Propagating waves of self-assembly in ... - Semantic Scholar

Propagating waves of self-assembly in organosilane monolayers Jack F. Douglas†‡, Kirill Efimenko§, Daniel A. Fischer¶, Fredrick R. Phelan†, and Jan Genzer‡§ †Polymers

and ¶Ceramics Divisions, National Institute of Standards and Technology, Gaithersburg, MD 20899; and §Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, NC 27695-7905 Communicated by Johanna M. H. Levelt Sengers, National Institute of Standards and Technology, Gaithersburg, MD, May 4, 2007 (received for review January 20, 2007)

Wavefronts associated with reaction– diffusion and self-assembly processes are ubiquitous in the natural world. For example, propagating fronts arise in crystallization and diverse other thermodynamic ordering processes, in polymerization fronts involved in cell movement and division, as well as in the competitive social interactions and population dynamics of animals at much larger scales. Although it is often claimed that self-sustaining or autocatalytic front propagation is well described by mean-field ‘‘reaction– diffusion’’ or ‘‘phase field’’ ordering models, it has recently become appreciated from simulations and theoretical arguments that fluctuation effects in lower spatial dimensions can lead to appreciable deviations from the classical mean-field theory (MFT) of this type of front propagation. The present work explores these fluctuation effects in a real physical system. In particular, we consider a high-resolution near-edge x-ray absorption fine structure spectroscopy (NEXAFS) study of the spontaneous frontal self-assembly of organosilane (OS) molecules into self-assembled monolayer (SAM) surface-energy gradients on oxidized silicon wafers. We find that these layers organize from the wafer edge as propagating wavefronts having well defined velocities. In accordance with two-dimensional simulations of this type of front propagation that take fluctuation effects into account, we find that the interfacial widths w(t) of these SAM self-assembly fronts exhibit a power-law broadening in time, w(t) ⬇ t␤, rather than the constant width predicted by MFT. Moreover, the observed exponent values accord rather well with previous simulation and theoretical estimates. These observations have significant implications for diverse types of ordering fronts that occur under confinement conditions in biological or materials-processing contexts. fluctuation-induced interfacial broadening 兩 frontal self-assembly 兩 mean-field Fisher–Kolmogorov equation 兩 reaction– diffusion fronts 兩 self-assembled monolayers

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n the early 1990s, Chaudhury and Whitesides developed an extremely facile method of creating surface energy gradients using self-assembled monolayers (SAMs) (1). Rather than using a solution as a carrier for adsorption or a rubber stamp for direct deposition of SAM molecules, these researchers simply placed a volatile fluid of organosilane (OS) molecules at the edge of a silica-covered substrate and enclosed the entire system in a container to avoid convection effects (Fig. 1). The symmetry-breaking perturbation associated with placing the source of diffusing material to the side of the wafer imparts a direction to the self-assembly of the SAM layer and leads to the spontaneous formation of a concentration gradient of OS molecules on the substrate. This gradient may be fixed into position at any point in its development by simply removing the wafer from OS exposure (2). These kinetically ‘‘frozen’’ SAM patterns are then ready for further measurements in which surface energy gradients are required. Although this method of creating surface energy gradients is relatively simple to implement, the physical processes governing the formation of such gradients are actually rather complex (2), involving adsorption–desorption equilibrium, molecular surface diffu10324 –10329 兩 PNAS 兩 June 19, 2007 兩 vol. 104 兩 no. 25

sion, self-assembly of OS molecules into clusters on the substrate that are probably different from those formed in the gas phase due to stabilizing effect of the substrate interactions, etc (2). It should thus not be surprising that the formation of the OS concentration gradient on the substrate generally involves more than the diffusive spreading of these molecules across the silica-coated wafer. Our investigation indicates that the formation of these SAM surface energy gradients occurs through a planar wavefront of self-assembly that propagates across the wafer at a constant velocity, where the gradient itself describes the interfacial region of the front separating the ordered from the disordered regions of the SAM layer. This process apparently arises from a nucleation of SAM self-assembly at the edge of the substrate due the relatively high concentration there, followed by a propagation of this selfassembly process into the thermodynamically unstable region ahead of the front. This wavelike phenomenon is evidently autocatalytic because the formation of the ordered regions catalyzes further ordering. The large geometrical dimensions of the vapor source ensure that unordered OS molecules and clusters on the substrate are not depleted during self-assembly and as the selfassembly front progressively moves across the wafer (2). In our measurements, we establish the surface composition gradients and the extent of SAM molecular ordering, including the detailed motion and characteristics of these fronts, using combinatorial near-edge x-ray absorption fine structure (NEXAFS) spectroscopy (2) (see Materials and Methods). Classical MFT Description of Self-Sustaining Front Propagation Under circumstances where the driving force for front movement is not too large (so that the front remains approximately planar and interfacial instabilities are avoided), the SAM front position can be described by the equations of motion for a one-component ‘‘order parameter’’ ⌽, quantifying the extent of the local SAM ordering in the plane of the silica-covered wafer [see supporting information (SI) Text and SI Figs. 4–6]. A reasonable model of the free energy governing the self-assembly process involves ascribing a higher free energy initial state to the unassembled molecules on the wafer and a lower free energy stable state to the assembled SAM; there are exact results for the deterministic MFT equations of motion governing this general type of ordering process (3, 4). In particular, the governing equations can be reduced to Fisher’s population model equation (FPME) (5, 6) (Eq. 2a indicates a special case of this equation). The application of the FPME equation to assembly Author contributions: J.F.D. and J.G. designed research; K.E., D.A.F., and J.G. performed measurements; F.R.P. performed numerical computations; K.E. and J.G. analyzed data; and J.F.D. and J.G. wrote the paper. The authors declare no conflict of interest. Abbreviations: SAM, self-assembled monolayer; OS, organosilane; NEXAFS, near-edge x-ray absorption fine structure; FPME, Fisher’s population model equation; FKE, Fisher– Kolmogorov equation; MFT, mean-field theory; UVO, UV/ozone. ‡To

whom correspondence may be addressed. E-mail: [email protected] or jan㛭[email protected].

This article contains supporting information online at www.pnas.org/cgi/content/full/ 0703620104/DC1. © 2007 by The National Academy of Sciences of the USA

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Fig. 1. Schematic illustration of surface energy gradient formation in SAM layers. Semifluorinated chlorosilane molecules are mixed with paraffin oil to control the rate of OS evaporation (2) and this mixture is placed in a small container positioned at the edge of a silicon wafer subjected to a UV-ozone treatment (Materials and Methods). The whole system is then enclosed in a Petri dish. The evaporated OS molecules deposit on the wafer and self-assembly initiates from the wafer where the OS concentration is higher. The average angular orientation of the molecules directly tracks the OS concentration (2) and thus the normalized local concentration is identified with an order parameter for SAM ordering.

⌽共x兲 ⫽ ⌽*关1 ⫺ 共1/2兲tanh共x f /w兲兴, where x f ⫽ x ⫺ ct,

[1]

which provides insight into the shape of these fronts. In Eq. 1, c is the front velocity, w is the front width, and xf defines the front position. ⌽(x) is obtained subject to the boundary conditions ⌽共x 3 ⫺⬁兲 ⫽ ⌽*, ⌽共x 3 ⬁兲 ⫽ 0 ⭸⌽共x 3 ⬁兲/⭸x ⫽ ⭸⌽共x 3 ⫺⬁兲/⭸x ⫽ 0. The sigmoidal hyperbolic tangent (tanh) function in Eq. 1 is a familiar mathematical form describing the order parameter variation at interfaces in a variety of contexts (10, 11). As a reference point for our discussion below of this type of ordering front, we consider the numerical solution of the classical Fisher–Kolmogorov equation (FKE) (a specialization of the FPME equation that exhibits many of its essential mathematical characteristics) (5) ⭸⌽共x兲/⭸t ⫽ Dⵜ2⌽共x兲 ⫹ k1⌽共x兲 ⫺ k2⌽2共x兲,

[2a]

where D is the molecular diffusion coefficient, k1 is the rate constant for the autocatalytic growth, and k2 describes the nonlinear interaction between the newly created species. The presence of the term k2 limits the extent of the local autocatalytic growth, ⌽(x). (In a population dynamics context (12), Eq. 2a is the well known ‘‘logistic equation’’ in the absence of the term Dⵜ2 ⌽(x) that accounts for the formation of spatial inhomogeneities by diffusion where k1 is the well known ‘‘Malthusian’’ growth term giving rise to exponential population growth on its own and k2 is the ‘‘Verhulst’’ term, describing the population Douglas et al.

interaction that limits growth at long times to a steady-state value, ⌽* ⫽ k1/k2). The more general FPME equation simply includes further nonlinear terms in the Eq. 2a, along with rate constants (k3, k4, . . . ) that specify the strength of these nonlinear interactions. For our illustrative calculations, we normalize ⌽(x) so that it ranges between 0 and 1. Specifically, we divide ⌽(x) by its steadystate value ⌽* ⬅ k1/k2 to obtain the reduced FKE (5) ⭸⌽共x兲/⭸t ⫽ Dⵜ2⌽共x兲 ⫹ k1⌽共x兲关1 ⫺ ⌽共x兲兴,

[2b]

which emphasizes the two main parameters governing the propagation of FKE fronts, the diffusion coefficient D and the linear growth rate k1. It has been established (5) for a wide range of initial conditions that the solution to Eq. 2b exhibits a unique steady-state propagating front solution with a velocity c that depends nonanalytically on D and k1, i.e., c ⫽ 2 (D/k1)1/2. Moreover, the ‘‘diffusion length’’ (D/c) determines the asymptotic width of the propagating interface, up to a constant of proportionality. In the ‘‘self-assembly front’’ analog of the FPME, the dimensionless variable ⌽(x) is the ‘‘order parameter’’ for the self-assembly process, D is replaced by a ‘‘mobility parameter’’ for the rate of ordering, and the polynomial in ⌽(x) and the corresponding rate constants in Eq. 2a, are determined by the free energy density describing the order–disorder transition. The self-assembly front analog of the FPME equation is described by Harrowell and Oxtoby (3) in the context of frontal crystallization. A numerical solution of the FKE is shown in Fig. 2, where we consider the representative parameter values, D ⫽ 0.01 and k1 ⫽ 1 (see Eq. 2b). The initial conditions relevant to our measurements correspond to the case where ⌽ ⬇ 1 near some localized region of space where growth initiates and we model this initially sharp interface by a step-function at the boundary edge (x ⫽ 0) where the OS molecules are presumed to be in an ordered state ⌽共t ⫽ 0; x兲 ⫽ 1 for x ⬍ 0 and

⌽共t ⫽ 0; x兲 ⫽ for x ⬎ 0.

Our illustrative solution for ⌽(x), which of course is not novel, propagates to the right (x ⬎ 0) as a wavefront with a constant velocity. The ⌽(x) curves in Fig. 2 neglect an initial solution transient at short times where the front spreads in a simple diffusive manner (it takes a certain induction time for the nonlinearities in Eq. 2a to ‘‘build up’’ to establish front propagation at constant velocity). If we transform to a coordinate system moving with the front (i.e., shifting the curves according to the relation xf ⫽ x ⫺ ct), we obtain a universal scaling curve for ⌽(x) (Fig. 2b). Interestingly, the tanh function (solid lines in Fig. 1a) fits the numerical data remarkably well, even though it is only a particular solution of the FPME (3). Below, we compare PNAS 兩 June 19, 2007 兩 vol. 104 兩 no. 25 兩 10325

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assumes that the amount of latent heat released upon ordering is small or vanishing and otherwise heat transport must also be included in the modeling. Recent simulations (7–9) of particle models exhibiting the same formal kinetics as the mean-field FPME model clarify the role of fluctuation effects on front propagation in low-dimensional systems (SAM layer formation is essentially a two-dimensional process, although diffusion from the vapor phase certainly influences the boundary conditions of this ordering process). Some of the consequences of these fluctuation effects that are relevant to our study are described below. We know from the extensive mathematical and physical literature relating to this class of ‘‘autocatalytic’’ fronts that the meanfield solution ⌽(x) to the FPME has a unique velocity c, related in the case of ordering fronts as in our measurements, to the thermodynamic driving force (undercooling or supersaturation) and to the interfacial energy of the front (3, 4). General exact analytic solutions for ⌽(x) do not exist, but there is a particular solution in one dimension (3, 4)

a

c

b

Fig. 2. Simulations of model autocatalytic fronts. (a) Solution ⌽(x) of Eq. 2a for a planar FKE front obtained by using the finite element package COMSOL (51) for D ⫽ 0.01 and k1 ⫽ 1. The planar solution profile was obtained for a long rectangular plate (modeling the silica coated substrate in Fig. 1), where the position (x) denotes the coordinate normal to the wafer edge at which the front initiates. Units in this figure are arbitrary. In b, the planar front interface position shown in a is shifted to a common origin and ⌽(x) is compared with a tanh function (solid thin curve). (c) Circular FKE front initiating from a small patch where ⌽ ⫽ 1. Numerical data in d show simulation front propagation data for a discrete particle model in two spatial dimensions exhibiting reaction kinetics consistent with the mean-field FKE. Data in d correspond to a range of times as in b, but the front shape is fitted to a tanh function with a timedependent width, w(t) ⬇ t␤.

d

this MFT estimate of ⌽(x) to data from discrete particle simulations consistent with the reaction kinetics of the FKE and to our self-assembly front measurements. This comparison is noted here because the quantification of self-assembly front properties is facilitated by having a good analytic approximation for ⌽(x). We can also introduce a circular ‘‘seed’’ (droplet of OS) onto a silica-coated wafer and follow the circular ordering front initiated by this perturbation (see SI Text for a description of these measurements). To model this situation approximately (Fig. 2c), we solve the FKE in the plane with the initial conditions: ⌽共t ⫽ 0; r兲 ⫽ 1 for r ⬍ 0 and ⌽共t ⫽ 0; r兲 ⫽ 0 for r ⬎ 0,

[3]

where r is the radial distance from the center of the initial circular spot of unit radius (one lattice site so the spot can be considered a delta function). Fig. 2c shows a top view of the solution, where the color-map describes the extent of ordering as a function of position x away from the water edge. After radial averaging of this front and shifting the interface position, the order parameter profile ⌽(t; r) reduces to an approximately universal function. Again, we find that the simple tanh function provides a good leading order description of the interfacial shape of these propagating fronts at the level of MFT. Predicted Breakdown of the MFT of Fisher–Kolmogorov Fronts The near two-dimensional nature of these autocatalytic reactiondiffusion and ordering fronts makes them particularly interesting from a theoretical standpoint. The MFT equations of motion in Eq. 2 are based on the assumption that the reactive species remains uniformly mixed (ordered) during front propagation. However, it has recently been shown that fluctuation effects (7, 8, 13–15), associated with the finite-size (discrete particle nature) of the organizing species, external environmental fluctu10326 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0703620104

ations (e.g., fluctuations in the local substrate energy and height, local temperature fluctuations, etc.) and composition fluctuations associated with the diffusion of the assembling particles, can lead to a breakdown in this idealized MFT description of the front propagation in low-dimensional systems where these fluctuation effects become more prevalent. Although rigorous theory or even formal renormalization group calculations do not exist for this problem, computer simulations have recently shown that fluctuation effects can lead to two distinct patterns of deviation from the mean-field FKE predictions in the twodimensional case relevant to our measurements. In particular, Riordin et al. (7) considered discrete particle analog simulations of the FKE in two dimensions [the autocatalytic reaction A ⫹ M 7 2A with A and M the reacted (‘‘assembled’’) and unreacted (‘‘unassembled’’) species, respectively] where there was an excess concentration of M (constant externally imposed concentration; ref. 9) and found propagating FKE wavefronts having ⌽(x) profiles qualitatively similar to the illustrative calculations in Fig. 2b. However, the interfacial width w(t) of these profiles exhibited a progressive spreading (‘‘roughening’’) in time t where w(t) ⬇ t␤, ␤ ⫽ 0.272 ⫾ 0.007. Given the superficial similarity of the data in Riordin et al. (7) to the data in Fig. 2b, we then compare their data to the expression for ⌽(x) given in Eq. 1a, where w(t) is taken to be time-dependent rather than the constant value predicted by MFT. [This comparison is not considered by Riordin et al. (7)]. An excellent fit of these data to the MFT interfacial profile function ⌽(x) is found where the exponent ␤ describing the fluctuation-induced interfacial broadening is the same as reported (7).We use this expression in our quantification of our SAM front experiments (see below). Warren et al. (8) also considered discrete particle simulation consistent with the FKE in MFT, but the concentration of the M species was not held fixed, so this reactive species could be depleted upon reaction. Douglas et al.

Douglas et al.

nonuniversal, we expect from the arguments of Tripathy et al. (14, 17) that only two patterns of interfacial broadening should be observed in 2D self-assembly fronts (the observed pattern depending on the analytic nature of the free-energy density function governing the self-assembly process). It seems likely that the two different values of ␤ observed by Riordin et al. (7) and Warren et al. (8) have a similar explanation in terms of pushed versus pulled fronts, although this possibility is hard to prove (see below). Here, we search for the experimental evidence for this type of interfacial broadening in a 2D geometry appropriate to comparing to the simulations and enabling high-resolution measurement of the selfassembly front propagation so that the different types of front propagation can be resolved if they occur. Fluctuation-Induced Interfacial Broadening in Organosilane Self-Assembly Wavefronts From the discussion above, we expect the interfacial spreading exponent ␤ to take one of two characteristic values that are difficult to predict a priori. Given this uncertainty, we consider two OS molecules that we designate mF8H2 and tF8H2, respectively (see Materials and Methods). As schematically indicated in Fig. 1 and described in ref. 2, the OS molecules that we employ in our SAM gradient measurements have rather different (monochloro and trichloro) end-group functionalities. These molecules have a different propensity to aggregate in the vapor phase and on the UV/ozone (UVO)-treated substrate when hydrolyzed through interaction with atmospheric water (2). This difference influences the free energy function describing the self-assembly process, and this variation could change the value of ␤. Consideration of this pair of OS molecules should then provide some insight into the universality of ␤ and a useful comparison to simulation studies of 2D reaction–diffusion fronts. We now take a quantitative look at how these selfassembly fronts propagate. NEXAFS (Materials and Methods) was used to determine the surface fractions of the SAM gradient patterns that are schematically shown grown in Fig. 1. Fig. 3 a and c show the surface fractions of the mF8H2 and tF8H2 molecules, respectively (standard relative uncertainty of the data points in Fig. 3 is less than the symbol size). The surface fraction of these molecules has been shown to track the extent of SAM ordering (2), so we identify this reduced concentration variable with the order parameter ⌽(x) of the self-assembly process. It is evident from Fig. 3a that a planar wave of self-assembly propagates across the wafer. The lines in Fig. 3 a and c represent best fits to the MFT expression for ⌽(x) given in Eq. 1a, but where fluctuation effects are accounted for by taking the interfacial width w(t) to be time dependent. Specifically, we can collapse the NEXAFS-derived OS concentration profiles onto master curves by rescaling the position x coordinate as: (x ⫺ ct)/t␤ where c ⫽ 0.010 ⫾ 0.002, ␤ ⫽ 0.37 ⫾ 0.01 and c ⫽ 0.018 ⫾ 0.002, ␤ ⫽ 0.28 ⫾ 0.01 for the mF8H2 and tF8H2 gradients, respectively [uncertainty intervals indicate (␹2) best-fit range determined by varying c and ␤ simultaneously]. As in the discrete particle reaction–diffusion simulations discussed above, the data in Fig. 3 are consistent with two values of ␤. Even the particular values ␤ found (i.e., 0.27 ⬇ 1/4 and 0.34 ⬇ 1/3) accord well with those suggested by the simulations and theoretical arguments discussed above. We can offer no specific explanation of the observed ␤ values for the mF8H2 and tF8H2 fronts shown in Fig. 3, and, despite the accord of our apparent exponent values with previous theoretical and simulation estimates, we must admit that a number of physical effects might contribute to the differing ␤ values in these SAM fronts (see discussion below and SI Text). Evidently, further experiments on other types of SAM molecules and for different experimental conditions will be required to clearly establish the extent of exponent universality. Nonetheless, it is evident that fluctuation effects lead to an interfacial broadening of our self-assembly fronts so that the MFT description of the front PNAS 兩 June 19, 2007 兩 vol. 104 兩 no. 25 兩 10327

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These simulations also led to propagating reaction–diffusion fronts, but the exponent ␤ was found to be somewhat larger, ␤ ⫽ 0.344 ⫾ 0.004, under these reaction conditions. Thus, fluctuation effects preserve the propagating nature of the reaction-diffusion fronts, but can lead to an evolution of the interfacial width w(t) in time. The nature of this interfacial broadening is evidently dependent on the details of the reaction–diffusion process, a phenomenon that cannot be understood within MFT and that is not generally understood theoretically. Although exact methods of calculating ␤ do not exist for autocatalytic reaction-diffusion fronts of the FPME type, there have been theoretical arguments for ␤ in two dimensions (2D). These arguments provide insights into the simulations mentioned above and offer some guidance about what to expect in our own measurements. Before discussing these predictions, we need to explain an essential concept relating to front propagation: ‘‘pulled’’ versus ‘‘pushed’’ fronts. In pulled fronts, the spreading front at long times is dominated by the linear growth rate k1 and D so that the unstable region ahead of the front draws it forward (hence the term ‘‘pulled’’) (14, 15). The solution shown in Fig. 2a provides a classic example of a pulled autocatalytic front. We can also have the situation where higher order nonlinear terms in the FPME strongly exert their influence so that the front movement also depends explicitly on the nonlinear rate parameters (k2, k3, k4, . . . ) so that the intrinsic nonlinearity of the ordering process helps drive the front forward. Such fronts are said to be “pushed” (14, 15). The crux of recent theoretical discussions of ␤ revolves around whether the front is pushed or pulled, a feature of no interest in MFT where the ‘‘anomalous dimension exponent’’ ␤ generally equals 0. We next briefly summarize arguments for ␤ that are relevant to our measurements. First, Moro (13) has questioned whether the apparent exponent ␤ found in the simulations of Riordin et al. (7) and others (14) represent a truly asymptotic ‘‘infinite time’’ scaling limit and he suggested that ␤ should instead become equal to ␤ ⬇ 1/3 at still longer times, the exponent predicted from the Kardar–Parisi– Zhang (KPZ) theory (16) of fluctuating fronts in 2D propagating into an unstable state. If nothing else, this study reminds us how difficult it is to prove ‘‘asymptotic’’ critical exponents through numeral study alone. Tripathy and Saarloos (14), however, reconfirmed the ␤ estimate of Riordin et al. (7), where they found ␤ ⫽ 0.29 ⫾ 0.012, and Tripathy et al. (14, 17) subsequently argued that ␤ should not be described by the standard KPZ model, but rather the intrinsically unstable nature of these (pulled) fronts should lead to a ‘‘dimensional shift’’ of the exponents of the KPZ model. Specifically, they formally argued that the exponents for the FKE reaction–diffusion process should correspond to the KPZ theory in a higher interface dimension, i.e., two surface dimensions ⫹ one time dimension or ‘‘2 ⫹ 1 dimensions’’ in the standard jargon. Tripathy et al. (14, 17) also indicated that ordinary KPZ theory provides an appropriate description for 1 ⫹ 1 dimensional pushed fronts where ␤ ⬇ 1/3. Beccaria and Curci (18) estimated ␤ ⬇ 0.25 for the KPZ model in 2 ⫹ 1 dimensions, which is consistent with the arguments of Tripathy et al. (14, 17). Moreover, Ojeda et al. (19) and Thompson et al. (20) have observed exponent values close to ␤ ⬇ 0.26 in the interfacial growth of silica and silver films grown by chemical vapor deposition where the 2 ⫹ 1 dimensional KPZ theory is thought to apply. Tripathy et al. (17) also performed instructive simulations of discrete particle simulations consistent with the FPME in MFT and showed that the predicted ␤ values for the pushed and pulled front cases could be recovered to a good approximation by adjusting kinetic parameters in their simulations. It has also been shown by Chen et al. (6) that the general FPME equation supports both pushed and pulled fronts, which in combination with the arguments of Tripathy et al. (14, 17), implies that ␤ for reaction–diffusion fronts in 2D should obtain one of two distinct values near 1/4 and 1/3, corresponding to pulled and pushed fronts, respectively. Thus, although the exponent ␤ is predicted to be

a

c

b

d

propagation fails. Tentatively, we conclude that these fronts exhibit two spreading patterns, characterized by distinct ␤ values. The consistency of our ␤ exponent estimates with previously observed values in the reaction–diffusion simulations and theoretical estimates is encouraging. We emphasize that the uncertainty in ␤ is not limited to our measurements. Even in reaction–diffusion simulations consistent with FKE in MFT, where the kinetic parameters are exactly known, there is no analytic method that allows for the prediction of ␤ under general conditions (14, 17). However, Chen et al. (6) have developed a method capable of determining whether continuum reaction–diffusion fronts of the FKME type are pushed or pulled based on a combination of renormalization group and variational methods. This methodology should provide a foundation for future investigations aimed at clarifying this important aspect of selfassembly front propagation. Implications of Observations Diverse molecular processes, and even social processes reflecting the population dynamics of humans and other organisms, can be described in terms of fundamental ‘‘entities’’ undergoing random local displacement events, but exhibit an emergent regular motion at large scales and associated pattern formation as these entities collectively move from some unstable situation to a relatively stable state of greater energetic or competitive ‘‘advantage.’’ The FKE and its various generalizations provide a highly successful minimal model for the resulting wavefronts of ordering, disordering, social change, life, disease, or death. These pattern formation processes are ubiquitous in the natural world and largely govern successive waves of evolutionary development (21, 22). Because the generality of this type of organizational process provides part of our motivation for our careful examination of the formation of OS layers by frontal self-assembly, we briefly summarize some of the phenomena that can be modeled by the FKE. The FKE has been used to model the spread of advantageous genes in animal populations (6), the growth of bacterial colonies (23–25), and the growth of complex cell organizations such as spreading of tumors (26), brain tumor growth (27), and wound healing (28), epidemics such as the spread of plague (29) and other pandemic 10328 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0703620104

Fig. 3. Time development of mF8H2 and tF8H2 surface energy gradients. Fractional coverage refers to the OS concentration relative that for the fully ordered layer. (a and c) mF8H2 and tF8H2 fractional coverage, respectively, as a function of substrate position z for a succession of times. (c and d) The same data for mF8H2 and tF8H2 where the data has been shifted in the x-coordinate to account for the constant velocity of the front movement and where the front width is time dependent, w(t) ⬇ t␤. Lines represent a best fit to a tanh function (see text).

diseases, social developments such as the spread of Neolithic farming practices (30), and the spread of Indo-European languages across Europe (31). Other important examples from biology include the frontal polymerization of actin (32) and microtubules (33). The FPME also arises in a variety of physical contexts, including the spreading wavefronts of superconductivity (34), the phenomenology of deep inelastic scattering in high-energy physics (35), chemical reaction fronts (36), nerve propagation (37), and flame propagation (38). There are also examples that directly pertain to self-assembly such as the wavelike replication RNA molecules initiated by a RNA ‘‘seed’’ molecule (39), virus proliferation in bacterial populations (‘‘plaques’’) (40). Other examples are discussed by Ferreiro et al. (41) as part of a general discussion of the similarities of nonequilibrium crystallization pattern formation in thin polymer films to autocatalytic chemical reaction fronts. Because the spreading of autocatalytic waves is a pervasive phenomenon, we expect that many further examples will be found in the future as other everyday ‘‘invasive’’ phenomena are subjected to quantitative investigation. For instance, increased foreign travel and trade has led to the introduction of foreign plant species into new environments (42), triggering waves of invasive growth in the countryside that can be tracked by satellite imaging. The growth of urban sprawl is another near 2D growth phenomenon for which detailed land use records and satellite imaging provide information about the front structure and dynamics (42). From a material science standpoint, it would be interesting to pursue studies of frontal polymerization (43), the corrosion of metallic films (44), and polymer dissolution (45), fronts in thin films to determine whether our results apply broadly to these phenomena as well. The general finding that fluctuation effects tend to make these fronts increasingly incoherent in time (diffuse) can be expected to have a large impact on the interaction of these frontal patterns under conditions where species (or different types of ordering) are competing for supremacy. Clearly, the interaction of autocatalytic wavefronts should be considered in future experimental and computer simulation work. We also mention the potential relevance of our work to modeling the origin of life. It has been seriously suggested (46) that life originated from surface-bound autocatalytic chemical reactions Douglas et al.

Conclusions The formation of organosilane layers provides an attractive model system for investigating frontal self-assembly because this process can be studied with a high degree of experimental precision and the phenomenon is useful in its own right. Here, we are particularly concerned with how confinement of the assembly process to two-dimensions alters the nature of selfassembly front propagation. Confinement has an important effect on this type of self-organizational process because these systems are more susceptible to the formation of concentration fluctuations arising from the recurrent nature of random walk motion and the enhanced hard-core excluded volume interactions between particles in 2D. In particular, computer simulations demonstrate that, although wavefronts still exist in these discrete particle systems exhibiting a reaction kinetics consistent with the mean-field FKE in the continuum limit, the simulated fronts of these models (7, 8, 14, 17) become progressively broadened in time. Our experiments on the self-organization of organosilane SAMs confirm this predicted pattern of behavior in a real physical system. The confinement effects that we observe should be even more pronounced if the self-organization process is confined to fractal regions of invasion (48) (e.g., cracks in the substrate or within bulk materials) that further reduce the ‘‘effective dimensionality.’’ These enhanced fluctuations arising from confinement are important because they can invalidate MFT kinetic modeling on which the description of reaction kinetics, evolutionary dynamics and phase ordering processes are traditionally based. Our experiments suggest that confinement preserves some of the predicted MFT features of the ordering wavefronts (i.e., front propagation with a constant velocity and the fronts have a typically sigmoidal shape), whereas new effects such as progressive interfacial broadening arise because of fluctuations. Although interfacial spreading with universal exponents describing this ‘‘roughening’’ phenomenon is perhaps the most important of these new features, the dependence of the front velocity c on system variables is also expected to change from the predictions of MFT because of 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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fluctuation effects (7–9). This interfacial broadening process has many potential implications for understanding the nature and interaction of autocatalytic fronts in the azoic and biological worlds. Materials and Methods Surface energy gradients of semifluorinated OS comprising 1H,1H,2H,2H Per f luoro-decyldi-methylchlorosilane [F(CF 2 ) 8 (CH 2 ) 2 Si(CH 3 ) 2 Cl, mF8H2] and 1H, 1H,2H,2HPerfluoro-decyl-trichlorosilane [F(CF2)8(CH2)2 SiCl3, tF8H2], both supplied by Lancaster (Windham, NH) and used as received, were formed by a variant of the method described by Chaudhury and Whitesides (4). Single-side polished, 300-␮mthick silicon wafers with a [100] orientation, supplied by Virginia Semiconductor (Fredericksburg, VA), were cut into ⬇ 1 cm ⫻ 5 cm and ⬇ 5 cm ⫻ 5 cm pieces, placed into a UVO cleaner [Jelight (Irvine, CA), Model 42, Suprasil lamp], and exposed to the UVO treatment for 30 min. This treatment produced a high concentration of -OH groups on the surface of the wafer (49). Evaporation of the OS solution was performed in an ambient air atmosphere at room temperature with a 30% relative humidity, a condition that favors end-group conversion to Si-OH groups that bind to the surface-bound hydroxyls (2). The OS ‘‘source’’ was placed at the edge of the oxidized wafer substrate, as illustrated in Fig. 1. Paraffin oil was mixed the OS with (1:2 relative mass) to control the rate of evaporation (2). Exposing the substrate to this source for various times (1 to 20 min) produced OS concentration gradients on the wafer. In previous work (2), we established that the orientation of the F8H2 molecules on the substrate is directly correlated to the average molecular orientation of the OS molecules relative to the surface (see figure 7 of ref. 2). After the deposition of OS gradients, the substrates were thoroughly cleaned with ethanol and then dried with N2 gas. We used combinatorial NEXAFS (2, 50) to establish the position-dependent concentration of OS on the substrate. NEXAFS involves the resonant soft x-ray excitation of a K or L shell electron to an unoccupied low-lying antibonding molecular orbital of ␴ symmetry, ␴*, or ␲ symmetry, ␲* (50). The initial K shell excitation gives NEXAFS its element specificity, whereas the final-state unoccupied molecular orbitals provide chemical selectivity. A measurement of the partial electron yield (PEY) intensity allows for identification of chemical bonds and the determination of their relative population densities within the sample. Further details related to the formation of SAM layers are described in ref. 2. J.G. and K.E. thank the National Science Foundation for supporting the research at North Carolina State University. We thank Anneke Levelt Sengers for her careful and constructive comments on our paper. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

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PHYSICS

occurring under similar conditions as the present work, where the substrate binding of the molecules is strong enough to confine them to the substrate while they are ‘‘organizing,’’ but not so strong as to inhibit their migration within the plane of the surface. It is interesting to consider the role of sexual and asexual reproduction on this type of self-organization because these processes influence the autocatalytic reaction order (47), thereby impacting the front growth velocities that control competitive advantage and species survival.