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Ring structure analysis of ethanol–water mixtures
Journal of Molecular Liquids 211 (2015) 812–820
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Ring structure analysis of ethanol–water mixtures Orsolya Gereben Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP HAS), Konkoly Thege út 29-33, H-1121 Budapest, Hungary
a r t i c l e
i n f o
Article history: Received 11 July 2015 Received in revised form 3 August 2015 Accepted 4 August 2015 Available online xxxx Keywords: Molecular dynamics simulation Hydrogen bond Morphological analysis Ring structure
a b s t r a c t Systematic molecular dynamics studies of ethanol–water mixtures with 20–80 mol% ethanol content and of pure ethanol and water, using three different water models and the OPLS-AA ethanol force field for ethanol, have been performed in our recent work. The systems displaying the best agreement with the experimental total scattering X-ray structure factors were selected at each composition for further analyses. Morphological characterization of the hydrogen-bonded clusters reveals that 60–95% of their molecules are involved in ring formation. The average number of rings necessary to describe the ring structure in the main, percolating cluster is ~2200 for pure water for the investigated system size. Only 6% of this is found for 80 mol% ethanol content, which is increasing with decreasing ethanol concentration to 56% for the 20 mol% ethanol–water mixture. In the mixtures rings with 5 molecules, while in pure water, 6-membered rings occur with the highest probability. A few pure ethanol rings exist only above ≥60 mol% ethanol concentration, with the most popular ring size of 5 molecules. Pure water rings exist in the whole concentration range, with the most frequent ring size increasing from 4 to 5 molecules with increasing water concentration. © 2015 Elsevier B.V. All rights reserved.
1. Introduction As aliphatic alcohols have an amphiphilic character, possessing both hydrophobic and hydrophilic parts, their behavior is rather complex. They are capable of forming hydrogen bonds themselves, as well as with other molecules, like water. The structure of alcohol–water solutions has been studied widely by both experimental [1–6] and theoretical methods [7,8]; the emerging picture is somewhat confusing and contradictory. Although lower alcohols are fully miscible with water over the entire concentration range, the homogeneity of the mixtures at the molecular level is at question [9–11]. Hydrogen bonding results in the formation of clusters of varying shapes, of which rings are perhaps the most attractive objects to recognize and investigate. Ring perception in chemistry has an extensive literature; the interested reader will find a short description of the terminology and methods used in Appendix A. Characterization of large networks with numerous rings is not routinely performed, as it is computationally demanding, also in terms of coding. To our best knowledge, no similar work for ethanol–water mixtures has been performed so far. In our preceding publication, morphological analyses of pure ethanol were provided [12]. Matsumoto [13] investigated the network topology of water and performed ring analysis: 6membered (6 molecules, 12 atoms) rings were found with the highest probability.
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http://dx.doi.org/10.1016/j.molliq.2015.08.010 0167-7322/© 2015 Elsevier B.V. All rights reserved.
Guo et al. [9] investigated the X-ray absorption and selectively excited X-ray emission spectra of an equimolar methanol–water mixture, and suggested that the mixture mainly consists of 6 and 8-membered rings and chains. We performed a comprehensive study of ethanol–water mixtures, containing 20 to 80 mol% ethanol. In the first part of the work [14] a series of molecular dynamics (MD) computer simulations for the mixtures, and for pure ethanol and water was conducted, applying one ethanol and three different water force fields. The aim was to find the models that provide the best agreement with the corresponding experimental total scattering X-ray structure factors (the data was provided by László Temleitner [15]). In each mixture the all-atom optimized potential for liquid simulations (OPLS-AA) [16] was used for ethanol, combined with the following water force fields: (1) the extended simple point charge (SPC/E) model [17], (2) the improved transferable intermolecular potential with four particles (TIP4P-2005) [18] and (3) the simple water model with four sites and Drude polarizability (SWM4DP) [19]. Water potential models were distinguished on the basis of deviations between calculated and measured total scattering X-ray structure factors, aided by ethanol–water pair binding energy comparison. No single water force field could be found to provide the best agreement with the experimental data over the entire concentration range: for the 80 mol% ethanol containing mixture the SWM4-DP, for 60 mol% the SWM4-DP and the TIP4P-2005, whereas for the 40 and 20 mol% ethanol containing mixtures the simulations with the TIP4P-2005 water force field provided the closest agreement with the experiment. Analyses of the partial radial distribution functions (PRDF), center of molecule distributions, coordination numbers, number of H-bonds, and several
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perception, whereas Appendix B is about the effect of percolation on hydrogen bond determination and ring perception.
Table 1 Simulation names and densities for the MD simulations. cE (%)a
Simulation name
nEb
nWc
ρE (Å−3)d
ρW (Å−3)e
ρ (Å−3)f
ρliq (g/cm3)g
100 80 60 40 20 0
Et100 Et80S Et60S, Et60T Et40T Et20T Et0S, Et0T
1331 1226 1093 889 571 0
0 316 715 1328 2280 3993
0.0103 0.0096 0.0088 0.0073 0.0047 0.0000
0.0000 0.0025 0.0057 0.0109 0.0188 0.0330
0.0930 0.0940 0.0960 0.0985 0.0990 0.0990
0.790 0.799 0.816 0.837 0.841 0.987
a b c d e f g
813
Ethanol concentration in mol%. Number of ethanol molecules. Number of water molecules. Molecular number densities of ethanol molecules. Molecular number densities of water molecules. Atomic number density of all the atoms in the simulation cell. Density of the liquids.
bivariate (distance-angle) distributions were performed. These analyses all concerned pairs of molecules. The occurrence of hydrogen bonds (HBs) determines the number and size of aggregations (‘clusters’), and therefore the cluster structure depends on the applied HB definition. There is not an exact rule for defining/identifying hydrogen bonds; several approaches can be found in the literature. The basis of the HB definition is mostly geometric: the Hbonding O⋯H and O⋯O distances should fall into a certain distance range, and sometimes there is an additional constraint on the O\\H⋯O angle, as well [20,21]. Alternatively, a cutoff value can be defined for the H-bond energy [22,23], and the two (geometric and energetic) approaches can be combined, as well [24]. Sometimes rather arbitrary definitions are used in combination with the abovementioned criteria [25]. According to recent considerations the transient breaks and bonds of the network are corrected based on the dynamic history of the HB [26,27]. As a continuation of our recent study [28], here detailed analyses of the hydrogen-bonded clusters/network were performed for the most successful MD simulations at each concentration, using four different geometric hydrogen bond definitions. These definitions differ in terms of the presence and/or the cutoff value of the angular constraint. Based on this work, the most feasible definition seemed to be the strictest one, with O\\H⋯O N 150°, similarly to the works of Noskov et al. [20] and Soper et al. [21]. This HB definition is used throughout the present study. This paper is organized in the following way: in Section 2, methods are discussed; in Section 3.1, morphological characterization of the clusters is found; in Section 3.2, connectivities of the atoms involved in cycle formation are given; while in Section 3.3, the ring size distributions are presented. Appendix A will give some details concerning ring
2. Methods 2.1. Molecular dynamics simulations Details of the MD simulations are given in our previous paper [14], here only the most important data will be repeated. All the MD simulations were performed by the GROMACS 4.0 simulation package [29], in NVT ensembles at T = 293 K, with time step dt = 2 fs. The temperature was maintained by a Nose–Hoover thermostat [30] with τT = 0.5 ps. The geometry of water molecules was maintained by the SETTLE algorithm [31]. The long-range electrostatic interactions were taken into account by the particle-mesh Ewald [32,33] (PME) algorithm. The cutoff value of 11 Å, used in the parameterization process for alcohols by Jorgensen et al. [16], was adopted. The simulation length was 2000 ps in each case. Particle configurations in the ‘production’-phase were collected 20 ps apart and in the end, 76 configurations were used for calculating the average morphology and ring size distributions. Cubic simulation boxes have been applied; the number of molecules is given in Table 1, along with the number densities matching the experimental values, for all the studied concentrations. In the MD calculations all-atom force fields were applied. For ethanol the OPLS-AA [16] force field was used; concerning water, results for two different water force fields, TIP4P-2005 [18] and/or SWM4-DP [19] were applied, depending on concentration. As mentioned before the basis for selection was the quality of agreement with the experimental total scattering X-ray structure factor [14]. All in all, five ethanol– water mixture simulations (with SWM4-DP force field for the 80 and 60 mol% ethanol concentrations, and with the TIP4P-2005 force field for the 60, 40 and 20 mol% ethanol concentrations), two pure water simulation with both force fields and one pure ethanol simulations were carried out. The pure ethanol and water simulations were needed as references. The equilibrium molecular geometries for the water force fields and full specification of the OPLS-AA force field parameters can be found in the literature [16,18,19,29]. Calculations reported here will be identified by their ethanol content and by the first letter of the water force field applied, so for example Et60S will refer to the 60 mol% ethanol containing mixture where the MD simulation was performed using the SWM4-DP water force field (Table 1). 2.2. Cluster analysis The upper limiting value of the distance ranges for the HB definition was determined from the minima occurring after the first peak of the O\\O and first intermolecular peak of the O\\H partial radial
Fig. 1. (a) The average number of main clusters/configuration versus the nullity, the labels are given in the order of increasing peak positions, the peaks of Et0T and Et0S are overlapping. (b) The distribution of the smaller clusters/configuration.
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distribution functions, and an additional angular restriction of O\\H⋯O N 150° was applied as well. The analysis of the hydrogenbonded clusters/network was performed by a C++ computer program specifically developed for this purpose. The term ‘cluster’ is applied only for molecules connected by hydrogen bonding (i.e., lone molecules are not considered as clusters). During the HB determination the simulation box was treated as an isolated system as in the work of Geiger et al. [22], namely that only one instant of the periodic box – and consequently each hydrogen bond – was used, but the hydrogen bonds were determined using the minimum image convention. Cycle perception was also performed by the same computer code; see Appendix A for some details. As it was found in another, separate part of this study [28], percolation occurs in all the ethanol–water mixtures and in pure water. The effect of percolation on building the HB network and on the ring perception is discussed in Appendix B. 3. Results and discussion First the hydrogen-bonds were determined, then the clusters (groups of molecules connected by HBs) were identified. Percolation occurs in all the mixtures and in pure water, where each configuration has a percolating main cluster, as well as smaller clusters beside that [28]. 3.1. Characterization of the cyclic and acyclic parts Ring structures can be described by graph theory, where the configuration is an undirected graph, ‘vertices’ are the atoms and ‘edges’ are the bonds between them. Cycles in chemistry are often referred to as rings (see Appendix A for a brief description of the terminologies and methods used). Percolation in the water-containing systems presented some difficulties regarding the definition of ‘nullity’ (see below) and ring perception: this is discussed in Appendix B. Nullity, μ, is the maximum number of linearly independent cycles, which form the basis of the cycle space (if there is any). μ was determined for each cluster according to μ = E − V + 1 [34], where E is the number of bonds (edges of the graph), and V is the number of atoms (vertices). The average number of main, percolating clusters/configuration with given nullity can be seen in Fig. 1a. Panel b displays the distribution of the smaller clusters, and it is important mainly for the simulations with high ethanol concentration. From Fig. 1 we can see that only pure ethanol has a substantial number of acyclic clusters (μ = 0) or clusters with only one cycle, and there
Fig. 2. The percentage of molecules in the main cluster involved in cycle formation in the ethanol–water mixtures and pure water, and the excess percentage of water (mol% water in cycles in the main cluster–mol% water in the main cluster).
Table 2 Classification of the clusters based on their ring structure and acyclic parts. Name Description LC BC OIR IR CR CRS ICR ICRS
Linear chain Branched chain Only isolated ring Isolated ring + acyclic side chain(s) Composite ring + acyclic side chain(s) Composite rings + possible acyclic side chain(s) Isolated ring(s) and one composite ring + possible acyclic side chain(s) Isolated ring(s) and more than one composite rings + possible acyclic side chain(s)
are no clusters with more than one cycle at all (μ = 4 × 10−4). The nullity for the main clusters forms peaks, and is increasing from μ = ~125 (Et80), ~2250 (Et0S) and ~2200 (Et0T), as it is visible from Fig. 1a. This means ~ 6% (Et80S), 15% (Et60T), 24% (Et60S), 30% (Et40t) and 56% (Et20T) of the averaged water nullity. The average nullity for Et0S and Et0T hardly differs, while Et60S has substantially larger value than Et60T. This is in agreement with our previous analysis of these mixtures [14,28], where we found that although for this concentration both SWM4-DP and TIP4P-2005 water force field simulations reproduce the experimental total scattering X-ray structure factor equally well, the structure created with the SWM4-DP water model has a higher connectivity. It has to be noted, that obviously the actual average nullity values depend on the size of the configuration used, but the tendency would be the same for other system sizes as well. The atoms involved in cycles were determined using the mechanical method of Welsh [35] and from this, the number of molecules participating in cycle formation was calculated. The participation in cycle formation in the main cluster is ~60–95% and increasing with increasing water content, with positive excess water involvement of b 10% (see Fig. 2) in cycles. For pure ethanol, less than 2% of the molecules in the simulation box were involved in cycle formation (not shown). The same trend that water is more prone to be involved in cycles was found in methanol–water mixtures as well [11].
Fig. 3. Simple examples of morphological characteristics of different cluster types (no depiction of real clusters).
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Fig. 4. Morphological properties of the clusters: (a) percentages of the cluster types in the main clusters; (b) the number of secondary cyclic clusters, as a function of their morphology; (c) the number of acyclic side chain/main cluster with a given morphology; and (d) the average (columns, left y-axis) and maximum sizes (marker + line, right y-axis) of acyclic clusters (acyc. clust.), or side chains for the main (main side ch.) or secondary cyclic (sec. side ch.) clusters. ‘Size’ is the number of molecules involved.
Fig. 5. Cyclic connectivity histograms (a) of the ethanol oxygen atoms as percentage of the total number of ethanol molecules at the given concentration; (b) of water oxygen atoms as percentage of the total number of water molecules at the given concentration; (c) of ethanol molecules having ethanol neighbors as percentage of the total number of ethanol molecules at the given concentration; and (d) of water molecules having water neighbors as percentage of the total number water molecules at the given concentration. The white ribbons above the other series are the total percentage of the cyclic atoms/molecules contributing to the non-zero histogram bins.
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Clusters were classified according to their basic morphological features (the presence and type of the ring structures and acyclic parts). Names and descriptions of the most popular ones are given in Table 2, whereas their schematic pictures can be seen in Fig. 3. ‘Composite ring’ means edge-joined rings. If a cluster contains cyclic parts separated by edges that are not part of a cycle then it contains more than one assembly of cycles, like CRS, ICR and ICRS. In the case of the systems with higher ethanol content the main cluster consists of several ring assemblies (see Fig. 4a), but with increasing water concentration more and more of the main clusters will contain only one large composite ring assembly (CR). Differences between the Et60S and Et60T calculations can be explained by the higher connectivity of Et60S (see also above). On the other hand, most of the smaller clusters are acyclic linear type (LC). Only Et100 and Et80S have measurable amounts of branched linear clusters (BC) (as for branching, at least three molecules are needed), and small cyclic clusters, see Fig. 4b. Considering the number of acyclic side chains for the main clusters (shown in Fig. 4c), the number is roughly the same for the different types of main clusters belonging to different configurations of the same simulation. The highest number of acyclic side chains can be found in the Et40T system: this is the sample with the lowest water content where more than one composite ring assemblies in the main cluster can be found. Seemingly, the higher number of hydrogen bonds compared to the Et60 systems not only increased the cyclic connectivity by joining cyclic assemblies, but also increased the number of acyclic side chains. For all the systems, 80–100% of the acyclic branches were linear, and only b20% were branched. Although there are a few larger acyclic clusters in the high ethanol content systems, the average size (number of molecules involved) is b8 even for these systems, see Fig. 4d. The average size of the side chains for the smaller, cyclic clusters (secondary cyclic) is usually higher than that of the side chains of the main cluster. On the other hand, their maximum size is considerably lower than the maximum side chain size of the main clusters; the latter is ~8 for high water content. Therefore we can conclude that there are many, but small side chains in all the systems. 3.2. Cyclic connectivities The connectivity of the atoms involved in cycle formation was determined. The connectivity of an atom is the total number of bonds (both covalent and hydrogen bonds) originating from it. Therefore the ideal connectivity for the hydrogen atoms is two (one covalent and one hydrogen bond), for ethanol oxygen is three (one covalent and two hydrogen bonds) and for water oxygen atoms it is four (two covalent and two hydrogen bonds). All the hydrogen atoms involved in cycle formation possess the ideal connectivity of two, so no hydrogen bifurcation was observed (not even in pure ethanol [12]). The percentage of ethanol and water oxygen atoms/particle configuration involved in cycle formation versus the ethanol concentration is increasing with decreasing ethanol concentration (Fig. 5a and b, white ribbons). This shows that the higher affinity of water for cycle formation due to its ideal connectivity of four increases the chances of ethanol molecules for participating in cycles. The value for Et60S is higher than that for Et60T, indicating higher cyclic connectivity, in agreement with the results of the combined cyclic + acyclic connectivities investigated previously [28]. While the majority of ethanol O atoms have a connectivity of two (see Fig. 5a), regarding water O atoms only the Et80S calculation brings about a connectivity value of two as the most popular. For samples with lower ethanol content the maximum is at connectivity of three, indicating composite ring structures; there are some atoms with higher than ideal connectivity, indicating complex ring systems where more than three rings are joined. The distribution of the connectivities of ethanol molecules having ethanol neighbors in the ring structure was calculated as well (see Fig. 5c). Having at least one ethanol neighbor, in terms of the percentage
Fig. 6. The observed and the theoretical (denoted as ‘t.’) probabilities of finding ethanol neighbors around an ethanol molecule and water neighbors around a water molecule in the ring structure.
of ethanol molecules at the given concentration (see white ribbon in Fig. 5c), has a maximum at 60 mol% ethanol concentration, Et60S being higher than Et60T. On the other hand, the percentage of water having water neighbor is monotonically increasing with increasing water concentration (see the white ribbon in Fig. 5d); the value for Et60S is, again, larger than that for Et60T. This difference can be explained the following way. (1) In the case of water the increasing water concentration increases the participation in cycle formation, as well as the chance of two water molecules being neighbors in the cycle, as water is the main agent causing the formation of ring systems in these mixtures. (2) Ethanol, on the other hand, in itself does not form extensive ring structures, and while the addition of some water (concentration range Et80, Et60) significantly increases ring formation, as well as the chance of neighboring ethanol molecules in rings, the decreasing ethanol concentration will diminish this chance for the lower ethanol concentration range. The number of ethanol molecules connected to ethanol in cycles is most frequently one, and, to a lesser extent, two, forming two or three molecule pure ethanol ‘islands’ in the ring structure. In the case of
Fig. 7. Size distribution (per particle configuration) of the relevant cycles. The first peak is shown in the inset. The x-axis displays the number of atoms, s, forming the rings.
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Fig. 8. Size distribution (per particle configuration) of the relevant cycles, (a) containing only ethanol molecules and (b) containing only water molecules. The x-axis displays the number of atoms, s, forming the rings.
water the maximum is moving from one water neighbor to three with increasing water concentration. Examining the ratio of ethanol molecules not having ethanol neighbor in the ring structure at all, the values increase from 6 to 55% with decreasing ethanol concentration (Fig. 5c). Analogously, for water molecules not having water neighbor in the ring structure the ratio is increasing from 0.4 to 25%, see Fig. 5d. From a comparison of the measured selectively excited X-ray emission spectra of equimolar methanol–water mixture to the spectra from theoretical models, Guo et al. [9] suggested that there are water molecules joining methanol chains together to form rings, although they did not make any assumption about their fraction. They concluded that despite the miscibility of methanol and water over the entire concentration range the mixing is incomplete at the molecular level. We calculated the theoretical probability of Et–Et and Wat–Wat neighbors in the ring structure, assuming random distribution, for each possible coordination number, based on the number of ethanol and water molecules in the cycles and the observed fraction of the different coordination numbers. Their total (summed according to the coordination numbers) was also calculated for each concentration, and compared to the observed values, see Fig. 6. The observed values are considerably higher, indicating that, indeed, water prefers water and ethanol prefers ethanol neighbors. Thus supports the view that there are molecular level inhomogeneities in the ring structure. 3.3. Ring size distributions Regarding the cyclic parts, the ‘relevant rings’ (the ones that cannot be divided into smaller rings, see Appendix A) were determined for
each system. Their distribution, expressed as the number of rings/particle configuration, is shown in Fig. 7. The peak position is at 10 atoms (5 molecules) for the ethanol containing systems, and at 12 atoms (6 molecules) for pure water, see Fig. 7. The 60 and 80 mol% ethanolcontaining mixtures have a second peak at 60–80 atoms, as well. The Et80S main cluster contains the largest rings, smax = 150 atoms, although the large rings occur with very low probability. For pure water the most popular ring size of twelve atoms is in accordance with previous findings [13,36], supposedly inheriting this feature from the 6 molecule membered rings of Ice-Ih and Ice-II [37]. The occurrence of 5 and 7 molecule membered rings (10 and 14 atoms) is the characteristic of Ice-III [37]. In pure ethanol, ~40% of the rings contain 10 atoms, and the dominance of this ring size prevails with 15– 20% probability in the ethanol–water mixtures, as well. Distributions for rings containing only ethanol or only water molecules were also determined (Fig. 8). The number of pure ethanol rings is very low (Fig. 8a), and they do not exist under 60 mol% ethanol concentration. The maximum is at a ring size of 10 atoms (5 molecules), similarly to mixed ethanol–water rings. The maximum ring size decreases from smax = 22 to 12 with decreasing ethanol concentration. On the other hand, pure water rings exist at even only 20 mol% water concentration (Fig. 8b). The most popular ring size is somewhat decreasing with decreasing water content: it is 10 atoms for Et20T, Et40T and Et60S, similarly to the ring sizes of the mixed rings, and it is 8 atoms for Et60T and Et80S. Interestingly, the largest pure water rings occur in the Et40T system, with smax = 142, and the distribution has an observable peak at s = 140 atoms with ~4 rings/configuration. Et40T is the lowest water content mixture among the systems examined in this work where monotype water percolation occurs [28]. Since
Fig. 9. Part of the HB network in the (a) Et80S and (b) Et20T ethanol–water mixtures displaying the ethanol oxygen (online red, in print dark), water oxygen atoms (online light green, in print light) and hydrogen atoms (small gray) involved in ring formation and the covalent and hydrogen bonds connecting them. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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the largest mixed rings were found in the Et80S system, which is just above the ethanol–water HB network percolation threshold, the occurrence of large rings seems to be the characteristic of systems not far above their percolation threshold, while the HB network is still relatively sparse. The ring structures in equal portion of the simulation box for the highest and lowest ethanol content mixtures are shown in Fig. 9. As it is visible, in the higher ethanol content Et80S mixture the ring structure is relatively sparse, and the individual rings can mostly be identified, while Et20T possesses rather extensive ring network. In both cases the clumping of like molecules can even visually be observed. 4. Conclusions The participation of molecules in cycle formation in the main, percolating clusters has changed between 60 and 95%. The number of rings forming the cycle base was 6% of the average cycle base size of water for 80 mol% ethanol and increased to 56% in the case of 20 mol% ethanol. Morphological analyses revealed that the main clusters contain extensive ring structures. The number of cyclic assemblies (separate ring structures in a cluster) decreases with increasing water content. For pure water only one composite ring assembly can be found with N90% probability, but even in this case, numerous smaller acyclic side chains are present. The size distributions of relevant rings were calculated: the main peak for the ethanol containing systems is at 10 atoms (5 molecules), while for pure water, it is at 12 atoms (6 molecules), in accordance with the characteristic feature of Ice-Ih. Larger rings, up to 150 atoms for Et80S, and 40 atoms for water can also occur with low probability. Rings formed solely from ethanol molecules only occur in small number in the systems with ≥60 mol% ethanol, mostly with a size of 5 molecules. Pure water rings, however, exist over the whole concentration range, with, again, 5 molecule rings as for the mixed rings in the higher water content (higher connectivity) systems, and somewhat smaller, 4 molecule rings, for Et60T and Et80S. The clumping of like molecules in the ring structure occurs, which is even visible from the visual inspection of the configurations. This supports the view that these mixtures are not homogenous at molecular level, at least regarding the ring structures. Acknowledgments This work was supported by the National Basic Research Fund (OTKA; Hungary), under contract no. 083529. Appendix A. The basics of ring perception The structure of atomic ensembles (‘aggregates’, or ‘clusters’) can be described by graph theory, where the configuration is an undirected graph, vertices are the atoms and edges are the bonds between them. Mainly the terminology from the review of Downs et al. [38] is used here. The set of vertices is V = {v1,…vN} and the set of edges is E = {e1,…eN}. Connectivity of a vertex (also called degree) is the number of edges which incident with it. A ‘walk’ is an alternating sequence of vertices and edges; if a walk is closed then it is a circuit. If all edges and vertices occur only once in the walk then it is called a path. A closed path is a cycle. A set of edges and vertices are called an elementary cycle, if it contains all the edges and vertices of the set, and one could trace a closed path crossing each vertex only once [39]. (In chemistry cycles are referred to as rings.) The graph is connected, if for any two vertices there is a path connecting them [40], and the number of connected components is the number of the graph's maximal connected subgraphs. In our context, each cluster or lone molecule is a connected component. From now on for sake of simplicity graph refers to a connected (sub)graph.
First the connection table [38] with dimensions V × Cmax (Cmax is the maximum connectivity, in other words the maximum number of Hbonded neighbors) was generated, which is a compact form of the V × V adjacency matrix A (having aij = 1 if vi and vj are connected by an edge, zero otherwise). The clusters were determined by depth-first random walking from the connection table, and the vertices and edges belonging to them were recorded. The cycles (C) of a connected graph form a vector space called the cycle space [41]. Let E be the number of edges and V the number of vertices. Each cycle can be represented by a vector of E dimension, having 1 at the nth dimension if the given nth edge is in the cycle, and 0 if it is not. The maximal number of linearly independent cycles, the nullity, μ is predicted by the Cauchy formula [34], μ = E − V + 1, and they form the cycle base for the connected graph. (Nullity is also called the set's cardinality, cyclomatic number, first Betti number or Frerejacque number.) As these linearly independent cycles are basis vectors, all cycles of the graph can be expressed by them. The addition of cycles is called the symmetric difference, and it is defined as C1 ⊕ C2 = (C1 ∪ C2)\(C1 ∩ C2). Finding a cycle set is usually done by growing a spanning tree, T of the cyclic graph (G) [38], where T is an acyclic graph containing all the vertices of the graph, but not all the edges, and obviously not containing any cycles. The edges left out are called the chords, which are the minimum number of edges required to be removed from G to turn the cyclic graph G to the spanning tree. According to Kirchoff [42] for each edge of G, where e ∉ T there is a unique cycle in G, which is called a fundamental cycle with respect to T. The fundamental cycles belonging to a spanning tree form a cycle base for the graph G, and called the Kirchofffundamental. Ring perception plays an important role in graph and network theory and has an important application for example in chemistry, therefore several methods were developed [35,43–45]. In this work the mechanical manipulation of the V × E incidence matrix developed by Welsh [35] was applied for the determination of the fundamental cycle basis. It has to be noted that the fundamental basis in most of the cases is not unique, it depends on the order the edges and vertices are represented in the incidence matrix, and it most probably not the basis set (B) with minimal length, L(B) = ∑μi = 1|Ci|. As each fundamental cycle by definition contains one unique edge, for complicated ring structures containing joined (composite), embedded cycles, some of the fundamental cycles will definitely be direct sums of smaller cycles. Therefore the fundamental basis in itself is not a good characterization of the system, and their determination gives us the atoms involved in cycle formation. In the literature of chemical graphs several concepts are introduced regarding which rings are important or meaningful, and this is in some cases subjective, see the review of Downs et al. [38]. A practical and clear concept is the smallest set of smallest rings, SSSR and it is defined originally as the minimum length Kirchoff-fundamental basis [38], though in the chemical literature the definition of SSSR is shifted to the description of the minimum length cycle basis, regardless it can be created from a spanning tree, or not. A program was created to determine the SSSR using the method of Gibbs [44], for the creation of all the cycles by forming 2μ–μ–1 linear combinations of the fundamental basis cycles, and then selecting the μ smallest, linearly independent cycles, but unfortunately, this only could be applied for pure ethanol. In the case of the ethanol–water mixtures the nullity for the main cluster even for the Et80 simulation was N 100, which made this type of calculations impossible both memory and time-wise, in accordance with Deo et al. [46], who showed that finding a Kirchoff-fundamental cycle basis with minimum length is NP-complete. Even if an SSSR could have been found, it is not unique for systems with symmetrically equivalent rings, and bridged systems. For example for adamantane, μ = 12 − 10+1 = 3, meaning that any three of the elementary rings with length 6 can provide an SSSR. Therefore the
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concept of K-rings [41], although called relevant rings [40] were introduced, which are those elementary rings, which appear in one of the SSSR of the graph. The relevant rings are basically elementary rings, which cannot be formed by the direct sum of smaller rings, but some of them can be created by the direct sum of rings of the same size. In this context, all the four size 6 elementary rings of adamantane are relevant rings. Therefore an algorithm was devised to find the relevant rings, which detailed description is out of the scope of this paper, briefly it is using a two-directional breadth-first search starting from each cord of the spanning tree (the unique edges of each fundamental cycle determined by the method of Welsh [35] discussed before) collecting elementary rings that can be potentially relevant rings. The number of rings then was reduced, keeping only the relevant rings. This was done by determining all the brides (series of edges crossing the cycle), and determining, whether the original ring can be formed as the direct sum of a (sub)set of smaller cycles. This approach was computationally more feasible, than the usual way of creating all the possible linear combination of all the available smaller cycles initially found. Appendix B. The effect of the periodic boundary condition on the hydrogen bond determination and ring perception During the HB determination the system was treated as an isolated system as in the work of Geiger et al. [22], namely that only one instant of the periodic box – and consequently each hydrogen bond – was used, but the hydrogen bonds were determined using the minimum image convention. The presence of percolations in most of the investigated systems presents some problems regarding the ring perception. Due to the above-mentioned handling of hydrogen bonds, the infinite open clusters are handled as finite ones, closing on themselves in the cases, where the atoms of either a hydrogen- or a covalently bonded atom pair are located near the opposite side of the simulation box. We could describe this conversion from the infinite open cluster to a finite one as an inverse single cycle unwrapping (SCU) applied by Hamilton and Pryadko [47]. The consequence of this can be the easiest demonstrated for a one-dimensional percolation, which could be perceived as a chain running through the periodic images, and obviously no ring is present due to this infinite chain in the system. Applying the inverse SCU the bond A–B′, which is connecting atom A residing in the simulation box to atom B′, which is the periodic image of atom B situated close to the opposite side of the box, and B–A′ will be broken, and topologically A will be connected to B closing a cycle in the process. Therefore during the ring perception one cycle will be detected. Although in this simple case it would be easy to recognize this false cycle, but in the case of two and three-dimensional percolations with complicated embedded ring systems this is not the case. As from our view point the emphasis is not so much on the absolute number of rings, or the size of the cycle base, than on the dependence of these properties with the concentration, no attempt was made to eliminate these false rings. References [1] K. Nishikawa, T. Iijima, Small-angle X-ray scattering study of fluctuations in ethanol and water mixtures, J. Phys. Chem. 97 (1993) 10824–10828, http://dx.doi.org/10. 1021/j100143a049. [2] N. Nishi, S. Takahashi, M. Matsumoto, A. Tanaka, K. Muraya, T. Takamuku, T. Yamaguchi, Hydrogen bonding cluster formation and hydrophobic solute association in aqueous solution of ethanol, J. Phys. Chem. 99 (1995) 462–468, http://dx. doi.org/10.1021/j100001a068. [3] A. Ben-Naim, Inversion of the Kirkwood–Buff theory of solutions: application to the water–ethanol system, J. Chem. Phys. 67 (1977) 4884–4890, http://dx.doi.org/10. 1063/1.434669. [4] G. Onori, Compressibility and structure of aqueous solutions of ethyl alcohol, J. Chem. Phys. 89 (1988) 4325–4332, http://dx.doi.org/10.1063/1.454816. [5] M. D'Angelo, G. Onori, A. Santucci, Self-association of monohydric alcohols in water: compressibility and infrared absorption measurements, J. Chem. Phys. 100 (1994) 3107–3113, http://dx.doi.org/10.1063/1.466452.
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Ring structure analysis of ethanol–water mixtures Orsolya Gereben Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences (Wigner RCP HAS), Konkoly Thege út 29-33, H-1121 Budapest, Hungary
a r t i c l e
i n f o
Article history: Received 11 July 2015 Received in revised form 3 August 2015 Accepted 4 August 2015 Available online xxxx Keywords: Molecular dynamics simulation Hydrogen bond Morphological analysis Ring structure
a b s t r a c t Systematic molecular dynamics studies of ethanol–water mixtures with 20–80 mol% ethanol content and of pure ethanol and water, using three different water models and the OPLS-AA ethanol force field for ethanol, have been performed in our recent work. The systems displaying the best agreement with the experimental total scattering X-ray structure factors were selected at each composition for further analyses. Morphological characterization of the hydrogen-bonded clusters reveals that 60–95% of their molecules are involved in ring formation. The average number of rings necessary to describe the ring structure in the main, percolating cluster is ~2200 for pure water for the investigated system size. Only 6% of this is found for 80 mol% ethanol content, which is increasing with decreasing ethanol concentration to 56% for the 20 mol% ethanol–water mixture. In the mixtures rings with 5 molecules, while in pure water, 6-membered rings occur with the highest probability. A few pure ethanol rings exist only above ≥60 mol% ethanol concentration, with the most popular ring size of 5 molecules. Pure water rings exist in the whole concentration range, with the most frequent ring size increasing from 4 to 5 molecules with increasing water concentration. © 2015 Elsevier B.V. All rights reserved.
1. Introduction As aliphatic alcohols have an amphiphilic character, possessing both hydrophobic and hydrophilic parts, their behavior is rather complex. They are capable of forming hydrogen bonds themselves, as well as with other molecules, like water. The structure of alcohol–water solutions has been studied widely by both experimental [1–6] and theoretical methods [7,8]; the emerging picture is somewhat confusing and contradictory. Although lower alcohols are fully miscible with water over the entire concentration range, the homogeneity of the mixtures at the molecular level is at question [9–11]. Hydrogen bonding results in the formation of clusters of varying shapes, of which rings are perhaps the most attractive objects to recognize and investigate. Ring perception in chemistry has an extensive literature; the interested reader will find a short description of the terminology and methods used in Appendix A. Characterization of large networks with numerous rings is not routinely performed, as it is computationally demanding, also in terms of coding. To our best knowledge, no similar work for ethanol–water mixtures has been performed so far. In our preceding publication, morphological analyses of pure ethanol were provided [12]. Matsumoto [13] investigated the network topology of water and performed ring analysis: 6membered (6 molecules, 12 atoms) rings were found with the highest probability.
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.molliq.2015.08.010 0167-7322/© 2015 Elsevier B.V. All rights reserved.
Guo et al. [9] investigated the X-ray absorption and selectively excited X-ray emission spectra of an equimolar methanol–water mixture, and suggested that the mixture mainly consists of 6 and 8-membered rings and chains. We performed a comprehensive study of ethanol–water mixtures, containing 20 to 80 mol% ethanol. In the first part of the work [14] a series of molecular dynamics (MD) computer simulations for the mixtures, and for pure ethanol and water was conducted, applying one ethanol and three different water force fields. The aim was to find the models that provide the best agreement with the corresponding experimental total scattering X-ray structure factors (the data was provided by László Temleitner [15]). In each mixture the all-atom optimized potential for liquid simulations (OPLS-AA) [16] was used for ethanol, combined with the following water force fields: (1) the extended simple point charge (SPC/E) model [17], (2) the improved transferable intermolecular potential with four particles (TIP4P-2005) [18] and (3) the simple water model with four sites and Drude polarizability (SWM4DP) [19]. Water potential models were distinguished on the basis of deviations between calculated and measured total scattering X-ray structure factors, aided by ethanol–water pair binding energy comparison. No single water force field could be found to provide the best agreement with the experimental data over the entire concentration range: for the 80 mol% ethanol containing mixture the SWM4-DP, for 60 mol% the SWM4-DP and the TIP4P-2005, whereas for the 40 and 20 mol% ethanol containing mixtures the simulations with the TIP4P-2005 water force field provided the closest agreement with the experiment. Analyses of the partial radial distribution functions (PRDF), center of molecule distributions, coordination numbers, number of H-bonds, and several
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perception, whereas Appendix B is about the effect of percolation on hydrogen bond determination and ring perception.
Table 1 Simulation names and densities for the MD simulations. cE (%)a
Simulation name
nEb
nWc
ρE (Å−3)d
ρW (Å−3)e
ρ (Å−3)f
ρliq (g/cm3)g
100 80 60 40 20 0
Et100 Et80S Et60S, Et60T Et40T Et20T Et0S, Et0T
1331 1226 1093 889 571 0
0 316 715 1328 2280 3993
0.0103 0.0096 0.0088 0.0073 0.0047 0.0000
0.0000 0.0025 0.0057 0.0109 0.0188 0.0330
0.0930 0.0940 0.0960 0.0985 0.0990 0.0990
0.790 0.799 0.816 0.837 0.841 0.987
a b c d e f g
813
Ethanol concentration in mol%. Number of ethanol molecules. Number of water molecules. Molecular number densities of ethanol molecules. Molecular number densities of water molecules. Atomic number density of all the atoms in the simulation cell. Density of the liquids.
bivariate (distance-angle) distributions were performed. These analyses all concerned pairs of molecules. The occurrence of hydrogen bonds (HBs) determines the number and size of aggregations (‘clusters’), and therefore the cluster structure depends on the applied HB definition. There is not an exact rule for defining/identifying hydrogen bonds; several approaches can be found in the literature. The basis of the HB definition is mostly geometric: the Hbonding O⋯H and O⋯O distances should fall into a certain distance range, and sometimes there is an additional constraint on the O\\H⋯O angle, as well [20,21]. Alternatively, a cutoff value can be defined for the H-bond energy [22,23], and the two (geometric and energetic) approaches can be combined, as well [24]. Sometimes rather arbitrary definitions are used in combination with the abovementioned criteria [25]. According to recent considerations the transient breaks and bonds of the network are corrected based on the dynamic history of the HB [26,27]. As a continuation of our recent study [28], here detailed analyses of the hydrogen-bonded clusters/network were performed for the most successful MD simulations at each concentration, using four different geometric hydrogen bond definitions. These definitions differ in terms of the presence and/or the cutoff value of the angular constraint. Based on this work, the most feasible definition seemed to be the strictest one, with O\\H⋯O N 150°, similarly to the works of Noskov et al. [20] and Soper et al. [21]. This HB definition is used throughout the present study. This paper is organized in the following way: in Section 2, methods are discussed; in Section 3.1, morphological characterization of the clusters is found; in Section 3.2, connectivities of the atoms involved in cycle formation are given; while in Section 3.3, the ring size distributions are presented. Appendix A will give some details concerning ring
2. Methods 2.1. Molecular dynamics simulations Details of the MD simulations are given in our previous paper [14], here only the most important data will be repeated. All the MD simulations were performed by the GROMACS 4.0 simulation package [29], in NVT ensembles at T = 293 K, with time step dt = 2 fs. The temperature was maintained by a Nose–Hoover thermostat [30] with τT = 0.5 ps. The geometry of water molecules was maintained by the SETTLE algorithm [31]. The long-range electrostatic interactions were taken into account by the particle-mesh Ewald [32,33] (PME) algorithm. The cutoff value of 11 Å, used in the parameterization process for alcohols by Jorgensen et al. [16], was adopted. The simulation length was 2000 ps in each case. Particle configurations in the ‘production’-phase were collected 20 ps apart and in the end, 76 configurations were used for calculating the average morphology and ring size distributions. Cubic simulation boxes have been applied; the number of molecules is given in Table 1, along with the number densities matching the experimental values, for all the studied concentrations. In the MD calculations all-atom force fields were applied. For ethanol the OPLS-AA [16] force field was used; concerning water, results for two different water force fields, TIP4P-2005 [18] and/or SWM4-DP [19] were applied, depending on concentration. As mentioned before the basis for selection was the quality of agreement with the experimental total scattering X-ray structure factor [14]. All in all, five ethanol– water mixture simulations (with SWM4-DP force field for the 80 and 60 mol% ethanol concentrations, and with the TIP4P-2005 force field for the 60, 40 and 20 mol% ethanol concentrations), two pure water simulation with both force fields and one pure ethanol simulations were carried out. The pure ethanol and water simulations were needed as references. The equilibrium molecular geometries for the water force fields and full specification of the OPLS-AA force field parameters can be found in the literature [16,18,19,29]. Calculations reported here will be identified by their ethanol content and by the first letter of the water force field applied, so for example Et60S will refer to the 60 mol% ethanol containing mixture where the MD simulation was performed using the SWM4-DP water force field (Table 1). 2.2. Cluster analysis The upper limiting value of the distance ranges for the HB definition was determined from the minima occurring after the first peak of the O\\O and first intermolecular peak of the O\\H partial radial
Fig. 1. (a) The average number of main clusters/configuration versus the nullity, the labels are given in the order of increasing peak positions, the peaks of Et0T and Et0S are overlapping. (b) The distribution of the smaller clusters/configuration.
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distribution functions, and an additional angular restriction of O\\H⋯O N 150° was applied as well. The analysis of the hydrogenbonded clusters/network was performed by a C++ computer program specifically developed for this purpose. The term ‘cluster’ is applied only for molecules connected by hydrogen bonding (i.e., lone molecules are not considered as clusters). During the HB determination the simulation box was treated as an isolated system as in the work of Geiger et al. [22], namely that only one instant of the periodic box – and consequently each hydrogen bond – was used, but the hydrogen bonds were determined using the minimum image convention. Cycle perception was also performed by the same computer code; see Appendix A for some details. As it was found in another, separate part of this study [28], percolation occurs in all the ethanol–water mixtures and in pure water. The effect of percolation on building the HB network and on the ring perception is discussed in Appendix B. 3. Results and discussion First the hydrogen-bonds were determined, then the clusters (groups of molecules connected by HBs) were identified. Percolation occurs in all the mixtures and in pure water, where each configuration has a percolating main cluster, as well as smaller clusters beside that [28]. 3.1. Characterization of the cyclic and acyclic parts Ring structures can be described by graph theory, where the configuration is an undirected graph, ‘vertices’ are the atoms and ‘edges’ are the bonds between them. Cycles in chemistry are often referred to as rings (see Appendix A for a brief description of the terminologies and methods used). Percolation in the water-containing systems presented some difficulties regarding the definition of ‘nullity’ (see below) and ring perception: this is discussed in Appendix B. Nullity, μ, is the maximum number of linearly independent cycles, which form the basis of the cycle space (if there is any). μ was determined for each cluster according to μ = E − V + 1 [34], where E is the number of bonds (edges of the graph), and V is the number of atoms (vertices). The average number of main, percolating clusters/configuration with given nullity can be seen in Fig. 1a. Panel b displays the distribution of the smaller clusters, and it is important mainly for the simulations with high ethanol concentration. From Fig. 1 we can see that only pure ethanol has a substantial number of acyclic clusters (μ = 0) or clusters with only one cycle, and there
Fig. 2. The percentage of molecules in the main cluster involved in cycle formation in the ethanol–water mixtures and pure water, and the excess percentage of water (mol% water in cycles in the main cluster–mol% water in the main cluster).
Table 2 Classification of the clusters based on their ring structure and acyclic parts. Name Description LC BC OIR IR CR CRS ICR ICRS
Linear chain Branched chain Only isolated ring Isolated ring + acyclic side chain(s) Composite ring + acyclic side chain(s) Composite rings + possible acyclic side chain(s) Isolated ring(s) and one composite ring + possible acyclic side chain(s) Isolated ring(s) and more than one composite rings + possible acyclic side chain(s)
are no clusters with more than one cycle at all (μ = 4 × 10−4). The nullity for the main clusters forms peaks, and is increasing from μ = ~125 (Et80), ~2250 (Et0S) and ~2200 (Et0T), as it is visible from Fig. 1a. This means ~ 6% (Et80S), 15% (Et60T), 24% (Et60S), 30% (Et40t) and 56% (Et20T) of the averaged water nullity. The average nullity for Et0S and Et0T hardly differs, while Et60S has substantially larger value than Et60T. This is in agreement with our previous analysis of these mixtures [14,28], where we found that although for this concentration both SWM4-DP and TIP4P-2005 water force field simulations reproduce the experimental total scattering X-ray structure factor equally well, the structure created with the SWM4-DP water model has a higher connectivity. It has to be noted, that obviously the actual average nullity values depend on the size of the configuration used, but the tendency would be the same for other system sizes as well. The atoms involved in cycles were determined using the mechanical method of Welsh [35] and from this, the number of molecules participating in cycle formation was calculated. The participation in cycle formation in the main cluster is ~60–95% and increasing with increasing water content, with positive excess water involvement of b 10% (see Fig. 2) in cycles. For pure ethanol, less than 2% of the molecules in the simulation box were involved in cycle formation (not shown). The same trend that water is more prone to be involved in cycles was found in methanol–water mixtures as well [11].
Fig. 3. Simple examples of morphological characteristics of different cluster types (no depiction of real clusters).
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Fig. 4. Morphological properties of the clusters: (a) percentages of the cluster types in the main clusters; (b) the number of secondary cyclic clusters, as a function of their morphology; (c) the number of acyclic side chain/main cluster with a given morphology; and (d) the average (columns, left y-axis) and maximum sizes (marker + line, right y-axis) of acyclic clusters (acyc. clust.), or side chains for the main (main side ch.) or secondary cyclic (sec. side ch.) clusters. ‘Size’ is the number of molecules involved.
Fig. 5. Cyclic connectivity histograms (a) of the ethanol oxygen atoms as percentage of the total number of ethanol molecules at the given concentration; (b) of water oxygen atoms as percentage of the total number of water molecules at the given concentration; (c) of ethanol molecules having ethanol neighbors as percentage of the total number of ethanol molecules at the given concentration; and (d) of water molecules having water neighbors as percentage of the total number water molecules at the given concentration. The white ribbons above the other series are the total percentage of the cyclic atoms/molecules contributing to the non-zero histogram bins.
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Clusters were classified according to their basic morphological features (the presence and type of the ring structures and acyclic parts). Names and descriptions of the most popular ones are given in Table 2, whereas their schematic pictures can be seen in Fig. 3. ‘Composite ring’ means edge-joined rings. If a cluster contains cyclic parts separated by edges that are not part of a cycle then it contains more than one assembly of cycles, like CRS, ICR and ICRS. In the case of the systems with higher ethanol content the main cluster consists of several ring assemblies (see Fig. 4a), but with increasing water concentration more and more of the main clusters will contain only one large composite ring assembly (CR). Differences between the Et60S and Et60T calculations can be explained by the higher connectivity of Et60S (see also above). On the other hand, most of the smaller clusters are acyclic linear type (LC). Only Et100 and Et80S have measurable amounts of branched linear clusters (BC) (as for branching, at least three molecules are needed), and small cyclic clusters, see Fig. 4b. Considering the number of acyclic side chains for the main clusters (shown in Fig. 4c), the number is roughly the same for the different types of main clusters belonging to different configurations of the same simulation. The highest number of acyclic side chains can be found in the Et40T system: this is the sample with the lowest water content where more than one composite ring assemblies in the main cluster can be found. Seemingly, the higher number of hydrogen bonds compared to the Et60 systems not only increased the cyclic connectivity by joining cyclic assemblies, but also increased the number of acyclic side chains. For all the systems, 80–100% of the acyclic branches were linear, and only b20% were branched. Although there are a few larger acyclic clusters in the high ethanol content systems, the average size (number of molecules involved) is b8 even for these systems, see Fig. 4d. The average size of the side chains for the smaller, cyclic clusters (secondary cyclic) is usually higher than that of the side chains of the main cluster. On the other hand, their maximum size is considerably lower than the maximum side chain size of the main clusters; the latter is ~8 for high water content. Therefore we can conclude that there are many, but small side chains in all the systems. 3.2. Cyclic connectivities The connectivity of the atoms involved in cycle formation was determined. The connectivity of an atom is the total number of bonds (both covalent and hydrogen bonds) originating from it. Therefore the ideal connectivity for the hydrogen atoms is two (one covalent and one hydrogen bond), for ethanol oxygen is three (one covalent and two hydrogen bonds) and for water oxygen atoms it is four (two covalent and two hydrogen bonds). All the hydrogen atoms involved in cycle formation possess the ideal connectivity of two, so no hydrogen bifurcation was observed (not even in pure ethanol [12]). The percentage of ethanol and water oxygen atoms/particle configuration involved in cycle formation versus the ethanol concentration is increasing with decreasing ethanol concentration (Fig. 5a and b, white ribbons). This shows that the higher affinity of water for cycle formation due to its ideal connectivity of four increases the chances of ethanol molecules for participating in cycles. The value for Et60S is higher than that for Et60T, indicating higher cyclic connectivity, in agreement with the results of the combined cyclic + acyclic connectivities investigated previously [28]. While the majority of ethanol O atoms have a connectivity of two (see Fig. 5a), regarding water O atoms only the Et80S calculation brings about a connectivity value of two as the most popular. For samples with lower ethanol content the maximum is at connectivity of three, indicating composite ring structures; there are some atoms with higher than ideal connectivity, indicating complex ring systems where more than three rings are joined. The distribution of the connectivities of ethanol molecules having ethanol neighbors in the ring structure was calculated as well (see Fig. 5c). Having at least one ethanol neighbor, in terms of the percentage
Fig. 6. The observed and the theoretical (denoted as ‘t.’) probabilities of finding ethanol neighbors around an ethanol molecule and water neighbors around a water molecule in the ring structure.
of ethanol molecules at the given concentration (see white ribbon in Fig. 5c), has a maximum at 60 mol% ethanol concentration, Et60S being higher than Et60T. On the other hand, the percentage of water having water neighbor is monotonically increasing with increasing water concentration (see the white ribbon in Fig. 5d); the value for Et60S is, again, larger than that for Et60T. This difference can be explained the following way. (1) In the case of water the increasing water concentration increases the participation in cycle formation, as well as the chance of two water molecules being neighbors in the cycle, as water is the main agent causing the formation of ring systems in these mixtures. (2) Ethanol, on the other hand, in itself does not form extensive ring structures, and while the addition of some water (concentration range Et80, Et60) significantly increases ring formation, as well as the chance of neighboring ethanol molecules in rings, the decreasing ethanol concentration will diminish this chance for the lower ethanol concentration range. The number of ethanol molecules connected to ethanol in cycles is most frequently one, and, to a lesser extent, two, forming two or three molecule pure ethanol ‘islands’ in the ring structure. In the case of
Fig. 7. Size distribution (per particle configuration) of the relevant cycles. The first peak is shown in the inset. The x-axis displays the number of atoms, s, forming the rings.
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Fig. 8. Size distribution (per particle configuration) of the relevant cycles, (a) containing only ethanol molecules and (b) containing only water molecules. The x-axis displays the number of atoms, s, forming the rings.
water the maximum is moving from one water neighbor to three with increasing water concentration. Examining the ratio of ethanol molecules not having ethanol neighbor in the ring structure at all, the values increase from 6 to 55% with decreasing ethanol concentration (Fig. 5c). Analogously, for water molecules not having water neighbor in the ring structure the ratio is increasing from 0.4 to 25%, see Fig. 5d. From a comparison of the measured selectively excited X-ray emission spectra of equimolar methanol–water mixture to the spectra from theoretical models, Guo et al. [9] suggested that there are water molecules joining methanol chains together to form rings, although they did not make any assumption about their fraction. They concluded that despite the miscibility of methanol and water over the entire concentration range the mixing is incomplete at the molecular level. We calculated the theoretical probability of Et–Et and Wat–Wat neighbors in the ring structure, assuming random distribution, for each possible coordination number, based on the number of ethanol and water molecules in the cycles and the observed fraction of the different coordination numbers. Their total (summed according to the coordination numbers) was also calculated for each concentration, and compared to the observed values, see Fig. 6. The observed values are considerably higher, indicating that, indeed, water prefers water and ethanol prefers ethanol neighbors. Thus supports the view that there are molecular level inhomogeneities in the ring structure. 3.3. Ring size distributions Regarding the cyclic parts, the ‘relevant rings’ (the ones that cannot be divided into smaller rings, see Appendix A) were determined for
each system. Their distribution, expressed as the number of rings/particle configuration, is shown in Fig. 7. The peak position is at 10 atoms (5 molecules) for the ethanol containing systems, and at 12 atoms (6 molecules) for pure water, see Fig. 7. The 60 and 80 mol% ethanolcontaining mixtures have a second peak at 60–80 atoms, as well. The Et80S main cluster contains the largest rings, smax = 150 atoms, although the large rings occur with very low probability. For pure water the most popular ring size of twelve atoms is in accordance with previous findings [13,36], supposedly inheriting this feature from the 6 molecule membered rings of Ice-Ih and Ice-II [37]. The occurrence of 5 and 7 molecule membered rings (10 and 14 atoms) is the characteristic of Ice-III [37]. In pure ethanol, ~40% of the rings contain 10 atoms, and the dominance of this ring size prevails with 15– 20% probability in the ethanol–water mixtures, as well. Distributions for rings containing only ethanol or only water molecules were also determined (Fig. 8). The number of pure ethanol rings is very low (Fig. 8a), and they do not exist under 60 mol% ethanol concentration. The maximum is at a ring size of 10 atoms (5 molecules), similarly to mixed ethanol–water rings. The maximum ring size decreases from smax = 22 to 12 with decreasing ethanol concentration. On the other hand, pure water rings exist at even only 20 mol% water concentration (Fig. 8b). The most popular ring size is somewhat decreasing with decreasing water content: it is 10 atoms for Et20T, Et40T and Et60S, similarly to the ring sizes of the mixed rings, and it is 8 atoms for Et60T and Et80S. Interestingly, the largest pure water rings occur in the Et40T system, with smax = 142, and the distribution has an observable peak at s = 140 atoms with ~4 rings/configuration. Et40T is the lowest water content mixture among the systems examined in this work where monotype water percolation occurs [28]. Since
Fig. 9. Part of the HB network in the (a) Et80S and (b) Et20T ethanol–water mixtures displaying the ethanol oxygen (online red, in print dark), water oxygen atoms (online light green, in print light) and hydrogen atoms (small gray) involved in ring formation and the covalent and hydrogen bonds connecting them. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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the largest mixed rings were found in the Et80S system, which is just above the ethanol–water HB network percolation threshold, the occurrence of large rings seems to be the characteristic of systems not far above their percolation threshold, while the HB network is still relatively sparse. The ring structures in equal portion of the simulation box for the highest and lowest ethanol content mixtures are shown in Fig. 9. As it is visible, in the higher ethanol content Et80S mixture the ring structure is relatively sparse, and the individual rings can mostly be identified, while Et20T possesses rather extensive ring network. In both cases the clumping of like molecules can even visually be observed. 4. Conclusions The participation of molecules in cycle formation in the main, percolating clusters has changed between 60 and 95%. The number of rings forming the cycle base was 6% of the average cycle base size of water for 80 mol% ethanol and increased to 56% in the case of 20 mol% ethanol. Morphological analyses revealed that the main clusters contain extensive ring structures. The number of cyclic assemblies (separate ring structures in a cluster) decreases with increasing water content. For pure water only one composite ring assembly can be found with N90% probability, but even in this case, numerous smaller acyclic side chains are present. The size distributions of relevant rings were calculated: the main peak for the ethanol containing systems is at 10 atoms (5 molecules), while for pure water, it is at 12 atoms (6 molecules), in accordance with the characteristic feature of Ice-Ih. Larger rings, up to 150 atoms for Et80S, and 40 atoms for water can also occur with low probability. Rings formed solely from ethanol molecules only occur in small number in the systems with ≥60 mol% ethanol, mostly with a size of 5 molecules. Pure water rings, however, exist over the whole concentration range, with, again, 5 molecule rings as for the mixed rings in the higher water content (higher connectivity) systems, and somewhat smaller, 4 molecule rings, for Et60T and Et80S. The clumping of like molecules in the ring structure occurs, which is even visible from the visual inspection of the configurations. This supports the view that these mixtures are not homogenous at molecular level, at least regarding the ring structures. Acknowledgments This work was supported by the National Basic Research Fund (OTKA; Hungary), under contract no. 083529. Appendix A. The basics of ring perception The structure of atomic ensembles (‘aggregates’, or ‘clusters’) can be described by graph theory, where the configuration is an undirected graph, vertices are the atoms and edges are the bonds between them. Mainly the terminology from the review of Downs et al. [38] is used here. The set of vertices is V = {v1,…vN} and the set of edges is E = {e1,…eN}. Connectivity of a vertex (also called degree) is the number of edges which incident with it. A ‘walk’ is an alternating sequence of vertices and edges; if a walk is closed then it is a circuit. If all edges and vertices occur only once in the walk then it is called a path. A closed path is a cycle. A set of edges and vertices are called an elementary cycle, if it contains all the edges and vertices of the set, and one could trace a closed path crossing each vertex only once [39]. (In chemistry cycles are referred to as rings.) The graph is connected, if for any two vertices there is a path connecting them [40], and the number of connected components is the number of the graph's maximal connected subgraphs. In our context, each cluster or lone molecule is a connected component. From now on for sake of simplicity graph refers to a connected (sub)graph.
First the connection table [38] with dimensions V × Cmax (Cmax is the maximum connectivity, in other words the maximum number of Hbonded neighbors) was generated, which is a compact form of the V × V adjacency matrix A (having aij = 1 if vi and vj are connected by an edge, zero otherwise). The clusters were determined by depth-first random walking from the connection table, and the vertices and edges belonging to them were recorded. The cycles (C) of a connected graph form a vector space called the cycle space [41]. Let E be the number of edges and V the number of vertices. Each cycle can be represented by a vector of E dimension, having 1 at the nth dimension if the given nth edge is in the cycle, and 0 if it is not. The maximal number of linearly independent cycles, the nullity, μ is predicted by the Cauchy formula [34], μ = E − V + 1, and they form the cycle base for the connected graph. (Nullity is also called the set's cardinality, cyclomatic number, first Betti number or Frerejacque number.) As these linearly independent cycles are basis vectors, all cycles of the graph can be expressed by them. The addition of cycles is called the symmetric difference, and it is defined as C1 ⊕ C2 = (C1 ∪ C2)\(C1 ∩ C2). Finding a cycle set is usually done by growing a spanning tree, T of the cyclic graph (G) [38], where T is an acyclic graph containing all the vertices of the graph, but not all the edges, and obviously not containing any cycles. The edges left out are called the chords, which are the minimum number of edges required to be removed from G to turn the cyclic graph G to the spanning tree. According to Kirchoff [42] for each edge of G, where e ∉ T there is a unique cycle in G, which is called a fundamental cycle with respect to T. The fundamental cycles belonging to a spanning tree form a cycle base for the graph G, and called the Kirchofffundamental. Ring perception plays an important role in graph and network theory and has an important application for example in chemistry, therefore several methods were developed [35,43–45]. In this work the mechanical manipulation of the V × E incidence matrix developed by Welsh [35] was applied for the determination of the fundamental cycle basis. It has to be noted that the fundamental basis in most of the cases is not unique, it depends on the order the edges and vertices are represented in the incidence matrix, and it most probably not the basis set (B) with minimal length, L(B) = ∑μi = 1|Ci|. As each fundamental cycle by definition contains one unique edge, for complicated ring structures containing joined (composite), embedded cycles, some of the fundamental cycles will definitely be direct sums of smaller cycles. Therefore the fundamental basis in itself is not a good characterization of the system, and their determination gives us the atoms involved in cycle formation. In the literature of chemical graphs several concepts are introduced regarding which rings are important or meaningful, and this is in some cases subjective, see the review of Downs et al. [38]. A practical and clear concept is the smallest set of smallest rings, SSSR and it is defined originally as the minimum length Kirchoff-fundamental basis [38], though in the chemical literature the definition of SSSR is shifted to the description of the minimum length cycle basis, regardless it can be created from a spanning tree, or not. A program was created to determine the SSSR using the method of Gibbs [44], for the creation of all the cycles by forming 2μ–μ–1 linear combinations of the fundamental basis cycles, and then selecting the μ smallest, linearly independent cycles, but unfortunately, this only could be applied for pure ethanol. In the case of the ethanol–water mixtures the nullity for the main cluster even for the Et80 simulation was N 100, which made this type of calculations impossible both memory and time-wise, in accordance with Deo et al. [46], who showed that finding a Kirchoff-fundamental cycle basis with minimum length is NP-complete. Even if an SSSR could have been found, it is not unique for systems with symmetrically equivalent rings, and bridged systems. For example for adamantane, μ = 12 − 10+1 = 3, meaning that any three of the elementary rings with length 6 can provide an SSSR. Therefore the
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concept of K-rings [41], although called relevant rings [40] were introduced, which are those elementary rings, which appear in one of the SSSR of the graph. The relevant rings are basically elementary rings, which cannot be formed by the direct sum of smaller rings, but some of them can be created by the direct sum of rings of the same size. In this context, all the four size 6 elementary rings of adamantane are relevant rings. Therefore an algorithm was devised to find the relevant rings, which detailed description is out of the scope of this paper, briefly it is using a two-directional breadth-first search starting from each cord of the spanning tree (the unique edges of each fundamental cycle determined by the method of Welsh [35] discussed before) collecting elementary rings that can be potentially relevant rings. The number of rings then was reduced, keeping only the relevant rings. This was done by determining all the brides (series of edges crossing the cycle), and determining, whether the original ring can be formed as the direct sum of a (sub)set of smaller cycles. This approach was computationally more feasible, than the usual way of creating all the possible linear combination of all the available smaller cycles initially found. Appendix B. The effect of the periodic boundary condition on the hydrogen bond determination and ring perception During the HB determination the system was treated as an isolated system as in the work of Geiger et al. [22], namely that only one instant of the periodic box – and consequently each hydrogen bond – was used, but the hydrogen bonds were determined using the minimum image convention. The presence of percolations in most of the investigated systems presents some problems regarding the ring perception. Due to the above-mentioned handling of hydrogen bonds, the infinite open clusters are handled as finite ones, closing on themselves in the cases, where the atoms of either a hydrogen- or a covalently bonded atom pair are located near the opposite side of the simulation box. We could describe this conversion from the infinite open cluster to a finite one as an inverse single cycle unwrapping (SCU) applied by Hamilton and Pryadko [47]. The consequence of this can be the easiest demonstrated for a one-dimensional percolation, which could be perceived as a chain running through the periodic images, and obviously no ring is present due to this infinite chain in the system. Applying the inverse SCU the bond A–B′, which is connecting atom A residing in the simulation box to atom B′, which is the periodic image of atom B situated close to the opposite side of the box, and B–A′ will be broken, and topologically A will be connected to B closing a cycle in the process. Therefore during the ring perception one cycle will be detected. Although in this simple case it would be easy to recognize this false cycle, but in the case of two and three-dimensional percolations with complicated embedded ring systems this is not the case. As from our view point the emphasis is not so much on the absolute number of rings, or the size of the cycle base, than on the dependence of these properties with the concentration, no attempt was made to eliminate these false rings. References [1] K. Nishikawa, T. Iijima, Small-angle X-ray scattering study of fluctuations in ethanol and water mixtures, J. Phys. Chem. 97 (1993) 10824–10828, http://dx.doi.org/10. 1021/j100143a049. [2] N. Nishi, S. Takahashi, M. Matsumoto, A. Tanaka, K. Muraya, T. Takamuku, T. Yamaguchi, Hydrogen bonding cluster formation and hydrophobic solute association in aqueous solution of ethanol, J. Phys. Chem. 99 (1995) 462–468, http://dx. doi.org/10.1021/j100001a068. [3] A. Ben-Naim, Inversion of the Kirkwood–Buff theory of solutions: application to the water–ethanol system, J. Chem. Phys. 67 (1977) 4884–4890, http://dx.doi.org/10. 1063/1.434669. [4] G. Onori, Compressibility and structure of aqueous solutions of ethyl alcohol, J. Chem. Phys. 89 (1988) 4325–4332, http://dx.doi.org/10.1063/1.454816. [5] M. D'Angelo, G. Onori, A. Santucci, Self-association of monohydric alcohols in water: compressibility and infrared absorption measurements, J. Chem. Phys. 100 (1994) 3107–3113, http://dx.doi.org/10.1063/1.466452.
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