Melting temperatures deduced from molar volumes: a consequence of the combination of enthalpy/entropy compensation with linear cohesive free-energy densities †

Enthalpy/entropy compensation is a general issue of intermolecular binding processes when the interaction between the partners can be roughly modelled with a single harmonic potential. Whereas linear H / S correlations are wished for by experimentalists, and often graphically justi ﬁ ed, no in ﬂ exible law of thermodynamics supports the latter statement. On the contrary, the non-directional Ford's approach suggests logarithmic H / S relationships, which can be linearized only over narrow enthalpy/entropy ranges. Predictions covering larger domains require mathematical mapping obeying speci ﬁ c boundary conditions which are not compatible with linear plots. The analysis of solvent-free melting processes operating in six di ﬀ erent classes of organic and inorganic materials shows that reciprocal Hill plots are acceptable functions for correlating melting enthalpies and entropies. The combination of H / S compensation with the observed linear dependence of the cohesive free energy densities with respect to the melting temperature eventually provides an unprecedented interdependence between melting temperatures and molar volumes. This procedure is exploited for the prediction of melting temperatures in substituted cyanobiphenyls.


Introduction
A rational control and programming of the various contributions to the changes in standard thermodynamic free energy DG 0 asso , enthalpy DH 0 asso and entropy DS 0 asso of association accompanying the binding process depicted in eqn (1) is at the heart of self-assemblies occurring in biology, 1 in physics 2 and in chemistry 3 (k refers to the activities of various partners).
A þ B#½AB DG 0 asso ¼ DH 0 asso À TDS 0 asso ¼ ÀRT ln j½ABj jAjjBj (1) Beyond the theoretically justied Gibbs free energy relationship (eqn (1), right-hand side), 4 the simple modelling of the interaction operating between the partners in the [AB] pair by using a harmonic potential (i.e. a spring) suggests that the stronger the binding energy (measured as the potential well depth u min f DH 0 asso ), the larger its force constant k (measured as its mean vibrational frequency n ¼ ð1=2pÞ ffiffiffiffiffiffiffiffiffiffi ffi k=m r p f À DS 0 asso where m r is the reduced mass of the harmonic oscillator). 5 Consequently, enthalpy and entropy changes are usually correlated for a simple intermolecular binding event with a propensity for compensation, i.e.DS 0 asso and DH 0 asso concomitantly decrease or increase. 6Recurrent, but empirical observations collected from series of association processes investigated in biology, in physics and in chemistry show that minor structural perturbations of the two partners lead to apparent linear correlations (eqn (2), le-hand side).The slope a of the DH i versus DS i plot has Kelvin units and is oen referred to as the compensation temperature T comp , i.e. the temperature at which all association processes i within a homogeneous series display the same compensation free energy change DG comp ¼ g (eqn (2), center).Its re-writing at the right-hand side of eqn (2) is reminiscent of the Gibbs relationship, which may explain the rather common belief in the statement that linear enthalpy/ entropy compensation corresponds to a 'fourth law' of thermodynamics. 6,7 Theoretical approaches based on statistical thermodynamics include both attempts to refute the relevance of enthalpyentropy correlation, 8 as well as to establish its signature as the result of (i) minor perturbations of equilibrium constants, 9 (ii) partition functions governed by a Gaussian density of states 6b or (iii) underlying hidden thermodynamic processes.6c In 2005, Ford derived a non-quantum justication (see next section) claiming that enthalpy/entropy compensation (T comp > 0 in eqn (2)) occurs when the minimum host-guest separation r 0 in the [AB] i pairs remains constant within a series of intermolecular binding events. 10However, the linearity proposed in eqn (2) does not result from Ford's model, 6d and alternative physical justications for parabolic 5a,11 or rectangular-hyperbolic 12 correlation for H/S compensation have been proposed.In this context, Liu and Guo documented that the general emergence of linear H/S compensation is oen the consequence of statistical and mathematical artefacts arising from the data analysis.6a The dimerization process depicted in Fig. 1a illustrates this statement since the convincing linear relationship found for H/S compensation (Fig. 1b) is signicantly discredited when one considers the alternative, but equivalent van't Hoff plots which should cross at a common hT comp ; DG comp i couple (red rectangle in Fig. 1c). 13ince complexation processes occurring in dilute solutions are complicated by unavoidable solvation changes, 11,14 which may induce additional H/S compensation phenomenon, 15 we resorted to the melting of solids into isotropic liquid conducted in absence of solvents or of additives for getting a rough estimation of the strength of the intermolecular interactions occurring in a solid. 16These processes can be idealized as reversible n th order chemical reactions, in which n identical monomeric units A associate into fully assembled entities A n (eqn (3)). 17

nA ) *
DG 0 asso ¼DH 0 asso ÀTDS 0 asso At the melting temperature T ¼ T m , the solid-liquid phase equilibrium implies that DG asso ¼ DG m ¼ 0 and T m ¼ DH 0 m /DS 0 m ¼ DH 0 asso /DS 0 asso ¼ DH m /DS m when one reasonably assumes that the melting enthalpies DH m and entropies DS m operating at the melting temperatures are satisfyingly estimated by those dened in the standard conditions (i.e.DH m z DH 0 m and DS m z DS 0 m ). 17Expressed at a common reference temperature T 0 , each melting process is further characterized by its specic standard free energy change DG 0 m ¼ DH 0 m À T 0 DS 0 m z DH m À T 0 DS m , which is further used to compute the standard cohesive free energy density CFED ¼ DG 0 m /V mol (V mol is the molar volume), 18 a parameter estimating the average cohesive forces operating in the solid. 19Interestingly, CFED were found to be linearly correlated with the melting temperatures for series of association/dissociation processes obeying H/S compensation (Fig. 2a), 17 a phenomenon at the origin of the use of molar volumes for predicting the melting temperatures of linear alkanes (Fig. 2b). 17However, the exclusive consideration of linear H/S correlations limits this predictive model to the exploration of narrow domains of melting temperatures. 17

Theoretical background
Following the formalism of molecular association proposed by Luo and Sharp, 20 the equilibrium constant K A,B asso associated with the simple binding event shown in eqn (1) can be written as eqn (4), where c q is the standard concentration of the reference state (xed to 1 M in this contribution), 21 H(r, U) is a bonding function depending on the separation (r) and orientation (U) of the two partners in the [AB] pair (H(r, U) ¼ 1 when the complex [AB] exists and H(r, U) ¼ 0 otherwise), b ¼ (k b T) À1 stands for the thermal factor and u(r, U) is the potential mean force between A and B. 10 ð Hðr; UÞe Àbuðr;UÞ drdU (4) Solving eqn (4) within the frame of the van't Hoff isotherm for a non-directional harmonic potential u(r, U) ¼ u A,B min + (k A,B /2) r 2 operating between A and B, whereby u A,B min is the minimum potential energy and k A,B is the force constant, leads to eqn (5) and (6), where N Av is Avogadro number. 10 Interestingly, the enthalpy change DH A,B asso mainly depends on the magnitude of the interaction energy u A,B min , while the entropy change DS A,B asso is controlled by the force constant k A,B , the latter term being an estimation of the capacity of the bound system to gain residual degrees of freedom.The development of the harmonic potential at the minimum of a standard Lennard-Jones potential V L-J shows that the absolute minimum energy of the attractive well depth corresponds to u A,B min when the equilibrium A/B separation amounts to 2 1/6 r 0 (r 0 is the critical intermolecular A/B distance at which the interaction potential is zero: V L-J (r ¼ r 0 ) ¼ 0, see Fig. S1 in the ESI †).6d Consequently, the total energy of the harmonic oscillator for the special motion amplitude (2 1/6 r 0 À r 0 ) exactly corresponds to the well depth u A,B min and eqn (7) results.6d For a minor structural perturbation affecting a series of A and B partners, the minimum contact distance r 0 is constant within the resulting [AB] pairs and Ford's model (eqn (7), righthand side) predicts that the force constants k A,B (which affects the entropy changes, eqn (6)) are linearly correlated with the potential well depths u A,B min (which measures the enthalpy change, eqn (5)).Since the coefficient f ¼ 2/(1 À 2 1/6 ) 2 (r 0 ) 2 is positive, a larger cohesive energy between A and B in the [AB] pair (i.e.u A,B min and DH A,B asso become more negative) produces an increase in the force constant k A,B , hence in the mean vibrational frequency in the deeper potential.6d,10 Since DS A,B asso /N Av f À3/2 ln[k A,B ] in eqn (6), larger k A,B induces more negative association entropies and H/S compensation occurs.This model is nothing but the Einstein model for crystals, 16,18b,22 from which Lindemann postulated that the melting of a solid occurs when the amplitude of the atomic thermal vibrations reaches some critical fraction of the equilibrium lattice spacing. 23ntroducing eqn (7) into eqn (5) provides a simple correlation between the enthalpy change of the association process and the force constant in the bound state (eqn (8); R ¼ k b N Av is the ideal gas constant).Further introduction into eqn (6) gives the searched (logarithmic) dependence between DH A,B asso and DS A,B asso (eqn ( 9)).6d Its application to the melting of the A n assembly shown in eqn (3) requires DH m ¼ ÀDH asso and DS m ¼ ÀDS asso , and eqn (10) eventually results Applied to the melting of a series of similar compounds, 17 the force constant k A,B can be approximated by a rstorder Taylor series around its average magnitude k A,B 0 , in eqn ( 10) can be replaced with , where DH m,0 is the average melting enthalpy in the series.6d However, the rapid divergence of the logarithmic function from a linear approximation restricts the use of linear H/S compensation within this model to a narrow thermodynamic range of melting processes.In this contribution, we propose to improve the predictive capacity of the latter thermodynamic model via a thorough investigation of the melting processes occurring in metals, in various organic compounds and in inorganic oxides.

Results and discussion
Modelling H/S compensation Thanks to Yaws' efforts for gathering melting enthalpies and entropies (Fig. 3), 24 we were able to select six families of compounds, linear alkanes (Table S1 †), linear alkanoic acids (Table S2 †), organosilanes (Table S3 †), lanthanide metals (Table S4 †), transition metals (Table S5 †) and transition metal oxides (Table S6 †), for which (i) the internal cohesion forces result from various types of intermolecular interactions (permanent and/or induced electric multipolar, H-bonds, covalency) and (ii) the melting temperatures span a broad domain.Assuming the justied approximation that DH m and DS m are essentially constant over a reasonable temperature range around a reference temperature T 0 taken as the average melting temperature within each series (Tables S1-S6 in the ESI †), 17 the experimental DS m versus DH m plots of the six individual series were tted with Ford's model by tuning f in eqn (10), but only a poor match  and S7-S11) †.The use of linear and parabolic ts is reminiscent of the use of rst-order, respectively second-order Taylor polynomials for approaching eqn (10). 17Higher-order Taylor series are precluded by the well-known Runge's phenomenon, which strongly limits the approximation of logarithmic mapping by using power polynomials of increasing orders. 25On the contrary, the use of a Hill function has no mathematical justication for tting logarithmic plots, but its stability over a large enthalpy range appeared to be a considerable advantage for tting H/S compensation (vide infra).A simple look at Fig. 3 indicates that DS m / 0 when DH m / 0 for melting processes recorded at normal pressure (in other words, the melting temperature T m ¼ DH m /DS m does not diverge for DS m / 0), 26 a boundary condition which is not satised by the tting traces collected in Fig. 4 and S2-S6 (ESI †).Including the latter constraint into H/S compensation results in the loss of one degree of freedom by xing one parameter during the tting process (Tables 1 and S7-S11, † column 1).Consequently, Ford's approach lacks of acceptable correlation (magenta traces in Fig. 5 and S7-S11, † no tuneable parameter), while linear approximations (red traces in Fig. 5 and S7-S11, † one tuneable parameter, DS m ¼ bDH m ) are dismissed because only a single melting temperature T m ¼ b À1 is tolerated along the complete series of compounds.
Therefore, among the four mathematical functions selected for extending the logarithmic H/S dependence suggested by the simple Ford's model, only parabolic (blue traces, two tuneable parameters) and Hill (green traces, three tuneable parameters) plots provide satisfying tted traces compatible with the inclusion of the extra boundary condition DS m / 0 when DH m / 0 (Fig. 5 and S7-S11, Tables 1 and S7-S11, ESI †).6a Cohesive free energy densities and three-dimensional DH m , DS m , V mol plots Taking Trouton's rule into account for the vaporization of liquids, 27 Hildebrand introduced the concept of cohesion energy density CED ¼ (DH v À RT)/V mol for estimating the average cohesive forces operating within liquids and for predicting their vaporization temperatures (DH v is the vaporization enthalpy). 19n absence of valuable alternative to Trouton's rule xing the melting entropy in solids, 18 we have resorted to standard melting Gibbs free energy DG 0 m ¼ DH m À T 0 DS m computed at a   Constrained: DS m / 0 when DH m / 0 Eqn (10) 11.39 reference temperature T 0 for estimating the standard cohesive free energy densities CFED ¼ DG 0 m /V mol along each homogeneous series of compounds (Tables S1-S6, ESI †).We are aware that the melting enthalpies (DH m ) and entropies (DS m ) are indeed collected at different temperatures for each compound (i.e. at T m ) but their dependence on temperature is limited, 17 and the choice of a reference temperature T 0 around the center of each series obeying H/S compensation is thus acceptable for computing DG 0 m .The experimental linear correlations observed between CFED and the associated melting temperatures T m in Fig. 6 provide eqn (11), which interconnects DH m , DS m and V mol within each series (note that T m ¼ DH m /DS m ) Taken separately, eqn ( 11) is of limited interest for tuning melting temperatures by molecular design since only the molar volume V mol can be easily computed for unknown compounds, 28 while satisfying estimations and/or modelling of melting    entropies or enthalpies are difficult. 29However, the consideration of H/S compensations operating in each series (Fig. 5 and S7-S11, ESI †) provides a correlation DS m ¼ g i (DH m ) which can be introduced into eqn (11) to give eqn (12) when one reminds that T m ¼ DH m /DS m ¼ DH m /g i (DH m ) (g i represents either a parabolic or a reciprocal Hill function).
For quadratic ts, DS m ¼ g quadra (DH m ) ¼ bDH m + c(DH m ) 2 and eqn (12) corresponds to a polynomial, which can solved to give an estimation of the melting enthalpy DH m as soon as the molar volumes V mol of a target compound belonging to the series is at hand.The associated melting entropy DS m immediately results from H/S correlation.The same procedure holds when using reciprocal Hill ts characterized by The quality of the correlations can be estimated in the three-dimensional V mol , DH m , DS m plots for each series (Fig. 7 and S12-S16, ESI †).
As an ultimate step, the combination of melting enthalpies and entropies into the melting temperatures T m ¼ DH m /DS m reduces the dimension of the correlation shown in Fig. 7 to give an easy-manageable T m versus V mol plot (Fig. 8 and S17-S21, ESI †).The careful inspection of the latter two-dimensional plots indicate that, for any series under investigation, reciprocal Hill functions provide the best ts.
Combination of H/S compensation with linear cohesive free energy densities (CFED) for predicting the melting temperatures of substituted lipophilic cyanobiphenyls Lipophilic cyanobiphenyls with variable lengths (n-CB) and substitutions (12Me m -CB) have been considered as precursors for inducing liquid crystalline mesophases (Fig. 9a, Table S12,

ESI †)
. 17 Both constrained quadratic and reciprocal Hill correlations satisfyingly model H/S compensation along the cyanobiphenyl series (Fig. 9b), which further displays a linear CFED versus T m plot according to eqn (11) (Fig. 9c).The threedimensional V mol , DH m , DS m plot tted with eqn (12) conrms the comparable quality of quadratic (blue trace) or Hill (green trace) functions (Fig. 9d).However, the trends predicted beyond the range of available experimental data signicantly differ for the quadratic (blue trace) and Hill (green trace) plots, a feature highlighted in the reduced two-dimensional T m versus V mol plot (Fig. 10). 30he ultimate step of our model offers the opportunity to predict the melting temperature for the unknown permethylated 4 0 -(dodecyloxy)-4-cyanobiphenyl compound (12-Me 8 -CB), for which a molecular volume of 688.5 Å3 , hence a molar volume of V mol ¼ N Av Â 688.5 Â 10 À24 ¼ 414.2 cm 3 mol À1 can be estimated. 17,28Depending on the choice of the tting curve used for modelling H/S compensation, an increase (quadratic t) or a stagnation (Hill t) of the melting temperature is expected, and one can therefore predicts T m (12-Me 8 -CB) ¼ 380.5 K based on the quadratic t and 340.7 K based on the Hill t (Fig. 10).Since the experimental number of methyl groups bound to the cyanobiphenyl core in 12-Me m -CB is currently limited to m ¼ 2, 17 no denitive preference between quadratic and Hill predictions can be set.However, the stagnation of the melting temperature detected upon successive methylation (T m (12-CB) $ T m (12-Me-CB) z T m (12-Me 2 -CB) in Fig. 10) has been conrmed by the systematic preparation of all the possible regioisomers of 12-Me-CB (4 isomers, average melting temperature T m (monomethyl-CB) ¼ 326(11) K), and those of 12-Me 2 -CB (4 isomers with one methyl group connected to each aromatic group, average melting temperature T m (dimethyl-CB) ¼ 327(16) K). 17 This trend represents a strong support in favor of the Hill plot as the best predictive tool within this novel series, in line with the previous analysis of linear alkane and alkanoic acids, organosilanes, lanthanide metals, transition metals and transition metal oxides.

Conclusion
The non-directional Ford's approach uses a single harmonic potential with a unique minimum contact distance for catching any intermolecular interaction.It predicts that, for a series of binding reactions involving structurally similar partners, the thermodynamic of association/dissociation is dominated by H/S compensation and logarithmic DS versus DH dependences are expected.Applied to the melting processes occurring in six (arbitrarily selected) sets of chemical compounds, this rough model justies the emergence of H/S compensation, but it fails in reproducing accurate trends because the modelling of the intermolecular interactions operating in solid with the help of a single harmonic potential is a too big oversimplication.Increasing the number of degrees of freedom for approaching H/S correlations restores satisfying ts when using linear or quadratic polynomials, or reciprocal Hill functions.The inclusion of the boundary condition DS m / 0 when DH m / 0 recuses linear H/S correlations and only quadratic or Hill plots are able to give satisfying DS m versus DH m ts.Standard empirical analysis of enthalpy/entropy compensation within the frame of this simple binding thermodynamics stops at this point, and no predictive tool is at hand for estimating the strength of intermolecular interactions and for programming melting temperatures.The additional concept of standard cohesive free energy density (CFED ¼ DG 0 m /V mol ) and its observed linear correlation with the melting temperatures (eqn (11)) introduces an unprecedented relationship between the molar volume, V mol , a parameter accessible to supramolecular chemists and the correlated melting enthalpies and entropies.For the six different series of compounds selected in this contribution, i.e. linear alkanes, linear alkanoic acids, organosilanes, lanthanide metals, transition metals and transition metal oxides, the ultimate reduced T m versus V mol plots satisfyingly reproduce the experimental data.We notice that the use of reciprocal Hill plots provide the best ts for H/S correlations in melting processes, and its application for rationalizing melting temperatures recently determined for substituted cyanobiphenyls provides reasonable predictions for currently unknown compounds in the series.

Experimental
The mathematical analyses were performed by using Igor Pro® (WaveMetrics Inc.) and Excel® (Microso) sowares.

Fig. 2
Fig. 2 Plots of (a) the cohesive free energy densities (CFED) versus the melting temperature (T m ) and (b) the melting temperature (T m ) versus the molar volumes (V mol ) for saturated linear hydrocarbons C n H 2n+2 (n ¼ 2-20).The dotted red traces show the theoretical curves.Adapted from ref.17.

Fig. 3
Fig.3Plot of melting entropies DS m versus melting enthalpies DH m for 764 randomly selected organic compounds (red markers) and 230 inorganic compounds (blue markers).24

Fig. 5
Fig. 5 (a) Full representation, (b) low enthalpy inset and (c) high enthalpy inset for plots of melting entropies DS m versus melting enthalpies DH m in linear alkanes C n H 2n+2 (n ¼ 1-30) and fitted constrained correlations including DS m / 0 when DH m / 0 and using (i) Ford's approach (eqn (10), magenta trace), (ii) a linear H/S function (red trace), (iii) a parabolic H/S function (blue trace) or (iv) a reciprocal Hill plot (green trace).

Fig. 7
Fig. 7 Three-dimensional V mol , DH m , DS m plot for linear alkanes fitted with a parabolic H/S function (blue trace) or a reciprocal Hill plots (green trace).An asymptotic behaviour is expected to occur when the melting temperature approaches the selected reference temperature T 0 ¼ 298.15 K.

Fig. 8
Fig.8Correlations between molar volumes V mol and melting temperatures T m for linear alkanes C n H 2n+2 (n ¼ 1-30) fitted using a parabolic H/S function (blue trace) or a reciprocal Hill plots (green trace).The horizontal dotted red trace corresponds to the asymptotical behaviour occurring when the melting temperature approaches the selected reference temperature T 0 ¼ 298.15 K.

Fig. 9
Fig. 9 (a) Chemical structures of substituted cyanobiphenyls with associated plots corresponding to (b) melting entropy versus melting enthalpy changes, (c) cohesion free energy densities (CFED) versus the melting temperature (T m ) and (d) three-dimensional V mol , versus DH m and DS m fitted with a parabolic H/S function (blue trace) or a reciprocal Hill function (green trace).Reference temperature T 0 ¼ 325.92 K.

Fig. 10
Fig.10Correlations between molar volumes V mol and melting temperatures T m for substituted lipophilic cyanobiphenyls fitted using a parabolic H/S function (blue trace) or a reciprocal Hill plots (green trace).The horizontal dotted red trace corresponds to the asymptotical behaviour occurring when the melting temperature approaches the selected reference temperature T 0 ¼ 325.95 K.The vertical dotted magenta trace shows the predictions obtained for the permethylated cyanobiphenyl 12-Me 8 -CB.

Table 1
Entropy-enthalpy correlations fitted for linear alkanes C n