UniÐed stability concept of mixed molecular lattices : random alloys or complexes Alain Marbeuf,¤a Didier Mika• litchenko,a Alois Wu rger,a Harry A. J. Oonkb and Miquel-Angel Cuevas-Diartec a Centre de Physique Mole culaire Optique et Hertzienne, CNRS-Universite Bordeaux I, F-33405, T alence Cedex, France b Department of Interfaces and T hermodynamics, Faculty of Chemistry and Petrology Group, Faculty of Earth Sciences, Utrecht University Budapestlaan 4, 3584 CD, Utrecht, T he Netherlands c Departament de CristallograÐa, Universitat de Barcelona Marti i Franque` s, E-080828, Barcelona, Spain Received 30th September 1999, Accepted 5th November 1999
By using a pairwise-interaction approach in which homomolecular energies depend on the composition, and heteromolecular energies on a long-range order parameter, a uniÐed model able to explain the thermodynamic properties of random and ordered molecular alloys is obtained. The resulting Gibbs free energy describes both the asymmetry of mixing properties of random alloys and the coexistence of the immiscibility and long-range order through three independent parameters. The model is tested successfully in more than thirty binaries belonging to aromatic or aliphatic families.
1 Introduction Binary molecular systems A ] B, as for inorganic binaries, exhibit usually three di†erent kinds of high temperature T [ x phase diagrams (Fig. 1) : eutectic or peritectic systems with nearly complete immiscibility (type a), systems where mixing is sufficient for yielding random molecular alloys (type b) and systems showing at least one ordered mixed alloy with a deÐned composition (type c). In Fig. 1a, the crystalline isomorphism degree ei1 is too low to allow the van der Waals m stabilization of a mixed lattice ; moreover, molecules A and B are often not homeomorphous (a classical example lies in a system where A is a planar aromatic molecule and B a linear aliphatic one) : alloys do not exist in such systems or without ¤ CPMOH, CNRS-Universite Bordeaux I, F-33405 Talence Cedex, France.
a very large immiscibility gap. In Fig. 1b, ci is generally m greater than 0.90 : 1 A and B molecules interact through van der Waals forces and are randomly distributed in the mixed lattice of A B alloys or short-range ordering (such as 1~x x clustering) may be present. Fig. 1c corresponds, as in the Ðrst case, to non-homeomorphous A and B molecules ; but the weak van der Waals interactions are enhanced by other speciÐc interactions (charge transfer, hydrogen bonding2) : the mixed lattice of the A B alloys is completely ordered and 1~x x the solid phase acts in these systems as a line compound A B m m where m \ 1, 2 and n \ 1, 2 are generally equal. The purpose of this paper is to establish a model to understand (i) the formation or (ii) not of mixed molecular alloys A B and (iii) to Ðnd a criterion for long-range order 1~x x (complexation) in such lattices. For that, we shall use a statistical thermodynamic approach. Because molecular alloys and complexes behave with respect to their thermodynamic
Fig. 1 Main kinds of binary molecular systems AÈB : (a) eutectic or peritectic systems (type a), (b) systems with random molecular alloys A B 1~x x (type b) and (c) systems showing at least one ordered mixed alloy with a deÐned composition (type c).
Phys. Chem. Chem. Phys., 2000, 2, 261È268 This journal is ( The Owner Societies 2000
261
properties as their inorganic counterparts, we try to start from a Bragg-Williams-type model3 in which intermolecular interactions will lead to the mixing Gibbs free energy * G mix all (disordered alloy case) or to the Gibbs free energy of complexation * G (complex case). Finally, we will show that the f com present approach is sufficient to investigate molar thermodynamic properties of di†erent A ] B binary systems in terms of intermolecular interactions AÉ É ÉA, BÉ É ÉB, AÉ É ÉB and BÉ É ÉA.
2 The pairwise-interaction model Let us consider a molecular alloy with the chemical formula A B . e (T ), e (T ), e (T ) and e (T ) are the intermolecu1~x x AA BB AB BA lar interaction energies at the temperature T corresponding, respectively, to the molecular pairs AA, BB, AB and BA. If we have in mind that structural properties in molecular alloys are well described in the hard sphere approximation through van der Waals interactions, even in the case of chain-type molecules, then e (T ) and e (T ) are equal : therefore AÉ É ÉB and AB BA BÉ É ÉA heteromolecular interactions are identical. In fact, the equality between c (T ) and e (T ) comes from the summaAB BA tion of all atomÈatom mÉ É Én interactions, V , between atoms mn belonging either to A or to B molecules (respectively m and n atoms) and from the equality V \ V . Now, if z is the coormn nm i dination number (number of neighbouring molecules of a given molecule in the crystal lattice) and N the Avogadro number, one mole of the crystal lattice of a pure component contains z N/2 molecular pairs. Therefore, the internal energy i of a component i is given by : E \ 1/2z Ne0(T ), (1) i i ii where e0(T ) is the intermolecular interaction energy in the ii crystal of the component i. For further comparison between di†erent binary systems, the pure solid constituent i in its stable state is taken as the reference state. In alloys, the total number of pairs is zN/2. A and B are often isomorphous molecules : that means in particular that A B , A and B are isostructural and, therefore, that z has 1~x x i the same value (z) in the three lattices. In fact, the present approach corresponds to a mean Ðeld for interactive van der Waals forces with the same mean value of z in the three lattices and, then, z has the same value in the three lattices, even if A and B are not isomorphous molecules. 2.1 Order parameter in alloys The crystal lattice is built by putting A and B molecules onto two equivalent sublatices, a and b. In such a way (sublattice model4,5), the mixed compound A B corresponds to the 1~x x general formula [A, B]a : [B, A]b where m \ n \ 1. m@m`n n@m`n Let Nk (i \ A, B, k \ a, b) be the number of molecule i on the i sublattice k in such a case. They must satisfy the relations : ; Na \ ; Nb, i i i/A, B i/A, B ; Nk \ N(1 [ x), A k/a, b ; Nk \ Nx. (2) B k/a, b Introducing an order parameter p, eqn. (2) can be solved in the following form : Na \ ya N/2 \ N(1 [ x ] p/2)/2, A A Na \ ya N/2 \ N(x [ p/2)/2, B B Nb \ yb N/2 \ N(1 [ x [ p/2)/2, A A Nb \ yb N/2 \ N(x ] p/2)/2, B B 262
Phys. Chem. Chem. Phys., 2000, 2, 261È268
(3)
where yk is the site fraction of i in the k sublattice ; the domain i of p is given by : [ 2x O p O 2x [ 2(1 [ x) O p O 2(1 [ x)
(0 O x O 1/2), (1/2 O x O 1).
(4)
p \ 0 means random mixing (disordered alloys), whereas p \ ^ 2x or p \ ^ 2(1 [ x) means maximum order at a given composition x. The order parameter p can take ^ 1 when and only when x \ 1/2, that which corresponds to the A : B complexes. The number of pairs P (P ) are equal to AA BB the total number of pairs multiplied by the probability of Ðnding one pair AA(BB). Because neighbouring pairs correspond to molecules located on di†erent sublattices, then from eqn. (3) we obtain : P \ zN[(1 [ x)2 [ p2/4]/2, AA P \ zN[x2 [ p2/4]/2, BB and one deduces the number of pairs P : AB P \ zN[2x(1 [ x) ] p2/2]/2. AB
(5)
(6)
2.2 Internal energy The internal energy, *E, referred to the pure components, A and B, will be expressed as a function of temperature T , N, x and p. In the van der Waals picture, *E is the sum of intermolecular energies, that is : *E \ P e (T , x) ] P e (T , x) ] P e (T , x) AA AA BB BB AB AB [ (1 [ x)E [ xE . A B
(7)
2.3 Composition- and order-dependent intermolecular energies All of the e (T , x) quantities are assumed to be temperature ij and composition dependent. For homomolecular interaction energies, this composition dependence corresponds to an alloying e†ect : the perturbation terms in e (T , x) and in AA e (T , x) expressions, f (T , x) and f (T , x), can give either a BB A B stabilizing or a destabilizing e†ect on the pure homomolecular interations. In such a way, molecular alloys will behave as metallic or semiconductor alloys.6 Secondly, a long-range stabilization is introduced, acting in the complex formation through an heteromolecular parameter, / . Such a stabilizaAB tion e†ect (positive values of / ) is related to the relative AB contractions of the ““ bonds ÏÏ observed in complex lattices ; if we compare the intermolecular distances between sites belonging to di†erent kinds, both in pure compounds (do , do ), and AA BB in the corresponding complex (d , d ), a strong distance AA BB contraction is found [in 1,4-C H (CH ) (\A), do ^ 5.02 A 7 6 4 32 AA and in C F (\B), do ^ 4.20 A ,8 whereas in 1,46 6 BB C H (CH ) : C F d \ 3.66 A ,9 which leads to shorter dis6 4 32 6 6 AB tances d \ d ^ d and then to the relative contractions, AA BB AB *d /d \ [0.27 and *d /d \ [0.13]. AA AB BB AB In the present approximation, we assume that f (T , x) and A f (T , x) are x-polynomia and that the x-dependence of the B f (T , x) must be proportional to the mole fraction x of the i j compound j : f (T , x) \ 2xg (T , x), A A f (T , x) \ 2(1 [ x)g (T , x). (8) B B (The factor 2 is introduced in order to simplify the Ðnal equations.) The stabilizing (destabilizing) functions f (T , x) vanish i when the alloy reduces to a pure compound, as it must. Moreover, we expand e (T , x) (\e (T , p)) by considering AB AB only its lowest terms (the odd-order terms must be equal to zero in order to take into account the symmetry property of the van der Waals bonds in the lattice) : e (T , p) \ e (T , AB AB [p). Then, because our purpose is not to explain orderÈ
disorder transitions of the Ðrst kind, the expansion of e (T , p) AB is restricted to second-order and fourth-order terms in p : e (T , p) ^ eo (T ) [ / p2 ] j p4. (9) AB AB AB AB Ordering and distance contraction will result from a stabilizing e†ect, if e (T , p) \ eo (T ). All equimolar complexes AB AB (x \ 1/2) are stable under the given value of T , when p \ ^1. In other words, the interaction energy, e (T ), is minimal. If AB / is positive, this requirement leads to the constraint : AB j \ / /2. By using eqns. (8) and (9), we obtain Ðnally : AB AB e (T , x) ^ eo (T ) [ 2xg (T , x), AA AA A e (T , x) ^ eo (T ) [ 2(1 [ x)g (T , x), BB BB B e (T , p) ^ eo (T ) [ / p2 ] / p4/2. (10) AB AB AB AB 2.4 Expression for the internal energy : the interaction parameter Eqn. (7) can be transformed with the help of eqns. (5) and (6) and eqn. (10) into the following form : *E \ zNx(1 [ x)Meo (T ) [ 1/2[eo (T ) ] eo (T )] AB AA BB ] (1 [ x)g (T , x) ] xg (T , x)N ] zNMeo (T ) A B AB [ 1/2[eo (T ) ] eo (T )] ] xg (T , x) ] (1 [ x) AA BB A ] g (T , x) [ 4x(1 [ x)/ Np2/4 B AB ] zN[2x(1 [ x) [ 1]/ p4/4. (11) AB We can now deÐne an interaction parameter, X (T , x) : AB X (T , x) \ zNMeo (T ) [ 1/2[eo (T ) ] eo (T )] AB AB AA BB ] (1 [ x)g (T , x) ] xg (T , x)N. (12) A B If the phenomenological description of random alloys is restricted to the sub-regular approximation for evaluating the excess Gibbs free energy, * G : 10 xs all 2 * G \ x(1 [ x) ; [H(l) [ T S(l)](1 [ 2x)(l~1), xs all l/1 then, at given T , we can identify the set of eo (T ) terms with ij the constant part, H(1) [ T S(1), and the set of g -terms with i (1 [ 2x)[H(2) [ T S(2)], if we assume that g (T , x) and g (T , x) A B are opposite constants. This last assumption leads to the deÐnition here of a new constant related to the AÉ É ÉB interactions, c (T ) \ g (T ) \ [ g (T ), and, then, a parameter, AB A B C (T ) \ zNc (T ). By rearranging eqn. (12) : AB AB X (T , x) \ zNMeo (T ) [ 1/2[eo (T ) ] eo (T )]N AB AB AA BB ] (1 [ 2x)C (T ). (13) AB We can verify immediately that the second term, (1 [ 2x) C (T ), vanishes for x \ 1/2. AB 2.5 Equilibrium Gibbs free energy Now, by inserting into eqn. (11) the ideal molar entropy of conÐguration calculated from ref. 4 with the site fractions, yk i (see eqn. (3)) : * S \ [R/2 ; ; ( yk ln yk), conf i i k/a, b i/A, B the non-equilibrium Gibbs free energy of the alloy, *G ^ *E [ T * S, can be obtained (R is the gas constant). conf After writing zN/ as U , *G is given by : AB AB a*G \ x(1 [ x)X (T , x) ] [X (T , x) [ 2(1 [ 2x) AB AB ] C (T ) [ 4x(1 [ x)U ]p2/4 AB AB ] [2x(1 [ x) [ 1]U p4/4 AB ] RT /2M(1 [ x ] p/2)ln(1 [ x ] p/2) ] (x [ p/2) ] ln(x [ p/2) ] (1 [ x [ p/2)ln(1 [ x [ p/2) ] (x ] p/2)ln(x ] p/2)N.
(14)
The Ðrst term, x(1 [ x)X (T , x), represents the excess AB energy of mixing for disordered alloys in the sub-regular approximation ; because this quantity is often positive, random alloys can exhibit immiscibility ; both the second and third terms, X (T , x) [ 2(1 [ 2x)C (T ) [ 4x(1 [ x)U ]p2/4 AB AB AB and [2x (1 [ x) [ 1]U p4/4, correspond to a stabilizing conAB tribution of long-range ordering to the energy : ordered alloys can exist, even if X (T , x) is positive. Eqn. (14) shows also AB that *G is not altered by the inversion of the sign of p, as it must. Because all solid phases studied here are prepared starting from the liquid phase, they are in thermodynamic equilibrium. Therefore, p takes a value that minimizes *G, under the given values of T , N and x. Then, the stable state corresponds to the equilibrium condition d(*G)/dp \ 0 : [X (T , x) [ 2(1 [ 2x)C (T ) [ 4x(1 [ x)U ] AB AB AB ] p ] 2[2x(1 [ x) [ 1]U p3 AB RT (1 [ x ] p/2)(x ] p/2) ] ln \ 0. 2 (1 [ x [ p/2)(x [ p/2)
(15)
3 Random alloys or complexes ? 3.1 The particular case of x = 1/2 For comparison between random alloys and complexes, let us consider the case where x \ 1/2. For this particular composition, eqn. (14) reduces to : *G \ X (T , 1/2)/4 ] [X (T , 1/2) [ U ] AB AB AB ] p2/4 [ U p4/8 ] RT /2[(1 ] p)ln(1 ] p) AB ] (1 [ p)ln(1 [ p) [ 2 ln 2],
(16)
and eqn. (15) to : (1 ] p) [X (T ) [ U ]p [ U p3 ] RT ln \ 0. AB AB AB (1 [ p)
(17)
For random alloys where p \ 0, *G is the Gibbs free energy of mixing and then : * G \ X (T , 1/2)/4 [ RT ln 2. mix all AB From eqn. (13), we see that :
(18)
X (T , 1/2) \ H(1) [ T S(1). (19) AB An orderÈdisorder transition of the second kind may occur at T \ T when eqn. (17) is satisÐed. Because p ^ 0 near T , trs trs after expanding ln[(1 ] p)/(1 [ p)] in power of p up to the second-order, T is found to be equal to : trs [ H(1) ] U AB . T \ (20) trs 2R [ S(1) In a complex phase, A : B, for which o p o ^ 1, the entropy of conÐguration vanishes and, because the derivative of entropy is not deÐned at o p o \ 1, eqn. (17) can be used only if o p o D 1. Eqn. (16) becomes : *G ^ X (T , 1/2)/2 [ 3U /8 \ * G /2, (21) AB AB f com by introducing the Gibbs free energy of complexation for one mole of A : B, * G . Which means that X (T , 1/2) can be f com AB expressed as a function of the enthalpy and of the entropy of complexation, * H and * S , if capacity e†ects are f com f com neglected : X (T , 1/2) ^ * H [ T * S ] 3U /4. (22) AB f com f com AB One may conclude that eqns. (19) and (22) are not formally identical, which would not be the case if heteromolecular Phys. Chem. Chem. Phys., 2000, 2, 261È268
263
interaction energies were p independent (U \ 0). NevertheAB less, by the present statistical model, the stability point of view between random alloys and complexes is uniÐed : (i) the same quantity X (T , x) takes into account all possible intermolecAB ular interactions in mixed lattices ; (ii) the same approach allows an understanding of both random alloys and complexes ; (iii) the di†erence between random and ordered alloys lies, for complexes, in the importance of the stabilization of the lattice due to the long-range ordering terms, [X (T , 1/2) AB [ U ]p2/4 and [ U p4/8, in the Gibbs free energy expresAB AB sion. But the interesting point of the model lies in the nature of X (T , x). This molar quantity is related to the mean values of AB intermolecular interaction energies, eo (T ), which act as van ij der Waals potentials in a mean Ðeld and, therefore, are insensitive to the molecular geometry. Consequently, the model must be valid both for planar molecules, such as aromatic compounds, and linear molecules, such as normal alkanes and their derivatives. Moreover, the stabilization and/or the destabilization of homomolecular interactions by alloying, which are balanced in X (T , x) only for equimolar alloys, may AB explain the sub-regular behaviour of some alloys.
lower p values should give a negative value of U for the AB most stable complexes in contradiction to the stabilization of the ““ bonds ÏÏ AÈB. Finally, the heteromolecular interaction energy, eo (T ), will be deduced from eqn. (13). AB Now, we will apply the model to di†erent binaries. We have chosen aromatic families, such as the benzene14h20 and the naphthalene ones,13,21,22 because intensive works show binary phase diagrams of the three kinds a, b and c allowing the study of the evolution of the eo (T ) energies within the ij same aromatic family at a given temperature T (here T \ 298 K). With other kinds of binary systems, such as the one built with n-alkanes,23h25 comparison will also be useful in showing the versatility of the present model. It will be of interest to test also the present approach in binaries where alloys A B are present both under random form and under 1~x x ordered state (x \ 1/2), such as systems where A and B are enantiomers of the same substance.
4 Results and discussion For each pure substance, enthalpy sublimation value, * H sub i (directly from the literature or deduced by adding known melting and vaporization enthalpy values, * H and * H ) fus i vap i are reported in Table 1 and gives the corresponding homomolecular quantity, zHeo(T ), through the equation : ii 1/2zNeo(T ) \ [(* H ] 2RT ). ii sub i 4.1 Interaction- and long-range order parameters
3.2 Evaluation of the model parameters Starting from eqn. (1) and by giving the enthalpies of the components related themselves to the lattice energies, we are able to estimate the homomolecular interaction energies, eo (T ) AA and eo (T ). All the binary systems, where thermodynamic BB properties (excess Gibbs free energy, * G , complexation xs all Gibbs free energy, * G ) are known and assessed, may lead f com then to the evaluation of X (T , 1/2) and U under the given AB AB values of T and x. For that, eqn. (18) for the random case and eqns. (17) and (21) (assuming p ^ ^0.98) for the complex case will be used. This procedure and this p value appear to be correct because : (i) crystal structures11h13 show always perfect long-range ordering between A and B molecules in complex lattices in the limits of the X-ray di†raction accuracy ; (ii)
For the entire set of systems investigated in the present study, values of the * G function in disordered alloys, and then xs all the interaction parameter values at 298 K, come from the assessed excess parameters, H(1) and S(1) (Table 2) : we can see that X (298 K, 1/2) is generally positive, but may be negaAB tive. The asymmetry parameter, C (298 K), may be obtained AB from the assessed excess parameters, H(2) and S(2) : for naphthalene alloys between 2-C H X (X \ Cl and Br) and 210 7 C H CH with a maximum on the X-side of the liquidÈsolid 10 7 3
Table 1 Thermodynamic data of pure molecular substances (in J mol~1 or K) Substance
* H fus
T
C H 6 6 C H F 6 5 C H Cl 6 5 C H Br 6 5 H Cl 1,4-C 1,4-C6H4Br2 6 H 4 Cl 2 1,3,5-C 1,3,5-C6H3Br3 6 3 3 C H CH 6 5 1,4-C H 3CH Cl 1,4-C6H4(CH3 ) 6 H 4 (CH 3 2) 1,3,5-C 33 C F 6 3 6 6 C H 10 H 8 Cl 2-C 2-C10H7Br 2-C10H7CH 3 C 10 F 7 C10H8 C11H24 C12H26 C13H28 C14H30 C15H32 C16H34 C17H36 C18H38 C19H40 C20H42 C21H44 C22H46 23 48
9950 11 400 10 380 10 620
278.7 232.7 228.0
6850 13 600 17 350 9500 11 200 18 970 13 940
178.1 280.6 286.4 228.5 277.5 352.8 330.7
38 160 38 765 42 980 48 360 33 170
11 800 17 550 22 500 35 800 31 100 42 700 34 200 53 000 39 400 60 100 42 700 68 100 46 600 49 100 52 600
306.9 358.8
52 754 40 100 56 500 61 300 66 300 71 100 76 200 75 200 81 300 90 900 95 900 100 840 105 780 110 730 115 700
264
fus
Phys. Chem. Chem. Phys., 2000, 2, 261È268
*
vap
H
34 625 40 100 45 510
* H sub
zNe (298 K) ii
Ref.
46 800 46 025 50 480 56 130 61 910 70 147 65 385 80 421 45 010 52 365 60 280 57 860 44 369 72 350 75 100 78 000 64 554 57 650 79 000 97 100 97 400 113 800 110 400 128 200 120 700 151 000 138 600 168 900 152 380 159 830 168 300
[102 500 [102 000 [110 900 [122 200 [136 400 [152 900 [144 200 [213 200 [99 940 [119 150 [130 470 [125 630 [99 800 [154 880 [160 200 [165 920 [128 000 [125 200 [168 000 [204 000 [204 800 [237 600 [230 800 [267 800 [253 200 [312 200 [287 200 [347 800 [314 600 [329 600 [346 600
26, 27, 18, 26, 17 17 17 17 26, 27, 26, 26, 18, 22, 13, 29 27, 22, 24, 24, 24, 24, 23, 23, 23, 24, 24, 24, 24, 24, 24,
27 28 27 27
27 28 27 27 27 29 29 28 27 30 30 30 30 30 30 30 30 30 30 30 30 30
Table 2 Thermodynamic data of random molecular alloys at 298 K (in J mol~1 or J mol~1 K~1) Alloy
H(1)
S(1)
X AB
Ref.
C H ClÈC H Br 6 5 H Cl6 È1,4-C 5 1,4-C H Br 6 4 2 6 4 2 1,3,5-C H Cl È1,3,5-C H Br 6 3 3 2-C H ClÈ2-C H Br6 3 3 10 7 10 7 2-C H ClÈ2-C H CH 10 7 10 2-C H BrÈ2-C H7 CH3 10 7 10 7 3 C H ÈC H C11H24ÈC12H26 12 26 13 28 C H ÈC H C13H28ÈC14H30 14 30 15 32 C H ÈC H C15H32ÈC16H34 16 34 17 36 C H ÈC H 17 36 18 38 C H ÈC H 18 38 19 40 C H ÈC H C19H40ÈC20H42 20 42 21 44 C H ÈC H 21 44 22 46 C H ÈC H 14 30 16 34 C H ÈC H C16H34ÈC18H38 18 38 20 42 C H ÈC H 20 42 22 46 C H ÈC H C13H28ÈC15H32 C15H32ÈC17H36 17 36 19 40 C H ÈC H 19 40 21 44 C H ÈC H 21 C H44ÈC23H48 19 40 23 48
0 2760 4690 240 [2600 [2670 6400 5600 4840 6500 5000 4800 5200 3600 4700 4750 4240 13 400 12 200 11 200 8400 14 800 13 000 9800 10 300 7000 26 100
0 2.8 10 0.2 [2.5 [2.3 21.2 17.6 14.8 20.1 14.9 14.0 16.4 11.2 15.9 16.2 15.2 40.9 37.6 35.2 25.6 47.6 40.0 29.6 32.3 22.0 79.2
0 1930 1710 170 [1860 [1930 80 360 430 510 560 630 310 260 [40 [80 [290 1210 1000 710 770 610 1080 980 670 440 2500
31 17 17 21 21 21 24 24 24 25 25 23 24 24 25 25 25 25 24 24 24 24 23 24 25 24 25
T Èx phase diagram,21 C (298 K) \ 0.350 kJ mol~1 and AB 0.200 kJ mol~1, respectively ; if we compare this with the pure substances, these values correspond to a destabilization of the 2-C H CH É É É2-C H CH interactions and to a stabiliza10 7 3 10 7 3 tion of the 2-C H XÉ É ÉC H X interactions. 10 7 10 7 In the complexation case, the assessed Gibbs free energy of complexation * G values, and then the values of X (298 f com AB K, 1/2) and U , are deduced from the corresponding Gibbs AB free energy of formation from the pure liquid components, * Ho [ 298* So, by using both the enthalpy and the temf f perature of melting of the pure components : * G \ * Ho [ 298* So ] * H (1 [ 298/T ) f com f f fus A fusA ] * H (1 [ 298/T ). fus B fusB * Ho and * So functions (required for the evaluation of X (T , f f AB 1/2) and U and the deduced values are reported in Table 3 : AB the X (298 K, 1/2) values are negative (except for the AB benzene complex 1,4-C H CH Cl : C F ), whereas U corre6 4 3 6 6 AB sponds to a enhancement of the interactions AÉ É ÉB. For more convenient comparison of properties between the two types of mixed structural alliances, we have plotted the variation of X (298 K, 1/2) vs. the half-sum zN[eo (298 K) AB AA ] eo (298 K)]/2 in Fig. 2. The left part of this Ðgure is relaBB tive to the aromatic families : some random alloys lie near the zero line and correspond to the case of halogen-substituted ring (Cl and Bl) ; when the number of halogen atoms increases,
X (298 K, 1/2) becomes more positive (X (298 K, 1/2) ^ 2 AB AB kJ mol~1) ; negative values (X (298 K, 1/2) ^ [2 kJ mol~1) AB are present for naphthalene binaries 2-C H XÈ2-C H CH 10 7 10 7 3 (X \ Cl, Br). The values of T , deduced from eqn. (20) where trs U \ 0 are 136 K (X \ Cl) and 140 K (X \ Br). Because difAB fusion processes are very slow under these temperature conditions, a complex cannot form at low temperatures. Complexes can correspond either to strong negative values of the interaction parameter (X (298 K, 1/2) \ [11.7 kJ mol~1 for 1,3, AB 5-C H (CH ) : C F , X (298 K, 1/2) \ [10.8 kJ mol~1 for 6 3 33 6 6 AB the complex C H : C F ), or positive values (X (298 K, 10 8 10 8 AB 1/2) \ 1.2 kJ mol~1 for the complex 1,4-C H CH Cl : C F ). 6 4 3 6 6 Finally, it is not surprising that U has the highest value for AB this complex (6.5 kJ mol~1) and a zero one for complex 1,3,5C H (CH ) : C F : this is a consequence of the balance 6 3 33 6 6 between X (T , 1/2) and U in eqn. (21). The existence of AB AB complexes and random alloys in the same family cannot be understood by X (T , 1/2) only, as shown by the dispersive AB values of this parameter, but can be explained in connection with U : the physical meaning of this parameter lies in the AB global stabilization of ““ bonds ÏÏ AÈB when long-range ordering. On the other hand, the aliphatic familyÏs points are near the zero line for the n-alkane binary systems C H n 2n`2 [C H (*n \ 1) with o X (298 K, 1/2) o O 0.6 kJ n`1 2n`4 AB mol~1) ; for the systems C H [C H (*n \ 2), n 2n`2 n`2 2n`6 either n is even or n is odd, X (298 K, 1/2) is positive and AB
Table 3 Thermodynamic data of ordered molecular alloys (complexes) at 298 K (in J mol~1 or J mol K~1) Complex
* Ho f
* So f
X
C H :C F 6 6F C6H6F : C C6H5Cl : C6 F6 C6H5CH : 6C 6F 6 5 H 3CH Cl 6 6: C F 1,4-C 1,4-C6H4(CH3 ) : C6F6 6 H 4 (CH 3 2) : C 6 6F 1,3,5-C 6 6 C H 6: C3 F 3 3 10 8H Cl10: C8 F 2-C 2-C10H7CH : 10 C 8F 10 7 3 10 8
[21 120 [17 430 [12 440 [16 680 [16 930 [26 500 [25 000 [30 950 [28 930 [23 100
[57.3 [56.0 [38.3 [42.4 [49.9 [70.4 [57.0 [60.0 [65.7 [45.0
[1800 [500 [900 [8100 1100 [4200 [11 700 [10 800 [1200 [4000
AB
U AB
Ref.
5000 5700 5500 1800 6500 3800 ^0 400 5300 3900
20 28 18 20 32 20 28 22 13 28
Phys. Chem. Chem. Phys., 2000, 2, 261È268
265
Fig. 2 Evolution of the interaction parameter, X , vs. the negative AB of the homonuclear interaction energy quantity, [zN (eo ] eo )/2, at AA BB 298 K for : benzene complexes (=) : 1 (C H : C F ), 2 (C H CH : 6 6 6 6 6 5 3 C F ), 3 (1,4-C H (CH ) : C F ), 4 (1,3,5-C H (CH ) : C F ), 5 6 6 6 4 32 6 6 6 3 33 6 6 (C H Cl : C F ), 6 (C H F : C F ) and 7 (1,4-C H CH Cl : C F ) ; 6 5 6 6 6 5 6 6 6 4 3 6 6 naphthalene complexes (…) : 8 (C H : C F ), 9 (2-C H Cl : 10 8 10 8 10 7 C F ) and 10 (2-C H CH : C F ) ; benzene random alloys (K) : 10 8 10 7 3 10 8 (C H ClÈC H Br, 1,4-C H Cl È1,4-C H Br and 1,3,5-C H Cl È 6 5 6 5 6 4 2 6 4 2 6 3 3 1,3,5-C H Br ), naphthalene random alloys (L) : (2-C H ClÈ26 3 3 10 7 C H Br, 2-C H ClÈ2-C H CH and 2-C H BrÈ2-C H CH ) ; 10 7 10 7 10 7 3 10 7 10 7 3 random alloys between normal alkanes : C H (|) *n \ 1 (n \ 11, n 2n`2 . . . , 21), ()) *n \ 2 even (n \ 14, . . . , 20), (È) *n \ 2 odd (n \ 13, . . . , 21) and (]) C H ÈC H . 19 40 23 48
maximal for the shortest length chain (C H ÈC H , 14 30 16 34 X (298 K, 1/2) \ 1.2 kJ mol~1) ; and Ðnally, for the system AB C H ÈC H (*n \ 4), X (298 K, 1/2) takes the highest 19 40 23 48 AB positive value (2.5 kJ mol~1) : in this system, this fact can be interpreted as the consequence of the noticeable length di†erence between two pure compounds. 4.2 Heteromolecular interaction energy Another way of presenting the results is to follow the evolution of the heteromolecular interaction energy, zNeo (298 K), AB vs. the half-sum : zN[eo (298 K) ] eo (298 K)]/2, AA BB as shown in Fig. 3. Fig. 4 Except when long-range ordering occurs in complexes, points are located near the bisecting line : zNeo (298 K) \ zN[eo (298 K) ] eo (298 K)]/2 AB AA BB which means that homomolecular and heteromolecular interactions in the random alloys studied here are nearly balanced.
Fig. 3 Evolution of the negative of the heteromolecular interaction energy, [ zNeo , vs. the negative of homomolecular interaction energy quantity,AB[zN(eo ] eo )/2, at 298 K for alloys. (Symbols have the same meaning as in AA Fig. 2.)BB
266
Phys. Chem. Chem. Phys., 2000, 2, 261È268
Fig. 4 Evolution of the AÈB stabilization vs. the negative of the Gibbs free energy of complexation, [* G , at 298 K for benzene f com complexes. (Symbols have the same meaning as in Fig. 2.)
Finally, X (298 K, 1/2), U and zNeo (298 K) values AB AB AB have been reported in Table 4 for all systems studied in this paper for earlier comparison between them. Of course, zNeo AB (298 K) is always negative. In the aromatic family, in both the benzene and naphthalene systems, heteromolecular interactions appear weaker ([162.8 kJ mol~1 for the system 2C H ClÈ2-C H Br) than in the alkane family, even for the 10 7 10 7 shortest alkane lengths ([180.0 kJ mol~1 for the system C H ÈC H ). The e†ect of the long-range order stabiliza11 24 12 26 tion on the interactions AÉ É ÉB can be seen by comparison between zNe (298 K) and zNeo (298 K) deduced from eqns. AB AB (10) and (13) ; the corresponding stabilization values, given by :
A B
A
B
*e U AB AB \[ , eo 2zNeo AB 298 K AB 298 K are also reported in Table 4. These values show that stabilization e†ect is the highest (3.0%) for the less stable complex (1,4C H CH Cl : C F ), whereas this e†ect can be neglected in 6 4 3 6 6 the description of stable complexes (1,3,5-C H (CH ) : C F , 6 3 33 6 6 C H : C F ). 10 8 10 8 4.3 The model in systems between enantiomeric compounds In binary systems where A and B are the two enantiomers L and D of the same substance, the phase diagram is symmetrical. The system M(1 [ x) L-carvoxime ] x D-carvoximeN (carvoxime \ C H NO) displays unique thermo10 15 dynamical33 and structural properties : 34 the solidÈliquid phase diagram has a complete miscibility region (random alloys L D or conglomerate) with a maximum at x \ 1/235 1~x x and the following properties33,36 T (1/2) \ (365.1 ^ 0.2) K, fus * H(x \ 1/2) \ (22.70 ^ 0.06) kJ mol~1, * G \ x(1 [ x) fus mix all [ [ 22.72 ] 52.23 ] 10~3T ] ] RT [ [ x)ln(1 [ x) ] xln x], a \ 10.24 A , b \ 11.67 A , c \ 8.54 A , b \ 103.1¡ at 298 K (P2 , Z \ 4), (x \ 0 or 1), and an ordered phase at x \ 1/2 1 (the so-called racemate, D : L),34 a \ 9.856 A , b \ 11.848 A , c \ 8.480 A , b \ 98.95¡ at 298 K, (P2 /c, Z \ 4). Because the 1 P2 /c space group is a supergroup of the P2 space 1 1 group,37,38 a transition of the second kind is allowed below T . The model was tested in this binary system : the racemate fus is taken as a nearly perfectly ordered phase (p ^ ^0.98) ; from eqn. (17), we obtain : U \ 2.28 kJ mol~1 ; with this value LD reported in eqns. (21) and (22), we have T \ 363 K and trs * G \ [8.9 kJ mol~1. f com Both the proximity of T and of T (*T ^ 2 K) and the fus trs kind of transition are consistent with the experimental phase diagram : (i) no transition was clearly detected by calorimetry at x \ 1/2, (ii) structural continuity between the two phases at room temperature as shown by X-ray di†raction.
Table 4 Model parameters at 298 K (in J mol~1) and long-range stabilization (in %) System
X AB
U AB
zNeo AB
*e /e AB AB
C H ÈC F 6 6F C6H6FÈC 6 5 6 6 C H ClÈC F C6H5ClÈC6H6 Br 6 5 6 5 1,4-C H (Cl) È1,4-C H (Br) 6 5 2 È1,3,5-C 6 5 H (Br) 2 1,3,5-C H (Cl) 6 5 3 6 5 3 C H CH ÈC F 6 5 H CH 3 6ClÈC 6 F 1,4-C 6 4 3 6 6 1,4-C H (CH ) ÈC F 6 4 3 2 6 1,3,5-C H (CH ) ÈC 6F 6 4 33 6 6 C H ÈC F 10 8H ClÈC 10 8 F 2-C 10 7 10 8 2-C H ClÈ2-C H Br 10 7 10 7 2-C H ClÈ2-C H CH 10 7 10 7 3 2-C H BrÈ2-C H CH 3 2-C10H7CH ÈC10 F 7 10 7 3 10 8 C H ÈC H 11 24 12 26 C H ÈC H 12 26 13 28 C H ÈC H C13H28ÈC14H30 14 30 15 32 C H ÈC H 15 32 16 34 C H ÈC H C16H34ÈC17H36 C17H36ÈC18H38 18 38 19 40 C H ÈC H 19 40 20 42 C H ÈC H C20H42ÈC21H44 C21H44ÈC22H46 14 30 16 34 C H ÈC H C16H34ÈC18H38 C18H38ÈC20H42 C20H42ÈC22H46 C13H28ÈC15H32 C15H32ÈC17H36 C17H36ÈC19H40 C19H40ÈC21H44 C21H44ÈC23H48 19 40 23 48
[1800 [500 [900 0 1930 1710 [8100 1100 [4200 [11 700 [10 800 [1200 170 [1860 [1930 [4000 80 360 430 510 560 630 310 260 [40 [80 [290 1210 1000 710 770 610 1080 980 670 440 2500
5000 5700 5500
[102 900 [101 400 [106 200 [116 500 [142 800 [157 500 [108 000 [108 300 [119 400 [124 500 [150 800 [143 900 [162 800 [145 900 [148 900 [130 700 [186 000 [203 800 [220 800 [233 700 [248 700 [259 800 [282 300 [296 900 [315 100 [331 300 [322 400 [251 500 [288 900 [329 200 [337 900 [217 200 [240 900 [266 800 [297 800 [330 200 [314 400
2.4 2.8 2.6
The present approach does not allow a disorder ] order transition of the Ðrst kind, because the Gibbs free energy, *G, given by eqn. (16) does not contain any sixth-order term of p. In order to avoid this fact, according to LandauÏs theory,39 a p6-dependence of the heteromolecular interaction energy, e (T ), would be taken into account. But, in view of the AB absence of any known disorder ] order transitions of the Ðrst kind in aromatic alloys, such news terms are not necessary.
5 Conclusion By using a pairwise-interaction approach in which homomolecular energies are assumed to depend on the composition, and the heteromolecular interaction energy is assumed to depend on a long-range order parameter, a thermodynamic model both for random and ordered molecular alloys is achieved. The obtained Gibbs free energy describes the asymmetry of mixing properties and the coexistence of immiscibility and long-range order through only three independent parameters, as shown in di†erent families. The existence of random alloys, with or without immiscibility, originates in the strong homomolecular interaction energies, which cannot be compensated by the heteromolecular interaction energy, whereas the asymmetry of mixing properties comes from the unbalanced stabilizationÈdestabilization of homomolecular interactions by alloying. Long-range ordering is the consequence of the dependence of the heteromolecular interaction energy on the order parameter, which enhances interactions. The model has been also tested successfully on a binary system between two enantiomers involving both random and ordered alloys.
1800 6500 3800 ^0 400 5300
3900
0.8 3.0 1.6 0.0 0.1 1.8
1.5
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