Spontaneous chiral resolution in two-dimensional systems of patchy particles J. A. Martínez-González, G. A. Chapela, and J. Quintana-H Citation: The Journal of Chemical Physics 140, 194505 (2014); doi: 10.1063/1.4876575 View online: http://dx.doi.org/10.1063/1.4876575 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/19?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optical purification of a mixture of chiral forms by dimer formation J. Chem. Phys. 135, 124304 (2011); 10.1063/1.3641283 Influence of chirality on the interfacial bonding characteristics of carbon nanotube polymer composites J. Appl. Phys. 103, 044302 (2008); 10.1063/1.2844289 Two-dimensional theory of chirality. II. Relative chirality and the chirality of complex fields J. Math. Phys. 41, 5986 (2000); 10.1063/1.1285981 Nonlinear dependence of Maxwell displacement current across chiral phospholipid mixed monolayers on molar ratio J. Chem. Phys. 113, 2880 (2000); 10.1063/1.1305868 Chiral symmetry breaking. II. Synthesis in cooperative systems J. Chem. Phys. 110, 10923 (1999); 10.1063/1.479004
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THE JOURNAL OF CHEMICAL PHYSICS 140, 194505 (2014)
Spontaneous chiral resolution in two-dimensional systems of patchy particles J. A. Martínez-González,1 G. A. Chapela,1 and J. Quintana-H2,a) 1
Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 México D.F., Mexico 2 Instituto de Química, Universidad Nacional Autónoma de México - Apdo. Postal 70213, 04510 Coyoacán, México D.F., Mexico
(Received 29 January 2014; accepted 2 May 2014; published online 21 May 2014) Short ranged potentials and their anisotropy produce spontaneous chiral resolution in a two dimensional model of patchy particles introduced in this paper. This model could represent an equimolar binary mixture (racemic mixture) of two kinds of chiral molecules (enantiomers) adsorbed to a bidimensional domain where only lateral short ranged interactions are present. Most racemic mixtures undergo chiral resolution due to their spatial anisotropy, the combined effect of long range forces and the thermodynamic conditions. The patchy particles are modeled as a hard disk and four different bonding sites located to produce chirality. Phase behavior and structural properties are analysed using Discontinuous Molecular Dynamics in the canonical ensemble. When the four patchy particles are separated by the angles {60◦ , 120◦ , 60◦ , 120◦ }, spontaneous chiral resolution is produced, given by the formation of homochiral clusters, if started from the corresponding racemic mixture. Gel behavior is also obtained in all the systems for low temperatures and low densities. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4876575] I. INTRODUCTION
Since living organisms are composed of mainly one of the enantiomers of chiral biomolecules, chiral resolution is of great importance in the production of biologically active molecules. As up today, only about 10% of the referenced chiral compounds, biologically active or otherwise, have been resolved by crystallization.1 Thus the experimental and theoretical study of chiral resolution is of fundamental importance. A given enantiomer has a different optical response2 than its corresponding mirror image but all other physical properties are exactly the same. According to the theoretical results of Arnett and Steward3 chiral resolution is enhanced in two-dimensional domains. Experimental results4–7 and molecular simulations8, 9 support this fact. D.A. Huchaby is one of the pioneers in studying this complex problem using Statistical Mechanics rigorous schemes to analyse various types of lattice models. In one of their very first papers, Huckaby et al.10 considered a tetrahedral molecule adsorbed to a triangular lattice where the substrate-molecule interactions did not favor any of the two enantiomers. They found that spontaneous chiral resolution was developed if the chemical potential and temperature were sufficiently high and low, respectively. Despite the three dimensional nature of the tetrahedral molecule,10 there is similarity with the 2D model introduced here, because their molecules are adsorbed on to a lattice. Also, in both cases the interactions are short ranged and do not favor any of the enantiomers, while in the case presented in this work the shape of
a) Electronic mail: [email protected]
0021-9606/2014/140(19)/194505/5/$30.00
the patchy particles plays a role because they are not anchored to any lattice, they are free to wander around. Later on Huckaby et al.11 analysed the same system more exhaustively using two order parameters, resulting in a more complex phase diagram. Their approach provided a way to study the liquidvapor phase transition, the enantiomeric phase separation in the form of two symmetric condensed phases and the racemic condensed solid. Recently, they studied the chiral separation of the Andelman de Gennes7 model adsorbed to a lattice using different kinds of interactions, including van der Waals and electrostatics,12 and different ranges of interactions.13 In both cases, they obtained chiral separation under specific conditions. Something remarkable is that in many of these results, spontaneous chiral resolution occurs as a consequence of positional and orientational correlations. Actually, liquidcrystalline mesophases are produced in several chiral compounds and molecular models because of their molecular shape anisotropy.8, 9, 14, 15 In real systems, chiral resolution has been found in asymmetrically shaped molecules6, 16 interacting with long ranged potential.17 The aim of this work is the study of this phenomenon in molecules with isotropic shape where the chirality is defined through sites with attractive short range potentials. In this light, patchy particles, first introduced by Wertheim,18 have successfully been used to add selectivity and directionality in the intermolecular interaction, allowing a deeper understanding of 3D and 2D phenomena like self-assembly into polymer chains,19–21 gelation,22–24 and the formation of ordered structures,25–27 to name just a few. The paper is organized as follows: Sec. II contains the definition of the model and the simulation details, Sec. III
140, 194505-1
© 2014 AIP Publishing LLC
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194505-2
Martínez-González, Chapela, and Quintana-H
J. Chem. Phys. 140, 194505 (2014)
contains the results and their explanations and finally the conclusions are given in Sec. IV. II. THE MODEL AND SIMULATION DETAILS
A chiral patchy particle model is proposed in this work, consisting of a hard disk (HD) of diameter σ = 1 decorated with four distinguishable square well (SW) sites (s). Each site is situated at a distance 0.4σ from the center of the HD, and their SW parameters are: diameter σ s = 0.1σ , well depth = −1, and well size λ = 0.3σ . The inter-particle interaction potential between patchy particles m and n, is the sum of the interactions of the two HD belonging to each particle, plus the contribution of the interactions among the sites of both particles. Besides this inter-particle interaction, an intra-particle interaction is required to keep each particle composed as a whole. Therefore, it is necessary to define intra-particle interactions UIN as follows. Within a given particle, each site interacts with the two adjacent sites and with the hard disk at the center of this particle. Therefore the full potential writing in detail is: U (m, n) = UHD (rmn ) +
4 4
Us rijmn
i=1 j =1
+
4 k=1
+
4 k=1
UI N rk0 mm +
3
+ UI N r41 UI N rkk+1 mm mm
k=1 3 UI N rk0 UI N rkk+1 + + UI N r41 nn nn nn , k=1
(1) where UHD denotes the hard-disk potential and rmn is the distance between the centers of the hard-disks m and n. Us is ij the site potential with rmn the distance between sites i = 1, 4 and j = 1, 4 of particles m and n, respectively, with i = j. The second line of Eq. (1) gives the intra-particle trapping potential well UIN acting between each site in particle m, with rk0 mm the distance between sites k = 1, 4 and the HD of particle m, with rkk+1 mm the distance between sites k = 1, 3 and sites k + 1 and r41 mm the distance between sites 4 and 1. The last line of Eq. (1) gives the intra-particle potential for parti√ ij ij cle n. If rmn ≤ λ, then Us = (1 − κij ) i j for rmn > λ, Us = 0. In this work, i = j = = −1 and κ 13 = κ 24 = 0 and 1 otherwise. In order to simulate the patchy particles with the sites kept in their places, vibrating wells (vw), as intraparticle potentials,28 are used. If Lvw is the range of the intraparticle vibrating SW centered at L and Lvw = 0 is its depth, then its definition is: If L − Lvw /2 ≥ rkl mm , then UIN = ∞. < L + L /2, then UI N = Lvw . If If L − Lvw /2 < rkl vw mm ≥ L + L /2, U = ∞. There are 8 trapping wells rkl vw IN mm in each patchy particle. The study is done using Discontinuous Molecular Dynamics (DMD)29 in the canonical ensemble. The simulations start by assigning initial positions and velocities to the hard disks and their sites, collision times for each pair of particles and sites are then calculated. The minimum collision time is
FIG. 1. (a) The studied patchy particles differ from the distribution of the sites. In the P60 model, the angle between sites is represented by the set {60◦ , 120◦ , 60◦ , 120◦ }, while for the P90 model the set is {90◦ , 90◦ , 90◦ , 90◦ }. In both cases the numbers represent a site. (b) Mirror images of the P60 and P90 models.
selected and the particles and their sites are moved, at constant velocities, for this minimum time up to the collision position of the corresponding pair. Once the collision is solved, new possible collision times are calculated for the pair that just collided. The new minimum collision time is selected and the procedure is repeated again. In this work, a block consists of 2.5 × 107 collisions, depending on the thermodynamical conditions between 150 and 300 blocks are necessary to get equilibration and about 100 blocks are used to obtain statistics. Two molecular conformations defined by the angles among the patchy sites are considered. They are depicted in Fig. 1(a) and are referred as P90 and P60, respectively. Their corresponding mirror images are shown in Fig. 1(b) and are denoted by P90* and P60*, respectively. Distances are measured in units of σ while temperatures are in units of . The number of sites and disks used in all the simulations is 2880 (576 disks). In each case, the initial configuration is a racemic mixture with a fcc crystalline array in a rectangular box (the lengths of the simulation box in the x and y direction satisfy Lxx = 2Lyy ). The calculated properties are the density profile, surface tensions, vapor pressures, and internal energies. The density profile is given by ρ(x) =
N (x, x + x)
, V
(2)
where N(x, x + x) is the average number of particles with positions between x and x + x that in the slab with volume V . The components of the pressure tensor were obtained by using the Clausius’ virial theorem30, 31 Pxx = ρkB T +
1 ml vxi xij , Vt coll ’s
(3)
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J. Chem. Phys. 140, 194505 (2014)
where Pxx is the time average of the instantaneous diagonal component Pxx of the pressure tensor, ρ is the number density, V the volume, T is the temperature, and kB is the Boltzmann constant. vxi is the velocity difference vector, before and after the collision, xij is the x component of the vector joining the center of the two sites or disks that collided, mi is the mass of the particle i (in this case mi = 1for all i), t is the sum of all individual collision times and coll ’s indicates that the sum runs through the total number of collision times. For a planar interface, the surface tension is given by30 γ = L2x [ Pxx − Pyy ]. The vapor pressure corresponds to the time average of the pressure tension component normal to the interface. The structure of the solid phases was characterized by means of the bond orientational order parameter, n =
Nb N 1 1 einθj k , N k=1 Nb j =1
(4)
where Nb is the number of nearest neighbors given by the Voronoi construction and θ jk is the angle between particles j and k, in an hexagonal solid n = 6 while for a square solid n = 4. For well ordered solids | n |2 ≈ 1. III. RESULTS
For the racemic mixture of P60 and P60* molecules, the atomic number densities ρ = 2, 2.5, 3, 4, and 4.5 are analysed in the temperature range (0.100, 0.330) for increments of T = 0.010. The corresponding mesophases diagram is given in Fig. 2. As can be appreciated from the figure, a solid-fluid phase transition takes place in the studied isochores. Chiral resolution is obtained along the solid phase, and it is manifested in solid homochiral clusters with a triangular structure. For densities below 3, when the temperature is small enough, due to the strength of the attraction the system is in an arrested state, and molecules assembly in polymeric chains, as the thermal energy increases bonded particles can be separated allowing entropically more favorable states like
FIG. 2. Phase diagram T* vs. ρ* for the P60-P60* racemic mixture. It shows a colloid fluid phase for high temperatures, an arrested fluid for low temperatures and low densities. In the solid branch of the solid-fluid coexistence region the chiral resolution is obtained.
FIG. 3. Internal energy and vapor pressure as a function of the temperature.
a solid-fluid coexistence. For higher temperatures, thermal fluctuations avoid bonded molecules and the system is a colloidal fluid. Coexistence lines in Fig. 2 are drawn through the simulated points of the different isochors in between the temperatures where a change of phase is evident from the snapshots. They are smoothed lines to guide the eye in order to delimitate the structure of different regions. The insets give examples of the type of mesophases encountered. Figure 3 shows vapor pressures and internal energies. As expected for all cases, vapor pressure and internal energy increase with temperature. A change in the slope of this graphs occurs along the solid-fluid phase transition. An analysis of the cluster sizes in the fluid-solid coexistence region was made to determine the ratio in which the enantiomers solidify. Practically both enantiomers were found in the same proportion in homochiral clusters, which means that the chiral resolution is always symmetric. These results show that chiral resolution is possible in symmetric particles with an asymmetric short ranged potential. However, the chirality of the potential in this kind of particles does not guarantee chiral resolution. One example of this fact is treated below for racemic mixtures of P90 and P90* molecules. For these racemic mixtures, the simulation results for the site number densities ρ* = 1, 2, 3, 4, and 4.5 in the temperature range (0.050, 0.225) for increments of T* = 0.025, show again a solid-fluid phase transition but the system solidifies in a heterochiral square arrangement (see Fig. 4). However, the solid phase is not homogeneous as the particles
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FIG. 4. Phase diagram T* vs. ρ* for P90-P90* racemic mixtures. It shows a colloid fluid phase for high temperatures, an arrested fluid for low temperatures and low densities. In the solid-fluid coexistence region, the alternatedhomochiral chains form a solid in an heterochiral square arrangement.
assemble in homochiral chains instead of fully resolving the chirality. The alternation of P90 and P90* chains make the solid achiral. An explanation of this lack of chiral resolution might be that, although the heterochiral cluster made by alternated-homochiral chains and the homochiral cluster have the same energy and structure (see Fig. 5), there are much more accessible configurations of the first kind, thus forming a solid in an heterochiral square arrangement. In Fig. 4, the coexistence lines are drawn through the simulated points of the different isochors in between the temperatures where a change of phase is evident from the snapshots. They are smoothed lines to guide the eye in order to delimitate the structure of different regions. The insets give examples of the type of mesophases encountered. Arrested states are also found in the P90-P90* racemic mixture. The presence of gelation in the patchy particles systems studied here, is in agreement with Bianchi et al.23 which states the fact that systems of small valences states (small number of patchy sites), at low densities and temperatures have a large bond lifetime and does not present a phase transition in these conditions. The large bond lifetimes are a consequence of the directionality, selectivity, and the short potential range of the patchy sites. In the P60-P60* racemic mixture, the distribution of the patchy sites allow the particles to be bonded in pairs for small densities and temperatures, while for the P90-P90* racemic mixture, four particles can be bonded
FIG. 5. Graphical representation of a P90-P90* racemic mixture in the homochiral solid configuration and the heterochiral solid configuration made by homochiral chains. Both configuration are thermodynamically equivalent.
J. Chem. Phys. 140, 194505 (2014)
at the same conditions. Larger values of λ also frustrate chiral resolution in the P60-P60* racemic mixture. The anisotropy of the short ranged attractive part of the sites is responsible for the chiral segregated state. This desired property can be affected by the interaction of several patchy sites producing possible states with lower energy than the effect accomplished by the short ranged potential. In the case of homochiral systems, a solid-fluid phase transition and gel behavior is found for P90 and P60 systems. In each case, the density and temperature values in which phase transitions occur are practically the same than in the corresponding heterochiral case. As represented in Fig. 5, the P90 system and the P90-P90* racemic mixture are thermodynamically equivalent. Something similar happens with the P60 system and the P60-P60* racemic mixture because chiral resolution is equivalent to two homochiral phases coexisting. However, the demixing process causes the equilibration time to be longer in the heterochiral than in the homochiral case. Additionally, the P60-P60* racemic mixture shows surface tension between the coexisting homochiral solids, which is the main difference between the homochiral and heterochiral cases. IV. CONCLUSIONS
Chiral resolution is achieved with anisotropy of short ranged potentials in isotropic molecules modeled by a HD decorated with four patchy sites. The properties of directionality and selectivity that can be defined in patchy particle models can be successfully used to study complex phenomena like chiral resolution and gelation. Particular attention should be paid to the fact that shape anisotropy and long range potentials are not necessary conditions to produce enantioselectivity. The chiral resolution is always symmetric, both enantiomers are found in the same proportion in homochiral solid clusters. These results show that chiral resolution is possible in symmetric particles with an asymmetric short ranged potential. However, the chirality of the potential in this kind of particles does not guarantee chiral resolution. One example of this fact is treated below for racemic mixtures of P90 and P90* molecules. For these racemic mixtures, a solid-fluid phase transition appears as a heterochiral square arrangement with the solid phase assembled in homochiral chains instead of fully resolving the chirality. Although the heterochiral and homochiral clusters have the same energy and structure, entropy considerations might favor the former. Arrested states are found in both racemic mixture, the P90-P90* and the P60-P60*. Larger values of λ also frustrate chiral resolution in the P60-P60* racemic mixture. In the case of homochiral systems, a solid-fluid phase transition and gel behavior is found for P90 and P60 systems. In each case, the density and temperature values in which phase transitions occur are practically the same than in the corresponding heterochiral case. The demixing process causes the equilibration time to be longer in the heterochiral than in the homochiral case. The appearance of surface tension between the coexisting homochiral solids is the main difference between the homochiral and heterochiral cases.
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194505-5
Martínez-González, Chapela, and Quintana-H
ACKNOWLEDGMENTS
This research was financially funded by CONACYT grant 168001 “Efecto de la quiralidad, polaridad y anisotropía en el auto-ensamblaje molecular en dos dimensiones.” J.A. Martinez-Gonzalez is grateful to CONACYT for the support given by this grant. We also thank UAM Iztapalapa for the computing time made available by the Super Computing Center UAM-I. 1 I.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 194505 (2014)
Spontaneous chiral resolution in two-dimensional systems of patchy particles J. A. Martínez-González,1 G. A. Chapela,1 and J. Quintana-H2,a) 1
Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 México D.F., Mexico 2 Instituto de Química, Universidad Nacional Autónoma de México - Apdo. Postal 70213, 04510 Coyoacán, México D.F., Mexico
(Received 29 January 2014; accepted 2 May 2014; published online 21 May 2014) Short ranged potentials and their anisotropy produce spontaneous chiral resolution in a two dimensional model of patchy particles introduced in this paper. This model could represent an equimolar binary mixture (racemic mixture) of two kinds of chiral molecules (enantiomers) adsorbed to a bidimensional domain where only lateral short ranged interactions are present. Most racemic mixtures undergo chiral resolution due to their spatial anisotropy, the combined effect of long range forces and the thermodynamic conditions. The patchy particles are modeled as a hard disk and four different bonding sites located to produce chirality. Phase behavior and structural properties are analysed using Discontinuous Molecular Dynamics in the canonical ensemble. When the four patchy particles are separated by the angles {60◦ , 120◦ , 60◦ , 120◦ }, spontaneous chiral resolution is produced, given by the formation of homochiral clusters, if started from the corresponding racemic mixture. Gel behavior is also obtained in all the systems for low temperatures and low densities. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4876575] I. INTRODUCTION
Since living organisms are composed of mainly one of the enantiomers of chiral biomolecules, chiral resolution is of great importance in the production of biologically active molecules. As up today, only about 10% of the referenced chiral compounds, biologically active or otherwise, have been resolved by crystallization.1 Thus the experimental and theoretical study of chiral resolution is of fundamental importance. A given enantiomer has a different optical response2 than its corresponding mirror image but all other physical properties are exactly the same. According to the theoretical results of Arnett and Steward3 chiral resolution is enhanced in two-dimensional domains. Experimental results4–7 and molecular simulations8, 9 support this fact. D.A. Huchaby is one of the pioneers in studying this complex problem using Statistical Mechanics rigorous schemes to analyse various types of lattice models. In one of their very first papers, Huckaby et al.10 considered a tetrahedral molecule adsorbed to a triangular lattice where the substrate-molecule interactions did not favor any of the two enantiomers. They found that spontaneous chiral resolution was developed if the chemical potential and temperature were sufficiently high and low, respectively. Despite the three dimensional nature of the tetrahedral molecule,10 there is similarity with the 2D model introduced here, because their molecules are adsorbed on to a lattice. Also, in both cases the interactions are short ranged and do not favor any of the enantiomers, while in the case presented in this work the shape of
a) Electronic mail: [email protected]
0021-9606/2014/140(19)/194505/5/$30.00
the patchy particles plays a role because they are not anchored to any lattice, they are free to wander around. Later on Huckaby et al.11 analysed the same system more exhaustively using two order parameters, resulting in a more complex phase diagram. Their approach provided a way to study the liquidvapor phase transition, the enantiomeric phase separation in the form of two symmetric condensed phases and the racemic condensed solid. Recently, they studied the chiral separation of the Andelman de Gennes7 model adsorbed to a lattice using different kinds of interactions, including van der Waals and electrostatics,12 and different ranges of interactions.13 In both cases, they obtained chiral separation under specific conditions. Something remarkable is that in many of these results, spontaneous chiral resolution occurs as a consequence of positional and orientational correlations. Actually, liquidcrystalline mesophases are produced in several chiral compounds and molecular models because of their molecular shape anisotropy.8, 9, 14, 15 In real systems, chiral resolution has been found in asymmetrically shaped molecules6, 16 interacting with long ranged potential.17 The aim of this work is the study of this phenomenon in molecules with isotropic shape where the chirality is defined through sites with attractive short range potentials. In this light, patchy particles, first introduced by Wertheim,18 have successfully been used to add selectivity and directionality in the intermolecular interaction, allowing a deeper understanding of 3D and 2D phenomena like self-assembly into polymer chains,19–21 gelation,22–24 and the formation of ordered structures,25–27 to name just a few. The paper is organized as follows: Sec. II contains the definition of the model and the simulation details, Sec. III
140, 194505-1
© 2014 AIP Publishing LLC
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194505-2
Martínez-González, Chapela, and Quintana-H
J. Chem. Phys. 140, 194505 (2014)
contains the results and their explanations and finally the conclusions are given in Sec. IV. II. THE MODEL AND SIMULATION DETAILS
A chiral patchy particle model is proposed in this work, consisting of a hard disk (HD) of diameter σ = 1 decorated with four distinguishable square well (SW) sites (s). Each site is situated at a distance 0.4σ from the center of the HD, and their SW parameters are: diameter σ s = 0.1σ , well depth = −1, and well size λ = 0.3σ . The inter-particle interaction potential between patchy particles m and n, is the sum of the interactions of the two HD belonging to each particle, plus the contribution of the interactions among the sites of both particles. Besides this inter-particle interaction, an intra-particle interaction is required to keep each particle composed as a whole. Therefore, it is necessary to define intra-particle interactions UIN as follows. Within a given particle, each site interacts with the two adjacent sites and with the hard disk at the center of this particle. Therefore the full potential writing in detail is: U (m, n) = UHD (rmn ) +
4 4
Us rijmn
i=1 j =1
+
4 k=1
+
4 k=1
UI N rk0 mm +
3
+ UI N r41 UI N rkk+1 mm mm
k=1 3 UI N rk0 UI N rkk+1 + + UI N r41 nn nn nn , k=1
(1) where UHD denotes the hard-disk potential and rmn is the distance between the centers of the hard-disks m and n. Us is ij the site potential with rmn the distance between sites i = 1, 4 and j = 1, 4 of particles m and n, respectively, with i = j. The second line of Eq. (1) gives the intra-particle trapping potential well UIN acting between each site in particle m, with rk0 mm the distance between sites k = 1, 4 and the HD of particle m, with rkk+1 mm the distance between sites k = 1, 3 and sites k + 1 and r41 mm the distance between sites 4 and 1. The last line of Eq. (1) gives the intra-particle potential for parti√ ij ij cle n. If rmn ≤ λ, then Us = (1 − κij ) i j for rmn > λ, Us = 0. In this work, i = j = = −1 and κ 13 = κ 24 = 0 and 1 otherwise. In order to simulate the patchy particles with the sites kept in their places, vibrating wells (vw), as intraparticle potentials,28 are used. If Lvw is the range of the intraparticle vibrating SW centered at L and Lvw = 0 is its depth, then its definition is: If L − Lvw /2 ≥ rkl mm , then UIN = ∞. < L + L /2, then UI N = Lvw . If If L − Lvw /2 < rkl vw mm ≥ L + L /2, U = ∞. There are 8 trapping wells rkl vw IN mm in each patchy particle. The study is done using Discontinuous Molecular Dynamics (DMD)29 in the canonical ensemble. The simulations start by assigning initial positions and velocities to the hard disks and their sites, collision times for each pair of particles and sites are then calculated. The minimum collision time is
FIG. 1. (a) The studied patchy particles differ from the distribution of the sites. In the P60 model, the angle between sites is represented by the set {60◦ , 120◦ , 60◦ , 120◦ }, while for the P90 model the set is {90◦ , 90◦ , 90◦ , 90◦ }. In both cases the numbers represent a site. (b) Mirror images of the P60 and P90 models.
selected and the particles and their sites are moved, at constant velocities, for this minimum time up to the collision position of the corresponding pair. Once the collision is solved, new possible collision times are calculated for the pair that just collided. The new minimum collision time is selected and the procedure is repeated again. In this work, a block consists of 2.5 × 107 collisions, depending on the thermodynamical conditions between 150 and 300 blocks are necessary to get equilibration and about 100 blocks are used to obtain statistics. Two molecular conformations defined by the angles among the patchy sites are considered. They are depicted in Fig. 1(a) and are referred as P90 and P60, respectively. Their corresponding mirror images are shown in Fig. 1(b) and are denoted by P90* and P60*, respectively. Distances are measured in units of σ while temperatures are in units of . The number of sites and disks used in all the simulations is 2880 (576 disks). In each case, the initial configuration is a racemic mixture with a fcc crystalline array in a rectangular box (the lengths of the simulation box in the x and y direction satisfy Lxx = 2Lyy ). The calculated properties are the density profile, surface tensions, vapor pressures, and internal energies. The density profile is given by ρ(x) =
N (x, x + x)
, V
(2)
where N(x, x + x) is the average number of particles with positions between x and x + x that in the slab with volume V . The components of the pressure tensor were obtained by using the Clausius’ virial theorem30, 31 Pxx = ρkB T +
1 ml vxi xij , Vt coll ’s
(3)
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J. Chem. Phys. 140, 194505 (2014)
where Pxx is the time average of the instantaneous diagonal component Pxx of the pressure tensor, ρ is the number density, V the volume, T is the temperature, and kB is the Boltzmann constant. vxi is the velocity difference vector, before and after the collision, xij is the x component of the vector joining the center of the two sites or disks that collided, mi is the mass of the particle i (in this case mi = 1for all i), t is the sum of all individual collision times and coll ’s indicates that the sum runs through the total number of collision times. For a planar interface, the surface tension is given by30 γ = L2x [ Pxx − Pyy ]. The vapor pressure corresponds to the time average of the pressure tension component normal to the interface. The structure of the solid phases was characterized by means of the bond orientational order parameter, n =
Nb N 1 1 einθj k , N k=1 Nb j =1
(4)
where Nb is the number of nearest neighbors given by the Voronoi construction and θ jk is the angle between particles j and k, in an hexagonal solid n = 6 while for a square solid n = 4. For well ordered solids | n |2 ≈ 1. III. RESULTS
For the racemic mixture of P60 and P60* molecules, the atomic number densities ρ = 2, 2.5, 3, 4, and 4.5 are analysed in the temperature range (0.100, 0.330) for increments of T = 0.010. The corresponding mesophases diagram is given in Fig. 2. As can be appreciated from the figure, a solid-fluid phase transition takes place in the studied isochores. Chiral resolution is obtained along the solid phase, and it is manifested in solid homochiral clusters with a triangular structure. For densities below 3, when the temperature is small enough, due to the strength of the attraction the system is in an arrested state, and molecules assembly in polymeric chains, as the thermal energy increases bonded particles can be separated allowing entropically more favorable states like
FIG. 2. Phase diagram T* vs. ρ* for the P60-P60* racemic mixture. It shows a colloid fluid phase for high temperatures, an arrested fluid for low temperatures and low densities. In the solid branch of the solid-fluid coexistence region the chiral resolution is obtained.
FIG. 3. Internal energy and vapor pressure as a function of the temperature.
a solid-fluid coexistence. For higher temperatures, thermal fluctuations avoid bonded molecules and the system is a colloidal fluid. Coexistence lines in Fig. 2 are drawn through the simulated points of the different isochors in between the temperatures where a change of phase is evident from the snapshots. They are smoothed lines to guide the eye in order to delimitate the structure of different regions. The insets give examples of the type of mesophases encountered. Figure 3 shows vapor pressures and internal energies. As expected for all cases, vapor pressure and internal energy increase with temperature. A change in the slope of this graphs occurs along the solid-fluid phase transition. An analysis of the cluster sizes in the fluid-solid coexistence region was made to determine the ratio in which the enantiomers solidify. Practically both enantiomers were found in the same proportion in homochiral clusters, which means that the chiral resolution is always symmetric. These results show that chiral resolution is possible in symmetric particles with an asymmetric short ranged potential. However, the chirality of the potential in this kind of particles does not guarantee chiral resolution. One example of this fact is treated below for racemic mixtures of P90 and P90* molecules. For these racemic mixtures, the simulation results for the site number densities ρ* = 1, 2, 3, 4, and 4.5 in the temperature range (0.050, 0.225) for increments of T* = 0.025, show again a solid-fluid phase transition but the system solidifies in a heterochiral square arrangement (see Fig. 4). However, the solid phase is not homogeneous as the particles
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FIG. 4. Phase diagram T* vs. ρ* for P90-P90* racemic mixtures. It shows a colloid fluid phase for high temperatures, an arrested fluid for low temperatures and low densities. In the solid-fluid coexistence region, the alternatedhomochiral chains form a solid in an heterochiral square arrangement.
assemble in homochiral chains instead of fully resolving the chirality. The alternation of P90 and P90* chains make the solid achiral. An explanation of this lack of chiral resolution might be that, although the heterochiral cluster made by alternated-homochiral chains and the homochiral cluster have the same energy and structure (see Fig. 5), there are much more accessible configurations of the first kind, thus forming a solid in an heterochiral square arrangement. In Fig. 4, the coexistence lines are drawn through the simulated points of the different isochors in between the temperatures where a change of phase is evident from the snapshots. They are smoothed lines to guide the eye in order to delimitate the structure of different regions. The insets give examples of the type of mesophases encountered. Arrested states are also found in the P90-P90* racemic mixture. The presence of gelation in the patchy particles systems studied here, is in agreement with Bianchi et al.23 which states the fact that systems of small valences states (small number of patchy sites), at low densities and temperatures have a large bond lifetime and does not present a phase transition in these conditions. The large bond lifetimes are a consequence of the directionality, selectivity, and the short potential range of the patchy sites. In the P60-P60* racemic mixture, the distribution of the patchy sites allow the particles to be bonded in pairs for small densities and temperatures, while for the P90-P90* racemic mixture, four particles can be bonded
FIG. 5. Graphical representation of a P90-P90* racemic mixture in the homochiral solid configuration and the heterochiral solid configuration made by homochiral chains. Both configuration are thermodynamically equivalent.
J. Chem. Phys. 140, 194505 (2014)
at the same conditions. Larger values of λ also frustrate chiral resolution in the P60-P60* racemic mixture. The anisotropy of the short ranged attractive part of the sites is responsible for the chiral segregated state. This desired property can be affected by the interaction of several patchy sites producing possible states with lower energy than the effect accomplished by the short ranged potential. In the case of homochiral systems, a solid-fluid phase transition and gel behavior is found for P90 and P60 systems. In each case, the density and temperature values in which phase transitions occur are practically the same than in the corresponding heterochiral case. As represented in Fig. 5, the P90 system and the P90-P90* racemic mixture are thermodynamically equivalent. Something similar happens with the P60 system and the P60-P60* racemic mixture because chiral resolution is equivalent to two homochiral phases coexisting. However, the demixing process causes the equilibration time to be longer in the heterochiral than in the homochiral case. Additionally, the P60-P60* racemic mixture shows surface tension between the coexisting homochiral solids, which is the main difference between the homochiral and heterochiral cases. IV. CONCLUSIONS
Chiral resolution is achieved with anisotropy of short ranged potentials in isotropic molecules modeled by a HD decorated with four patchy sites. The properties of directionality and selectivity that can be defined in patchy particle models can be successfully used to study complex phenomena like chiral resolution and gelation. Particular attention should be paid to the fact that shape anisotropy and long range potentials are not necessary conditions to produce enantioselectivity. The chiral resolution is always symmetric, both enantiomers are found in the same proportion in homochiral solid clusters. These results show that chiral resolution is possible in symmetric particles with an asymmetric short ranged potential. However, the chirality of the potential in this kind of particles does not guarantee chiral resolution. One example of this fact is treated below for racemic mixtures of P90 and P90* molecules. For these racemic mixtures, a solid-fluid phase transition appears as a heterochiral square arrangement with the solid phase assembled in homochiral chains instead of fully resolving the chirality. Although the heterochiral and homochiral clusters have the same energy and structure, entropy considerations might favor the former. Arrested states are found in both racemic mixture, the P90-P90* and the P60-P60*. Larger values of λ also frustrate chiral resolution in the P60-P60* racemic mixture. In the case of homochiral systems, a solid-fluid phase transition and gel behavior is found for P90 and P60 systems. In each case, the density and temperature values in which phase transitions occur are practically the same than in the corresponding heterochiral case. The demixing process causes the equilibration time to be longer in the heterochiral than in the homochiral case. The appearance of surface tension between the coexisting homochiral solids is the main difference between the homochiral and heterochiral cases.
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ACKNOWLEDGMENTS
This research was financially funded by CONACYT grant 168001 “Efecto de la quiralidad, polaridad y anisotropía en el auto-ensamblaje molecular en dos dimensiones.” J.A. Martinez-Gonzalez is grateful to CONACYT for the support given by this grant. We also thank UAM Iztapalapa for the computing time made available by the Super Computing Center UAM-I. 1 I.
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