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The entropy of mixing of athermal monomer-dimer liquid mixtures

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1964 Proc. Phys. Soc. 84 969

(http://iopscience.iop.org/0370-1328/84/6/315)

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?ROC. PHYS. SOC., 1964, VOL. 84

The entropy of mkhg of athermal monomer-her liquid mixtures

D. H. TREVENA Physics Department, University College of Wales, Aberystwyth

MS. received 29th July 1964

Abstract. In this paper the properties of athermal monomer-dimer liquid mixtures are considered on the basis of the quasi-crystalhe model. The treatment is a development of the previous work of Rushbrooke, Scoins and Wakefield and the lattice considered is the face-centred cubic. The results obtained are compared with those of other workers by considering the molar entropy of mixing and it is shown that the present results are an improvement on these previous treatments.

1. Introduction The first theoretical treatment of different-sized molecules was given by Fowler

and Rushbrooke (1937). Their work was based on the quasi-crystalline model of a liquid and they investigated the properties of a mixture of molecules occupying one site on the lattice with molecules occupying two sites, i.e. a monomer-dimer mixture. This theory was subsequently developed by Chang, Miller, Flory and Guggenheim and an excellent survey of this work has been given by Guggenheim (1952). The various results obtained are restricted to athermal mixtures, that is, to mixtures having zero energy of mixing.

More recently Rushbrooke, Scoins and Wakefield (1953) have studied the properties of an athermal mixture of monomers and r-mers, that is, molecules occupying r lattice sites. They derived expressions for the partial vapour pressures of the two components as power series in the volume concentration tJ of the r-mers and they showed that the coefficients of successive powers of tJ were given by direct analogues of the functions 81, occurring in Mayer’s theory of imperfect gases. In their work they also considered the special case of Y = 2, that is, dimers, and they gave the values of ,Bb for the simple cubic and body-centred cubic lattices as far as k = 5 and for the diamond and plane square lattices as far as k = 6 .

In the present work monomer-dimer mixtures have been considered and the values o f j k for the face-centred cubic lattice have been obtained as far as K = 5. The evalua- tion of P6 and the succeeding fir’s would have been prohibitively difficult. The values Ofthe entropy of mixing have also been obtained and it is shown that the results are an mprovement on those of previous treatments.

2. A brief summary of the resdts of Rushbrooke, Scoins and Wakefield In their work Rushbrooke et al. consider an assembly of Nl monomers and Nr rigid

?-mers occupying N (= Nl+rNr) lattice sites. The volume concentration 0 and the 969

970 D. H. Trevena

mole fraction x of the r-mers are then given by

and

These authors then give the equation

for the partial vapour pressure pl of monomers. Here p l o is the partial vapour pressure of the pure monomer liquid and w = bl (the direct analogue of the first bI occurring in Mayer’s imperfect gas theory).

For the r-mer component the corresponding result is

wherep? is the partial vapour pressure of the pure r-mer liquid and $,N. is the number of ways of arranging N, r-mers on rN, lattice sites such that they do not in any way overlap.

In these equations the p i s are the cluster sums discussed by Rushbrooke and his co-workers. The evaluation of f!!k involves two different geometrid concepts, namely (i) the number of ways in which k+ 1 mutually overlapping r-mers can be placed on the lattice and (5) the set of irreducible cluster diagrams connecting k + 1 points. These / l i s will be discussed more fully in the next section.

Considering now the case r = 2 (dimers) the authors show that equations (1) and (2) can be reduced to

and 9

In(*) = e+ln(:)-(l-O)g(e)- 1 Q(8)dO 42P2O 0

(4)

where z = 2w is the co-ordination number of the lattice concerned and

The quantities 6, in this equation will be discussed in the next section.

leads to the result 42 in equation (4) is determined by the condition that when 0 = 1, p , = Pzo; this

The entropy of mixing of athermal monomer-dim liquid mixtures 971

3, m e quantities Bk and 8, and their values for the face-centred cubic lattice Rushbrooke et al. show how the previous results of Flory and those of Miller and

Gugge&eim fit quite naturally into the mathematical scheme. The cluster sum Pk in- volves putting k+ 1 mutually overlapping dimers on the lattice. Now let us mite

f l k = flk’ f 6k* (7) It then emerges from their discussion that flk’ only takes into account those dimer configurations in which the A + 1 dimers all overlap at the same site. In general there are other arrangements of k+ 1 mutually overlapping dimers which contribute to pk, i.e. arrangements in which the dimers do not all have a point in common; the sum of h e contributions from these remaining configurations is 8,. The Flory a n d ’ m e r - Guggenheim equations rest on the assumption that the only contributions are those coming from the configurations in which the k+ 1 dimers all overlap at the same point, i.e. putting flk = Bk’. The .Miller-Guggenheim approximation evaluates correctly while the Flory approximation does not.

The value of Bk in equation (7) is the Correct value and 8, may be regarded as a correction term to the Miller-Guggenheim value P i . At &st sight it appears that many dimer codgurations will contribute to 8k but a detailed study shows that the only remaining configurations which do contribute to 8, are those involving closed circuits of dimers with no ‘tails’. The results obtained for the face-centred cubic lattice are given in the table. In this table the numbers are exact. It is interesting to note that k jk (but not f l k itself) is always a negative integer while 8, is invariably an integer.

It was found possible to go as far as k = 5, that is, codgurations involving six overlapping dimers. The difficulties of dealing with seven or more overlapping dimers would be prohibitive. (To give some idea of the situation it can be stated that 99 different dimer codgurations were involved in the evaluation of lq5). As far as the present work has gone the results are exact.

One final word about the corrections 6, at this point. At this stage they appear to be quite insignificant but, as we shall see later, they make quite a difference when the Flory terms are subtracted out.

Vdues of Pk and 5, obtained for the face-centred cubic lattice (2 = 12)

k Pis Bk

1 - 23 0 2 -29512 -4 3 - 343713 +6 4 - 4143114 + 10 5 -49844315 -156

4. Results for the molar entropy of mixing It is interesting to compare the results of the present work with those of other

workers by considering the molar entropy of mixing A,$, for which we have the general result

for a monomer-dimer mixture.

972 D. H. Trevem

If we consider Flory’s results this equation takes the form

-- AmS’ 1 {eInO+2(l-e)ln(l-e)}. - -- R 2-8 (9)

In our comparison we shall consider McGlashan’s (1951) ‘second approximation’. This approximation implies that some, but not all, of the dimer configurations neglected in the Miller-Guggenheim approximation are accounted for. It is therefore a better approximation than the Miller-Guggenheim one and leads to the result

AmS 1 8 1 -- - -{ - e In e - 2(i -e) ln(1 -e) - j In @(e) de + fi j In @(e) de} (io)

0 0 R 2-e

where e e2 02 e 4 @(e) = 1--+-+-+-+ ... . 12 36 144 864

(The coefficient of B5 in this series is zero.)

to substitute into equation (8) we have Now let us turn to the results of the present work. Using equations (3) and (4)

since x = e/@-e). equivalent form

Using the fact that when 0 = 1, AmS/R = 0, equation (12) may be written in the

- = -{-elnt3-2(1-e)In(l-8)+ Q(e)de -e Q(O)dO}. (120) AmS 1 s: s: R 2-e

Comparison of equations (10) and (12a) show that the results of McGlashan and those of the present work are of the same form if we identify Q(8) with -lnO(fl). We shall therefore take the expression on the right-hand side of equation (12) as being the ~GT- which edxaces both the resdts of McGIashm zx! those of the present work; it will also embrace the Miller-Guggenheim results which are a special case of the present work (i.e. putting 6, = 0 (see equation (7)). Correct substitution for Q(0) will then yield the corresponding value of A,S/R.

For brevity we shall refer to McGlashan’s results by the letters MLM, those of Floq by PJF, those of Miller-Guggenheim by MG and those of the present work bY DHT.

For the MLM results we have, using equation (ll),

as far as t5 where t = O/z = 8/12.

The entropy of mixing of athermal mono"-dimer liquid mixtures 973

T~ obtain the value of Q(e) for the present theory we substitute the values of the s i s in the table into equation ( 5 ) with z = 12, whence

7t2 19t3 41t4 779t5 2 3 4 5

Q(O) = t--+-+--- + ... (14)

(DHT)

far as our calculations on the face-centred cubic lattice allow us. The form of Q(0) appropriate to the MG theory is obtained by putting each 6 ,

t2 t3 t4 t5 2 3 4 5

(MG)

equal to zero and z = 12 in equation (5 ) , whence we obtain

Q(e) = t+-+-+-+-+ ... (15)

as far as t5.

0

Curves showing deviations from Flory's results: curve 1, McGlashan's 'second approximation'; curve 2, results of present work; curve 3, Miller-Guggenheh

approximation.

A convenient way of comparing these varioiis resiiks is to consider the Merexxe given by Flory's formula (equation (9)) and those given between the value of

by the other three. using equations (9) and (12) we then have on subtraction

A,S(PJF) - AmS = 1[8(ln(:)+l)- /'Q(O)dO] R 2 - e 0

where the value of dZ is determined by the condition that the left-hand side of this equation is zero when 0 = 1. Using equation (16) with the appropriate form of Q(e) as &en by (13), (14) or (E), we find that

where A(e) is a power series in 8.

974 D. H. Trevenu

For the MLM results

4 4 = Adel = 462 95 o - 159 1308 - 3 8 17082 - 43 3083 + 24284.

For the DHT results

4 4 = &(e) = 514718--107362e+i3 598e2-82~- 15586'4.

For the MG results

-44 = 4 6 ) = 640118+18 0 3 8 e + 7 5 8 ~ + 3 8 ~ + 2 e 4 .

In the graphs of the figure the values of e(l-B)A(8)/(2-8) have been plotted against the volume fraction 6' of the dimers. The ordinates of these graphs are &en proportional to the difference between the Flory value of AmS/R and those given by other theories.

5. Discussion of results and conclusions The first claim to be made is that the present work takes into account dimer con-

figurations which were ignored by previous workers, and in this sense it is an improvement on the Miller-Guggenheim and McGlashan treatments. From the graphs of the figure we may conclude that the McGlashan theory underestimates the true deviations from the Flory results by about 15 to 25 per cent while the Miller-Guggenheim theory overestimates it by about 40 per cent.

It is also fair to claim that the results of the present work are substantially correct in the sense that they do justice to the model adopted for the treatment of monomer- dimer mixtures. The p i s have been evaluated as far as &,. The evaluation of ,!$ and and the succeeding p i s becomes prohibitively difficult if not impossible. But & for example, is the coefficient of (6 ' /2~)~ , i.e. (e/,@ in equation (l), and, since 8 < 1, the term /3e(8/z)6 is very small; a similar argument holds for succeeding terms in the series in equation (1). Thus rhe evaluation of the pk)s as far as p5 gives a pretty fair indication of the results of the present theory. If all the p i s were known the results would do full justice to the model. As far as they go, therefore, the results of the present work can be regarded as correct and an improvement on previous treatments.

Acknowledgments

I am most grateful to Professor G. S. Rushbrooke of the Physics Departmen6 The University, Newcastle on Tyne, for suggesting this problem and for his constant encouragement. I am also very indebted to Dr. H. I. Scoins of the University of "- castle for countless helpful discussions.

References FLORY, P. J., 1942, J . Chm. Phys., 10, 51. FOWLER, R. H., and RUSHBROOKE, G. S., 1937, Trans. Faraday Soc., 33,1272. GUGGENHEIM, E. A., 1952, Mixtures (London: Oxford University Press). MCGLASHAN, M. L., 1951, Trans. Faraday Soc., 47,1042. RUSHBROOKE, G. S., SCOINS, H. I., and WAKEFIELD, A. J., 1953, Disc. Faraday Soc., NO. 15,