Asymmetric charge patterning on surfaces and interfaces: Formation of hexagonal domains

THE JOURNAL OF CHEMICAL PHYSICS 127, 164707 2007 Asymmetric charge patterning on surfaces and interfaces: Formation of hexagonal domains Sharon M. Loverde Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208-3108, USA Monica Olvera de la Cruz a Department of Materials Science and Engineering, Department of Chemistry, and Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208-3108, USA Received 23 July 2007; accepted 10 September 2007; published online 24 October 2007 The structure of soft matter systems at interfaces is of utmost importance in the fields of nanopatterning and self-assembly. It has been shown that lamellar and hexagonal patterns can form on interfaces, for a wide variety of systems. The asphericity of charged domains is considered here for different strengths of the electrostatics, determined by the interface media, relative to the short range van der Waals interactions between the molecular components. The phase behavior of the surface structure is explored by using molecular dynamics simulations, including some dynamical aspects of the interaction between neighboring domains, using the Lindemann criterion F. Lindemann, Z. Phys. 11, 609 1910. The charge ratio of the electrostatic components influences the shape of the domains, as well as the degree of local order in the interdomain structure. 2007 American Institute of Physics. DOI: 10.1063/1.2793038 I. INTRODUCTION The competition of long and short range interactions in a multitude of systems leads to arrested nanostructures with various properties. For example, two-dimensional structures of mixed monolayer systems at the air-water interface have been observed using epiflourescence techniques. 1 3 Depending on the experimental conditions such as ph values, composition of the monolayer mixture, and the compression of the lipid monolayer, hexagonal domains, lamellar stripes, and more complex patterns have been observed. In addition to monolayers at the air-water interface, hexagonal ordering of domains has also been observed in various systems, including semiconductors, 4 6 metal alloys, 7 membranes, 8 block copolymer melts, 9,10 and reactive blends. 11 Arrested periodic metastable phases often result from the competition between a line tension which drives coarsening and a long range field which opposes growth such as a strain field, as explained in eutectics 12 and in gels undergoing shrinking 13,14 where the various observed patterns depend on the size 13 and shape 14 of the gel. On the other hand, the competition of a line tension in a phase segregating multicomponent system of incompatible oppositely charged particles and the electrostatic penalty of accumulating charge leads to stable nanopatterns. 5 17 Transitions between lamellar and hexagonal structures are observed in many experimental systems including monolayers at an air-water interface, biomembranes and grafted charged chains with hydrophobic backbones. These transitional phases are temperature and concentration dependent. In many cases, the domains themselves are not perfectly spherical, but slightly elliptical. 8 The asphericity of the domains may result from an in-plane molecular dipolar a Electronic mail: m-olvera@northwestern.edu moment, 18 depending on the specific chemistry of the lipid molecules. The shape asymmetry for finite two-dimensional 2D domains has been examined theoretically, considering dipolar interactions 19 and charged interactions 8,20 between the molecular head groups. It is suggested that the electrostatic interactions promotes the shape asymmetry of charged domains. The understanding of their contribution to the individual domain shape and interdomain structure is crucial. Charged lipids, such as PIP2, have been shown to be influential in the regulation of many cell processes. 21 In this paper, the asphericity of charged domains is considered for different strengths of the electrostatics relative to the short range van der Waals interactions between the components by using molecular dynamics simulations, including some dynamical aspects of charge pattern formation. We characterize the transition from lamellar phases to hexagonal domains as a function of charge ratio of the molecular components and the strength of the electrostatic interactions, including the formation of hexagonal phases. A model system composed of oppositely charged, immiscible molecular components is used, that has been previously studied and shown to form lamellar phases 17 and coexistence with a gas phase at low interfacial densities of charged components. 22 In this paper, we extend the model to asymmetric charge ratios and choose to study the formation of domains, as well as the degree of local ordering in the structure. The degree of ordering in the local interdomain structure is determined using the Lindemann criterion, a parameter well known in solid state physics community for the characterization of the melting temperature of solids. 23 Most studies of arrested structures have considered a singular ionic domain. In this case, the free energy is the sum of the charged or dipolar interactions within the domain, as well 0021-9606/2007/127 16 /164707/6/$23.00 127, 164707-1 2007 American Institute of Physics

164707-2 S. M. Loverde and M. Olvera de la Cruz J. Chem. Phys. 127, 164707 2007 FIG. 1. Color online Snapshots of the equilibrium domain shapes as l B / decreases in a =0.5, l B / =0.5 and b =0.5, l B / =0.1, and as the strength of the Lennard-Jones potential increases in b =3.5, l B / =0.1 and f =3.5, l B / =0.5. The corresponding domain distributions, N cl vs N, are shown in c and g. The radius of gyration R g vs N is shown in d and h. as a contribution due to the line tension at the domain edge. It has been shown that the free energy can be written in terms of the anisotropy of the domains such as a square, an ellipse, or a torus. 19 Moreover, in a multiple domain system, the free energy can be minimized with respect to the periodic length scale of the interdomain spacing, for both the lamellar and hexagonal phases, incorporating the long range interactions between the domains as explicitly described in Ref. 15. For a symmetrically charged system, that is, an equal number of oppositely charged monomers, the lowest free energy state is lamellar while, for an asymmetric ratio of positively and negatively charged groups, the lowest free energy state consists of hexagonal domains. For systems with dipolar interactions, the coexistence between lamellar and hexagonal phases has also been examined, finding a small region of coexistence between the two phases. 24 In this paper we describe the possible phases as a function of charge ratio, degree of incompatibility and the effective dielectric constant, or strength of the electrostatics, at the film surface. In Sec. II we describe the molecular dynamic simulations. In Sec. III we analyze and characterize the structure of the periodic patterns. In Sec. IV we discuss the results, including spatial correlations, of the various systems studied. In Sec. V we give conclusions. II. MOLECULAR DYNAMICS The model system is composed of a mixture of a number of N + positively z + and N negatively z charged molecular units in a simulation box of size L 3. The molecules are confined to a two-dimensional plane perpendicular to the Z axis, with periodic boundary conditions in the X and Y directions. NVT molecular dynamics simulations were performed using ESPRESSO, simulation code developed by the MPIP-Mainz Group of Polymer Theory and Simulation. 25,26 The phase behavior of the molecules at different charge fractions of head groups is explored by varying z + and z, while maintaining the overall electroneutrality of the system, z + + =z. We explore the phase diagram at fixed surface densities, = N + +N 2 /4L 2 =0.6. The van der Waals interac- FIG. 2. Structure factor for hexagonal case. Structure factor S k for /k B T=4.0 for different strengths of the electrostatics, l B /, equal to 0.1 in a, 0.2 in b, and 0.5 in c. The local ordering of the domains is not perfectly hexagonal. For higher values of the electrostatics l B / =0.5 the ordering at larger length scales displays a cubic or square structure, which is a finite size effect.

164707-3 Asymmetric charge patterning J. Chem. Phys. 127, 164707 2007 FIG. 3. Normalized root mean square displacement r of the center of masses of domains for increasing values of. The Lindemann criterion indicates that the local structure of the domains are liquid for these values of the parameters l B and. tions between like molecules are represented by the classic Lennard-Jones potential, U LJ =4 r 12 r 6, r r c, 1 where is the molecular radius. The potential is cut at a radius r c of 2.5 for oppositely charged molecules and r c =2 1/6 for similarly charged molecules, where is an effective molecular radius. This selection of cutoffs produces a net immiscibility between species of magnitude per contact between oppositely charged particles. The potential between charges is a full Coulomb potential, U C =l B Tq 1 q 2 /r, calculated using the electrostatic layer correction method, 27 which is a 2D correction to P3M Ewald summation. 28 A Langevin thermostat is used and l B =e 2 /4 k B T, where is the average dielectric permittivity of the surrounding media. III. FORMATION OF HEXAGONAL PHASE The strength of the electrostatic interactions plays a significant role in the formation of the domains, as well as the resulting equilibrium shape. In this section, the results are presented for 900 N + positively and 300 N 3 negatively charged monomer units of charge 3. The charge ratio is f =N + /N =z /z + =3/1. The formation of domains at two different strengths of electrostatic interactions is shown in Fig. 1. As the value of is increased, for a constant value of the Bjerrum length, the negatively charged particles begin to cluster into domains as shown in Figs. 1 a and 1 e. Notice that the distribution of clusters and domain sizes is nearly dependent of the Bjerrum length when is small. For stronger values of the Bjerrum length Fig. 1 a, the negatively charged particles possess less of a tendency to cluster and are more disperse than for lower values of the Bjerrum length Fig. 1 e. Figures 1 b and 1 f are typical domain distributions obtained when the short range interaction increases. Increasing results in the formation of finite domains. The domains are smaller for high values of the Bjerrum length Fig. 1 b than for lower values of the Bjerrum length Fig. 1 f since the resulting structures are a result of the competition between the short range interactions, which promotes the growth of the domains, with electrostatics, which favors small domains. Electrostatics leads to elongated domains. In certain cases, these domains also display a preference for curvature perpendicular to their axis. The cluster size distribution for these respective images are shown in Figs. 1 c and 1 g. The average number of clusters N cl per simulation size box L 2 is exponentially decreasing with the number of particles per cluster N for small values Figs. 1 c and 1 g. As the value of increases, the distribution function develops a small peak at finite size N, corresponding to domain formation. Lower values of Bjerrum length Fig. 1 g give broader peaks for the size distribution and larger domain sizes. The average radius of gyration, R g, as a function of the cluster size N is plotted in Figs. 1 d and 1 h. At a critical cluster size value, the radius of gyration displays a transition between two different dependencies upon increasing. Below this length scale, R g scales as N 1/2 which is to be expected for circular domains. For larger clusters above this critical cluster size, R g scales linearly with N. This is to be expected for stretched, linear domains. This critical size transition from circular to noncircular domains has been coupled theoretically with the line tension, charge density, and the area of the individual domain, 8 which is consistent with these simulation results. The order of the domains at the interface can be further examined using the 2D structure factor S k. In Fig. 2, contour plots of the structure factor for three different strengths of the electrostatics l B / =0.1, 0.2, and 0.5 in Figs. 2 a 2 c, respectively are shown. Notice that the local ordering of the domains does not display a perfect sixfold sym- FIG. 4. Color online Sphericity of the domains as a function of the charge ratio of the components. The most spherical domains are present for charge ratios f close to 3:1, while for increasingly asymmetric mixtures, there is a gradual increase in the symmetry of the domains.

164707-4 S. M. Loverde and M. Olvera de la Cruz J. Chem. Phys. 127, 164707 2007 FIG. 5. Normalized root mean square displacement r of the center of masses of domains as the asymmetry of the system is changed. metry. There is a ring of peaks at small values of the wave vector k when l B / =0.1. The domains themselves are not perfectly spherical, as previously shown. This does not allow for perfect hexagonal packing of the domains. For increased values of the electrostatic interactions, the ring of peaks corresponding to the local domain structure shifts to larger values of the wave vector k. In addition, ordering at larger length scales smaller wave vectors indicates cubic or square structures close to the center of S k. Examination at larger system indicates that this is a finite size effect. The structure factor S k for /k B T=4.0 and l B / =0.5 shows that increasing the total area of the box by a factor of 2 from L =66 to L=93.4, these peaks shift relative to the box size, indicating that this ordering at large wavelengths is an artifact of the finite box size. The degree of local ordering between the domains can be further characterized using the Lindemann criterion. The Lindemann criterion has been used to study melting and condensation for two-dimensional Coulombic systems 29 and, in this case, is applied to the study of the underlying lattice structure of the domains of the solid phase. Peierls and Landau argued that there is no long range order for a two- FIG. 6. Snapshots of simulations for values of short range attraction /k B T=4.0 for strength of the electrostatics l B / =0.5, for various charge ratios of the components. a 2:1, b 3:1, c 5:1, and d 10:1. FIG. 7. Mean square displacement as a function of time step at l B / =0.5 and /k B T=4.0, for increasing charge ratios of the components. For charge ratios f 5:1, diffusion behavior of the components indicates strong local correlations. dimensional solids. 30 32 The local thermal fluctuations of the atoms induces a displacement with respect to the equilibrium position which increases logarithmically with the system size. For two-dimensional Coulombic systems, it has been suggested that, although long range translational order cannot exist, long range orientational order can be extended infinitely far. 33 Melting in two dimensions is described by the Kosterlitz and Thouless transition, 34,35 which was further extended by Halperin and Nelson, in which the local structure melts through dislocations. 36,37 For two-dimensional systems, the melting of the structure according to the Lindemann criterion described below is when the root mean square value of the lattice fluctuations exceeds 0.1 of the lattice spacing. The center of masses of the charged domains represent the lattice structure for the two-dimensional Wigner crystal formed by the solid phase. For perfect circular domains at low temperatures, the center of masses of the neighboring domains should occupy equilibrium positions within an underlying hexagonal lattice. The fluctuations of the center of masses of the domains about this equilibrium lattice increase with temperature, and, at a certain ratio of the fluctuation length scale relative to the periodicity of the lattice, the structure should melt. The root mean square displacement for the center of mass, R, of each cluster denoted by cl, is defined as r = 1 1/N N cl,i cl,j cl,j R cl,i R cl,j 2 1/2. 2 cl,i R cl,i R cl,j Upon examination of the root mean square displacement of the domain center of masses, r, it is found that the local fluctuations exceed the Lindemann criterion limit for solid behavior 0.1 normalized by the lattice spacing. In Fig. 3, the peak represents the first appearance of neighboring domains, followed by a minimum which indicates the local structure. The increase after the minimum is due to the appearance of non-neighboring domains. Upon increasing, the fluctuations in the local lattice of domains is decreasing, but not enough to form a crystalline phase. The local domain

164707-5 Asymmetric charge patterning J. Chem. Phys. 127, 164707 2007 FIG. 8. Structure factor S k for value of short range attraction /k B T=4.0 for strength of the electrostatics l B / =0.5, for varying charge ratios of the components. From left to right, a 2:1, b 5:1, and c 10:1. For 2:1 mixtures, two strong peaks indicate lamellar order. With increase in charge ratio, lamellar phase is lost and ring of peaks reminiscent of hexagonal order appears. Cubic structure is indicated at long wavelengths, which is a finite size effect. structure is thus still liquid. This agrees with the diffusion of the particles, from the mean square displacement, as a function of time step which is further described in the next section in Fig. 7. IV. TRANSITION FROM LAMELLAR TO HEXAGONAL PHASES The transition from the lamellar to hexagonal phases is studied here as shown in Fig. 4 varying the charge ratio of the components, f, while still maintaining the overall electroneutrality of the system. This is analogous to changing the ph of the solution, depending on the specific pk a s of the molecular components at the interface. In this section, the results for zn + positively and N z negatively charged monomer units of charge z are presented, where charge ratio is defined as f =N + /N =z /z +. When varying the charge ratios of the components, the magnitude of the electrostatic and short range contributions to the free energy shifts. Therefore, increasing the charge asymmetry may result in the formation of local crystalline order. To examine this supposition, we start by determining the magnitude of the sphericity of the individual domains as a function of the charge ratio of the components. The matrix of inertia of the domains is defined as T x,y = 1 N Cl N C r i x r x r i y r y, 3 i where N Cl represents the number of of ions in one cluster or domain and r i x represents the component of the position of ith ion. The eigenvalues of the matrix represent R g 2 and R g 2. When the ratio R, ofr g to R g, is close to 1, this signifies a circular domain. As one might expect, the distribution for the sphericity of the formed domains changes as a function of the charge ratio of the components, as seen in Fig. 5. Beginning with a symmetric system, this ratio increases towards spherical domains for charge ratios f of 3:1. Increasing f, the charge ratio of the components, there is a gradual increase in the sphericity of the domains, until the domains are simply single particles, arranged in a hexagonal lattice. For a fixed ratio of the electrostatics to the short range interactions l B / =0.5, =4.0, the root mean square fluctuations of the domain positions, relatively normalized by the average radius of the interdomain distances, are greater than 0.1 of the interdomain distance for values of f 7. As in Fig. 6, the peak represents the first appearance of neighboring domains, followed by a minimum which indicates the local ring structure. However, upon increasing the charge ratio of the components, the peak shifts to the left. The interdomain distance is shrinking. The minimum, corresponding to the fluctuations in the local ordering of the domains, decreases until it passes beneath the Lindemann criterion ratio 0.1 for a charge ratio f between 7:1 and 10:1. The short range order of the local hexagonal lattice is increasing as a function of increased electrostatic interactions between the minority components. As the charge ratio increases, the domains mainly consist of single particles. In addition, the solid phase is compressing; additional vacancies are expelled from the solid phase. In examining Fig. 6 c, this is signified by the region of empty space that is segregated from the solid phase. If the local structure is freezing or forming a solid, this should be signified in the thermodynamic quantities such as the heat capacity of the system. Indeed, as a function of the charge asymmetry of the system, the heat capacity per molecule displays a peak for charge ratios of 5:1. This indicates the onset of stronger local order of the system. Upon examination of the diffusion of the particles, from the root mean square displacement, for charge ratios greater than 5:1, diffusion of the components, as shown in Fig. 7, indicates strong local correlations in the charged components. Mean square displacement as a function of time step at l B / =0.5 and /k B T=4.0, for increasing charge ratios of the components, shows that, for charge ratios f 5:1, diffusion behavior of the components depends on strong local correlations and is no longer linear. The order of the domains can be further examined using the structure factor S k. As a function of increasing charge ratio of the components, there is a clear shift from elongated, stretched domains to the local ordering at smaller length scales, a strongly correlated liquid. In Fig. 8 on the left, S k displays two peaks that correspond to lamellar phases at angles slightly tilted from the horizontal, as seen in snapshot in Fig. 8 a. With an increase in the charge ratio of the components, the lamellar peaks disappear and a ring of peaks

164707-6 S. M. Loverde and M. Olvera de la Cruz J. Chem. Phys. 127, 164707 2007 reminiscent of hexagonal order appears at smaller length scales. Cubic structure is indicated at longer wavelengths, which consists of finite size effects. V. DISCUSSION This transition from the lamellar to the hexagonal phase is relevant to many systems. For example, domain shape instabilities from circular domains to branched phases have been shown to be induced upon compression of Langmuir monolayers. 1 The electrostatic contribution to the pattern formation in these systems, as it pertains to the monolayer films, at high densities, is decidedly relevant to the melting transitions of these films. It is shown here that increasing the electrostatic contribution, or charging the charge ratio of the components, strongly increases correlations and ordering between domains. In this paper, theoretical results for lamellar and hexagonal phases were compared with molecular dynamics simulations at intermediate temperatures. When considering the formation of hexagonal domains, at finite temperatures, there are deviations from both the shape and packing within the domain structures that digress from the phase behavior described from strong segregation theory at low temperatures. A critical domain nucleus is found at which the domains change their shape from circular to noncircular behavior. Moreover, it is found that the charge ratio of the electrostatic components influences the shape asymmetry, as well as the degree of local order in the interdomain structure. 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