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Theoretical and Computational Analysis of Static and Dynamic Anomalies in Water-DMSO Binary Mixture at Low DMSO Concentrations Susmita Roy, Saikat Banerjee, Nikhil Biyani, Biman Jana, and Biman Bagchi* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India ReceiVed: October 7, 2010; ReVised Manuscript ReceiVed: December 2, 2010
Experiments have repeatedly observed both thermodynamic and dynamic anomalies in aqueous binary mixtures, surprisingly at low solute concentration. Examples of such binary mixtures include water-DMSO, water-ethanol, water-tertiary butyl alcohol (TBA), and water-dioxane, to name a few. The anomalies have often been attributed to the onset of a structural transition, whose nature, however, has been left rather unclear. Here we study the origin of such anomalies using large scale computer simulations and theoretical analysis in water-DMSO binary mixture. At very low DMSO concentration (below 10%), small aggregates of DMSO are solvated by water through the formation of DMSO-(H2O)2 moieties. As the concentration is increased beyond 10-12% of DMSO, spanning clusters comprising the same moieties appear in the system. Those clusters are formed and stabilized not only through H-bonding but also through the association of CH3 groups of DMSO. We attribute the experimentally observed anomalies to a continuum percolation-like transition at DMSO concentration XDMSO ≈ 12-15%. The largest cluster size of CH3-CH3 aggregation clearly indicates the formation of such percolating clusters. As a result, a significant slowing down is observed in the decay of associated rotational auto time correlation functions (of the SdO bond vector of DMSO and O-H bond vector of water). Markedly unusual behavior in the mean square fluctuation of total dipole moment again suggests a structural transition around the same concentration range. Furthermore, we map our findings to an interacting lattice model which substantiates the continuum percolation model as the reason for low concentration anomalies in binary mixtures where the solutes involved have both hydrophilic and hydrophobic moieties. I. Introduction Binary mixtures, both aqueous and nonaqueous, have received considerable attention in recent years because of their unusual and often spectacularly deviating properties from what can be observed for the pure components they are made of. For example, many substances dissolve easily in mixed solvents, while they are hardly soluble in either of the pure components. It is of no surprise then that the properties of these binary mixtures depend on the mutual concentration of the components. The composition-dependent properties of binary mixtures have been exploited to tune relevant properties of proteins, enzymes and other biopolymers.1-10 The aqueous solution of dimethyl sulfoxide (DMSO) provides a striking example of a mixed solvent system and has aroused much interest over the last few decades. It was not until recently that the interesting properties of DMSO were explored.4 In fact, this compound has rapidly grown in use and, presently, this small molecule has wide applications in diverse fields like cell biology, cryoprotection, pharmacology, etc.14 The interest in this compound is due not only to its unique biological properties but also to the varying properties of water-DMSO binary mixture as a solvent and as a reaction medium. The oxygen atom of the DMSO molecule can interact with water through H-bonding, while on the other hand, the methyl groups of DMSO may induce cooperative ordering in the system by hydrophobic hydration effects.15 The aqueous solution of DMSO, therefore, exhibits strongly nonideal behavior. The nonideality has been reflected in a number of physical properties such as viscosity, density, dielectric constant, translational and rotational diffusion constants, excess mixing volume, surface
tension and heat of formation, to mention a few.16-19 It has been found that the deviation in most of these physical properties occurs at compositions with mole fraction of DMSO in the range XDMSO ) 0.3-0.4. There have been several theoretical studies toward explaining the molecular structure and interactions in aqueous solution of DMSO in this composition range. Neutron diffraction experiments followed by simulation studies have proved the existence of 1-DMSO-2-H2O aggregations in this binary mixture.20 However, several experiments have revealed an appealing anomaly of water-DMSO binary mixture in the low DMSO concentration regime, especially when the mole fraction of DMSO (XDMSO) is in the range 0.10-0.20. DMSO, as a cosolvent of water, increases enzymatic activity at low concentrations, but decreases the same at higher concentrations. Similarly, it is a protein stabilizer at low concentrations, whereas it is a protein destabilizer at higher concentrations.8,9 This crossover occurs typically below XDMSO e 0.20. However, the structural aspects of this mixed solvent at low DMSO content are still unknown. In fact, very few studies have focused on the structure and energetics of water-DMSO mixture in the low concentration regime. Several experiments have also found anomalies in thermodynamic and transport properties in this concentration range. Hernandez-Perni and Leuenberger have detected deviation in properties like relative permittivity, dielectric constant, Kirkwood g-values and Ei/E parameter.21 They have also predicted the presence of a lower percolation threshold at a composition with DMSO volume fraction 34% (i.e., mole fraction XDMSO ) 0.125).
10.1021/jp109622h 2011 American Chemical Society Published on Web 12/27/2010
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On the contrary, infrared spectroscopy as well as density measurements show that small quantities of DMSO have little effect on the water hydrogen bonding. Many studies even lead to the conclusion that small amounts of DMSO act as a “structure breaker” in water.22 Specially designed mass spectrometric experiments to understand the microscopic structure of water-DMSO binary mixture have revealed the existence of clusters of DMSO which starts forming near XDMSO ≈ 0.10. Shin et al. used a specially designed mass spectrometric technique that offers a direct way to isolate the clusters from the solvent mixture. They measured the changes in the degree of the cluster size distribution as a function of the mixing ratio of the binary mixture. They found that the clustering distribution varies nonlinearly with the solvent composition, exhibiting the existence of a critical value of the mixing ratios, at XW ≈ 0.91-0.93 (i.e., XDMSO ≈ 0.07-0.09), where drastic changes occur in the cluster distribution.23 They did not, however, realize the continuum percolation phenomenon. Several simulation studies have been directed toward the understanding of such anomalous behavior. Vaisman and Berkowitz simulated diluted mixtures (XDMSO ) 0.005, 0.04, 0.2) of DMSO. They observed that the lone pair of electrons on the oxygen atom of DMSO forms two hydrogen bonds with two water molecules.15 This association leads to certain local ordering. These hydrogen bonds are found to be more stable than the hydrogen bonds among two water molecules in the bulk. Soper and Luzar investigated the structure of water in aqueous solution of DMSO (XDMSO ) 0.21 and 0.35) by neutron diffraction. They determined the partial structure factors and pair correlation functions, and found that the water structure is not strongly affected by the presence of DMSO.24 However, the percentage of water molecules which are hydrogen bonded to themselves is substantially reduced compared to pure water, with a large proportion of the hydrogens available for bonding associated with the lone pairs on the DMSO. Following the neutron diffraction study, Luzar and Chandler performed molecular dynamics simulations at those two compositions (XDMSO ) 0.21 and 0.35). They studied the structural properties, H-bond distribution and H-bond dynamics.20 They found that the local tetrahedral ordering of the water molecule is preserved at both the concentrations. They also found the presence of 1-DMSO-2-H2O complexes of nearly tetrahedral geometry. Borin and Skaf presented simulation results of water-DMSO mixtures over the whole range of composition, focusing on the study of local structures, H-bond distribution and dynamical properties.25 Other groups40,41 have also concentrated on the dynamical properties of this binary mixture. While these studies relate explicitly the structural changes with the associated anomalies, none of these groups have investigated the local structure at low DMSO content. Previous studies from our group have shown that there is a sharp increase in pair hydrophobicity as small amounts of DMSO are added to water up to a composition of XDMSO ≈ 0.15. The increase in pair hydrophobicity was measured by an increase in the depth of the first minimum in the potential of mean force (PMF) between two methane molecules.26-28 However, this enhanced hydrophobicity weakens at higher DMSO concentration. We have also shown that several other properties like diffusion coefficient of DMSO, local composition fluctuation of water, etc. show a transition at XDMSO ≈ 0.10-0.20 in the pure binary mixture of water and DMSO.29 Although the signature of a weak phase transition in the low DMSO concentration regime is found to be quite prominent from this
Roy et al. simulation result, as discussed above, the microscopic details of the system are still elusive and demand a more detailed study. In this article, we report some fascinating results which elucidate the underlying reason of the anomalous behavior of water-DMSO binary mixture at XDMSO ≈ 0.10-0.15. We find a continuum percolation transition in an explicit model of water and DMSO at the same concentration range. Our simulation study envisages the existence of “micromicelle” like arrangements in the water-DMSO binary mixture at around the 5-10% DMSO concentration range that disappear with further addition of DMSO. The formation of large spanning clusters of DMSO molecules starts to percolate through the association of CH3 groups of DMSO. As a result, we find a significant slowing down in the decay of rotational auto time correlation functions (SdO vector of DMSO and O-H vector of water). Furthermore, we map our findings to a suitable lattice model calculation which allows us to understand the relevant reason(s) of low concentration binary mixture anomalies. In a series of recent experimental studies, Biswas and coworkers found that for both water-ethanol and water-TBA11-13 binary mixtures, striking dynamic anomalies occur at a low solute concentration range. The anomalies can be captured by the spectroscopic techniques. They have also shown that the rotational anisotropy in such systems has a fast component which becomes faster and a slow component which becomes slower with increasing solute concentration. The structural and dynamic anomalies were found to occur at 10% mole fraction for water-ethanol mixture and 4% for water TBA mixture. Biswas and co-workers attributed these anomalies to a “structural transition” in the solvent. The precise nature of the transition, however, was left unclear. Thus, all these three binary mixtures, water-DMSO, water-ethanol, and water-TBA, show similar kinds of anomalies. Here we show that for water-DMSO these anomalies have their origin in a continuum percolation type transition that leads to the formation of clusters of DMSO spanning through the entire solution. The transition is driven partly by the hydrophobic interaction between the two methyl groups of DMSO but also facilitated by the formation of the DMSO-(H2O)2 association complex observed earlier. II. Simulation Details All the simulations in this study have been done at 300 K and 1 bar pressure. The extended simple point charge model (SPC/E) model30 was employed to study the water-water interaction. Full atomistic details have been retained for every molecule in the simulation except for the hydrogen atoms bound to carbon atoms in DMSO. The hydrogen atoms attached to the carbon atoms were treated as united atoms within the GROMOS96 53a6 force field.31 To perform the molecular dynamics simulation, we have chosen the Groningen Machine for Chemical Simulation (GROMACS, version 4.0.5) software package, which is a highly efficient and scalable molecular dynamics simulation engine.32-35 We prepared water-DMSO mixtures at various concentrations in cubic boxes, with sides 3.0 nm. We took a total of ∼1000 molecules, but as we varied the composition, the number ratio of DMSO molecules and water molecules changed accordingly. After steepest descent energy minimization, the NVT system was equilibrated for 2 ns. The temperature was kept constant using the Nose-Hoover thermostat.43,44 It was followed by an NPT equilibration for 2 ns using the ParinelloRahman barostat.42 The production run was then carried out with the NPT system for 20 ns. All properties were extracted from the trajectory of the final production run.36
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Figure 2. Snapshot of spatial orientation of water-DMSO binary mixture at XD ) 0.15. In this snap, one DMSO is randomly selected and its neighboring DMSO and water molecules are captured within 10 Å cutoff. The molecules that can form an H-bond (1.98 Å) are connected. Here methyl groups are colored as blue, oxygen as red, hydrogen as white, and sulfur atom as yellow. Note that the “micromicellization” has been started clustering three methyl groups of DMSO.
Figure 1. Rotational auto time correlation function (TCF) in semilog plot at various DMSO contents. (a) TCF for SdO bond vector of DMSO and (b) TCF for O-H bond vector of water. Note the slow dynamics for both the TCFs and the emergence of long time tail for water TCF at 20% DMSO concentration.
All the above simulations used a time-step of 2 fs. Periodic boundary conditions were applied, and nonbonded force calculations employed a grid system for neighbor searching. Neighbor list generation was performed after every 5 steps. A cutoff radius of 1.3 nm was used both for neighbor list and van der Waals interaction. To calculate the electrostatic interactions, we used particle mesh Ewald (PME)48 with a grid spacing of 0.16 nm and an interpolation order of 4. III. Numerical Results and Analysis A. Rotational Dynamics of O-H and SdO Vectors. As the dynamical properties provide us significant information about the structural properties of the system, we first analyze the average time autocorrelation functions of the SdO bond vector of DMSO and O-H bond vector for water at different concentrations of the mixed solvent. The correlation function Ci(t) (i ) O-H or SdO) is defined as
Ci(t) )
〈ei(0) · ei(t)〉 〈ei(0) · ei(0)〉
(1)
Here ei(t) is the unit vector of the corresponding bond at time ) t. The autocorrelation functions are shown in Figures 1a and 1b for SdO and O-H bond vectors, respectively in semilog plot. We find a gradual slowing down of the dynamics for the SdO vectors O-H bond vectors with increasing DMSO
concentration (Figure 1a and 1b) that originates from the formation of a stable structure comprising DMSO molecules. A significant slowing down of the dynamics starting from 10% DMSO concentration (Figure 1a) suggests that at this particular composition range the cavity in the H-bonding network of water is occupied by the alkyl groups of the clustered DMSO molecules, resulting in a more compact solvation environment. The increased compactness naturally stabilizes the waterDMSO binary system. We find a similar slowing down in the orientational dynamics of the O-H bond vector of water molecules in the system with increasing DMSO concentration. The most appreciable change occurs in the composition range, XDMSO ≈ 0.20, where it shows a considerable slowing down of the dynamics with a long time tail in the relaxation behavior. It indicates the involvement of water molecules through hydrogen bonding in the spatial arrangement of DMSO molecules and a dramatic nonexponential behavior becomes prominent after XDMSO ≈ 0.20. These results clearly suggest the emergence of an enhanced solution structure with cooperative stabilization by both components in the concentration range of XDMSO ≈ 0.10-0. 20. B. Micromicellization. In order to investigate the reason for slow dynamical behavior even at 5% of DMSO concentration, microscopic structural orientation of water-DMSO complex is explored. Upon selecting one DMSO molecule randomly we have looked into the spatial arrangement of the neighboring DMSO and water molecules within a certain cutoff (10 Å). Interestingly here we observe the tendency of DMSO molecules to form a “micromicelle” like structure involving three or four methyl groups (Figure 2). At low concentration micromicellization occurs when a few of the DMSO molecules form aggregates which are solvated by water molecules and continues until 10-12%. Since each aggregate uses a lot of water molecules for solvation, beyond a certain DMSO concentration, there are not enough water molecules for solvation of each micromicelle. So free energy consideration leads us to expect that these microdroplets of DMSO break up their micelle-like structure and form an extended structure. This is the reason why
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Roy et al. decreases monotonically with increasing DMSO concentration. This indicates, with increase in DMSO concentration, the solution structure generally becomes more stable with a continuum percolation like transition at the concentration range of 10-15%. The onset of such structural transition is also evident in the composition dependence of the diffusion coefficient of DMSO, local compressibility, average CH3-CH3 contact pairs (by figures similar to Figure 3),29 etc. All these quantities show two separate branches in their composition dependence with a deviation in the concentration range of 10-15%.
Figure 3. Mean square deviation of total dipole moment as a function of DMSO concentration. Note the two separate branches in the plot, one below XDMSO ≈ 0.10 and the other one above XDMSO ≈ 0.15. Deviation in the concentration range XDMSO ≈ 0.10-0.15 region is notable. Here the X-axis label, xD, represents the mole fraction of DMSO (XDMSO).
at low concentration of DMSO, the effect is strong because each DMSO added goes on to make its surrounding water molecules slow. This visual inspection of micromicellization is indeed a new scenario captured here but has yet to develop by further analysis. Moreover, after XDMSO ) 0.15, cluster size increases rapidly and string formation through methyl group aggregation is found which penetrates H-bond network of water. Near XDMSO ) 0.20, the chain length becomes appreciably large and a significant number of water molecules are involved in making strong H-bond with DMSO. This can give rise to long time tail as observed in Figure 1b. C. Fluctuation of Total Dipole Moment. Next, we calculate the root-mean-square fluctuation of the total dipole moment of the system. The fluctuation of total dipole moment of the system is obtained at ith step as
δMi ) |Mi | - 〈|M|〉
(2)
and the mean square fluctuation is obtained by
〈(δM)2〉 )
1 Nstep
∑ (δMi)2
(3)
IV. Toward a Microscopic Model A. CH3-CH3 Aggregation and Percolation Transition. Soper and Luzar have demonstrated through a neutron diffraction study of DMSO/water mixtures that although there is clearly some disordering of the water structure, the broadly tetrahedral coordination of water molecules remains intact. This is because a part of the hydrogen bonding has simply been transferred from the water/water complex to water/DMSO complex, and the proportion of this transfer increases with increasing concentration of DMSO.24 We have already mentioned that this binary mixture forms DMSO · 2H2O complexes. At the same time, due to enhanced hydrophobicity, DMSO molecules tend to come closer through methyl-methyl aggregation.29 We have shown in an earlier study that, in this binary mixture, the potential of mean force between two methane molecules exhibits a pronounced enhancement in the attractive force between them around mole fraction equal to 0.15. We now combine all of these factors to develop a microscopic explanation of the observed anomalies. In order to quantify the connection between this structural deformation and those observed anomalies we have studied clusters formed by the 1-DMSO-2-H2O complexes across the anomalous concentration range. A considerably large span of cluster formation has been detected at around 15% DMSO concentration range. In order to explain the anomalies of the said range after every observation, a simple lattice model of water-DMSO binary mixture has been proposed in the framework of the continuum percolation theory. The model is based on site percolation where DMSO-(H2O)2 species is the candidate which forms an infinite cluster in a sea of small water molecules (Figure 4). In a tetrahedral system, the percolation transition occurs at a critical concentration (Pc) given by Flory formula,37
Pc ) 1/(z - 1)
(4)
i
where Mi is the total dipole moment of the system at the ith configuration/step, and its magnitude is given by |Mi| ) (Mx,i2 + My,i2 + Mz,i2)1/2. The root-mean-square fluctuation of the total dipole moment at various compositions of the binary solvent system is plotted in Figure 3. Composition dependence of the mean square fluctuation shows the presence of two separate, almost linear, branches with a notable break at the composition XDMSO ≈ 0.10-0.15. Below 10% concentration of DMSO, mean square fluctuation decreases monotonically with increasing DMSO concentration indicating the formation of gradually stable solution structure. A sudden increase in the mean square fluctuation in the concentration range of 10-15% implies an enhanced fluctuation in the solution structure. This suggests the emergence of continuum percolation kind of transition in the system. Above 12.5%, the composition dependence again
Now the species shown in Figure 4a has coordination number z ) 8. Hence if that undergoes a percolation-like transition, then it should occur at Pc ) 14.28% with respect to that species. Thus in water-DMSO binary mixture such a transition should occur at around 12-15% DMSO concentration region. The predicted transition region is thus very close to the concentration range of the observed anomalies. The above Flory estimate of the percolation probability is a mean field estimate which neglects formation of rings among the particles and is strictly valid only when the dimensionality is infinity. Usually, the Flory estimate provides a lower bound for the percolation probability. B. Detection of Percolation Threshold. In essence, the percolation based explanation presented here needs to be substantiated by a more accurate method to explicitly detect the percolation threshold. In the following, therefore, we attempt
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Figure 5. Average maximum cluster size as a function of DMSO concentration. Note the two separate branches in the plot. A significant increase in the 10-12.5% composition range designates the occurrence of a continuum percolation transition induced by CH3-CH3 aggregation.
m(r) ∼ rdf Figure 4. Schematic view of hydrogen bonding in water-DMSO binary mixture. (a) 1:2 DMSO-water complexes. (b) Network formation through H-bonding and methyl-methyl aggregation of DMSO molecules among the species shown in Figure 4a.
to establish the percolation phenomenon by using several efficient strategies, such as evaluation of maximum cluster size and fractal dimension analysis. We have observed that at said, rather small, composition range the average coordination number of methyl groups of distinct DMSO molecules varies between 2.3 and 2.5, indicating the formation of a chainlike extended connectivity.29 Note that this does not also rule out separate aggregation of few DMSO molecules in a distinct place, so one needs to establish the exclusivity of the chainlike structure required for percolation. 1. EWaluation of Maximum Cluster Size. To quantify the proposition of chain formation, we examined the clusters of CH3 groups in the system. In our present calculation, two CH3 groups separated by less than 4.2 Å belong to a single cluster. Here the cutoff distance of 4.2 Å is taken as the Me-Me distance which corresponds to the minimum of Lennard-Jones interaction energy between them. We then calculated the aVerage of the maximum cluster size as a function of composition which is presented in Figure 5. We find again two branches separated by the composition range 10-12.5%. Both in low and in high concentration ranges, the maximum cluster size increases with increasing DMSO concentration. Interestingly, one significant increase in the maximum cluster size is evident going from 10 to 12.5%. This is a typical scenario of continuum percolation transition49 induced by the CH3-CH3 aggregation, and the percolation threshold exists in the concentration range of 10-12.5%. 2. Fractal Dimension Analysis. Next, we exploit the fact that the largest cluster that appears in the system can be characterized as a fractal object, and it has now been wellestablished that an infinite cluster can exist in the system only when the fractal dimension reaches a particular critical Value of 2.53 at the percolation threshold.45-47 The fractal dimension of the largest cluster can be estimated by fitting the cumulative radial distribution function of its constituents as
(5)
where m(r) is the number of constituents that belong to the largest cluster and are located closer than the distance r from a given constituent of this cluster. The m(r) functions resulting from the largest cluster in our present system are shown in Figure 6a. The data of m(r) are fitted to eq 5, and those wellfitted lines are shown in the same figure by solid lines for different DMSO compositions. The obtained df values from the fit are plotted for various mole fraction of DMSO, as shown in Figure 6b. Around XDMSO ≈ 0.15, it just crosses the critical Value of 2.53. This important result conclusively proves that for the present system a true percolation threshold exists at XDMSO ≈ 0.15. Snapshots of the molecular dynamics trajectory reveal the structural features of the aggregation clearly as shown in Figure 7. We indeed find the aggregation of CH3 groups of DMSO and formation of a percolating string which penetrates through the extended hydrogen bonded network formed by water molecules at 15% DMSO concentration. Here DMSO molecules are pinned to the water network by the oxygen atom of the SdO end through a hydrogen bond and aggregate through CH3 groups in the center of a percolating string. On the other hand, snapshot at 10% DMSO concentration shows small clusters of CH3 groups without the formation of a connecting network. This indicates that a percolation threshold exists above 10% for this system. In the next section, a lattice model is proposed considering the structural aspects of the water-DMSO binary system. C. Lattice Model. In order to further substantiate the numerical evidence presented above toward a percolation transition as a general explanation, here we construct a lattice model to mimic and to capture the behavior of the water-DMSO binary mixture. The salient features of the model are listed below, and the model is illustrated pictorially in Figure 8. (i) Here we construct a square lattice where the sites are occupied by a mixture of large and small particles. The sizes are so chosen that the solvent (considered as water) molecules have a coordination number of four while the large (considering the size of DMSO-2H2O species) particles have a coordination number of eight. (ii) The equilibrium properties of this model are determined by its Hamiltonian which considers only the nearest neighbor interactions. Thus, three different types of interactions are
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Figure 6. (a) m(r), the cumulative radial distribution function of methyl groups of DMSO molecules represented at different DMSO concentrations. All the curves are fitted to the eq 5. (b) Fractal dimension of the largest cluster (df) obtained from the fitted curves in panel a for various DMSO concentrations. For a true percolation threshold, crossover of the critical value (2.53) is found to exist around the DMSO concentration range XDMSO ≈ 0.15.
present, i.e., solute-solute interaction, εuu; solute-solvent interaction, εuv; solvent-solvent interaction, εvv. The energy for one solute molecule can be expressed by the Hamiltonian as38,39
Hu,i ) -εuv
∑ nuv,i - 21 εuu ∑ nuu,i
(5a)
where nuv,i is solvent which is in nearest neighbor of the solute i and similarly nuu,i is one lattice site of solute which is in the nearest neighbor of the solute. (iii) The energy of the solvent molecule is given as
Hv,j ) -εuv
∑ nvu,j - 21 εvv ∑ nvv,j
(6)
where nvu,j is one lattice site of solute which is in the nearest neighbor of the solvent j and nuu,j is the solute which is in the nearest neighbor of the solvent. The energy of the whole configuration is given by
Figure 7. Snapshot of the molecular dynamics trajectory at (a) XDMSO ) 0.10 and at (b) XDMSO ) 0.15. In this surface representation all methyl groups of DMSO molecules are colored in blue and the water surface is colored in silver. Penetration through the well-defined hydrogen bonded network of water structure and the percolating string formation is evident from panel b, but only small cluster formation has been observed for panel a.
E)
1 2
(∑H
u,i
i
+
∑ Hv,j)
(7)
j
This energy is further used to move the particles by the Metropolis Monte Carlo algorithm. The solute particles undergo a random walk which would be biased by this energy. The initial configuration is generated by placing randomly the solute molecules in the empty lattice with an occupational probability p, and then the remaining part is filled up by the solvent molecules. Monte Carlo simulations are carried out on this box with periodic boundary conditions. One Monte Carlo cycle consisted of Nsou such random moves, where Nsou is the number of solute molecules present in the lattice. The box is first equilibrated by running 1000 cycles. In the production run, after every 100 cycles one configuration is generated and the basic properties are calculated as described in the next section. The properties are averaged for such 200 configurations.
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Figure 8. Sketch of the lattice model containing large solute particles (violet disks) and small solvent particles (black disks). Each solute particle occupies four lattice sites, and the solvent particle occupies one lattice site. Note the 4 and 8 coordination of the small and large particles, respectively.
The temperature of the simulations is kept at 1.0 (in reduced units), and the interaction parameters as explained in the previous section are assumed as follows: solute-solute ) 1.5, solvent-solvent ) 1.0 and solute-solvent ) 0.8. The behavior of the continuum percolation depends on the choice of interaction parameters. We consider that the solute particles as well as the solvent particles attract one another and they try to aggregate in the absence of the other component. This is assumed as we see that there is attraction between the particles of solvent like water (H-bonding) and also solute particles like DMSO-(H2O)2 species also do attract each other (H-bonding and hydrophobic interaction). The interaction between solute and solute is considered to be more than the solvent-solvent interaction. In the model, the solute has twice the number of nearest neighbors as the solvent has, so the solute-solvent interaction is kept at approximately half the value of the solute-solute interaction. 1. Susceptibility Plot. We have calculated the basic quantities involved in the percolation, i.e., s and ns. The cluster is defined as a group of solute particles connected together. The quantity s is the size of the cluster. ns or the cluster number is defined as the average number of s-clusters per solute particle. Cluster number depends on the occupational probability and is generally written as ns(p). Figure 9 shows susceptibility or second moment with respect to occupational probability. The susceptibility is given by the following expression:
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Figure 9. Plot of susceptibility against the occupational probability. The plot shows a clear divergence around an occupational probability of 0.175.
Figure 10. Plot of percolation probability with the occupational probability of the solute. Percolation starts from an occupational probability of 0.14.
The plot is shown in Figure 10. It suggests that the percolation starts from an occupational probability of 0.14 and the system gets percolated by the occupational probability 0.30. The system starts building up the infinite cluster from the occupational probability 0.14, but all the configurations are not percolated below an occupational probability 0.30. At 0.30 the value of percolation probability is 0.96, which is very close to 1. V. Summary and Conclusions
M(p) )
∑ s ns 2
(8)
s
The summation is done over all possible values of s occurring in the lattice. The plot shows that it is diverging after the occupational probability 0.175, which suggests a value of critical percolation for the system. 2. Percolation Probability. The percolation probability is defined as the fraction of entire cluster taken by infinite cluster for a given occupational probability p. It can be expressed as
P(p) )
number of sites in infinite cluster total number of occupied sites
(9)
Understanding the appearance of anomalies in binary mixtures especially at low concentration range is indeed a challenging problem. In addition to the inherent complexity of studying the structural association in binary liquids, the contribution due to composition variation gives rise to additional difficulty. While this problem has attracted a large number of experimental studies before, we are not aware of any comprehensive computer simulation studies that focus to low concentration anomalies. The main finding of the present work is the observation of a continuum percolation transition which can provide a microscopic origin of these anomalies. At the percolation threshold (XDMSO ≈ 12-15%), a spanning cluster is formed comprising DMSO-(H2O)2 species through methyl-methyl aggregation and hydrogen bonding. The sudden increase in the average
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maximum cluster size at the said composition range is in agreement with the continuum percolation phenomenon. We also observe an appreciable slowing down in the decay of rotational auto time correlation functions of SdO bond vector of DMSO and O-H bond vector of water and an unusual behavior in the mean square fluctuation of total dipole moment around the same concentration range. More interestingly disappearance of “micromicelle” structure after a certain concentration of DMSO followed by the appearance of long chain comprising DMSO molecule facilitates the continuum percolation transition. We have constructed an interacting lattice model to capture the essence of the continuum percolation in the binary mixture at low DMSO concentrations in the viewpoint of lattice model formalism. In our model, the calculated susceptibility and percolation probability against occupational probability show divergent-like behavior (typical of continuum percolation transition) in the similar solute composition range. The main aspect of our lattice model calculation is the solute induced percolation transition. Therefore, many fluctuations in the environment surrounding the solute molecule may obscure the signature of continuum percolation. The stronger solute-solute interaction is the only factor that drives the solvent rearrangement around the solute. In contrast, we can draw a similar analogy specifically in water-DMSO binary mixture that the interplay between the hydrophobicity and H-bonding interactions governs the structural association and the dynamics in binary mixture. Thus, lattice model calculations of binary mixture provide a significant conceptual unification. It is demonstrated that the use of percolation theory gives new insights that can resolve the mystery of the anomalous behavior of aqueous binary mixtures at low solute concentrations. We have already observed that several other aqueous binary mixtures display striking dynamical anomalies in the low solute concentration range. In all the known mixtures that display such anomalies, solute molecules contain both hydrophobic and hydrophilic parts. We tend to believe that the mechanism proposed here is universal in the sense that all these low concentration anomalies are driven by a continuum percolationlike transition which gives rise to a structural morphogenesis in the solution.
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Acknowledgment. We thank Mr. M. Santra, Mr. R. S. Singh and Mr. R. Biswas for useful discussions. This work was supported in parts by grants from DST and CSIR, India. B.B. acknowledges support from a JC Bose fellowship. We acknowledge the help from the anonymous reviewer for pointing out the efficient techniques of percolation detection. References and Notes (1) (2) 372. (3) (4) (5) (6) 9726. (7)
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