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Corresponding states relations of dimethylsiloxanes: comparison of short and long chain linear- and cyclic-dimethylsiloxanes
LTUIDPHAS[ [QUIHBRIA ELSEVIER
Fluid Phase Equilibria 128 (1997) 115-130
Corresponding states relations of dimethylsiloxanes: comparison of short and long chain linear- and cyclic-dimethylsiloxanes J u a n - M a n u e l B a r b a r l n - C a s t i l l o a, *, I a n A. M c L u r e b a
Facultad de Ciencias Qufmicas, Unioersidad Aut6noma de Nueoo Le6n, Monterrey, Mdxico b Department of Chemistry, The University, Sheffield $3 7HF, UK
Received 5 February 1996; accepted 22 May 1996
Abstract A previous analysis of the pressure-volume-temperature behaviour showed that the linear dimethylsiloxanes conform to a phenomenological principle of corresponding states resembling but not identical to that of the normal alkanes. A similar analysis for tetramethylsilane--regarded as the first member of the dimethylsiloxane homologous series: two cyclic dimethylsiloxanes, octamethylcyclotetrasiloxane and decamethylcyclopentasiloxane, and two polydimethylsiloxanes, one linear and one cyclic--confirms that they conform to a similar principle. The results are used to predict with fair success the excess volumes of mixing of a series of six related mixtures of cyclic substances--cyclohexane, cyclooctane and cyclodecane + either octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane--and two linear + cyclic substance mixtures--either n-hexane or decamethyltetrasiloxane + octamethyicyclotetrasiloxane. Keywords: Theory; Corresponding states relations; Reduction parameters; Excess volumes of mixing; Linear- and cyclic-dimethylsiloxanes; Cycloalkanes; Tetramethylsilane
I. I n t r o d u c t i o n In a previous paper [1] we gave an account of the conformance o f the linear dimethylsiloxanes to a simple phenomenological principle o f corresponding states o f the kind described by Patterson and Bardin [2]. Here we turn to tetramethylsilane (TMS), regarded as the forerunner o f the series, and to four different dimethylsiloxane liquids: two cyclic oligomers, octamethylcyclotetrasiloxane and decamethylcyclopentasiloxane, and two polymers, one linear ( M S 2 0 0 / 2 0 c s ) containing around 26 silicon atoms and the other a carefully-fractionated cyclic polymer containing 79 number-average skeletal bonds. It is our purpose first to seek the degree of conformity of these materials with the
* Corresponding author. 0378-3812/97/$17.00 Copyright© 1997 ElsevierScienceB.V. All rights reserved. PII S0378-3812(96)03132-9
116
J.-M. Barbar{n-Castillo, 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
earlier oligomeric linear dimethylsiloxane corresponding-states principle and, secondly, to use the results to calculate the excess volumes of mixing of six binary liquid mixtures containing a cycloalkane with six, eight, or ten carbon atoms with either octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane, and of two linear+ cyclic substance mixtures, either n-hexane or decamethyltetrasiloxane with octamethylcyclotetrasiloxane. In this paper we review first the advantages of the phenomenological approach for the validation of a principle of corresponding states and demonstrate its applicability to this extended range of dimethylsiloxanes with particular reference to the normal alkanes, we follow with the evaluation of the temperature, volume, pressure and entropy reduction parameters, and we close with an application of the principle to the prediction of excess volumes of mixing of some mixtures containing dimethylsiloxanes.
2. Phenomenological corresponding-states principle Principles of corresponding states are written in terms of reduced quantities ) ( = X / X * , where X is some appropriate quantity and X * its reduction parameter, characteristic of the particular substance concerned, which in some formulations of the principle is a molecular quantity or in others phenomenological although related directly or otherwise to a molecular quantity or combinations of molecular quantities. Applied to equations of state, the principle is written as /5 =/3(7~,~7), where T is temperature, V is volume--usually molar v o l u m e - - a n d ~ ( T , V ) is a universal function for the class of substances concerned. The main problems in this application of the principle of corresponding states are the unequivocal identification of the reduction factors for different substances and the formulation of the universal function. The essence of the Patterson and Bardin phenomenological approach to these difficulties rests on the recognition that if a corresponding-states principle exists for a series of related substances, such as here a homologous series in the liquid state for which the reduction factors might be expected to depend heavily on chain length, then at vanishing pressure the quantity (anT), where at, is the isobaric thermal expansivity (01n V/OT)p, equals the reduced quantity (SpT~ and is therefore an effective measure of the reduced temperature thus reducing the first problem for temperature at least - - t h e need to identify T *. It follows then by simple geometry that if the curves of the logarithm of X as functions of either ( a p T ) or, for some choices of X to take advantage of essentially-accidental linearity, its reciprocal, are parallel, then a demonstration of the conformity of the family of substances concerned to the principle is achieved. The actual reduction factors themselves emerge in this procedure from simple single-axis shifts of the curves for each substance on to the curve for a reference substance. Procedures of this kind permit the calculation of T* and V * from simple density-temperature data and of p * from data on a pressure-related quantity--usually the isochoric thermal pressure coefficient ( 0 p / 0 T ) v as a function of temperature. The combined reduction factor ( p * V * / T * ) has the nature of an entropic reduction parameter S* and this quantity assumes particular importance when the principle is applied to chain-molecule substances. The genesis of this procedure was recognized inter alia by Prigogine and coworkers [3] and later by Simha and Havlik [4] but the approach employed required a double mapping of log V - log T plots which undermined the achievable accuracy of the reduction factors which were obtained.
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
117
3. Conformity of dimethylsiloxanes The data on which this analysis chiefly rests are the result of the accurate determination of the orthobaric densities of the substances identified in the introduction over a fairly wide range of temperature using a hydrostatic density balance. The results have been published elsewhere [5]. The most searching test of the validity of a phenomenological principle of corresponding states, quite independent of the discussion of reduction factors, probably rests on the superimposability of the dimensionless reduced quantity - { 0 ( a p ) - l / OT )}p = (ap) - 2 ( OOlp/OT)p as a function of (apT) for all the members of the class of substance which might be expected to conform to it. Fig. 1 shows the result of the calculation. There is fair conformity with the principle. It must be borne in mind that the calculation is incorporating a double-differentiation of the density and this procedure places very heavy weight both on the accuracy of the original density measurements and on the form of curve-fitting expression used to describe them. The test involves double differentiation of smoothed specific volume data, with the result that the exact form of the curves--especially at the ends--is somewhat artificial and depends upon particularities of the data fitting procedure. The inclusion of an extra coefficient in the polynomial fitting expression for the molar volume changes the shapes of the curves a little [6] not affecting the final broad conclusions; the use of only three coefficients, however, making d a p / d T a temperature independent quantity, leads to absurd results. In this study few of the results refer to temperatures close to the gas-liquid critical point and so a simple power series in temperature for the natural logarithm of the specific volume was employed. In more detailed studies where the behaviour of the density close to the gas-liquid critical point is 2 "r5
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Fig. 1. Test for the conformity of the dimensionless reduced quantity (Otp) -2 as a function of (apT)" (a) tetramethylsilane, (b) to (g) the linear oligomers dimer to hexamer, and the linear polymer MS200/20cs. The dashed lines (i) to (k) refer to the octamethylcyclotetrasiloxane and decamethylcyclopentasiloxane, and the cyclic polymer, respectively. Curve (o) reflects a mean value of B = 8.70.
118
J.-M. Barbarfn-Castillo , I.A. McLur e / Fluid Phase Equilibria 128 (1997) 115-130
explored we have employed Wegner extended-scaling of an expression with non-classical exponents [7]. Although this is a superior process, it is not employed here since it is unnecessary for all but tetramethylsilane and even for that substance, the relevant portion of liquid range is closer to the triple point than to the critical point and so its introduction is an unnecessary sophistication.
4. Relative reduction parameters from corresponding states relations On the basis that the most stringent test for the existence of a principle of corresponding states for the cyclic dimethylsiloxanes has been satisfied we now move to the evaluation of the relative reduction parameters. Absolute values of temperature, volume, and pressure reduction parameters for chain-molecule liquids have been calculated by using theoretical equations of state based upon the cell partition function. Shih and Flory [8] used their theoretical equation of state in combination with experimental specific volume, thermal expansivity, and thermal pressure coefficient data over wide temperature ranges. Simha and Havlik [4] followed the route to evaluate the reduction parameters from the equation of state developed by Prigogine et al. [9], by using volume-temperature data at atmospheric pressure in combination with experimental enthalpies of vaporization at low temperatures. A treatment essentially phenomenological is followed in this work for the evaluation of the relative reduction parameters, in a procedure requiring no fluid model or empirical equation of state. In the process were also calculated the reduction parameters for the dimethylsiloxanes, after choosing a
36
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loglo(T/K) Fig. 2. Test for T* through the plot of ( a p T ) ] against logl0(T/K): (a) tetramethylsilane, (b) to (h) the linear oligomers dimer to hexamer, and the linear polymers MS200/20cs and PDMSO, respectively. Dashed lines (i) to (k) octamethylcyclotetrasiloxane, decamethylcyclopentasiloxane, and the cyclic polymer, respectively.
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
119
reference substance for which the reduction parameters were already known from calculations based on equation of state data. The temperature reduction parameter T* is most easily obtained from the plot of (OtpT)- ! against log l0 ( T / K ) . If a principle of corresponding states exists then the curves for each substance should be parallel. Fig. 2 demonstrates that this is so, not only for tetramethylsilane and the linear dimethylsiloxane oligomers considered previously, but also for octamethylcyclotetrasiloxane and decamethylcyclotpentasiloxane, the linear and cyclic polymers of this work for which density were measured, and the long-chain polymer of Shih and Flory, which was added to this test in order to extend the scope of the treatment and of the reduction parameters to be calculated. Since the (o~pT)-i against log l0 ( T / K ) curves are highly linear--probably particularly true only far from the gas-liquid critical p o i n t - - t h e results are fitted to expressions of the form:
(apT)-'= A -
(1)
Blog,o(T/K )
The values of the constants A and B for each substance are listed in Table 1. A useful measure of the parallelism of the lines for each s u b s t a n c e - - a n d hence their conformity with the principle--is afforded by the similarity of the B values. It is clear that B is not entirely constant but decreases somewhat with chain length as do the corresponding quantity in the normal alkanes. A comparison of parameters A and B for the dimethylsiloxanes with those reported by Patterson and Bardin for the n-alkanes (24.4 for A and 8.7 for B as averaged values against 27.7 and 10.7 respectively for the n-alkanes) shows that these series follow slightly different equations of state. The relative temperature reduction factor T
Table 1 The constants A to F or the various fitting expressions over temperature ranges AT of the the thermodynamic properties of tetramethylsilane, the oligomeric linear and cyclic dimethylsiloxanes, and the linear and cyclic polydimethylsiloxanes as linear functions of either or ( OtpT)- ! Substance AT (K) A B C D E F
(apT)
Linear TMS dimer trimer tetramer pentamer hexamer MS200/20cs PDMSO Cyclic tetramer pentamer PDMSO
224/327 278/358 300/412 300/412 297/408 293/412 292/470 293/473
26.78 25.62 25.11 24.76 24.47 24.26 22.40 22.41
10.06 9.38 9.07 8.85 8.68 8.56 7.64 7.57
2.03 2.24 2.38 2.48 2.56 2.62 3.20 4.91
0.21 0.22 0.21 0.25 0.27 0.28 0.35 0.39
5.52 5.55 5.56 5.59 5.61 5.63
0.48 0.58 0.65 0.75 0.83 0.91
5.69
1.25
292/403 302/414 302/455
24.90 24.97 22.27
8.94 8.87 7.53
2.41 2.52 3.38
0.24 0.23 0.37
5.78
1.13
J.-M. Barbarfn-Castillo, 1.,4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
120
Table 2 Chain length n, molecular weight MW, and the temperature, volume, pressure and entropy reduction parameters, T~,n) / T~f, PCs,,)/Pr*~f and Vcs,,)/Vr~f, respectively, relative to those for n-octane, for tetramethylsilane, the oligomeric linear and cyclic dimethylsiloxanes, and three long chain molecules, two linear and one cyclic polydimethylsiloxanes Substance Linear TMS dimer trimer tetramer pentamer hexamer MS200/20cs PDMSO Cyclic tetramer pentamer PDMSO
*
,
*
,
*
*
n
MW
T~s.,o / Tree
V~s.,o / Vref
P~S,,,) / Pref
3 5 7 9 11 13 47 2560
88.2 162.4 236.5 310.7 384.9 459.0 1900 100000
0.797 0.904 0.968 1.023 1.064 1.093 1.283 1.362
0.775 1.263 1.763 2.262 2.767 3.273 13.21 695.3
0.871 0.836 0.801 0.782 0.756 0.742 0.616 0.619
8 10 79
296.6 370.8 2935.
0.999 1.066 1.349
1.915 2.436 20.30
0.847 0.808 0.681
The relative v o l u m e reduction parameter ratio V{s,,,)/vr*~f is evaluated from the closely linear plot o f logl0 ( V / c m 3 m o l - l ) as a function o f (OtpT) shown as Fig. 3. From the linear fits o f the various curves we see that the data fairly well conform to expressions o f the form l o g , o ( V c m -3 m o l - ' ) = C
+ D(apT)
(2)
The coefficients C and D are listed in Table 1. The essential similarity o f D is again evidence for the conformity o f both tetramethylsilane and the cyclic dimethylsiloxanes to the same correspondingstates principle as the linear dimethylsiloxanes. The relative v o l u m e reduction factors V~s*,,)/Vref are evaluated at ( a p T ) - l = 2.0 and the values are listed in Table 2. Besides the linear dimethylsiloxanes, only for octamethylcyclotetrasiloxane [10] and the high molecular weight dimethylsiloxane p o l y m e r o f Shih and Flory has the thermal pressure coefficient been measured over a reasonable range o f temperature. For tetramethylsilane, usefully regarded as the first m e m b e r o f the linear dimethylsiloxane series [11], the thermal pressure coefficients were calculated from published ultrasonic speeds [12], heat capacities [13], and the results o f our own density measurements [5]. By straightforward thermodynamic manipulation we determine the isothermal compressibility fir and hence Yv via Yv Olp/J~V" Using these results together we have used the route to the relative pressure reduction factor P~S.,,)/Pr*~f by setting the values in the form ( T ' Y v / k P a ) in the linear expression =
log,o ( T y v / k P a )
= E - F(apT)
(3)
The values o f the coefficients E and F are listed in Table 1 and the relative pressure reduction parameters P~s,,,)/Pr~f, evaluated at ( a p T ) - l = 2.0, are in Table 2. The relative reduction parameters throughout this work and our previous paper on the subject have been calculated taking n-octane as the reference substance and at an arbitrary fixed value o f
J.-M. Barbarln-Castillo, LA. McLure / Fluid Phase Equilibria I28 (1997) 115-130 52
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(apT) Fig. 3. Test for V * through the plot of (or,T) against log10(V/cm 3 tool l): (a) to (h), tetramethylsilane, the linear oligomers dimer to hexamer and the linear polymers MS200/20cs and PDMSO, respectively. Dashed lines (i) to (k), octamethylcyclotetrasiloxane, decamethylcyclopentasiloxane, and the cyclic polymer, respectively.
( a p T ) - 1 _~ 2.0 at which there is good overlap of the different dimethylsiloxane reduced temperatures. However, any of the dimethylsiloxanes could act equally well as the reference substance.
5. Temperature, volume, pressure, and entropy reduction parameters A theoretical equation of state gives the functional dependence of ~pT on 7~. It is from it that absolute values of V *, and then T *, can be formally obtained. All the T * found by any one theory are arbitrary to within a constant factor. Patterson et al. [ 14], working with an equation of state obtained from the smoothed potential cell model of Prigogine, reported for the dimer a temperature reduction parameter T~s,2) = 4410 K at 30°C, while Shih and Flory, using a different equation of state, gave also for the dimer the temperature reduction parameter T~s,2~ = 4420 K at 25°C. Since there is no appreciable difference between those two values reported for the same substance, we took the second one as a reference for the calculation of the absolute values of T* for the rest of the oligomers and the polymers of this treatment. Hence, for T~s,2) = 4420 K, together with coefficients A and B, Eq. (1) gives ( a p T ) - 1 = 2.41 for which there is an excellent overlap of the reduced temperatures covered by all the substances involved in this study. The temperature reduction parameter can then be calculated from
T~s,, ) = T~s,:)T~s,,)/T~ s,z )
(4)
where T~s,z) is 298.15 K and T~s,,) is the temperature at which ( a p T ) -1 is equal to 2.41 for the species (S,n). Table 3 shows the values of the temperature reduction parameters for tetramethylsilane,
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
122 Table 3
The temperature, volume, pressure, and entropy reduction parameters T~s.,), V~s.,), P(s,,) and S(s,,), respectively, for tetramethylsilane, five linear and two cyclic oligomeric dimethylsiloxanes, and three long chain molecules, two linear and one cyclic polydimethylsiloxanes Substance T~.~.,) (K) V{.~.,) (cm 3 tool-') P{s.,) (J cm-3) St*s,.) (J mol-t) Linear TMS dimer trimer tetramer pentamer hexamer MS200/20cs PDMSO Cyclic tetramer pentamer PDMSO
3921. 4420. 4718. 4970. 5158. 5291. 6130. 6501.
99.1 161.5 225.5 289.3 353.8 418.6 1690.0 88910.
373. 358. 343. 335. 324. 318. 265. 265.
9.42 13.08 16.40 19.50 22.22 25.15 73.15 3624.
4860. 5180. 6433.
244.9 311.5 2595.7
363. 347. 293.
18.29 20.87 118.4
five linear and two cyclic dimethylsiloxane oligomers, and three long chain p o l y m e r s - - t w o linear and one cyclic. A similar procedure was followed for the calculation o f the specific v o l u m e v * and pressure reduction parameters p * after the known values o f V{s,2) = 0.9945 cm 3 g - l and P{s,2) 358 J cm -3 for the dimer also reported by Shih and Flory. Finally, the entropy reduction parameter S{s,, ) was calculated from the relation S{s,n} = p¢s,n)V{s,,}/T~s,,}, and the results are given in Table 3.
6. Dependence of reduction parameters on chain length or ring size The corresponding states principle for polymeric liquids introduces the concept o f a division in the degrees o f freedom o f the chain-molecule into internal and external categories. The ratio o f the thermal energy of the 3c external degrees o f freedom (pushing for expansion) to the cohesive or intermolecular contact energy qE * (pulling for contraction) gives a quantity o f dimension 1 defined as the r e d u c e d t e m p e r a t u r e T, a quantity that in fact characterizes the expansion or free v o l u m e o f a liquid. Thus, T = (c/q) (kT/e *) = T/T*. The molar configurational quantities o f liquids are those important for their mixing properties and they are related by reduction parameters to the reduced quantities, all functions o f chain length n and reduced temperature 7~:
V(n.T) = V *(n). V ( T ) ;
V* =
U(n,T) = U*(n). U ( T ) ;
U* =Nq(~}e*
S(n,T)= S*(n). S(T);
Nr(,}v*
S* =Nkc(.)
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
2.25
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200
123
2
1.75 1.50 <
125 1.00
0.50
o
0.25 0,00
I
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l
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2
3
4
5
6
7
8
9
I
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10 II 12 13 14 15
n
Fig. 4. Demonstration of chain-length dependence for S ~ through the relative number of the external degrees of freedom, C(s,n)/C(c,8 ) of the n-alkanes ( © ) and the linear ( × ) and cyclic (*) dimethylsiloxanes as a function of chain length. Full lines 1 and 2 are the plots of Eq. (8) and Eq. (9), respectively.
where N is Avogadro's number and k is the Boltzmann constant. The effective number of segments, r, q, and c are proportional to the molecular volumes, molecular surfaces, and number of external degrees of freedom of the chain-molecule liquid, respectively. From these relations results that T* = U * / S * and finally S* = p * V * / T * . From Simha and others there has been much discussion about the chain-length dependence of reduction parameters for chain molecules. We shall not rehearse the conclusions in detail but present our results in the most apt way. The temperature reduction parameters evaluated in this work are well-enough described by an expression of the Simha form T~s,., = 6583{1 - 3 . 3 9 / ( n + 5)}
(5)
for chain lengths n, taken as the number of atoms in the backbone of the longest chain, exceeding 7. Similarly the volume reduction parameters V(S,N) follow a Simha-like relationship, again for n > 7, V~s,.) = (n - 0 . 9 4 1 ) / 0 . 0 2 9
(6)
Turning now to the relative behaviour of the entropy reduction factor S* as a function of chain length, it is expected that the external degrees of freedom and relative chain flexibility for the dimethylsiloxanes be linear with chain length, as in the n-alkanes. The number of external degrees of freedom (3c) of a molecule of chain length n is found through the entropy reduction parameter by the relation S(*s,.)= Nkc(s.n ). With n-octane as the reference substance, this yields C~s../C~c.8 ) =
S~..)IS~c.8 ~
(7)
124
J.-M. Barbarin-Castillo. IA. McLure / Fluid Phase Equilibria 128 (1997) 115-130
A linear relationship was found between the term C
C(c,n)/c(c.8 ) = 0.42846 + 0.0714n
(8)
reported by Patterson and Bardin for the n-alkanes, and
C(s.,)/C(c.8 ) = 0.584
+ 0.1223 n
(9)
from this work and which gives the ratios to within + 0.05% for the linear dimethylsiloxanes from the trimer (n = 7) to the high polymer. For the cyclotetramer this ratio is estimated to within 1.6%. For tetramethylsilane and the dimer C(s,3)/C~c.8) = 0.816 and C(s,5)/cw.s) = 1.132. These values fall below the straight line in a very similar way, albeit less markedly than those for methane and propane. In this respect tetramethylsilane behaves better as the forerunner of the dimethylsiloxane series than does methane as that of the n-alkanes. The relative position of the two lines in Fig. 4 shows the higher flexibility of the dimethylsiloxane chain compared to that of the n-alkanes and this is manifested by the ratio C~s,n)/C~c,8) ranging from 1.93 for tetramethylsilane/methane to 2.58 for hexamer/hexane, although this goes up to 3.11 and 3.63 for the long-chain polymers showing to a good extent the apparent increase of flexibility for the dimethylsiloxanes relative to the n-alkanes as the chain length increases.
7. Corresponding-states principle treatment of excess volumes It has been shown by Simha and Havlik [4] that the molar volumes as a function of temperature,
V(T), for the n-alkanes, dimethylsiloxanes, and several other series of homologous series obey a particular form of the corresponding states principle, i.e.
(10)
V(n,T) = V *(.).V(T)
where V(n,T), V *(n), and 17(7~) are the molar volume of a chain-molecule liquid of chain length n at temperature T, the volume reduction parameter of that liquid, and the reduced volume respectively. According to this, the data can be reduced empirically to a single V - T plot well described by theoretical equations of state developed by Prigogine and co-workers. A phenomenological corresponding-states treatment, independent of theoretical models, was used here for the mixtures involving dimethylsiloxanes. This procedure is due to Patterson and Bardin [2] and was successfully applied to mixtures of n-alkanes. The reduced temperature (7~) of a binary mixture is given by the simple average
( T ) = T / ( T °) = X , T , + X 2 T 2
(11)
where 7~1 and 7~2 are the reduced temperatures of the pure components, T and ( T *) are the temperature and average temperature reduction parameter of the mixture respectively, and X~ is the surface fraction defined by
X, = p ; V , * x , / ( p ; V , * x
t + p 2 V2*x2)
(12)
J.-M. Barbarin-Castillo, 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
125
Table 4 Pressure, volume, and temperature reduction parameters P(s,.), V(s..) and T~s..), respectively, for n-hexane, c-hexane, c-octane and c-decane. Terms A and B are the parameters for Eq. (1) Substance P(c.,) (J cm- 3) V(c,,) (cm 3 mol- 1) T(c., ) (K) A B n-hexane 451 99.67 4450. 27.16 10.01 c-hexane 432 84.26 4720. 27.50 9.80 c-octane 448 115.19 5516. 27.56 9.75 c-decane 459 144.30 6050. 27.17 9.45
where p * and V * are the pressure and volume reduction parameters for the pure components. Any thermodynamic quantity of the mixture can be fou~0d from the reduced curve established for the pure liquids using the reduced temperature of the mixture (7~) obtained with an appropriate averaged reduction parameter. Thus the molar excess volume of mixing is given by
vE=(x,W + x2v; )~(~)-x,v,,17(~,)-x2v;~(g)
(13)
where x t is the mole fraction of component 1 in the mixture. The quantities 17(7~1) and i7(T2), the reduced volumes of the pure components, may be expanded around 17((T)) in terms of power series of (7~1 - (7~)) and ( T 2 - ( T ) ) respectively so that
~(~,) = ~(~)+ (~,- ~)(d~/d~)+
(~)(~,- ~)(d2~/d~T~ ) 2
-
2
+ (~1(7~, - (7~))3(d317/d(7~)3) + (1)(7~, - (7~))4(d417/d(7~)4) + ...
(14)
and similarly for component 2. Since (7~1 - (7~)) = X2(7~1 -- 7~2), the expression for the molar excess volume of mixing becomes
V E= - ( x,X2V, *- x2X,V2* )(7~, -
× (da17/d<~>~)- (+)(x,x~w --("~4)(xIX4VI
* -- x 2 X ? V 2 *
7~2)(dlT/d( 7 ~ ) ) - (½)( - x~x~W;
x,X~V,?- x2X)V2* )(T,- ~2)2
)(~,- ~2)-(d~17/d<~>-)
)(T,- T2)4(d4~7/d(~)4).
(15)
The term involving the second derivative of the reduced volume depends on the temperature variation of ap for the mixture but is related to ( a p T ) itself through the expression
a- 2(dap/dT)
= B/2.303 - (apT)-'
(16)
and the same applies to the higher derivatives, where ap is the isobaric thermal expansion coefficient. All the derivatives of 17((7~)) with respect to ( ( T ) ) can be evaluated from experimental data for the pure components. Full details of the equations involved are to be found elsewhere [15]. Eq. (15) has been used to predict the molar excess volumes of mixing of eight binary liquid mixtures containing either octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane. In one set of results octamethyicyclotetrasiloxane is mixed with cyclohexane, cyclo6ctane or cyclodecane, in the second set decamethylcyclopentasiloxane is mixed with each of the same cycloalkanes, and in the
J.-M. Barbarln-Castillo. 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
126
Table 5 Comparison between the equimolar experimental excess volumes V E(x = 0.50) of six (cycloalkane + cyclic dimethylsiloxane oligomer) mixtures and two (linear+cyclic substance) mixtures and the values predicted from the phenomenological principle of corresponding states
VE(x =
Mixture
c-tetramer: + c-hexane + c-octane + c-decane c-pentamer: + c-hexane + c-octane + c-decane c-tetramer: + n-hexane + n-tetramer
third
Experimental
Predicted
0.086 - 0.540 - 1.020
0.073 - 0.460 -0.900
O. 150 - 0.570 - 1.030
0.316 - 0.290 - 0.810
0.127 0.440
0.054 0.360
set o c t a m e t h y l c y c l o t e t r a s i l o x a n e
camethyltetrasiloxane. aries a r e t h o s e
0.50) (cm 3 mol-~)
is m i x e d
with
linear partners--n-hexane
or the linear de-
T h e v a l u e s o f T * , V *, a n d p * u s e d in t h e c a l c u l a t i o n s f o r t h e d i m e t h y l s i l o x -
of Table
3. T h o s e
for hexane
are from
Orwoll
and
Flory
[16].
For
the higher
c y c l o a l k a n e s t h e s a m e p r o c e d u r e w a s a p p l i e d as f o r c y c l o h e x a n e , u s i n g t h e d e n s i t y d a t a o f M e y e r a n d
0.2
I
r
I
I
I
J
~
r
I
a
01 0 -01 -02 -03
.7.
-0.4 -0.5 -06
> -07 -08 09 -1 -I1 -12
I
J
I
I
I
I
I
I
I
0.1
02
03
04
05
06
07
0.8
09
x
Fig. 5. Molar excess volumes V E as a function of mole fraction x of the second component for mixtures of octamethylcyclotetrasiloxane + cyclohexane at 298.2 K (a), + cyclooctane at 303.2 K (b), or cyclodecane at 298.15 K (c). Full lines are smoothed experimental values and the dashed lines are calculated by the theory described.
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
0.4
I
I
~
I
0.3
I
I
i
I
i
OH
09
127
~--------~----~ /
0.1
~
.'2
j~
-03
-o.4 -0.5 -06
-0.7
\\x x
s///
-0.8
-0.9 -I -I.L 0
I
I
I
I
I
t
I
01
02
03
0.4
05
06
07
I
X
Fig. 6. Molar excess volumes V e as a function of mole fraction x of the second component for mixtures of decamethylcyclopentasiloxane + c y c l o h e x a n e at 303.2 K (a), cyclooctane at 303.2 K (b), or cyclodecane at 298.15 K (c). Full lines are smoothed experimental values and the dashed lines are calculated by the theory described.
Oq)
() 45
i) 35
E.
030
O t~ 2S
>
~) 20
010
IJ 05
0(3O 0
I
I
I
J
1
1
O1
02
0.3
04
0.5
06
[ 07
0.8
0.9
X
Fig. 7. Molar excess volumes V e at 298.2 K as a function of mole fraction x of the second component for the mixtures of octamethylcyclotetrasiloxane + n-hexane (a) or decamethyltetrasiloxane (b). Full lines are smoothed experimental values and the dashed lines are calculated by the theory described.
128
J.-M. Barbarln-Castillo. 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
Hotz [17] for cyclooctane, and our own densities for cyclodecane [18]. The absolute values of the parameters for cyclohexane were those reported by Shih and Flory. The values of all the reduction parameters are listed in Table 4.
8. Comparison of theory and experiment Table 5 compares theory and experiment for eight dimethylsiloxane mixtures in a good variety of substances and this is also illustrated in Figs. 5-7. The experimental results for octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane + cyclohexane or cyclooctane were taken from a previous paper [19]. The experimental results for octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane + cyclodecane and for octamethylcyclotetrasiloxane + n-hexane or decamethyltetrasiloxane at 298.15 K are from our own measurements [20]. Full lines show experimental values of the molar excess volumes of mixing. The dashed lines were obtained after a careful fit was made to the results obtained from the theory and in order to present these results in the same form as Patterson and Bardin did before for n-hexane + n-hexadecane. The agreement between theory and experiment is good in that the magnitudes of V E are reproduced tolerably well--unlike the van der Waals theories which fail to do this [21]--despite the very small changes involved for some of the mixtures, particularly those containing cyclohexane or n-hexane. The relative trend from positive vF~ for mixtures containing cyclohexane to negative for those containing a larger cycloalkane is also satisfactorily predicted. It is interesting to note that the V E of mixtures of a cyclodimethylsiloxane + a linear substance are described fairly well by the theory. For completeness, it should be mentioned that we are leaving for a future paper a comparison of our results against those obtained by using the Flory theory for the prediction of the mixing properties, particularly because although the treatment for linear molecules is straightforward, it is unclear how to adapt it to mixtures containing cyclic molecules.
9. Conclusions Our analysis confirms that a straightforward phenomenological principle of corresponding states exists for dimethylsiioxane liquids of chain length varying from 1 to polymer values, whether cyclic or otherwise, and that the entropy reduction parameter reflects the modest reduction in chain flexibility occasioned by ring closure. Clearly, the phenomenological treatment outlined above does not offers a rigorous interpretation of the excess volumes of alkanes + dimethylsiloxanes, whether linear or cyclic, but it demonstrates that the apparently complex volumetric behaviour of these mixtures is usefully described by a simple application of the principle of corresponding states. Although the above evidence would seem to suggest that the n-alkanes and dimethylsiloxanes follow dissimilar reduced equations of state, we believe that the difference is not sufficient to preclude a meaningful treatment of mixtures of alkanes with dimethylsiloxanes using approaches which assume a strict obedience to a unique principle of corresponding states. That was the reasoning behind the treatment followed in this work in which both, alkanes and dimethylsiloxanes, were taken as being in the same homologous series.
J.-M. Barbarln-Castillo, I A. McLur e / Fluid Phase Equilibria 128 (1997) 115-130
129
The experimental equations o f state for the two series o f chain-molecule l i q u i d s - - a s deduced from line o in Fig. 1, or its equivalent in the corresponding curve for the alkane s e r i e s - - a r e in much greater conformity with each other than they are with a number o f theoretical equations o f state, as it is shown in figure 2 of Ref. [2]. W e have presented further evidence for the usefulness o f the phenomenological approach in predicting with fair certainty the properties o f mixtures o f these and related hydrocarbons.
10. List of symbols T*,p*,S*,V
Lp,
N k r,q,c Ve qE*
* temperature, pressure, entropy, and volume reduction parameters reduced values o f temperature, pressure, and v o l u m e Avogadro's number Boltzmann constant effective numbers o f chain segments proportional to the molar volumes, molecular surfaces and degrees o f freedom, respectively excess v o l u m e o f mixing intermolecular contact energy
G r e e k letters C~p
Yv
isobaric thermal expansivity isothermal compressibility thermal pressure coefficient
Acknowledgements J.-M.B.-C. acknowledges receipt o f financial support from the Consejo Nacional de Ciencia y Tecnologla, M6xico, and leave-on-absence from the Universidad Aut6noma de N u e v o Le6n, Monterrey, M~xico.
References [1] E. Dickinson, I.A. McLure, A.J. Pretty and P.A. Sadler, Chem. Phys., 10 (1975) 17-22. [2] D. Patterson and J.M. Bardin, Trans. Faraday Soc., 66 (1970) 321-334. [3] I. Prigogine, with the collaboration of A. Bellemans and V. Mathot, The Molecular Theory of Solutions, North Holland, Amsterdam, 1957, Chapter 16. [4] R. Simha and A.J. Havlik, J. Am. Chem. Soc., 86 (1964) 197-204. [5] I.A. McLure and J.-M. Barbarln-Castillo, J. Chem. Eng. Data, 39 (1994) 12-13. [6] P.A. Pretty, PhD thesis, University of Sheffield, 1973. [7] I.A. McLure and J.-M. Barbarln-Castillo, Int. J. Thermophys., 14 (1993) 1173-1186. [8] H. Shih and P.J. Flory, Macromolecules, 5 (1972) 758-761. [9] I. Prigogine, N. Trappeniers and V. Mathot, Disc. Faraday Soc., 15 (1953) 93-104. [10] M. Ross and J.H. Hildebrand, J. Phys. Chem., 67 (1963) 1301-1303. [11] I.A. McLure and J.F. Neville, J. Chem. Thermodyn., 9 (1977) 957-961. [12] I.A. McLure, J.-M. Barbarln-Castillo, J.F. Neville and R.A. Pethrick, Thermochim. Acta, 223 (1994) 325-328.
130
[13] [ 141 [ 151 [16] [ 171 [ 181 [19] [20] [21]
J.-M. Barbarin-Custillo,
IA. McLure/Fluid
Phase Equilibria
128 (1997) 115-130
M. Harada, T. Atake and H. Chihara, J. Chem. Thermodyn., 9 (1977) 523-534. D. Patterson, S.N. Bhattacharyya and P. Picker, Trans. Faraday Sot., 64 (1968) 648-661. J.-M. Barbarfn-Castillo, PhD thesis, University of Sheffield, 1984. R.A. Otwoll and P.J. Flory, J. Am. Chem. Sot., 89 (1967) 6814-6822. E.F. Meyer and C.A. Hotz, J. Chem. Eng. Data, 21 (1976) 274-279. LA. McLure and J.-M. Barbarin-Castillo, Chem. Eng. Data, 30 (1985) 253-2.57. E. Dickinson, DC. Hunt and LA. McLure, J. Chem. Thermodyn., 7 (1975) 731-740. J.-M. Barbarfn-Castillo and LA. McLure, J. Chem. Thermodyn., 25 (1993) 81 l-821. E. Dickinson, PhD thesis, University of Sheffield, 1972.
Fluid Phase Equilibria 128 (1997) 115-130
Corresponding states relations of dimethylsiloxanes: comparison of short and long chain linear- and cyclic-dimethylsiloxanes J u a n - M a n u e l B a r b a r l n - C a s t i l l o a, *, I a n A. M c L u r e b a
Facultad de Ciencias Qufmicas, Unioersidad Aut6noma de Nueoo Le6n, Monterrey, Mdxico b Department of Chemistry, The University, Sheffield $3 7HF, UK
Received 5 February 1996; accepted 22 May 1996
Abstract A previous analysis of the pressure-volume-temperature behaviour showed that the linear dimethylsiloxanes conform to a phenomenological principle of corresponding states resembling but not identical to that of the normal alkanes. A similar analysis for tetramethylsilane--regarded as the first member of the dimethylsiloxane homologous series: two cyclic dimethylsiloxanes, octamethylcyclotetrasiloxane and decamethylcyclopentasiloxane, and two polydimethylsiloxanes, one linear and one cyclic--confirms that they conform to a similar principle. The results are used to predict with fair success the excess volumes of mixing of a series of six related mixtures of cyclic substances--cyclohexane, cyclooctane and cyclodecane + either octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane--and two linear + cyclic substance mixtures--either n-hexane or decamethyltetrasiloxane + octamethyicyclotetrasiloxane. Keywords: Theory; Corresponding states relations; Reduction parameters; Excess volumes of mixing; Linear- and cyclic-dimethylsiloxanes; Cycloalkanes; Tetramethylsilane
I. I n t r o d u c t i o n In a previous paper [1] we gave an account of the conformance o f the linear dimethylsiloxanes to a simple phenomenological principle o f corresponding states o f the kind described by Patterson and Bardin [2]. Here we turn to tetramethylsilane (TMS), regarded as the forerunner o f the series, and to four different dimethylsiloxane liquids: two cyclic oligomers, octamethylcyclotetrasiloxane and decamethylcyclopentasiloxane, and two polymers, one linear ( M S 2 0 0 / 2 0 c s ) containing around 26 silicon atoms and the other a carefully-fractionated cyclic polymer containing 79 number-average skeletal bonds. It is our purpose first to seek the degree of conformity of these materials with the
* Corresponding author. 0378-3812/97/$17.00 Copyright© 1997 ElsevierScienceB.V. All rights reserved. PII S0378-3812(96)03132-9
116
J.-M. Barbar{n-Castillo, 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
earlier oligomeric linear dimethylsiloxane corresponding-states principle and, secondly, to use the results to calculate the excess volumes of mixing of six binary liquid mixtures containing a cycloalkane with six, eight, or ten carbon atoms with either octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane, and of two linear+ cyclic substance mixtures, either n-hexane or decamethyltetrasiloxane with octamethylcyclotetrasiloxane. In this paper we review first the advantages of the phenomenological approach for the validation of a principle of corresponding states and demonstrate its applicability to this extended range of dimethylsiloxanes with particular reference to the normal alkanes, we follow with the evaluation of the temperature, volume, pressure and entropy reduction parameters, and we close with an application of the principle to the prediction of excess volumes of mixing of some mixtures containing dimethylsiloxanes.
2. Phenomenological corresponding-states principle Principles of corresponding states are written in terms of reduced quantities ) ( = X / X * , where X is some appropriate quantity and X * its reduction parameter, characteristic of the particular substance concerned, which in some formulations of the principle is a molecular quantity or in others phenomenological although related directly or otherwise to a molecular quantity or combinations of molecular quantities. Applied to equations of state, the principle is written as /5 =/3(7~,~7), where T is temperature, V is volume--usually molar v o l u m e - - a n d ~ ( T , V ) is a universal function for the class of substances concerned. The main problems in this application of the principle of corresponding states are the unequivocal identification of the reduction factors for different substances and the formulation of the universal function. The essence of the Patterson and Bardin phenomenological approach to these difficulties rests on the recognition that if a corresponding-states principle exists for a series of related substances, such as here a homologous series in the liquid state for which the reduction factors might be expected to depend heavily on chain length, then at vanishing pressure the quantity (anT), where at, is the isobaric thermal expansivity (01n V/OT)p, equals the reduced quantity (SpT~ and is therefore an effective measure of the reduced temperature thus reducing the first problem for temperature at least - - t h e need to identify T *. It follows then by simple geometry that if the curves of the logarithm of X as functions of either ( a p T ) or, for some choices of X to take advantage of essentially-accidental linearity, its reciprocal, are parallel, then a demonstration of the conformity of the family of substances concerned to the principle is achieved. The actual reduction factors themselves emerge in this procedure from simple single-axis shifts of the curves for each substance on to the curve for a reference substance. Procedures of this kind permit the calculation of T* and V * from simple density-temperature data and of p * from data on a pressure-related quantity--usually the isochoric thermal pressure coefficient ( 0 p / 0 T ) v as a function of temperature. The combined reduction factor ( p * V * / T * ) has the nature of an entropic reduction parameter S* and this quantity assumes particular importance when the principle is applied to chain-molecule substances. The genesis of this procedure was recognized inter alia by Prigogine and coworkers [3] and later by Simha and Havlik [4] but the approach employed required a double mapping of log V - log T plots which undermined the achievable accuracy of the reduction factors which were obtained.
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
117
3. Conformity of dimethylsiloxanes The data on which this analysis chiefly rests are the result of the accurate determination of the orthobaric densities of the substances identified in the introduction over a fairly wide range of temperature using a hydrostatic density balance. The results have been published elsewhere [5]. The most searching test of the validity of a phenomenological principle of corresponding states, quite independent of the discussion of reduction factors, probably rests on the superimposability of the dimensionless reduced quantity - { 0 ( a p ) - l / OT )}p = (ap) - 2 ( OOlp/OT)p as a function of (apT) for all the members of the class of substance which might be expected to conform to it. Fig. 1 shows the result of the calculation. There is fair conformity with the principle. It must be borne in mind that the calculation is incorporating a double-differentiation of the density and this procedure places very heavy weight both on the accuracy of the original density measurements and on the form of curve-fitting expression used to describe them. The test involves double differentiation of smoothed specific volume data, with the result that the exact form of the curves--especially at the ends--is somewhat artificial and depends upon particularities of the data fitting procedure. The inclusion of an extra coefficient in the polynomial fitting expression for the molar volume changes the shapes of the curves a little [6] not affecting the final broad conclusions; the use of only three coefficients, however, making d a p / d T a temperature independent quantity, leads to absurd results. In this study few of the results refer to temperatures close to the gas-liquid critical point and so a simple power series in temperature for the natural logarithm of the specific volume was employed. In more detailed studies where the behaviour of the density close to the gas-liquid critical point is 2 "r5
~
i
i
25 2 25 2
i
i
~
a
i
0 i
,'-'--- =i
17 5
E--
15 L 25
I
0 75
i/
I,z /
iiIIJt/~//I/ /// I{i
05 O25 0
/
,/ /
I
I
-0.25 02
I
I
I
I
I
03
04
0.5
06
07
0.8
(apT)
(cgOtp/OT)
Fig. 1. Test for the conformity of the dimensionless reduced quantity (Otp) -2 as a function of (apT)" (a) tetramethylsilane, (b) to (g) the linear oligomers dimer to hexamer, and the linear polymer MS200/20cs. The dashed lines (i) to (k) refer to the octamethylcyclotetrasiloxane and decamethylcyclopentasiloxane, and the cyclic polymer, respectively. Curve (o) reflects a mean value of B = 8.70.
118
J.-M. Barbarfn-Castillo , I.A. McLur e / Fluid Phase Equilibria 128 (1997) 115-130
explored we have employed Wegner extended-scaling of an expression with non-classical exponents [7]. Although this is a superior process, it is not employed here since it is unnecessary for all but tetramethylsilane and even for that substance, the relevant portion of liquid range is closer to the triple point than to the critical point and so its introduction is an unnecessary sophistication.
4. Relative reduction parameters from corresponding states relations On the basis that the most stringent test for the existence of a principle of corresponding states for the cyclic dimethylsiloxanes has been satisfied we now move to the evaluation of the relative reduction parameters. Absolute values of temperature, volume, and pressure reduction parameters for chain-molecule liquids have been calculated by using theoretical equations of state based upon the cell partition function. Shih and Flory [8] used their theoretical equation of state in combination with experimental specific volume, thermal expansivity, and thermal pressure coefficient data over wide temperature ranges. Simha and Havlik [4] followed the route to evaluate the reduction parameters from the equation of state developed by Prigogine et al. [9], by using volume-temperature data at atmospheric pressure in combination with experimental enthalpies of vaporization at low temperatures. A treatment essentially phenomenological is followed in this work for the evaluation of the relative reduction parameters, in a procedure requiring no fluid model or empirical equation of state. In the process were also calculated the reduction parameters for the dimethylsiloxanes, after choosing a
36
\\xx\
32
2,
\
2.8
2.4
g
\d"i 1.2 23
I 24
25
I 26
2.7
loglo(T/K) Fig. 2. Test for T* through the plot of ( a p T ) ] against logl0(T/K): (a) tetramethylsilane, (b) to (h) the linear oligomers dimer to hexamer, and the linear polymers MS200/20cs and PDMSO, respectively. Dashed lines (i) to (k) octamethylcyclotetrasiloxane, decamethylcyclopentasiloxane, and the cyclic polymer, respectively.
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
119
reference substance for which the reduction parameters were already known from calculations based on equation of state data. The temperature reduction parameter T* is most easily obtained from the plot of (OtpT)- ! against log l0 ( T / K ) . If a principle of corresponding states exists then the curves for each substance should be parallel. Fig. 2 demonstrates that this is so, not only for tetramethylsilane and the linear dimethylsiloxane oligomers considered previously, but also for octamethylcyclotetrasiloxane and decamethylcyclotpentasiloxane, the linear and cyclic polymers of this work for which density were measured, and the long-chain polymer of Shih and Flory, which was added to this test in order to extend the scope of the treatment and of the reduction parameters to be calculated. Since the (o~pT)-i against log l0 ( T / K ) curves are highly linear--probably particularly true only far from the gas-liquid critical p o i n t - - t h e results are fitted to expressions of the form:
(apT)-'= A -
(1)
Blog,o(T/K )
The values of the constants A and B for each substance are listed in Table 1. A useful measure of the parallelism of the lines for each s u b s t a n c e - - a n d hence their conformity with the principle--is afforded by the similarity of the B values. It is clear that B is not entirely constant but decreases somewhat with chain length as do the corresponding quantity in the normal alkanes. A comparison of parameters A and B for the dimethylsiloxanes with those reported by Patterson and Bardin for the n-alkanes (24.4 for A and 8.7 for B as averaged values against 27.7 and 10.7 respectively for the n-alkanes) shows that these series follow slightly different equations of state. The relative temperature reduction factor T
Table 1 The constants A to F or the various fitting expressions over temperature ranges AT of the the thermodynamic properties of tetramethylsilane, the oligomeric linear and cyclic dimethylsiloxanes, and the linear and cyclic polydimethylsiloxanes as linear functions of either or ( OtpT)- ! Substance AT (K) A B C D E F
(apT)
Linear TMS dimer trimer tetramer pentamer hexamer MS200/20cs PDMSO Cyclic tetramer pentamer PDMSO
224/327 278/358 300/412 300/412 297/408 293/412 292/470 293/473
26.78 25.62 25.11 24.76 24.47 24.26 22.40 22.41
10.06 9.38 9.07 8.85 8.68 8.56 7.64 7.57
2.03 2.24 2.38 2.48 2.56 2.62 3.20 4.91
0.21 0.22 0.21 0.25 0.27 0.28 0.35 0.39
5.52 5.55 5.56 5.59 5.61 5.63
0.48 0.58 0.65 0.75 0.83 0.91
5.69
1.25
292/403 302/414 302/455
24.90 24.97 22.27
8.94 8.87 7.53
2.41 2.52 3.38
0.24 0.23 0.37
5.78
1.13
J.-M. Barbarfn-Castillo, 1.,4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
120
Table 2 Chain length n, molecular weight MW, and the temperature, volume, pressure and entropy reduction parameters, T~,n) / T~f, PCs,,)/Pr*~f and Vcs,,)/Vr~f, respectively, relative to those for n-octane, for tetramethylsilane, the oligomeric linear and cyclic dimethylsiloxanes, and three long chain molecules, two linear and one cyclic polydimethylsiloxanes Substance Linear TMS dimer trimer tetramer pentamer hexamer MS200/20cs PDMSO Cyclic tetramer pentamer PDMSO
*
,
*
,
*
*
n
MW
T~s.,o / Tree
V~s.,o / Vref
P~S,,,) / Pref
3 5 7 9 11 13 47 2560
88.2 162.4 236.5 310.7 384.9 459.0 1900 100000
0.797 0.904 0.968 1.023 1.064 1.093 1.283 1.362
0.775 1.263 1.763 2.262 2.767 3.273 13.21 695.3
0.871 0.836 0.801 0.782 0.756 0.742 0.616 0.619
8 10 79
296.6 370.8 2935.
0.999 1.066 1.349
1.915 2.436 20.30
0.847 0.808 0.681
The relative v o l u m e reduction parameter ratio V{s,,,)/vr*~f is evaluated from the closely linear plot o f logl0 ( V / c m 3 m o l - l ) as a function o f (OtpT) shown as Fig. 3. From the linear fits o f the various curves we see that the data fairly well conform to expressions o f the form l o g , o ( V c m -3 m o l - ' ) = C
+ D(apT)
(2)
The coefficients C and D are listed in Table 1. The essential similarity o f D is again evidence for the conformity o f both tetramethylsilane and the cyclic dimethylsiloxanes to the same correspondingstates principle as the linear dimethylsiloxanes. The relative v o l u m e reduction factors V~s*,,)/Vref are evaluated at ( a p T ) - l = 2.0 and the values are listed in Table 2. Besides the linear dimethylsiloxanes, only for octamethylcyclotetrasiloxane [10] and the high molecular weight dimethylsiloxane p o l y m e r o f Shih and Flory has the thermal pressure coefficient been measured over a reasonable range o f temperature. For tetramethylsilane, usefully regarded as the first m e m b e r o f the linear dimethylsiloxane series [11], the thermal pressure coefficients were calculated from published ultrasonic speeds [12], heat capacities [13], and the results o f our own density measurements [5]. By straightforward thermodynamic manipulation we determine the isothermal compressibility fir and hence Yv via Yv Olp/J~V" Using these results together we have used the route to the relative pressure reduction factor P~S.,,)/Pr*~f by setting the values in the form ( T ' Y v / k P a ) in the linear expression =
log,o ( T y v / k P a )
= E - F(apT)
(3)
The values o f the coefficients E and F are listed in Table 1 and the relative pressure reduction parameters P~s,,,)/Pr~f, evaluated at ( a p T ) - l = 2.0, are in Table 2. The relative reduction parameters throughout this work and our previous paper on the subject have been calculated taking n-octane as the reference substance and at an arbitrary fixed value o f
J.-M. Barbarln-Castillo, LA. McLure / Fluid Phase Equilibria I28 (1997) 115-130 52
~
i
h
i
121
i
48
4
] ,~
36
. . . . . . . . . . . . .
>
k
J g
m 32
~~ fe
2.8
----~=---- . . . . . .
j 22-2222-~- f--
c
b
2.4
2 02
I
[
I
I
I
03
0.4
0.5
06
07
08
(apT) Fig. 3. Test for V * through the plot of (or,T) against log10(V/cm 3 tool l): (a) to (h), tetramethylsilane, the linear oligomers dimer to hexamer and the linear polymers MS200/20cs and PDMSO, respectively. Dashed lines (i) to (k), octamethylcyclotetrasiloxane, decamethylcyclopentasiloxane, and the cyclic polymer, respectively.
( a p T ) - 1 _~ 2.0 at which there is good overlap of the different dimethylsiloxane reduced temperatures. However, any of the dimethylsiloxanes could act equally well as the reference substance.
5. Temperature, volume, pressure, and entropy reduction parameters A theoretical equation of state gives the functional dependence of ~pT on 7~. It is from it that absolute values of V *, and then T *, can be formally obtained. All the T * found by any one theory are arbitrary to within a constant factor. Patterson et al. [ 14], working with an equation of state obtained from the smoothed potential cell model of Prigogine, reported for the dimer a temperature reduction parameter T~s,2) = 4410 K at 30°C, while Shih and Flory, using a different equation of state, gave also for the dimer the temperature reduction parameter T~s,2~ = 4420 K at 25°C. Since there is no appreciable difference between those two values reported for the same substance, we took the second one as a reference for the calculation of the absolute values of T* for the rest of the oligomers and the polymers of this treatment. Hence, for T~s,2) = 4420 K, together with coefficients A and B, Eq. (1) gives ( a p T ) - 1 = 2.41 for which there is an excellent overlap of the reduced temperatures covered by all the substances involved in this study. The temperature reduction parameter can then be calculated from
T~s,, ) = T~s,:)T~s,,)/T~ s,z )
(4)
where T~s,z) is 298.15 K and T~s,,) is the temperature at which ( a p T ) -1 is equal to 2.41 for the species (S,n). Table 3 shows the values of the temperature reduction parameters for tetramethylsilane,
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
122 Table 3
The temperature, volume, pressure, and entropy reduction parameters T~s.,), V~s.,), P(s,,) and S(s,,), respectively, for tetramethylsilane, five linear and two cyclic oligomeric dimethylsiloxanes, and three long chain molecules, two linear and one cyclic polydimethylsiloxanes Substance T~.~.,) (K) V{.~.,) (cm 3 tool-') P{s.,) (J cm-3) St*s,.) (J mol-t) Linear TMS dimer trimer tetramer pentamer hexamer MS200/20cs PDMSO Cyclic tetramer pentamer PDMSO
3921. 4420. 4718. 4970. 5158. 5291. 6130. 6501.
99.1 161.5 225.5 289.3 353.8 418.6 1690.0 88910.
373. 358. 343. 335. 324. 318. 265. 265.
9.42 13.08 16.40 19.50 22.22 25.15 73.15 3624.
4860. 5180. 6433.
244.9 311.5 2595.7
363. 347. 293.
18.29 20.87 118.4
five linear and two cyclic dimethylsiloxane oligomers, and three long chain p o l y m e r s - - t w o linear and one cyclic. A similar procedure was followed for the calculation o f the specific v o l u m e v * and pressure reduction parameters p * after the known values o f V{s,2) = 0.9945 cm 3 g - l and P{s,2) 358 J cm -3 for the dimer also reported by Shih and Flory. Finally, the entropy reduction parameter S{s,, ) was calculated from the relation S{s,n} = p¢s,n)V{s,,}/T~s,,}, and the results are given in Table 3.
6. Dependence of reduction parameters on chain length or ring size The corresponding states principle for polymeric liquids introduces the concept o f a division in the degrees o f freedom o f the chain-molecule into internal and external categories. The ratio o f the thermal energy of the 3c external degrees o f freedom (pushing for expansion) to the cohesive or intermolecular contact energy qE * (pulling for contraction) gives a quantity o f dimension 1 defined as the r e d u c e d t e m p e r a t u r e T, a quantity that in fact characterizes the expansion or free v o l u m e o f a liquid. Thus, T = (c/q) (kT/e *) = T/T*. The molar configurational quantities o f liquids are those important for their mixing properties and they are related by reduction parameters to the reduced quantities, all functions o f chain length n and reduced temperature 7~:
V(n.T) = V *(n). V ( T ) ;
V* =
U(n,T) = U*(n). U ( T ) ;
U* =Nq(~}e*
S(n,T)= S*(n). S(T);
Nr(,}v*
S* =Nkc(.)
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
2.25
/
//
200
123
2
1.75 1.50 <
125 1.00
0.50
o
0.25 0,00
I
I
I
l
I
I
I
I
I
I
2
3
4
5
6
7
8
9
I
I
I
I
I
10 II 12 13 14 15
n
Fig. 4. Demonstration of chain-length dependence for S ~ through the relative number of the external degrees of freedom, C(s,n)/C(c,8 ) of the n-alkanes ( © ) and the linear ( × ) and cyclic (*) dimethylsiloxanes as a function of chain length. Full lines 1 and 2 are the plots of Eq. (8) and Eq. (9), respectively.
where N is Avogadro's number and k is the Boltzmann constant. The effective number of segments, r, q, and c are proportional to the molecular volumes, molecular surfaces, and number of external degrees of freedom of the chain-molecule liquid, respectively. From these relations results that T* = U * / S * and finally S* = p * V * / T * . From Simha and others there has been much discussion about the chain-length dependence of reduction parameters for chain molecules. We shall not rehearse the conclusions in detail but present our results in the most apt way. The temperature reduction parameters evaluated in this work are well-enough described by an expression of the Simha form T~s,., = 6583{1 - 3 . 3 9 / ( n + 5)}
(5)
for chain lengths n, taken as the number of atoms in the backbone of the longest chain, exceeding 7. Similarly the volume reduction parameters V(S,N) follow a Simha-like relationship, again for n > 7, V~s,.) = (n - 0 . 9 4 1 ) / 0 . 0 2 9
(6)
Turning now to the relative behaviour of the entropy reduction factor S* as a function of chain length, it is expected that the external degrees of freedom and relative chain flexibility for the dimethylsiloxanes be linear with chain length, as in the n-alkanes. The number of external degrees of freedom (3c) of a molecule of chain length n is found through the entropy reduction parameter by the relation S(*s,.)= Nkc(s.n ). With n-octane as the reference substance, this yields C~s../C~c.8 ) =
S~..)IS~c.8 ~
(7)
124
J.-M. Barbarin-Castillo. IA. McLure / Fluid Phase Equilibria 128 (1997) 115-130
A linear relationship was found between the term C
C(c,n)/c(c.8 ) = 0.42846 + 0.0714n
(8)
reported by Patterson and Bardin for the n-alkanes, and
C(s.,)/C(c.8 ) = 0.584
+ 0.1223 n
(9)
from this work and which gives the ratios to within + 0.05% for the linear dimethylsiloxanes from the trimer (n = 7) to the high polymer. For the cyclotetramer this ratio is estimated to within 1.6%. For tetramethylsilane and the dimer C(s,3)/C~c.8) = 0.816 and C(s,5)/cw.s) = 1.132. These values fall below the straight line in a very similar way, albeit less markedly than those for methane and propane. In this respect tetramethylsilane behaves better as the forerunner of the dimethylsiloxane series than does methane as that of the n-alkanes. The relative position of the two lines in Fig. 4 shows the higher flexibility of the dimethylsiloxane chain compared to that of the n-alkanes and this is manifested by the ratio C~s,n)/C~c,8) ranging from 1.93 for tetramethylsilane/methane to 2.58 for hexamer/hexane, although this goes up to 3.11 and 3.63 for the long-chain polymers showing to a good extent the apparent increase of flexibility for the dimethylsiloxanes relative to the n-alkanes as the chain length increases.
7. Corresponding-states principle treatment of excess volumes It has been shown by Simha and Havlik [4] that the molar volumes as a function of temperature,
V(T), for the n-alkanes, dimethylsiloxanes, and several other series of homologous series obey a particular form of the corresponding states principle, i.e.
(10)
V(n,T) = V *(.).V(T)
where V(n,T), V *(n), and 17(7~) are the molar volume of a chain-molecule liquid of chain length n at temperature T, the volume reduction parameter of that liquid, and the reduced volume respectively. According to this, the data can be reduced empirically to a single V - T plot well described by theoretical equations of state developed by Prigogine and co-workers. A phenomenological corresponding-states treatment, independent of theoretical models, was used here for the mixtures involving dimethylsiloxanes. This procedure is due to Patterson and Bardin [2] and was successfully applied to mixtures of n-alkanes. The reduced temperature (7~) of a binary mixture is given by the simple average
( T ) = T / ( T °) = X , T , + X 2 T 2
(11)
where 7~1 and 7~2 are the reduced temperatures of the pure components, T and ( T *) are the temperature and average temperature reduction parameter of the mixture respectively, and X~ is the surface fraction defined by
X, = p ; V , * x , / ( p ; V , * x
t + p 2 V2*x2)
(12)
J.-M. Barbarin-Castillo, 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
125
Table 4 Pressure, volume, and temperature reduction parameters P(s,.), V(s..) and T~s..), respectively, for n-hexane, c-hexane, c-octane and c-decane. Terms A and B are the parameters for Eq. (1) Substance P(c.,) (J cm- 3) V(c,,) (cm 3 mol- 1) T(c., ) (K) A B n-hexane 451 99.67 4450. 27.16 10.01 c-hexane 432 84.26 4720. 27.50 9.80 c-octane 448 115.19 5516. 27.56 9.75 c-decane 459 144.30 6050. 27.17 9.45
where p * and V * are the pressure and volume reduction parameters for the pure components. Any thermodynamic quantity of the mixture can be fou~0d from the reduced curve established for the pure liquids using the reduced temperature of the mixture (7~) obtained with an appropriate averaged reduction parameter. Thus the molar excess volume of mixing is given by
vE=(x,W + x2v; )~(~)-x,v,,17(~,)-x2v;~(g)
(13)
where x t is the mole fraction of component 1 in the mixture. The quantities 17(7~1) and i7(T2), the reduced volumes of the pure components, may be expanded around 17((T)) in terms of power series of (7~1 - (7~)) and ( T 2 - ( T ) ) respectively so that
~(~,) = ~(~)+ (~,- ~)(d~/d~)+
(~)(~,- ~)(d2~/d~T~ ) 2
-
2
+ (~1(7~, - (7~))3(d317/d(7~)3) + (1)(7~, - (7~))4(d417/d(7~)4) + ...
(14)
and similarly for component 2. Since (7~1 - (7~)) = X2(7~1 -- 7~2), the expression for the molar excess volume of mixing becomes
V E= - ( x,X2V, *- x2X,V2* )(7~, -
× (da17/d<~>~)- (+)(x,x~w --("~4)(xIX4VI
* -- x 2 X ? V 2 *
7~2)(dlT/d( 7 ~ ) ) - (½)( - x~x~W;
x,X~V,?- x2X)V2* )(T,- ~2)2
)(~,- ~2)-(d~17/d<~>-)
)(T,- T2)4(d4~7/d(~)4).
(15)
The term involving the second derivative of the reduced volume depends on the temperature variation of ap for the mixture but is related to ( a p T ) itself through the expression
a- 2(dap/dT)
= B/2.303 - (apT)-'
(16)
and the same applies to the higher derivatives, where ap is the isobaric thermal expansion coefficient. All the derivatives of 17((7~)) with respect to ( ( T ) ) can be evaluated from experimental data for the pure components. Full details of the equations involved are to be found elsewhere [15]. Eq. (15) has been used to predict the molar excess volumes of mixing of eight binary liquid mixtures containing either octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane. In one set of results octamethyicyclotetrasiloxane is mixed with cyclohexane, cyclo6ctane or cyclodecane, in the second set decamethylcyclopentasiloxane is mixed with each of the same cycloalkanes, and in the
J.-M. Barbarln-Castillo. 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
126
Table 5 Comparison between the equimolar experimental excess volumes V E(x = 0.50) of six (cycloalkane + cyclic dimethylsiloxane oligomer) mixtures and two (linear+cyclic substance) mixtures and the values predicted from the phenomenological principle of corresponding states
VE(x =
Mixture
c-tetramer: + c-hexane + c-octane + c-decane c-pentamer: + c-hexane + c-octane + c-decane c-tetramer: + n-hexane + n-tetramer
third
Experimental
Predicted
0.086 - 0.540 - 1.020
0.073 - 0.460 -0.900
O. 150 - 0.570 - 1.030
0.316 - 0.290 - 0.810
0.127 0.440
0.054 0.360
set o c t a m e t h y l c y c l o t e t r a s i l o x a n e
camethyltetrasiloxane. aries a r e t h o s e
0.50) (cm 3 mol-~)
is m i x e d
with
linear partners--n-hexane
or the linear de-
T h e v a l u e s o f T * , V *, a n d p * u s e d in t h e c a l c u l a t i o n s f o r t h e d i m e t h y l s i l o x -
of Table
3. T h o s e
for hexane
are from
Orwoll
and
Flory
[16].
For
the higher
c y c l o a l k a n e s t h e s a m e p r o c e d u r e w a s a p p l i e d as f o r c y c l o h e x a n e , u s i n g t h e d e n s i t y d a t a o f M e y e r a n d
0.2
I
r
I
I
I
J
~
r
I
a
01 0 -01 -02 -03
.7.
-0.4 -0.5 -06
> -07 -08 09 -1 -I1 -12
I
J
I
I
I
I
I
I
I
0.1
02
03
04
05
06
07
0.8
09
x
Fig. 5. Molar excess volumes V E as a function of mole fraction x of the second component for mixtures of octamethylcyclotetrasiloxane + cyclohexane at 298.2 K (a), + cyclooctane at 303.2 K (b), or cyclodecane at 298.15 K (c). Full lines are smoothed experimental values and the dashed lines are calculated by the theory described.
J.-M. Barbarln-Castillo, I.A. McLure / Fluid Phase Equilibria 128 (1997) 115-130
0.4
I
I
~
I
0.3
I
I
i
I
i
OH
09
127
~--------~----~ /
0.1
~
.'2
j~
-03
-o.4 -0.5 -06
-0.7
\\x x
s///
-0.8
-0.9 -I -I.L 0
I
I
I
I
I
t
I
01
02
03
0.4
05
06
07
I
X
Fig. 6. Molar excess volumes V e as a function of mole fraction x of the second component for mixtures of decamethylcyclopentasiloxane + c y c l o h e x a n e at 303.2 K (a), cyclooctane at 303.2 K (b), or cyclodecane at 298.15 K (c). Full lines are smoothed experimental values and the dashed lines are calculated by the theory described.
Oq)
() 45
i) 35
E.
030
O t~ 2S
>
~) 20
010
IJ 05
0(3O 0
I
I
I
J
1
1
O1
02
0.3
04
0.5
06
[ 07
0.8
0.9
X
Fig. 7. Molar excess volumes V e at 298.2 K as a function of mole fraction x of the second component for the mixtures of octamethylcyclotetrasiloxane + n-hexane (a) or decamethyltetrasiloxane (b). Full lines are smoothed experimental values and the dashed lines are calculated by the theory described.
128
J.-M. Barbarln-Castillo. 1.4. McLure / Fluid Phase Equilibria 128 (1997) 115-130
Hotz [17] for cyclooctane, and our own densities for cyclodecane [18]. The absolute values of the parameters for cyclohexane were those reported by Shih and Flory. The values of all the reduction parameters are listed in Table 4.
8. Comparison of theory and experiment Table 5 compares theory and experiment for eight dimethylsiloxane mixtures in a good variety of substances and this is also illustrated in Figs. 5-7. The experimental results for octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane + cyclohexane or cyclooctane were taken from a previous paper [19]. The experimental results for octamethylcyclotetrasiloxane or decamethylcyclopentasiloxane + cyclodecane and for octamethylcyclotetrasiloxane + n-hexane or decamethyltetrasiloxane at 298.15 K are from our own measurements [20]. Full lines show experimental values of the molar excess volumes of mixing. The dashed lines were obtained after a careful fit was made to the results obtained from the theory and in order to present these results in the same form as Patterson and Bardin did before for n-hexane + n-hexadecane. The agreement between theory and experiment is good in that the magnitudes of V E are reproduced tolerably well--unlike the van der Waals theories which fail to do this [21]--despite the very small changes involved for some of the mixtures, particularly those containing cyclohexane or n-hexane. The relative trend from positive vF~ for mixtures containing cyclohexane to negative for those containing a larger cycloalkane is also satisfactorily predicted. It is interesting to note that the V E of mixtures of a cyclodimethylsiloxane + a linear substance are described fairly well by the theory. For completeness, it should be mentioned that we are leaving for a future paper a comparison of our results against those obtained by using the Flory theory for the prediction of the mixing properties, particularly because although the treatment for linear molecules is straightforward, it is unclear how to adapt it to mixtures containing cyclic molecules.
9. Conclusions Our analysis confirms that a straightforward phenomenological principle of corresponding states exists for dimethylsiioxane liquids of chain length varying from 1 to polymer values, whether cyclic or otherwise, and that the entropy reduction parameter reflects the modest reduction in chain flexibility occasioned by ring closure. Clearly, the phenomenological treatment outlined above does not offers a rigorous interpretation of the excess volumes of alkanes + dimethylsiloxanes, whether linear or cyclic, but it demonstrates that the apparently complex volumetric behaviour of these mixtures is usefully described by a simple application of the principle of corresponding states. Although the above evidence would seem to suggest that the n-alkanes and dimethylsiloxanes follow dissimilar reduced equations of state, we believe that the difference is not sufficient to preclude a meaningful treatment of mixtures of alkanes with dimethylsiloxanes using approaches which assume a strict obedience to a unique principle of corresponding states. That was the reasoning behind the treatment followed in this work in which both, alkanes and dimethylsiloxanes, were taken as being in the same homologous series.
J.-M. Barbarln-Castillo, I A. McLur e / Fluid Phase Equilibria 128 (1997) 115-130
129
The experimental equations o f state for the two series o f chain-molecule l i q u i d s - - a s deduced from line o in Fig. 1, or its equivalent in the corresponding curve for the alkane s e r i e s - - a r e in much greater conformity with each other than they are with a number o f theoretical equations o f state, as it is shown in figure 2 of Ref. [2]. W e have presented further evidence for the usefulness o f the phenomenological approach in predicting with fair certainty the properties o f mixtures o f these and related hydrocarbons.
10. List of symbols T*,p*,S*,V
Lp,
N k r,q,c Ve qE*
* temperature, pressure, entropy, and volume reduction parameters reduced values o f temperature, pressure, and v o l u m e Avogadro's number Boltzmann constant effective numbers o f chain segments proportional to the molar volumes, molecular surfaces and degrees o f freedom, respectively excess v o l u m e o f mixing intermolecular contact energy
G r e e k letters C~p
Yv
isobaric thermal expansivity isothermal compressibility thermal pressure coefficient
Acknowledgements J.-M.B.-C. acknowledges receipt o f financial support from the Consejo Nacional de Ciencia y Tecnologla, M6xico, and leave-on-absence from the Universidad Aut6noma de N u e v o Le6n, Monterrey, M~xico.
References [1] E. Dickinson, I.A. McLure, A.J. Pretty and P.A. Sadler, Chem. Phys., 10 (1975) 17-22. [2] D. Patterson and J.M. Bardin, Trans. Faraday Soc., 66 (1970) 321-334. [3] I. Prigogine, with the collaboration of A. Bellemans and V. Mathot, The Molecular Theory of Solutions, North Holland, Amsterdam, 1957, Chapter 16. [4] R. Simha and A.J. Havlik, J. Am. Chem. Soc., 86 (1964) 197-204. [5] I.A. McLure and J.-M. Barbarln-Castillo, J. Chem. Eng. Data, 39 (1994) 12-13. [6] P.A. Pretty, PhD thesis, University of Sheffield, 1973. [7] I.A. McLure and J.-M. Barbarln-Castillo, Int. J. Thermophys., 14 (1993) 1173-1186. [8] H. Shih and P.J. Flory, Macromolecules, 5 (1972) 758-761. [9] I. Prigogine, N. Trappeniers and V. Mathot, Disc. Faraday Soc., 15 (1953) 93-104. [10] M. Ross and J.H. Hildebrand, J. Phys. Chem., 67 (1963) 1301-1303. [11] I.A. McLure and J.F. Neville, J. Chem. Thermodyn., 9 (1977) 957-961. [12] I.A. McLure, J.-M. Barbarln-Castillo, J.F. Neville and R.A. Pethrick, Thermochim. Acta, 223 (1994) 325-328.
130
[13] [ 141 [ 151 [16] [ 171 [ 181 [19] [20] [21]
J.-M. Barbarin-Custillo,
IA. McLure/Fluid
Phase Equilibria
128 (1997) 115-130
M. Harada, T. Atake and H. Chihara, J. Chem. Thermodyn., 9 (1977) 523-534. D. Patterson, S.N. Bhattacharyya and P. Picker, Trans. Faraday Sot., 64 (1968) 648-661. J.-M. Barbarfn-Castillo, PhD thesis, University of Sheffield, 1984. R.A. Otwoll and P.J. Flory, J. Am. Chem. Sot., 89 (1967) 6814-6822. E.F. Meyer and C.A. Hotz, J. Chem. Eng. Data, 21 (1976) 274-279. LA. McLure and J.-M. Barbarin-Castillo, Chem. Eng. Data, 30 (1985) 253-2.57. E. Dickinson, DC. Hunt and LA. McLure, J. Chem. Thermodyn., 7 (1975) 731-740. J.-M. Barbarfn-Castillo and LA. McLure, J. Chem. Thermodyn., 25 (1993) 81 l-821. E. Dickinson, PhD thesis, University of Sheffield, 1972.