General discussion

GENERAL DISCUSSIONProf. A. Silberberg (Weizmmn Institute of Science, Israel) said: In the usualmethod of deriving the Flory-Huggins expression,l a long-chain, linear, highlyflexible structure for the solute molecule is assumed. This, however, is not arequirement and a more general derivation can be given., One assumes mixing with-out volume change and divides the solute molecule, arbitrarily, into r subunits equalin size to the solvent molecule. This subdivision is completely conceptual and weassume that upon mixing with solvent each one out of the set of r sub-units, whileremembering its topological connection to the others, can occupy any position in thespace available for the mixture, i.e., can become arbitrarily far separated from theothers. It follows that n1 solvent molecules of volume V,, and n2 solute molecules ofvolume rV, (subdivided as above) can be distributed over a space of total volume,V = nlVl+m2Vl (1)in (n, +m2)! distinguishable ways if all the solvent molecules and all the solutemolecules and their sub-units are regarded as distinguishable.Dropping the distinction among solvent molecules and among solute moleculebut maintaining distinguishability within each set of sub-units composing a solutemolecule, i.e., retaining the topological, if not the physical, connectivity of eachsolute molecule, reduces the number of distinguishable arrangement to(2) (n, + m,) ! In, ! n, !To correct this expression for the dismemberment of the solute molecules we mayargue as follows.In (2) it is assumed that each solute subunit has the entire volumeVat its disposal irrespective of the real structural restrictions.This can be true onlyfor one, say the first, sub-unit in the solute molecule, the other (r - 1) sub-units arerestricted in some way to a parametric volume V* associated with the first sub-unitand not to the total volume V. Hence we have overestimated the number of con-figurations in (2) by a factor (V*/V\u003enz(\"-l). With this correction we arrive at thefollowing expression for the microcanonical partition function of the mixture :(3)where u), and u)= are mixture-independent factors associated with each solvent andsolute molecule respectively. We now use (3) to write the microcanonical partitionfunction for the unmixed species 1 and 2 respectively :R(n,, 11,) = o ~ ~ o ~ [ ( ~ ~ + rn,) !Inl !n, !](V*/V)nz(r- '1,which together with (3) give for the free energy of mixing (no internal energy change isassumed)AG = kTIn [a(n,)~(nl\u003e/~(n,, n2)l= kT In [nl ! (m2)!/(nl +rn2)!][(nl +rn2)/(rn,)]\"2(r- l )= (n, + rn2)kT[x1 In u1 + x2 In 24, (5)'see, e.g., C.Tanford, Physical Chemistry of Macromolecules (John Wiley, New York, 1961),p. 204.A. Silberberg, J. Chem. Phys., 1968, 48,2835, appendix.1 6GENERAL DISCUSSION 163where x1 and x2 are the mole fractions and u1 and u2 are the volume fraction of thecomponents.The Flory-Huggins mixing equation (5) was thus derived without detailed specifica-tion of how the solute molecules are built up. The derivation depends only on theassumption that a parameter V* exists that (V*/ V)n2(r- l) is an adequate correctionfactor for connectivity and most important that the parameter V* does not change ingoing from the unmixed state of pure 2 to the mixture.Mr.C. P . Hicks (Uniuersity of Bristol) (communicated): It has sometimes beensuggested that partial molar volumes should be used in the calculation of volumefractions in the place of the molar volumes of the pure components. While it isimpossible to say that it is correct to use pure data, it is certainly wrong to use partialmolar volumes instead. The relationship V, = xlvl +x2V2 must not be taken tosuggest that is the effective volume of component i in the mixture, i.e., the partof the total volume occupied by, or to be ascribed to,l component i.In fact, thepartial molar volume need not bear any particular relationship to the effective volume ;the former merely shows how the total volume will change for vanishingly smallincrements of the component. This is not a fine distinction : consider the extremesize difference case mentioned in Rowlinson’s paper where small molecules (1) arefitted into the spaces between large molecules (2). Here, until a certain criticalconcentration of 1 is reached, Vl = 0, but this is clearly not the volume which themolecules of 1 occupy. After this point, vl tends to the pure molar volume of 1,reaching it at once if the size ratio is large enough.This naive argument, based on a well-ordered structure, is difficult to extend to afluid, but there is experimental evidence to show that occasionally Vl\u003cO, as, e.g.,near the liquid-vapour critical point in methane + propane,2 methane + i-~entane,~methane + 2-methylpentane and ethane + n-~entane,~ all of which are “ simple ”systems.The use of partial molar volumes to assess effective volumes will sometimeslead to nonsensical values, and even when they are not so I suggest that it would bebetter to use the pure molar volumes as a not necessarily accurate estimate of theeffective volume of each component.Dr. H . Tompa (Union Carbide European Research Associates, Brussels) said :Flory has referred to an old suggestion of mine to consider the effects of a possibleconcentration dependence of the interaction parameter, though this suggestion hasalso been made by others.This encourages me to bring up another old suggestionof mine, also not original since it has also been made before by others, in particularby Munster : all theories of polymer solutions assume that the force field around themolecules or polymer segments is spherical, in other words, that there are noorientation effects. However, clearly the interaction energy, e.g., of two molecules ofbenzene depends on whether the molecules are “ face to face ” or ‘‘ face to edge ” or“ edge to edge ’’ or in any other mutual orientation, and the same applies, e.g., tothe phenyl group of polystyrene. The mixing of two components will interfere withthe state of orientation and will lead in particular to an additional term in the entropyof mixing.5 I would suggest that the discrepancies with experiment which remaineven in the greatly refined theory which Flory has just presented might well be due tosuch orientation effects.J.S. Rowlinson, Liquids and Liquid Mixtures, 2nd ed. (Butterworth, London, 1969), p. 106.* P. L. Chueh and J. M. Prauznitz, Ameu. Znst. Chem. Eng. J., 1967, 13, 1099.A. J. Davenport, J. S. Rowlinson and G. Saville, Trans. Faraduy SOC., 1966,62, 322.H. H. Reamer, V. Berry and B. H. Sage, J . Chem. Eng. Data, 1961, 6,185.H. Tompa, J. Chem. Phys., 1953, 21, 250164 GENERAL DISCUSSIONDr. R. F. T. Stepto (University of Mmchester) said: In agreement with Flory'sremarks, one believes that since the advent and acceptance of the rotational-isomericstate model for real polymer chains, the problem of excluded volume could beapproached more meaningfully by using this model than by using lattice models.Using the rotational-isomeric state model and Lennard-Jones expressions for theenergies of interaction between segments we have obtained preliminary results whichindicate that the value of the exponent y depends to a large extent on the actualLennard-Jones parameters used.Prof. R.L. Scott (University of California, Los Angeles) said: Flory has reporteddifficulties in fitting the thermodynamic properties of solutions of polydimethyl-siloxanes to his equation of state theory with a single parameter XI2. Similardifficulties were encountered by S. P. Koh working with Prof. G. N. Malcolm inNew Zealand.In their unpublished work on polydimethylsiloxane in CCl, andCHC13, they found it necessary to assume different values of X12 to fit GE, RE, and rE. I suggest that part of the difficulty may arise from the fact that polydimethyl-siloxane is a very bulky molecule with a chain cross-section much larger than thatof a solvent molecule. In such a case the usual Flory-Huggins configurational termis appreciably in error and contributes too large a term to the excess entropy.Adjusting X12, which is essentially an energy parameter, is insufficient and a secondparameter for the local configurational entropy (as distinct from equation of stateeffects) may be necessary.Prof. G. N. Malcolm (Massey University, New Zealand) (communicated): Theresults obtained by Dr.S. P. Koh for solutions of polydimethylsiloxane (PDMS)(with number average relative molecular mass of 1655+30) in CCl, and CHC13 aresummarized in the following table. The mixing functions are expressed as the totalchange in the property on mixing divided by the total volume of the unmixedcomponents.Scott has described our experience in attempting to fit these results to the equationof state theory of Flory.MAXIMUM VALUES OF MIXING FUNCTIONS AT 278.68 K AND 1 atm.system AmHmax/J cm-3 AmGmax/J cm-3 103AmV/cm3 cm- 3*0.10 f 0.25 f 008PDMS (A)+ CCI, (B) 0.06 at $A = 0.35PDMS (A)+ CHC13 (B) 0.14 at ( b ~ = 0.55- 5.89 at $A = 0.70- 5.50 at +A = 0.75- 2.13 at 4~ = 0.60- 3.83 at @A = 0.60Prof. P. J. Flory (Stanford University) said : I agree with Scott that a large disparitybetween the diameter of the polymer chain and that of the solvent should affect theentropy of mixing. The diameter of the PDMS chain is comparatively large, and insolutions of PDMS with CHC13 or CCl, this difference may act to diminish the entropyof mixing.But in the systems PDMS + cyclohexane and PDMS + chlorobenzenethis difference is not great. Moreover, the excess volume as well as the entropydepart from our theory. The former would not be altered by modification of thecombinatorial entropy. Hence, we must look elsewhere for the explanation of thediscrepancy between theory and experiment in this case.Dr. M. L. Huggins (Arcadia Institute for Scientijic Research l) said: For comparisonwith other theories, I outline briefly my new theory of the thermodynamic properties135 Northridge Lane, Woodside, Calif.94062GENERAL DISCUSSION 165of solutions.' It is based on concepts I have previously discussed with regard topolymer solutions, but is simplified by omission of some factors that are usuallyunimportant.Like Flory in his new theory, outlined in the Spiers Memorial Lecture, I deal withsegment surfaces, but the surface area that enters into the equations is the averagearea that makes contact with other surfaces. The energies in the equations are theaverage contact energies per unit area of contact, for each type of contact. For asystem containing two types of segments there are three types of contact. I assumethe relative contact areas for these three types to be governed by an equilibriumconstant, equal to unity for randomness of contacts.With these assumptions, equations are deduced for the total energy of the systemand derived quantities as functions of the variables mentioned.For excess enthalpiesof mixing the contact energies enter only as the energy change for replacement ofcontacts between like segments by contacts between unlike segments. Similarly, theequations for excess volumes of mixing involve the volume change when contactsbetween like segments are replaced by contacts between unlike segments. UnlikeFlory, I do not assume the energy and volume change parameters to be the same;each is deduced from experimental data.An important feature of my theory is the transferability of the magnitudes of theparameters from one system to another containing the same segment types.Thistheory gives excellent agreement with experiment, when tested by good data on avariety of systems.Prof. J. D. Ghmez-Ibhiiez ( WesZeyan University, Conn.) said : Relative to solutionsin which n-alkanes are one of the components, it may be of interest to report somerecent measurements on binary mixtures of cyclohexane with n-alkanes carried outin our laboratory which together with others, available in the literature, may permitmaking some generalizations about the effect of chain length on the excess thermo-dynamic functions. We have measured the enthalpies of mixing of cyclohexanewith n-nonane and with n-dodecane. Similar measurements have been made byLundberg with n-heptane and with n-hexadecane.In all four cases the enthalpiesof mixing are positive and increase in magnitude with increasing length of the alkane,being in fact a linear function of n, the number of carbons in the chain. For a givenalkane the magnitude of HE decreases with increasing temperature, this effect beingmore pronounced as the length of the alkane increases.Our recent measurements of the excess Gibbs free energy of mixing of cyclo-hexane with n-eicosane, together with our previous ones with n-hexadecane andwith n-dodecane show a similar linear dependence on chain length, and the threebinary systems exhibit also a relation of congruence. In all three cases GE is negative,and increases in absolute value with increasing length of the alkane.For a givenalkane, GE becomes more negative with increasing temperature. The excess volumesof these mixtures, on the other hand, are positive and linear function (around themaximum) not of n, but of l / n . 5It may be of interest to report also on some preliminary results obtained for the sys-tems formed by n-hexadecane with 2,2,4-trimethylpentane and with transdecaline. Forthe first system, GE is small but positive, also decreasing with increasing temperature,M. L. Huggins, J. Phys. Chem., 1970, 74, 371.G. W. Lundberg, J. Chem. Eng. Data, 1964,9, 193.J. D. G6mez-IbBfiez and J. J. C. Shieh, J. Phys. Chem., 1965,69, 1660.J. D. G6mez-IbAiiez, J. J. C. Shieh and E. M. Thorsteinson, J . Phys. Chem., 1966, 70, 1998.J.D. G6mez-IbAfiez and C. T. Liu, J . Phys. Chem., 1961, 65,2148166 GENERAL DISCUSSIONand HE has a value of about 200 J for the equimolar mixture at 25”. For thesecond system GE is about zero at 25” and HE is about 50 J at x = 0.5. We believethat these type of measurements may yield valuable empirical information which mayhelp to elucidate the effect size and shape, previously referred to in this Discussion,may have on the excess thermodynamic functions.For the mixtures formed by cyclohexane with n-alkanes we have calculated, fromequations of state, and also following the procedure of Abe and Flory,, theparameter X I , of Flory’s theory. We find that the value for X , , thus calculatedpredicts rather well the experimental values for GE, but the predictions for the excessenthalpies, entropies and volumes of mixing are much less satisfactory.Dr. C.L. Young (University of Leeds) said : Fig. 2 of Scott and van Konynenburgis in surprisingly good agreement with the experimental data for hydrocarbons butthe boundaries between type 11, I11 and IV behaviour are strongly dependent on thecombining rule for a12. The van der Waals combining rule for a,, appears to befairly accurate for mixtures of widely differing size molecules. It may be comparedwith the geometric mean and Hudson and McCoubrey combining rules for theinteraction parameter E , ~ :a,2/o:za = E12(vdW) = ~:/,’~~~~[o:/l”o:/2”/~~,]~, van der Waals;q2(G.M.) = ~:/1~~;/22, geometric mean;E~ ,(H.M.) = E: i2\u0026:(2, [o: /:o~/z”lo, ,16j”, Hudson and McCoubrey,TABLE COMPARISON OF CALCULATED AND EXPERIMENTAL VALUES OF \u003csystem no.van der WaalsOMCTS (a)+ cyclopentane 1 0.937OMCTS+ cyclohexane 2 0.953OMCTS+ cycloheptane 3 0.964OMCTS+ cyclo-octane 4 0.974OMCTS + 2,3-dimethylbutane 5 0.965OMCTS+ benzene 6 0.938OMCTS+ carbontetrachloride 7 0.943ethane+ n-heptane 8 0.955propane+ n-octane 9 0.968n-pentane + n-tridecane 10 0.966n-hexane+ nitrogen 11 0.921n-hexme+ methane 12 0.932n-pentane + methane 13 0.947argon + methane 14 0.978(a) OMCTS = octamethylcyclotetrasiloxane.(b) assuming the ionization potentials are the same.Hudson andMcCoubrey (h)0.878 (b)0.907 (6)0.929 (b)0.948 (6)0.930 (b)0.879 (b)0.889 (6)0.9100.9370.9340.8330.8620.8910.952experimental0.935 (c)0.961 (c)0.979 (c)0.981 (c)0.968 (c)0.940 (c)0.953 (c)0.945 (d)0.972 (d)0.983 (c)0.87 (e)0.96 (f 10.97 (f)0.981 (9)(c) unpublished results obtained from critical temperature data.(d) obtained from critical temperature data taken from sources given in C.P. Hicks and C. L.( e ) obtained from virial coefficients, C. L. Young, Ph.D. Thesis, (University of Bristol, 1967).cf) obtained from virial coefficients data of E. M. Dantzler, C. M. Knobler and M. L. Windsor,(9) obtained from free energy of mixing data, T. W. Leland, J. S. Rowlinson, G. A. Sather and(h) G. M. Hudson and J. C. McCoubrey, Trans. Furahy SOC., 1960,56,76