Two-Dimensional Aggregation of Species with Weak and Strong

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Two-Dimensional Aggregation of Species with Weak and Strong Bonding Interactions: Modeling the Growth of Self-Assembled Alkylsiloxane Monolayers Rafael Bautista, Nils Hartmann,* and Eckart Hasselbrink Fachbereich Chemie, Universita¨ t Essen, Universita¨ tsstr. 5, 45141 Essen, Germany Received March 11, 2003. In Final Form: May 30, 2003 A simple two-dimensional model for the growth of self-assembled alkylsiloxane monolayers on hydroxylated substrates is suggested, which, for the first time, takes the principal weak and strong bonding interactions of the aggregating species into account. The model considers the adsorption, diffusion, vertical grafting, and lateral cross-linking of preordered alkylsilanol clusters. All model parameters are adjusted to experimental estimates of the respective rate constants. Despite the simplicity of the model, the simulations qualitatively reproduce many morphological aspects of the experimentally observed island growth and provide a simple explanation for the deviations from first-order Langmuir adsorption kinetics observed in various ex situ investigations.

Computational simulations of the interplay of processes governing the growth of thin films are both of fundamental and of technological interest in many fields of research including the two-dimensional self-assembly of amphiphilic molecules, such as fatty acids, alkylphosphonic acids, alkanethiols, and alkylsilanes.1 Among these molecules, alkylsilanes stand out with respect to their distinct variety of bonding interactions that come into play during the formation of the self-assembled alkylsiloxane monolayer.2 Generally, weak interactions, that is, van der Waals interactions and hydrogen bonds, can be distinguished from strong interactions, that is, covalent bonds. Both types of interactions operate laterally and vertically. Even though numerous experimental investigations refer to these interactions to explain the experimental data,1 corresponding computational simulations addressing the growth of alkylsiloxane monolayers are still rare.3,4 In a previous contribution, Schwartz et al. proposed a simple diffusion-limited aggregation model incorporating the continuous adsorption of new species but largely neglecting the characteristic interactions of the aggregating species.3 The authors successfully modeled the experimentally observed transition from branched to more compact domains throughout the growth of alkylsiloxane monolayers. The simulation results, however, focused solely on the strongly bound species and showed noticeable deviations with respect to the experimental uptake curve. In this letter, we present an extended model based on a general reaction scenario, which, for the first time, takes the principal weak and strong bonding interactions into account. On the basis of kinetic Monte Carlo simulations, we discuss the impact of these interactions on the monolayer morphology and the overall growth kinetics. At room temperature and below, the growth of alkylsiloxane monolayers proceeds via the nucleation of ordered islands with vertically aligned hydrocarbon chains.5-8 An * Corresponding author. E-mail: [email protected]. (1) Ulman, A. Chem. Rev. 1996, 96, 1533. Schreiber, F. Prog. Surf. Sci. 2000, 65, 151. Schwartz, D. K. Annu. Rev. Phys. Chem. 2001, 52, 107. (2) Sagiv, J. J. Am. Chem. Soc. 1980, 102, 92. (3) Schwartz, D. K.; Steinberg, S.; Israelachvili, J.; Zasadzinski, J. A. N. Phys. Rev. Lett. 1992, 69, 3354. (4) In fact, most computational contributions addressing the growth of thin films focus on metal and semiconductor films, where generally widely different interactions apply. For a review, see: Brune, H. Surf. Sci. Rep. 1998, 31, 121.

image of such alkylsiloxane islands on oxidized silicon obtained by atomic force microscopy (AFM) is shown in Figure 1. According to a general reaction scenario of the monolayer formation, the alkylsilanes first become hydrolyzed in solution, thereby forming the corresponding silanols.9 With increasing time, more and more silanol molecules polymerize and form preordered silanol species.10,11 During immersion of the substrate, the silanol species adsorb on the surface, where they laterally diffuse and aggregate on a thin water layer.2,5 In the course of this process, the adsorbed species initially interact with each other and with the surface via silanol hydrogen bonds and van der Waals interactions before vertical grafting and lateral cross-linking via Si-O-Si bonds take place.5,12 The growth of the alkylsiloxane islands has been investigated in numerous experimental studies using various spectroscopic techniques as well as AFM.1 Several AFM studies identified two types of islands, that is, small circular islands with diameters ranging from 50 to 100 nm and larger islands with a sometimes branched shape.3,6,13,14 Schwartz et al. assigned the smaller islands to preordered polysilanol species that are built in solution prior to adsorption and the larger islands to aggregates of these species.3 A recent in situ AFM investigation by Resch et al. indeed demonstrates the predominant adsorption of large polymeric species.15 The model presented here comprises the adsorption, diffusion, grafting, and cross-linking of silanol clusters. (5) Parikh, A. N.; Allara, D. L.; Azouz, I. B.; Rondelez, F. J. Phys. Chem. 1994, 98, 7577. Brzoska, J. B.; Azouz, I. B.; Rondelez, F. Langmuir 1994, 10, 4367. (6) Britt, D. W.; Hlady, V. J. Colloid Interface Sci. 1996, 178, 775. (7) Goldmann, M.; Davidovits, J. V.; Silberzan, P. Thin Solid Films 1998, 327-329, 166. (8) Carraro, C.; Yauw, O. W.; Sung, M. M.; Maboudian, R. J. Phys. Chem. B 1998, 102, 4441. (9) McGovern, M. E.; Kallury, K. M. R.; Thompson, M. Langmuir 1994, 10, 3607. (10) Vallant, T.; Kattner, J.; Brunner, H.; Mayer, U.; Hoffmann, H. Langmuir 1999, 15, 5339. (11) Vallant, T.; Brunner, H.; Mayer, U.; Hoffmann, H.; Leitner, T.; Resch, R.; Friedbacher, G. J. Phys. Chem. B 1998, 102, 7190. (12) Sung, M. M.; Carraro, C.; Yauw, O. W.; Kim, Y.; Maboudian, R. J. Phys. Chem. B 2000, 104, 1556. (13) Kropman, B. L.; Blank, D. H. A.; Rogalla, H. Thin Solid Films 1998, 327-329, 185. (14) Balgar, Th.; Bautista, R.; Hartmann, N.; Hasselbrink, E. Surf. Sci., in press. (15) Resch, R.; Grasserbauer, M.; Friedbacher, G.; Vallant, T.; Brunner, H.; Mayer, U.; Hoffmann, H. Appl. Surf. Sci. 1999, 140, 168.

10.1021/la030100t CCC: $25.00 © 2003 American Chemical Society Published on Web 07/18/2003

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Langmuir, Vol. 19, No. 17, 2003 6591 Table 1. Rate Constants and Relative Probabilites process adsorption diffusion grafting cross-linking

Figure 1. AFM image of alkylsiloxane islands on an oxidized silicon substrate after immersion in a millimolar octadecyltrichlorosilane solution. Note that the gray scale of the image has been inversed, , dark islands appear on a bright background.

In view of an ongoing controversy on the details of the underlying mechanism of the alkylsiloxane monolayer formation1 and its dependence on various parameters, such as the temperature5,16,17 and the water content of the solution,12,11,17 we chose to keep the model as simple as possible. The clusters are treated as structureless entities of equal size.18 Weak interactions among the clusters and with the surface are taken into account via a simple reciprocal dependence of the diffusion probability on the number of nearest-neighbored clusters. As soon as grafting or cross-linking occurs, strong interactions take over, and mobile clusters turn permanently into immobile clusters. To minimize the computational effort, however, the crosslinking between mobile species and the collective diffusion of such mobile cluster assemblies have been neglected. The substrate surface is modeled by an n × n square lattice with periodic boundary conditions. Lattice sites are checked randomly, on the average once per time step. Each time a lattice site is selected, one of the following processes is chosen at random: (i) Empty sites become occupied by new mobile clusters with a relative probability pads. (ii) Mobile clusters diffuse to a randomly selected neighboring empty site with a relative probability pdiff,i ) pdiff/(i + 1), depending on the number of nearest-neighbored clusters i ) 0-4. All empty sites in the eight neighboring positions are considered. (iii) Mobile clusters graft to the surface with a relative probability pgraft and become permanently immobile. (iv) Mobile clusters permanently cross-link to an already immobile cluster with a relative probability pcross,j ) j/4pcross, depending on the number of nearest-neighbored immobile clusters j ) 0-4. To explore a reasonable range of the model parameters, we considered some rough experimental estimates of the respective rate constants, shown in Table 1. The effective rate constant for adsorption varies over several orders of magnitude, depending on the nature of the substrate and the solvent as well as other parameters, such as the water content and the temperature.1 An average value at room temperature, however, has been measured to be on the order of 1 s-1 M-1 on various substrates, that is, about 10-2 s-1 at a typical alkylsilane concentration of several (16) Rye, R. R. Langmuir 1997, 13, 2588. (17) Chen, L.-J.; Tsai, Y.-H.; Liu, C.-S.; Chiou, D.-R.; Yeh, M.-C. Chem. Phys. Lett. 2001, 346, 241. (18) Note, initially, the adsorbed clusters must not necessarily be either completely ordered or ordered in the same way as the final monolayer. We neglect, however, the influence of any vertical reorganization on the growth of the monolayer.

estimated rate constant 10-2

s-1

e103 s-1 e103 s-1 g10-3 s-1

relative probability 1 e105 e10-1 g10-1

millimoles per liter.10,13,19 Considering cluster diameters between 50 and 100 nm, a diffusion coefficient on the order of 10-7 cm2 s-1 can be estimated based on the StokesEinstein equation.20 On a respective square lattice with a lattice distance equal to the cluster diameter, this estimate corresponds to a hopping rate of 103 s-1 using the Einstein relation for a random walker.20 Note that depending on the thickness of the water layer the diffusion coefficient and, hence, the hopping rate might be substantially lower because of the interactions of the clusters with the surface. Only few experimental estimates of the time scale for grafting and cross-linking are available.8,12 Recently, Sung et al. reported both grafting and crosslinking to proceed on oxidized silicon with a decay halflife of several minutes, corresponding to an estimate of the rate constants of 10-3 s-1.12 Considering the aggregation on a thin water layer, it appears, however, reasonable to assume a much higher rate constant for cross-linking in comparison to the rate constant for grafting. On this account, the estimate of 10-3 s-1 might be regarded as a lower limit of the rate constant for cross-linking. Furthermore, whereas the oxidized silicon substrate generally bears a maximum number of surface reactive groups, that is, about 5 × 1014 hydroxyl groups/cm2,21 this density may vary depending on the preparation procedure.22 Also, other substrates show a substantially lower density of surface hydroxyl groups.23 Hence, the effective rate constant for grafting might be lower depending on the nature and the detailed preparation of the substrate. Various investigations indeed indicate a substantial cross-linking of the final alkylsiloxane monolayer,24 while grafting in some cases appears to be only minimal.25 Finally, the corresponding relative probabilities in Table 1 were derived by dividing each rate constant by the rate constant for adsorption, which was kept constant in the simulations discussed here. Taking into account the constraints for the respective probabilities given in Table 1, the model parameters have been varied to obtain a reasonable progression of the island formation. Generally, various growth scenarios with different island morphologies can be modeled by an appropriate set of model parameters. When weak interactions between the clusters and the surface are considered, the simulations presented here are based on a relative probability for diffusion of pdiff ) 104. Also, taking into account a substantially higher rate constant for crosslinking in comparison with those for grafting, the corresponding relative probabilities were initially set to pgraft ) 10-2 and pcross ) 1. Note, however, that as a result of the negligence of the cross-linking among the mobile species, a much higher probability for cross-linking of pcross ) 103 had to be adjusted. (19) Richter, A. G.; Yu, C.-J.; Datta, A.; Kmetko, J.; Dutta, P. Phys. Rev. E 2000, 61, 607. (20) Hamley, I. W. Introduction to soft matter; John Wiley & Sons: Chichester, U.K., 2000. (21) Zhuravlev, L. T. Langmuir 1987, 3, 316. (22) Le Grange, J. D.; Markham, J. L.; Kurkjian, C. R. Langmuir 1993, 9, 1749. (23) Carson, G. A.; Granick, S. J. Mater. Res. 1990, 5, 1745. (24) Maoz, R.; Sagiv, J.; Degenhardt, D.; Mo¨hwald, H.; Quint, P. Supramol. Sci. 1995, 2, 9. (25) Nakagawa, T.; Ogawa, K. Langmuir 1994, 10, 525. Tripp, C. P.; Hair, M. L. Langmuir 1995, 11, 1215.

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Figure 2. (a) Simulated progression of the island formation during the growth of an alkylsiloxane monolayer on a 200 × 200 square lattice using pads ) 1, pdiff ) 104, pgraft ) 10-2, and pcross ) 103. An effective rate constant for the adsorption of 10-2 s-1 has been considered to calculate the time scale. The corresponding time is displayed in the top left corner of each image. (b) Corresponding temporal evolution of the mobile cluster, immobile island, and total coverage are in units of monolayers (ML). For further details, see the text.

Figure 2a diplays a simulation of the island growth with similar island morphologies, as is indicated by the AFM image displayed in Figure 1. In this image sequence, mobile and immobile clusters appear on a white background as gray and black dots, respectively. The temporal evolution of the corresponding coverages, including the total coverage, is shown in Figure 2b. As is evident, the simulation results qualitatively reproduce many morphological aspects of the experimentally observed island formation, including the evolution of depletion zones,6 defect sites (pinholes),3,6 and compact islands with rough domain boundaries.6,11,14 The general growth scenario starts with the adsorption of new clusters, which simply results in an increasing mobile cluster coverage. During this time, the grafting of mobile clusters determines the onset of the island growth, that is, the formation of the first island nuclei. Subsequently, the growth of these nuclei is limited by the net diffusional flux of additional clusters to their boundaries. As a result, depletion zones are formed around the islands. With mobile clusters being continuously incorporated into the islands, the mobile cluster coverage eventually decreases and the coverage of the immobile islands strongly increases. At higher exposures, the islands start to coalesce. In the course of this process, small gaps between the islands are formed. With a decreasing gap width, the probability for any mobile cluster inside these gaps to hit the island boundaries and

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become incorporated in the islands steadily increases. Hence, the growth of the islands finally is solely limited by the adsorption of new clusters, and the curves of the total coverage and the immobile island coverage collapse. At low coverages, that is, in the diffusion-limited growth regime, the shape of the islands is mainly determined by the balance between the laterally operating weak and the strong interactions, that is, the probabilities for a mobile cluster adjacent to an island to diffuse away or to crosslink. Whereas fractal island shapes result in the case of dominating strong interactions, compact island shapes evolve in the case of dominating weak interactions. Between, compact islands with rather rough island boundaries are formed, as are shown in Figure 2a. Also, as a result of an imperfect packing process these islands show numerous defect sites. In the experiment, the diffusion coefficient will strongly vary with the thickness of the water layer, which is difficult to control and hardly ever known. It is not surprising then that different island morphologies have been observed in various experimental investigations.3,6,8,11,14 Additionally, the fractal dimension of the islands increases throughout the growth of the monolayer because of the continuous adsorption and incorporation of new clusters inside the islands, as is shown by Schwartz et al.3 At high coverages, that is, in the adsorption-limited growth regime, this results in compact polygonal islands. Besides the morphological aspects, the simulation results also qualitatively agree with the experimentally observed growth kinetics. Generally, somewhat conflicting results have been reported in various in situ and ex situ investigations. Despite the complexity of the reaction scenario, recent in situ studies using X-ray reflectivity,19,26 infrared spectroscopy,11 and AFM27 concordantly suggest an overall growth following first-order Langmuir adsorption kinetics. In contrast, some ex situ studies using AFM indicate either the existence of an induction period7 or the presence of an initial period of fast linear growth up to high coverages followed by a slower growth up to the saturation of the surface.3,14 In a previous contribution, Schwartz et al. suggested that the uptake of additional species at the solution/air interface might alter the growth curves observed in ex situ studies.3 However, experimental results by Vallant et al. using ellipsometry indicate that such a transfer of additional species onto the substrate surface can be neglected.10 Recently, the reason for the discrepancies between in situ and ex situ investigations as well as among ex situ investigations has been attributed to the partial removal of physisorbed species during the cleaning and rinsing steps prior to ex situ examination.14,19 It appears reasonable that the amount of the removed mobile clusters depends on the effective bonding strength of the weak interactions as well as on the detailed cleaning procedure. In this regard, Figure 3 displays the resulting evolution of the overall coverage, including the immobile island coverage and a varying percentage of the mobile cluster coverage shown in Figure 2b. Clearly, the curve comprising the total coverage simply results from the firstorder Langmuir adsorption kinetics, which has been implied in the model to describe the growth curves observed in the in situ studies. Additionally, however, the (26) Note, for very low alkylsilane concentrations, that is, below 0.025 mM, an induction period has been observed in ref 15. Under these conditions, however, desorption might compete with adsorption throughout the monolayer growth. Simulations on the basis of our model including a desorption probability for mobile clusters indeed reproduce this characteristic feature at low alkylsilane concentrations. (27) Leitner, T.; Friedbacher, G.; Vallant, T.; Brunner, H.; Mayer, U.; Hoffmann, H. Mikrochim. Acta 2000, 133, 331.

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Figure 3. Total coverage (fat line) and resulting overall coverages comprised of the immobile island coverage and an increasing percentage of the mobile cluster coverage shown in Figure 2b (thin lines). All together, the respective percentage of the mobile cluster coverage goes from 0% (bottom curve) to 100% (top curve) in steps of 20%. All coverages are in units of monolayers (ML). The inset displays the positional shift of the maximum mobile cluster coverage upon changing the relative probabilities for grafting, cross-linking, and diffusion. Starting with the values given in Figure 2, these relative probabilities have been changed by a factor of 0.1 to 10. For further details, see the text.

curves comprised of a varying percentage of the mobile cluster coverage exhibit the characteristics reported in the ex situ studies, that is, an induction period at a low and a nearly linear growth at a high percentage of the mobile cluster coverage. As is evident, the width of the affected regime correlates with the position of the maximum mobile cluster coverage, which again depends on the operating weak and strong interactions. On one hand, increasing the probability for grafting or cross-linking results in an overall higher probability for mobile clusters to become captured. On the other hand, increasing the probability for diffusion effectively leads to a higher net diffusional flux to the island boundaries. Hence, increasing the weight of the respective probabilities governing the strong and the weak interactions shifts the maximum to lower exposures. In conclusion, the growth of self-assembled alkylsiloxane monolayers on hydroxylated substrates can be qualitatively modeled considering the adsorption, diffusion,

grafting, and cross-linking of silanol clusters. The simulation results demonstrate the crucial dependence of the morphology and the overall growth kinetics on the operating weak and strong interactions, which in many respects contribute to the poor reproducibility of the experimental results. While detailed quantitative simulations remain a computational challenge for the future, simple simulations, such as those presented here, may be useful to help interpret the experimental data on a qualitative level. Acknowledgment. Financial support by the Deutsche Forschungsgemeinschaft (HA 1424/5-1) and the Universita¨t Essen (Forschungspool 2001/2002) is gratefully acknowledged. The authors are indebted to Th. Balgar and S. Franzka for providing the AFM image shown in Figure 1. LA030100T