmuir 1998, 14, 5127-5132. (12) Adamczyk, Z.; Szyk, L. Langmuir 2000, 16, 5730-5737. (13) Kleimann, J.; Lecoultre, G.; Papastavrou, G.; Jeannret, S.; Galletto, P.; Koper, G. J. M.; Borkovec, M. J. Colloid Interface Sci. 2006, 303, 460-471. (14) Kun, R.; Fendler, J. H. J. Phys. Chem. B 2004, 108, 3462-3468. (15) Pericet-Camara, R.; Papastavrou, G.; Borkovec, M. Langmuir 2004, 20, 3264-3270. (16) Shin, J.; Roberts, J. E.; Santore, M. J. Colloid Interface Sci. 2002, 247, 220-230.
Figure 1. Schematic representation of particle deposition on surface features of circular (a) and rectangular (b) shape.
proteins,20,21 or colloid nanoparticles.22-25 Recently, much interest has been focused on heterogeneous surfaces bearing patterned (17) Kozlova, N.; Santore, M. Langmuir 2006, 22, 1135-1142. (18) Adamczyk, Z.; Zembala, M.; Michna, A. J. Colloid Interface Sci. 2006, 303, 353-364. (19) Adamczyk, Z.; Michna, A.; Szaraniec, M.; Bratek, A.; Barbasz, J. J. Colloid Interface Sci. 2007, 313, 86-96. (20) Joscelyne, S.; Tragardh, Ch. J. Colloid Interface Sci. 1997, 192, 294305. (21) Garno, J. C.; Amro, N. A.; Wadu-Mesthrige, K.; Liu, G.-Y. Langmuir 2002, 18, 8186-8192.
10.1021/la702650n CCC: $40.75 © 2008 American Chemical Society Published on Web 01/17/2008
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surfaces features of regular shapes such as circles and dots,26,27 squares,28 long rectangles (stripes),29,30 and others.31 These features are usually produced on homogeneous surfaces by microcontact printing, photolithography, or laser ablation. The growing interest in such surface architecture stems from their practical significance as antireflecting and self-cleaning surfaces, biosensors (protein arrays), optical filters, masks, photonic crystals, and microfluidic and microelectronic devices. This concerns, for example, the problem of particle deposition on stripes pertinent to microcircuitry. Besides practical importance, irreversible adsorption of particles on surface features has a major significance for basic colloid science, representing an interesting mapping problem.32,33 Despite considerable physical and practical significance, the problem of determining particle distribution and the jamming coverage (defined as the coverage where no additional particles can be placed in the monolayer without overlapping) on surface features of various shapes, especially stripes, has not been analyzed quantitatively in the literature. Theoretical results are available for lines of negligible width and infinite length and linear segments of negligible width and finite length only.34,35 In ref 33, numerical results were obtained for linear segments of finite length and for curved segments (semicircles or circles) characterized by an arbitrary particle to collector length ratio L h. The goal of this work is to develop a quantitative description of particle deposition on circular and rectangular surface features with a special emphasis focused on the case of stripes, which has not been studied before. The theoretical results will be used for interpretation of existing experimental data obtained for monodisperse colloidal particles adsorbing on patterned surfaces.
II. Theoretical Model Since there are no analytical results pertaining to the adsorption of particles on circles and rectangles of various shapes and a finite length, we applied Monte Carlo simulations to derive information on the jamming coverage and distribution of particles. These simulations have been carried out according to the random sequential adsorption (RSA) model in two dimensions (2D).1,36-39 The RSA process consists of placing a particle (usually of a spherical shape) of the characteristic dimension l on a surface of finite size (collector), characterized by the dimension L. We assume that the adsorbing objects are spheres of radius a and diameter d ) 2a and the collectors are either circles of diameter 2R or rectangles of the dimensions L × b (width) (see Figure 1). For b/d f 0, the previously treated case of adsorption of line segments of finite length is reflected.33 On the other hand, for L f ∞ and a finite value of b/d, the limiting case of deposition on stripes is approached. If there is an empty space between previously adsorbed particles large enough to accommodate the virtual particle, it is irreversibly adsorbed. Otherwise, an adsorption attempt is repeated. The main feature of this RSA process is that the new adsorption attempt is totally uncorrelated with previous adsorption attempts and that the adsorption probability is uniform over the entire collector. Additionally, the adsorption of particles is assumed to be irreversible and localized. The averaged 2D density (coverage) of particles is defined as
θ ) NpSg/Sc
(1)
where Np is the number of adsorbed particles over the collector, Sg ) πa2 is the geometrical cross-sectional area of the particle, (22) Adamczyk, Z.; Jaszczo´łt, K.; Siwek, B.; Weron´ski, P. J. Chem. Phys. 2004, 120, 1155-1162.
and Sc is the collector surface area equal to πR2 for circular collectors and L × b for rectangles. In the case of lines (infinitelly thin), it is more natural to introduce the one-dimensional coverage θ1D defined as the number of particles per unit length of the collector, that is,
θ1D ) Npd/L
(2)
It is interesting to mention that, in the case of line segments, a recursion formula has been derived34,35 which expresses the averaged number of particles over the collector under the jamming state in the following form:
h )〉 ) 1 + 〈Np(L
2 L h
∫0Lh Np(L′) dL′
(3)
where 〈Np(L h )〉 is the function of L h , which can be found recursively if an initial value is known. h )〉 ) 1 for 0 < L h < 1, Because, from simple geometry, 〈Np(L one can predict that
{
〈Np(L h )〉 ) 3 -
for 1 < L h 4). As can be seen in Figure 7, for L h ) 1, the limiting value of 〈Np〉 equals 1 for the circular collector as expected from simple geometry. On the other hand, for squares, the limiting value of h ) 1 equals 1.66. This is much higher because, in this 〈Np〉 for L case of L h ) 1, there can be up to four particles adsorbed at the
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Figure 2. Monolayers of particles adsorbed on circular and square collectors derived from simulations for various L h.
jamming state. With increasing L h , the value of 〈Np〉 increased abruptly, which was described by the polynomial fitting functions valid for L h > 1:
〈Np〉 ) 0.149 + 0.955L h + 0.686 L h 2 + 1.4 × 10-4L h3 squares 〈Np〉 ) -0.548 + 1.02L h + 0.519L h 2 + 4.4 × 10-4L h3 circles (11) On the other hand, for lines of finite length, the numerical results agree well, for L h < 2, with the analytical predictions given by eq 4. For L h > 2, the fitting function
h 〈Np〉 ) 0.505 + 0.748L
(12)
was found to be adequate. In Figure 8, the dependence of 〈Np〉 on the L h parameter is shown for L h > 4. As can be seen, the numerical results can be well reflected for this range of L h by the analytical results derived from eqs 8-10. Similar results for stripes are shown in Figure 9. In part a of this figure, the dependence of the one-dimensional coverage θ1D (averaged number of particles adsorbed per unit length of the collector) is shown as a function of bh . In part b, the dependence of the two-dimensional coverage θ is shown. It is interesting to mention that for
Figure 3. Monolayers of particles adsorbed on stripes for various bh including the limiting case of bh ) 0 (adsorption on a line).
bh < 1.5 the numerical results can be well fitted by the function
θ1D ) 0.7476 + 0.36bh2 + 0.026bh3 for bh < 1.5
(13)
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Figure 4. Schematic representation of the 2D density distribution of adsorbed particles on dots (upper part) and stripes (lower part) for various values of the L h or bh parameters (in the case of stripes). The points show the positions of particle centers averaged from over 10 000 collectors.
Figure 5. Dependence of the local 2D coverage of particles θ(r/R) for dots (circular collectors) characterized by various L h on the reduced distance from the center r/R. The points show the numerical results averaged from over 10 000 collectors. (O) L h ) 3, (1) L h ) 3.5, (b) L h ) 4.5, and (4) L h ) 10.
On the other hand, for bh > 1, the numerical results can be well approximated by the dependencies
4 θ1D ) 0.45 + θ∞ bh π θ2D ) 0.547 +
0.6 bh
(14)
Hence, these numerical results indicate that the limiting values of two-dimensional coverage are approached asymptotically according to the hyperbolic dependence.
IV. Comparison with Experimental Results Although no systematic studies on particle deposition on patterned surfaces have been carried out yet, there exists some
Figure 6. Dependence of the local 2D coverage of particles θ(y/b) for stripes characterized by various bh on the reduced distance from the symmetry line y/(1/2)b. The points show the numerical results averaged from over 10 000 collectors. (4) bh ) 1, (2) bh ) 3, and (O) bh ) 5.
results for dots and stripes, which can be analyzed in terms of our theoretical model. Interesting experiments of this type have been carried out by Zheng et al.26 for polystyrene latex particles of 1 µm diameter adsorbing on surface features of a circular shape. These features, having an average diameter of 11 µm, have been produced by stamping on gold covered silicon bearing 10 bilayers of polyelectrolytes (negatively charged sulfonated polystyrene (SPS) and positively charged poly(diallyldimethyl) ammonium chloride). Particle deposition was carried out under diffusion controlled transport from latex suspensions of a concentration of 0.005 (0.5%) by weight. The usual deposition time was 1-3 h, which was assumed sufficient for completing particle monolayers. However, neither the pH nor ionic strength of the deposition bath, which exert a major effect on the effective size of latex particles43 and consequently on the maximum coverage, have been specified. Most likely, the pH was 5.5 (usual value for
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Figure 7. Average number of particles 〈Np〉 adsorbed on various collectors as a function of L h : (1) square collector, 〈Np〉 ) 0.149 + h 3; (2) circular collector (dot), 〈Np〉 0.955L h + 0.686L h 2 + 1.4 × 10-4L ) -0.548 + 1.02L h + 0.519L h 2 + 4.4 × 10-4L h 3; (3) rectangular h + 0.356L h 2 + 8 × 10-4L h 3; collector (bh ) 0.5), 〈Np〉 ) 0.147 + 0.623L h + (4) rectangular collector (bh ) 0.25), 〈Np〉 ) 0.373 + 0.748L h 3; and (5) linear collector (bh ) 0), 〈Np〉 ) 0.0634L h 2 + 3.3 × 10-3L h + 0.505 for L h > 2. 3 - 2/L h for L h < 2, 〈Np〉 ) 0.748L
Figure 9. (a) One-dimensional coverage θ1D as a function of bh determined numerically for stripes: the solid line denotes the fitting function θ1D ) 0.7476 + 0.36bh2 + 0.026bh3, and the dashed line denotes the fitting function θ1D ) 0.45 + (4/π)θ∞bh. (b) Twodimensional coverage θ as a function of bh determined numerically for stripes: the solid line denotes the fitting function θ2D ) 0.547 + (0.6/bh), and the dashed line denotes the jamming limit for infinite large collector. Figure 8. Averaged number of particles 〈Np〉 adsorbed at the jamming state on the collectors of various shapes as a function of L h derived from numerical simulations (points). The solid lines denote the analytical results derived from eqs 7-10. (1) square collector, 〈Np〉 ) (4/π)0.547L h 2; (2) circular collector (dot), 〈Np〉 ) 0.547L h 2; (3) h 2; (4) rectangular rectangular collector (bh ) 0.5), 〈Np〉 ) (2/π)0.547L h 2; and (5) linear collector collector (bh ) 0.25), 〈Np〉 ) (1/π)0.547L h + 0.505. (bh ) 0), 〈Np〉 ) 0.748L
polystyrene latex suspensions1,10) and the overall ionic strength was I ) 410-5-10-4 M. It has been determined from the pictures shown in Figure 5e of the work of Zheng et al.26 (see also Figure 10a) that the averaged value of particles adsorbed over these collectors 〈Np〉 was 39.3. For comparison, our theoretical data obtained for hard (noninteracting) particles for L h )11 give the value of 〈Np〉 ) 66, which is significantly larger. This deviation can be most probably attributed the repulsive interactions between adsorbing particles, which were rather long ranged because of the low ionic strength of the suspension. Indeed, assuming I ) 4 × 10-5 M, it can be calculated that the electrical double layer thickness Le ) 48.2 nm, which equals 0.096 of the particle radius. Hence, the effective range of the interactions, which can be calculated from eq 223 on p 679 of ref 42, was 2.6Le, that is, 125 nm (0.25 of the particle radius). Therefore, the effective radius of the latex particles in the experiments of Zheng et al. can be estimated as 625 nm,
giving an effective L h value of 9. For such L h value, our theoretical prediction gives 〈Np〉 ) 44, which is quite close to the experimental vale of 39.3 determined above. In an analogous way, one can interpret other experimental data obtained by Chen et al.29 for stripes. A similar stamping technique was used for producing long stripes having an averaged width of 5 µm on gold and polyelectrolyte multilayer covered silicon. In this study, sulfate latex particles were used, having an averaged diameter d ) 0.53 µm and whose surface charge was regulated by surfactant addition (DTAB). Particle deposition was also carried out under diffusion controlled transport from latex suspensions of a concentration of 0.005 (0.5%) by weight. The usual deposition time was 1 h, which was assumed sufficient for completing particle monolayers. However, the ionic strength of the deposition bath was not controlled. It has been determined from the pictures shown in Figure 5a of the work of Chen et al.29 (shown also in Figure 10b) that the averaged 2D coverage of particles adsorbed over stripes in the case of no surfactant added was about 0.4. For comparison, our theoretical data obtained for hard (non-interacting) particles for bh ) 10 give the value of 0.64. This deviation can be attributed, analogously as above, to the repulsive interactions between adsorbing particles. Assuming I ) 10-4 M, one can calculate (eq 223 on p 679 of ref 42) that (42) Adamczyk, Z. Particles at Interfaces, Interactions, Deposition, Structure; Elsevier/Academic Press: London, 2006.
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are quite similar to those of theoretical predictions. In particular, a definite tendency of the particles to concentrate preferably at the perimeters of the collectors is well pronounced, especially in the case of stripes. However, further experimental studies carried out for higher ionic strengths where the particles behave like hard spheres and for L h < 5 are needed to quantitatively verify the above theoretical predictions.
V. Concluding Remarks
Figure 10. Comparison of the theoretical (left-hand side) and experimental (right-hand side) results obtained for dots: (a) Zheng et al.26 experiments (dots) and (b) Chen et al.29 experiments (stripes).
the effective diameter of the latex particles d* was 670 nm. Using this value, one can estimate that the effective coverage in the experiments of Zheng et al.29 was θ* ) θ(d*/d)2 ) 0.62, which is quite similar to the theoretically predicted value of 0.64. It is interesting to mention that the significant role of the lateral electrostatic interactions was qualitatively confirmed in the above work of Chen et al.29 who observed a significant increase in the maximum coverage of particles upon addition of DTAB (see Figure 5b of ref 29). However, particle deposition upon addition of surfactant was complicated by aggregation and multilayer formation, which prohibits a more detailed theoretical analysis. A qualitative comparison of the theoretically predicted and experimentally measured particle monolayers for circular collectors (dots) and for stripes are presented in Figure 10. As can be seen, these monolayers having 2D coverages of 0.33 and 0.40
Theoretical results obtained in this work revealed that irreversible adsorption of particles on surface features in the form of dots and rectangles (including the case of squares and stripes) leads to highly nonuniform coverage distribution. In all cases, the local coverage at the perimeters of these collectors exceeds by many times the averaged coverage. Secondary maxima of particle coverage also appear at the central part of the collectors and are especially well pronounced for the collector length to particle size ratio L h ) L/d between 3 and 5, and in the case of stripes for bh ) b/d of about 3. It was also predicted theoretically that for L/d > 4 the averaged number of particles adsorbed on collectors under the jamming state can be well described by the analytical dependence 〈Np〉 ) θ∞Sc/Sg, where the 2D jamming coverage equals to 0.547. Hence, for square collectors, 〈Np〉 increases as the square of the L h parameter according to the formula 〈Np〉 ) (4/π)θ∞L h 2. On the other hand, for circular collectors, 〈Np〉 ) θ∞L h 2. In the case of stripes, the calculated values of the one-dimensional coverage (averaged number of particles per unit length) were reflected well by the dependence:
θ1D ) 0.7476 + 0.36bh2 + 0.026bh3 4 θ1D ) θ∞bh + 0.45 π
for bh < 1.5 for bh > 1
It was also demonstrated that our theoretical results are in reasonable agreement with the experimental results of Zheng et al.26 and Chen et al.29 obtained for latex particles adsorbing on dots and stripes produced by polymer stamping. Acknowledgment. This work was partially supported by MEiSW Grant N205 02331 112 and by COST Action D43. LA702650N
