Adsorption of Self-Assembled Rigid Rods on Two-Dimensional Lattices

Monte Carlo (MC) simulations have been carried out to study the adsorption on square and triangular lattices of particles with two bonding sites that, by decreasing temperature or increasing density, polymerize reversibly into chains with a discrete number of allowed directions and, at the same time, undergo a continuous isotropic-nematic (IN) transition. The process has been monitored by following the behavior of the adsorption isotherms for different values of lateral interaction energy/temperature. The numerical data were compared with mean-field analytical predictions and exact functions for noninteracting and 1D systems. The obtained results revealed the existence of three adsorption regimes in temperature. (1) At high temperatures, above the critical one characterizing the IN transition at full coverage Tc(\theta=1), the particles are distributed at random on the surface and the adlayer behaves as a noninteracting 2D system. (2) At very low temperatures, the asymmetric monomers adsorb forming chains over almost the entire range of coverage, and the adsorption process behaves as a 1D problem. (3) In the intermediate regime, the system exhibits a mixed regime and the filling of the lattice proceeds according to two different processes. In the first stage, the monomers adsorb isotropically on the lattice until the IN transition occurs in the system and, from this point, particles adsorb forming chains so that the adlayer behaves as a 1D fluid. The two adsorption processes are present in the adsorption isotherms, and a marked singularity can be observed that separates both regimes. Thus, the adsorption isotherms appear as sensitive quantities with respect to the IN phase transition, allowing us (i) to reproduce the phase diagram of the system for square lattices and (ii) to obtain an accurate determination of the phase diagram for triangular lattices.

Experimental realization of such systems is growing. An example of real patchy particles is presented in Ref. 21 Such particles offer the possibility to be used as building blocks of specifically designed self-assembled structures. 19,20,[22][23][24] The term "molecular self-assembly" may be used to refer to spontaneous formation of an ordered molecular overlayer on a surface. Molecular self-assembly on surfaces via weak but selective noncovalent interactions, offers a promising bottom-up approach to fabricate highly organized systems from instructed molecular building blocks. In this way, the engineering of supramolecular arrays, with desired functionalities, on metal surfaces can be performed. 25 If, in addition, such systems are capable of undergoing phase transitions, then their functional properties may be increased.
Recently, several research groups reported on the assembly of colloidal particles in linear chains. Selectively functionalizing the ends of hydrophilic nanorods with hydrophobic polymers, Nie et al. reported the observation of rings, bundles, chains, and bundled chains. 26 In another experimental study carried out by Chang et al., 27 gold nanorods were assembled into linear chains using a biomolecular recognition system. Another prominent case is the self-assembly of flexible one-dimensional coordination polymers on metal surfaces, i.e. in Ref., 28 where the researchers employed a de novo synthesized porphyrin module to construct one-dimensional (1D) Cu-coordinated polymers on Cu(111) and Ag(111) surfaces.
In a direct relation with the present work, Clair et al. 29 investigated the self-assembly of terephthalic acid (TPA) molecules on the Au(111) surface. Using scanning tunneling microscopy, the authors showed that the TPA molecules arrange in one-dimensional chains with a discrete number of orientations relative to the substrate. However, in the experimental studies of chains selfassembled on a surface with a discrete number of orientations, orientational ordering transitions were not studied.
In this context, the main objective of this paper is to investigate the adsorption process in a system composed of monomers with two attractive (sticky) poles that self-assemble reversibly into polydisperse chains and, at the same time, undergo an orientational transition. In a recent series of papers, 30-37 the critical behavior of this system has been widely studied. Except mean-field calculations, 34 these studies showed the existence of a continuous phase transition along the entire coexistence curve. However, the universality class of the model has been a subject of controversy. 36,37 It was shown that the system under study represents an interesting case where the use of different statistical ensembles (canonical or grand canonical) leads to different and well-established universality classes (q = 1 Potts-type or q = 2 Potts-type, respectively). 36 The present work goes a step further addressing the study of the adsorption isotherms in different regions of the phase diagram, emphasizing their behavior at critical and subcritical temperatures. The adsorption isotherm appears as a sensitive quantity to the phase transition, allowing a very accurate determination of the phase diagram. These findings may be of interest for theoretical and experimental studies on critical adsorption phenomena.
It is clear that the model considered here is highly idealized and is not meant to reproduce a particular experimentally studied system. However, the understanding of simple models with increasing complexity might be a help and a guide for future experimental investigations. This work represents an effort in that direction.
The paper is organized as follows. In Sec. 2, lattice-gas model and theoretical formalism (mean-field approximation and exact functions for non-interacting and 1D systems) are presented. Section 3 is devoted to describe the Monte Carlo simulation scheme. The analysis of the results and discussion are given in Sec. 4. Finally, the conclusions are drawn in Sec. 5.

Lattice-Gas Model and Theory
As in Refs., 30-37 a system of self-assembled rods with a discrete number of orientations in two dimensions is considered. The substrate is represented by a square or triangular lattice of M = L×L adsorption sites, with periodic boundary conditions. N particles are adsorbed on the substrate with m possible orientations along the principal axis of the array, being m = 2 for square lattices and m = 3 for triangular lattices (see Fig. 1). These particles interact with nearest-neighbors (NN) through anisotropic attractive interactions. Thus, a cluster or uninterrupted sequence of bonded particles is a self-assembled rod. Then, the grand canonical Hamiltonian of the system is given by where i, j indicates a sum over NN sites; w represents the NN lateral interaction between two neighboring i and j, here w < 0 and the energy is lowered by an amount |w| only if the NN monomers are aligned with each other and with the intermolecular vector r i j ; σ j is the occupation vector, with σ j = 0 if the site j is empty and σ j =x i if the site j is occupied by a particle with orientation along the x i -axis; and ε o is the adsorption energy of an adparticle on a site. In the present work, ε o was set equal to zero (without any loss of generality) and the chemical potential where T is the temperature, k B is the Boltzmann constant, θ = N/M is the surface coverage and K(c) represents the number of available configurations (per lattice site) for a monomer at zero coverage. The term K(c) is, in general, a function of the connectivity of the lattice c and the structure of the adsorbate (c = 4 and 6 for square and triangular lattices, respectively). It is easy demonstrate that K(c) = 2 for square lattices and K(c) = 3 for triangular lattices. 45 In the limit of high temperatures (w/k B T → 0), particles adsorb randomly on the surface and Eq. (??) reduces to the well-known Langmuir isotherm: 12 On the other hand, the phase diagram reported in Ref. 34 indicates that, by decreasing temperature or increasing density, the monomers polymerize reversibly and form linear chains on the surface. In this limit, it is more convenient to describe the system from the exact form of the adsorption isotherm corresponding a interacting monomers on 1D lattices: 12

Monte Carlo Simulation
The thermodynamic properties of the present model were investigated following a standard importance sampling MC method in the grand canonical ensemble. 46 The procedure is as follows. In this framework, total and partial adsorption isotherms can be obtained as simple averages: and where θ represents the total surface coverage; θ x i denotes the partial surface coverage referring to the adparticles with orientation along the

Results and Discussion
In the present section, the main characteristics of the thermodynamic functions given in ??, will be analyzed in comparison with simulation results for a lattice-gas of asymmetric monomers on 2D lattices. The computational simulations have been developed for square and triangular L × L lattices, with L = 100, and periodic boundary conditions. With this lattice size we verified that finite-size effects are negligible.
In order to understand the basic phenomenology, it is instructive to begin by discussing the behavior of the simulation adsorption isotherms for square lattices and different values of the magnitude of the lateral interactions [ Fig. 2]. Curves from right to left correspond to w/k B T =0, -1, -2, -3, -4, -5, -6, -7, -8, -9 and -10, respectively. As it can be observed, isotherms shift to lower values of the chemical potential and their slope increases as the ratio |w|/k B T increases. This behavior is typical of attractive systems. However, a notable difference is observed with respect to the case of symmetric monomers, 12 where two nearest-neighbors particles interact with an interaction energy w. In fact, a clear discontinuity (jump) is observed in the adsorption isotherms corresponding to symmetric monomers for interactions above a critical value w > w c (in absolute values). The inset of Fig. 2 shows a scheme of the described process. The marked jump, which has been observed experimentally in numerous systems, is indicative of the existence of a first-order phase transition.
In this situation, the only phase which one expects is a lattice-gas phase at low coverage, separated by a two-phase coexistence region from a "lattice-fluid" phase at higher coverage. 48 On the other hand, the surface coverage varies continuously with the chemical potential for all cases shown in Fig. 2, and there is no evidence of the existence of a first-order phase transition. In other words, the mechanism for which the asymmetric monomers polymerize reversibly into chains at low temperatures (high values of |w|/k B T ) is not associated with the existence of a first-order phase transition in the adlayer. This point will be discussed in the next.
In Ref., 34   w/k B T = 0, path not shown in Fig. 3
In the second case [(ii)], lines represent data from ??. From a first inspection of the curves it is observed that (a) the results corresponding to square and triangular lattices coincide in a unique curve; and (b) simulation data agree very well with theoretical results obtained for 1D lattices.
These findings contribute to the understanding of the polymerization transition. Namely, in the case of strongly interacting particles (very low temperature limit), the IN phase transition occurs at very low coverage [see points A and B in Fig. 3(a)] and the asymmetric monomers adsorb forming chains over almost all range of coverage. A typical configuration of the adlayer in this regime [θ = 0.6 and k B T /|w| = 0.2, point J in Fig. 3(a)] is shown in Fig. 3(c).
Then, (1) the adsorption process behaves as a 1D problem, (2) the shape of the triangular and square isotherms are indistinguishable, and (3) at variance with the behavior observed for symmetric monomers (see inset of Fig. 2), the system studied here does not show a first-order phase transition at low temperature (it is well-known that no phase transition develops in a 1D lattice-gas 12 ).
In Ref., 34 the phase diagram corresponding to square lattices was also obtained by mean-field calculations. The results showed the existence of a coexistence region between a low-coverage isotropic phase and a high-coverage nematic one at low temperatures. The presence of this coexistence region (first-order phase transition) was completely at variance with the observed numerical simulation results. The study in Fig. 4(a) allows now to understand the mean-field prediction. Namely, the system goes to a 1D adsorption as the temperature decreases and, as it is well-known, mean-field approximation incorrectly predicts a phase transition in one dimension and low temperatures. 12 The comparison in Fig. 4(b) confirms this argument. In the figure, simulation isotherms (symbols) obtained for square lattices and three different values of w/k B T (as indicated) are compared with the corresponding ones obtained from ?? (lines). The mean-field curves lead to the characteristic van der Waals loops and, consequently, to the prediction of a first-order phase transition. 12 Summarizing the discussion above, the adsorption process passes from a 2D problem (at high temperatures) to a 1D one (at very low temperatures). At the intermediate regime, the system behaves in a mixed regime and the simulation isotherms are not well fitted, neither by ?? nor by ?? (data not shown here). These findings provide very important information about the critical behavior of the system and, as it will be seen in the following, suggest a method to obtain the phase diagram of the system through adsorption measurements.
Let us suppose now that the system is found in the point F of the phase diagram (intermediate temperature, isotropic region); with increasing coverage (at constant k B T /|w|, in this case The existence of a singularity in the adsorption isotherms indicates that a dramatic change in the adsorption process takes place in the system. One way to visualize this situation is to consider the partial adsorption isotherms. As an example, total and partial adsorption isotherms are plotted in Fig. 5(b) for square lattices and k B T /|w| = 1/3. θ x 1 (µ) and θ x 2 (µ) represent the fraction of particles adsorbed along the x 1 -axis (horizontal axis) and the fraction of particles adsorbed along the x 2 -axis (vertical axis), respectively. Two well-differentiated adsorption regimes can be observed: (i) for θ < θ c , particles are adsorbed isotropically on the surface and θ x 1 (µ) = θ x 2 (µ), and (ii) for θ > θ c , particles are adsorbed forming chains along one of the lattice axes (vertical axis in the case of the figure).
The procedure illustrated in Fig. 5 was used to obtain the temperature-coverage phase diagram corresponding to a triangular geometry. The results are shown in Fig. 6. Full circles represent data obtained from the inflection points in the adsorption isotherms. The figure also includes recent MC data by Almarza et al. 35 (open circles). The rest of the curve separating isotropic and nematic stability (dashed line) was built on the basis of the previous results on the behavior of the system at low temperatures (see Fig. 4). Namely, the critical properties corresponding to square and triangular lattices coincide in the low-temperature (coverage) regime. The complete phase diagram of triangular lattices has been reported here for the first time. A more exact determination of T c based on Monte Carlo simulations and finite-size scaling theory is in progress.
For comparative purposes, Fig. 6 includes the critical line corresponding to square lattices (solid line). Even though the shapes of the curves are similar, the critical temperature corresponding to a given density (θ > 0.1) is higher for square lattices than for triangular lattices.
In summary, the existence of singularities in the adsorption isotherms, which are induced by the phase transition, provides a way to determine the temperature-coverage phase diagram from adsorption experiments. These findings may be instructive for future theoretical and experimental studies on adsorption.

Conclusions
In the present paper, the main adsorption properties of self-assembled rigid rods on square and triangular lattices have been addressed. The results were obtained by using Monte Carlo simulations, mean-field theory and exact calculations in one dimension. According to the present analysis, the behavior of the system is characterized by the following properties: 1) At high temperatures, above the critical one characterizing the IN transition at full coverage T c (θ = 1), the particles are distributed isotropically on the surface, the adlayer behaves as a 2D system and an excellent agreement is observed between Monte Carlo simulation and theoretical results from Langmuir isotherm.
2) At very low temperatures, asymmetric monomers adsorb forming chains over almost all range of coverage and the adsorption process passes from a 2D problem to a 1D one. Then, the results corresponding to square and triangular lattices coincide in a unique curve, simulation data agree very well with exact theoretical results obtained for 1D lattices, and the system studied here does not show a first-order phase transition at low temperatures (it is well-known that no phase transition develops in a 1D lattice-gas 12 ).
3) Mean-field calculations 34 showed the existence of a coexistence region (first-order phase transition) between a low-coverage isotropic phase and a high-coverage nematic one, at complete variance with the observed numerical simulation data. The result described in the item above allows now to understand the mean-field prediction. Namely, the system goes to a 1D adsorption as the temperature decreases and, as it is well-known, mean-field approximation incorrectly predicts a phase transition in one dimension. 12 4) At intermediate temperatures, the system exhibits a mixed behavior, the 2D and 1D adsorption processes are present in the adsorption isotherms and a marked singularity can be observed separating both regimes. The measurement of the point at which this singularity occurs (pronounced maximum in the derivative of θ with respect to µ) allows an accurate determination of the critical coverage characterizing the IN phase transition.
5) The simple analysis described in the last item reproduces the previously obtained phase diagram for square lattices 34 and provides a good approximation for the case of triangular lattices. A more detailed and accurate study, including the variation of critical exponents and the determination of the phase diagram through finite-size scaling analysis, is being undertaken for triangular lattices.