J. CHEM. SOC. FARADAY TRANS., 1995, 91(22), 4071-4077 407 1

Mutual-diffusion Coefficients and Viscosities for the Water-2-Methylpropan-2-ol System at 15 and 25 OC

Kenneth R. Harris* and Hanh Ngoc Lam Chemistry Department, University College, University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2601, Australia

Mutual-diffusion coefficients have been measured by the Taylor dispersion technique and viscosities by capil- lary viscometry at 15 and 25°C for dilute aqueous solutions of 2-methylpropan-2-01 (tert-butyl alcohol or TBA). Comparison is made with other water-alcohol systems and water-2-butoxyethanol. Ornstein-Zernicke corre- lation lengths, 5 , for this system are compared with those for mixtures of water with methanol, ethanol, propan- 1-01 and butan-1-01 and contrasted with those of systems representative of other classes of non-electrolyte mixtures. < for water-propan-1-01 rises to a maximum at a mole fraction (x) of 0.15, suggestive of the long-range correlation of molecular motion in this solution or pseudo-critical behaviour ; water-TBA appears to behave similarly with a rapid increase in 5: abovex = 0.05.

In this paper we continue our study of diffusion in aqueous alcohol systems, having previously reported mutual-diffusion coefficients for the ethanol and propan-1-01 systems.' Mutual-diffusion coefficients have been measured by the Taylor dispersion technique and viscosities by capillary vis- cometry at 15 and 25°C for aqueous solutions of 2- methylpropan-2-01 (t-butyl alcohol or TBA).

Inter-diffusion coefficients (DJ in a two-component system are defined by Fick's Law2 relating the flux (J) of either com- ponent to the concentration (molarity) gradient producing that flux,

Ji = - D i v c i ; i = 1 , 2 (1)

In the frame of reference based on the centre of volume of the system,

D, = D, = D,,

and D,, is then termed the mutual-diffusion coefficient. Most experimental methods for measuring mutual diffusion are designed so that the experimental cell frame of reference is coincident with the volume-fixed frame. One of the objectives of the first paper in this series' was to compare results for water-ethanol obtained by the diaphragm cell method with those obtained from the Taylor dispersion method. Dia- phragm cell mutual-diffusion experiments require relatively large concentration gradients and may be subject to system- atic error with systems where the volume change on mixing is also large, which is generally the case with water-alcohol mixtures. Taylor dispersion, on the other hand, uses very small gradients and is not usually subject to this particular problem. In this case, good agreement was found with the (better) diaphragm cell results.

Inter-diffusion, which describes mixing and separation, must be distinguished from intra- or self-diffusion, which is a measure of the thermal translational motion of molecules, and can be measured in systems very close to thermodynamic equilibrium.2 It can be defined in terms of non-equilibrium thermodynamics by considering the diffusion of a trace of solute 1 in a mixture of solute 2 and solvent in the limit that the chemical and physical properties of 1 approach those of 2, a situation approximated in radio-tracer and NMR spin- echo studies. It is not frame-of-reference dependent. The sta- tistical mechanical formulation in terms of the ensemble-average time auto-correlation function of the molecular velocities is possibly more familiar than the ther-

modynamic description : I f m

(3)

The self-diffusion coefficient is thus a measure of the rate of randomisation of molecular velocities through molecular col- lisions, including the effects of translational-rotational coup- ling.

The statistical-mechanical formulation for the mutual- diffusion coefficient is more complex: it includes a cross- correlation term,3*4

~ ~ v 1 ( t ) v 2 ( O ) ) dt

and a purely thermodynamic factor,

where x is the mole fraction andfthe activity coefficient. While diffusion in liquids is primarily governed by molecu-

lar size and shape, in aqueous systems it can be a sensitive measure of changes induced in the water structure by solutes or high pressure.

Aqueous solutions of non-polar molecules are of particular interest as many thermodynamic and transport properties seem to show anomalous behaviour. To quote H ~ i d t : ~ ' Partly non-polar (amphipatic or amphiphilic) molecules, for example the alcohols, may be highly soluble in water, but effects that are the result of interactions with the polar part of the molecules are predominant in the solutions'. In the par- ticular case of water-alcohol mixtures, such anomalies occur, for example, as extrema in the concentration dependence of the partial molar volume (a minimum) and the partial molar heat capacity (a maximum). Such effects are not shown by other small non-polar solutes lacking alkyl groups such as sugars, formamide or urea.

Of the water-alcohol systems, water-TBA has been exten- sively studied by both experiment and computer simulation as the TBA molecule is often regarded as one that may form water clathrates in solution, perhaps induced by hydrophobic interaction. In this paper we present new data for mutual dif- fusion for this system and make comparison with other water-alcohol mixtures.

Bender and Pecora6 have pointed out that the models used in the literature to describe the thermodynamic and transport properties of the water-TBA system can be divided into three types.

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4072 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

In the first (model A), very dilute solutions of TBA are con- sidered to form hydrated species based on the solid hexahy- drate.’ At higher concentrations, these form clusters, which eventually break up as more and more alcohol molecules are directly in contact with one another.

Such a model was used by Ito et aL8 in an examination of light-scattering mutual-diffusion coeficients and they con- cluded that the moving unit contained from 11 to 20 water molecules per TBA molecule, the actual size increasing with increasing temperature. Their argument was based on the application of the Hartley-Crank equationg relating water and TBA self-diffusion coeficients (Dsi) to the mutual- diffusion coefficient and the activity term B , :

The equation was empirically modified to allow for associ- ation (water-water, TBA-TBA and water-TBA), with the degree of each type of association being calculated from a best fit with such activity data as were available at that time.

The validity of this approach is doubtful. It is well known that the Hartley-Crank equation is only exact under certain ideal conditions’ which are rarely met in practice. In essence, it neglects the correlation of the relative velocity of a given pair of molecules with those of other pairs3 While this equa- tion is sometimes a useful approximation for mixtures of mol- ecules of similar size that interact weakly, it is quite inappropriate for systems showing strong directional inter- actions such as hydrogen bonding, and more so for this case, given its complexity.

Bender and Pecora6 found a single relaxation time suffi- cient to describe scattering at each state point in their photon correlation spectroscopic (PCS) study, and concluded that their results did not provide any evidence for identifiable, long-lived water-TBA aggregates, in contradiction to the conclusions of Ito et al.

The second approach (B) is not really a model, but assumes the system is making a close approach to a consolute point. Consequently, there are large concentration fluctuations in the solution with alcohol- and water-rich regions, consistent with the very small values obtaining for the mutual-diffusion coefficient, and that were the molecular interactions slightly different, one would see the phase separation shown by aqueous solutions of butan-1-01 and butan-2-01. This inter- pretation has been used by Sorenson and co-workers,10-’2 in cloud-point and PCS studies of water-TBA and water-TBA- butan-2-01 mixtures. Ito et aL8 have also postulated pseudo- critical behaviour, for which aggregation of hydrated TBA clusters might be the basis.

Green and Jacksonl3 have used perturbation theory in a study of such phase equilibria in water-alcohol mixtures, the water molecules being modelled as hard-spheres with four off-centre square-well bonding sites and the alcohol mol- ecules as a chain of fused hard-spheres with two H-bonding sites on the OH group. They concluded that the symmetry of the TBA molecule allowed it to fit into the water structure, whereas for the other butanols, the conformations of the alkyl chains lead to ‘an unfavourable disruption of the water-water hydrogen-bonding structure promoting the immiscibility’ that is dependent on temperature and pressure.

The third model (C) assumes that TBA works as a ‘structure-maker’, enhancing the water structure around the alcohol molecules in very dilute solution and leading to a hydrophobic association of the alcohol molecules at slightly higher concentrations. This is the familiar Frank and Evans model for the general case of aqueous solutions of non-polar and amphiphilic molecules. Hvidt has discussed this model,

treating it as a simple equilibrium,

R + n H 2 0 e R(H20), (6)

where R is the solute. For this equilibrium, the standard enthalpy (AH*) and entropy (AS*) changes are negative, the standard volume change (AVe) is positive and the equi- librium constant ( K e ) is of the order of unity.

Hvidt interprets the negative excess volume of aqueous solutions of alcohols and other amphiphiles as being pri- marily due to packing of the water molecules in the voids between the solute molecules together with the formation of dense water structures around the polar group. In these cases, the apparent molar volume of the solute, which can be regarded as the sum of the volume of the solute in a pure liquid state (hypothetical in the case of solid solutes) and the excess volume per mole of the is less than the molar volume of the pure substance. [The apparent molar volume (Qv) is defined in terms of the solvent molar volume, V:, and solute and solvent amounts, n,, by

(7)

It is a quantity more conveniently and more accurately obtainable from experimental densities than partial molar volumes.] Whereas, say, urea or formamide have apparent molar volumes which vary monotonically with concentration, those of the alcohols show a sharp minimum. As the concen- tration is increased, Qv initially falls below its infinite dilution value and it is argued that this is due to hydrophobic hydra- tion in the case of the alcohols. At higher concentrations this effect is masked by the change in packing as more solute mol- ecules come into direct contact with one another and the apparent molar volume then increases.

This model has been questioned by Sakurai14 whose density results extend over the whole mole-fraction range and show that there is also a similar, but less marked, minimum in the partial molar volume of water in TBA-rich solutions. Thus it may be simply the interruption of the like-like H bonding of ‘solvent’ water or alcohol by the introduction of an H-bonding alcohol or water ‘solute’, respectively, that results in these volumetric effects.

Nonetheless, support for the hydration of the methyl groups of TBA and hydrophobic-interaction driven associ- ation of TBA molecules in dilute solution has come from the molecular dynamics simulations of Nakanishi’s group’ 5-1 ’ and the X-ray diffraction studies of Nishikawa and co- workers.’8,’9 Near mole fraction, x = 0.04, Nishikawa and Iijima” concluded that the hydration number of TBA is about 28 at 20°C and that their results were consistent with the formation of a cage structure in very dilute solution and the presence of solvent-separated TBA clusters (TBA),(H20), at higher concentrations.

Further support for this model has come from the dielec- tric relaxation experiments of Fioretto et carried out at 5°C. They interpreted their data below x = 0.02 on the basis of a two-state model involving exchange between hydration and bulk water, with a discontinuity in the dielectric relax- ation times suggestive of hydrophobically driven aggregation of TBA-water clusters above x = 0.045. Mashimo and Miura” have also studied dielectric relaxation times of a number of non-electrolyte systems by time-domain reflec- tometry. A discontinuity found in the log(re1axation time)- mole fraction plots was observed at the same composition, xSolute = 0.17, for a number of aqueous solutions, including those of alcohols (methanol, ethanol, propan-1 -01 and TBA), p-dioxane and acetone, but not those of glucose and other sugars. This was interpreted as a break-point of a water

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Table 1 Coefficients of eqn. (1) for water-TBA densities, di

407 3

Table 3 Mutual-diffusion coefficients for water-TBA at 25 "C

15 "C 25 "C

0 < cp < 2 0 < c < 1.9 1.9 < c < 4.2 3.5 < c < 5.8

do

1034 104d4 1044

lOd, 102d,

lo5 standard deviation/ g cm-3

M / g mol -

0.999100 0.997048 1.006541 0.979807 - 0.134516 - 0.134375 -0.401479 0.110251 0.257533 0.227385 2.79000 -0.853671

-0.302933 -0.763519 - 11.2940 -0.558526 2.88603 19.7309 - 0.543550

1.07291

0 -0.894657 - 1.29455 0 5 2 8 3

- - - 74.1288

Units for the concentrations, c, are mol dm-3.

structure, such as a tetrahedral pentamer or a hexagonal ring, observable on the very short timescale of the measurements (ca. 10 ps). The water-TBA system shows an additional dis- continuity at x = 0.034, close to that observed by Fioretto et

Table 2 Mutual-diffusion coefficients for water-TBA at 15 "C

X ?"/mol dm-3 Acb/mol dm-3 D/10-9 m2 s - l

0.000002 0.000002 0.000002 0.000002 0.00247 0.00247 0.00247 0.00247 0.00526 0.00526 0.00526 0.00526 0.00526 0.007 3 8 0.00738 0.00738 0.00738 0.01 170 0.01170 0.01 170 0.01170 0.01 170 0.01555 0.01555 0.01555 0.01555 0.01555 0.01555 0.02 1 1 3 0.02701 0.02844 0.02844 0.02844 0.03225 0.03225 0.03225 0.03225 0.03744 0.03744 0.04949 0.04949 0.04949 0.04949 0.06041 0.0604 1 0.06041 0.0604 1

o.oO0 1 o.oO0 1 o.oO0 1 o.oO0 1 0.1358 0.1358 0.1358 0.1358 0.28618 0.2861 8 0.286 18 0.2861 8 0.28618 0.39787 0.39787 0.39787 0.39787 0.62 123 0.62 123 0.62 123 0.62 123 0.62 123 0.81459 0.81459 0.81459 0.81465 0.81459 0.81465 1 .W08 1.3617 1.4273 1.4273 1.4273 1.5993 1.5993 1.5993 1.5993 1.8227 1.8227 2.3234 2.3234 2.3234 2.3234 2.7435 2.7435 2.7435 2.7435

0.01 1 0.01 1 0.01 1 0.01 1

-0.011 -0.011 -0.011 -0.01 1 -0.015 -0.015 -0.015 -0.015 -0.015 -0.017 -0.017 -0.017 -0.017 -0.015 -0.015 - 0.01 5 -0.015 -0.015 - 0.023 - 0.023 - 0.023 -0.016 - 0.023 -0.016 -0.012 - 0.020 - 0.032 -0.032 - 0.032 -0.014 -0.014 -0.014 -0.014 -0.019 -0.019 -0.018 -0.018 -0.018 -0.018 -0.015 -0.015 -0.015 -0.015

0.66 0.68 0.66 0.67 0.64 0.67 0.63 0.62 0.60 0.6 1 0.61 0.63 0.59 0.58 0.60 0.59 0.60 0.55 0.55 0.54 0.55 0.55 0.53 0.53 0.52 0.52 0.52 0.52

0.469 0.465

0.457

0.425 0.437 0.417 0.395 0.403 0.294 0.304 0.309

0.478

0.438

0.442

0.316 0.214 0.214 0 .20~ 0.212

a ? is the mean concentration calculated from eqn. (109) of ref. 24: it is flow rate dependent. x is the corresponding mole fraction. Ac is the difference between the concentration of the injected solution and that of the flowing stream.

~~~~

X F/mol dm-3 Acb/mol dmP3 D/10-9 m2 s - l

0.000002 0.000002 0.000002 0.00249 0.00249 0.00249 0.00249 0.00249 0.00483 0.00483 0.00483 0.00483 0.00483 0.00728 0.00728 0.00728 0.00728 0.01053 0.01053 0.0 1053 0.01053 0.0 1053 0.01573 0.01573 0.01966 0.029 1 1 0.029 1 1 0.03621 0.0362 1 0.03621 0.03621 0.047 18 0.047 18 0.047 18 0.06046 0.06046 0.073 17

o.oO0 1 o.oO0 1 o.oO01 0.1366 0.1366 0.1366 0.1366 0.1366 0.2625 0.2625 0.2625 0.2625 0.2625 0.3920 0.3920 0.3920 0.3920 0.5602 0.5602 0.5602 0.5602 0.5602 0.82 15 0.821 5 1.0129 1.4530 1.4530 1.7655 1.7655 1.7655 1.7655 2.2200 2.2200 2.2200 2.7255 2.7256 3.1677

0.010 0.0 10 0.010

-0.010 -0.010 -0.010 -0.010 -0.010 - 0.022 - 0.022 - 0.0229 - 0.022 - 0.022 -0.010 -0.010 -0.010 -0.010 - 0.020 - 0.020 - 0.020 - 0.020 - 0.020 -0.016 -0.016

0.022 - 0.03 1 - 0.03 1 - 0.02 1 - 0.02 1 - 0.02 1 - 0.02 1 -0.019 -0.019 -0.019 -0.0319 - 0.03 19 - 0.03 12

0.95 0.93 0.93 0.86 0.87 0.88 0.86 0.87 0.86 0.85 0.85 0.83 0.86 0.83 0.85 0.83 0.80 0.80 0.78 0.77 0.77 0.74 0.73 0.71 0.68 0.62 0.63 0.55 0.54 0.56 0.56 0.41 7 0.427 0.415 0.264 0.262 0.189

C is the mean concentration calculated from eqn. (109) of ref. 24: it is flow rate dependent. x is the corresponding mole fraction. Ac is the difference between the concentration of the injected solution and that of the flowing stream.

al. (0.045), this being interpreted as the point where all the TBA molecules participate in hydrophobic aggregation.

Models A and C have much in common, in that both pos- tulate water-caged TBA entities in very dilute solution: the difference is that these are assumed on the one hand to be long-lived structures (A) and on the other a short-lived hydrophobic ordering of water around the TBA molecules, with rapid exchange of water molecules between the hydra- tion region and the bulk.

In this paper we report mutual-diffusion coefficients in dilute aqueous solutions of TBA at 15 and 25°C and compare these with diffusion data for mixtures of water with methanol, ethanol, propan- 1-01 and butan- 1-01 using corre- lation lengths derived from combining the diffusion coeffi- cients with the viscosities.

Experimental The Taylor dispersion apparatus and the technique used in this laboratory have been described b e f ~ r e . ' . ~ ~ . ~ ~

TBA (99.5%) and ethanol (AR grade) were obtained from Aldrich Chemical Co. and Rh6ne-Poulenc, respectively, and used without further purification. Solutions were prepared gravimetrically from these materials using de-ionized water. The eluent solutions were outgassed by warming under the vacuum produced by a water pump and the compositions

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4074 J. CHEM. SOC. FARADAY TRANS., 1995, VOL. 91

then checked by comparison of the density, measured by pyc- nometry at 25 "C with a precision of & 10 pg cm-3, with liter- ature data.14 The latter were fitted to polynomials in the molarity (c/mol dm-3).

N

p = C d i c i i = O

The fitted coefficients, d i , have been given previously for water-ethanol:' those for the water-TBA system, derived from the results of S a k ~ r a i , ' ~ are given in Table 1. In the case of diffusion runs at 15 "C, the density determined at 25 "C for the eluent was used to calculate the molarity and mole frac- tion, and then the molarity at 15°C determined from the mole fraction using the appropriate density polynomial. The molarities are necessary for the small corrections required to obtain the mean concentration for each run,24 from which the final mean mole fractions reported in Tables 2 and 3 are derived.

Viscosities were also measured with an Ubbelohde viscom- eter, as previously described.'

Results Diffusion results are given in Tables 2 and 3 and are plotted in Fig. 1. They were fitted to simple polynomials in the mean mole fraction, x, weighted by an assumed experimental error of 1%.

Y

I

N

w

E m

z 6 1

Fig. 1 15 "C

I

1 .o

0.8

0.6

0.4

0.2

0.00 0.02 0.04 0.06 0.08 0.1 0

x (TBA)

Mutual diffusion coefficients for water-TBA: (0) 25 "C, (U)

0.0

D(15"C)/10-9 m2 s - l = 0.6647 - 12.010~

+ 2 2 7 . 5 8 ~ ~ - 2 5 4 9 . 1 ~ ~ (94

D(25"C)/10-9 m2 s-' = 0.9285 - 20.510~

+ 694.43~' - 15 349x3 + 106 670x4

(9b) The standard deviations are 0.009 and 0.015 x m2 s-l , respectively, and the average percentage deviations are 1.6 and 1.5.

Viscosity results are given in Table 4: they were also fitted to polynomials.

~ ( 1 5 "C)/mPa s = 1.139 + 27.947~

+ 2 6 9 . 9 8 ~ ~ + 4 3 5 0 . 4 ~ ~ - 7 2 6 8 8 ~ ~ (1Ou)

~ ( 2 5 "C)/mPa s = 0.8904 + 18.066~

+ 2 3 6 . 9 1 ~ ~ - 2 0 2 6 . 1 ~ ~

The standard deviations are 0.006 and 0.005 mPa s, respec- tively, and the average percentage deviations are 0.2 at both temperatures.

Some viscosity measurements were made on water-ethanol for the calculation of the correlation lengths (to be described below), good literature data being sparse in the region of interest near the viscosity maximum. These measurements are reported in Table 4. Excess viscosity coefficients, tf, defined in a similar way to thermodynamic excess functions,

(1 1)

( 1 Ob)

'IE = V - (4lVl + 4 2 t72)

were fitted to an orthogonal polynomial series in the volume fraction 4:'

N

vE = C aiPi(4142) (12) i = 2

with

p2 = 4142 ( 1 3 4

p3 = 4142u - 241) ( 1 3 4

P, = [(2n - 1)(1 - 24l)Pn-1 - (n - 3)P, -J(n + 2)-' ( 1 3 ~ )

The viscosities of the pure substances used in the fit were fixed at the literature values of 0.8904 and 1.0826 mPa s : 2 5 9 2 6

Table 4 Viscosities for water-TBA at 15 and 25 "C and water-ethanol at 25 "C

water-TBA water-ethanol

25 "C 15 "C 25 "C

X q/mPa s X q/mPa s

0.0 0.00247 0.00527 0.00738 0.01 170 0.0 15 55 0.02 122 0.02701 0.02845 0.03227 0.03738 0.04945 0.06048

1.139 1.21 1 1.293 1.361 1.505 1.652 1.882 2.132 2.212 2.398 2.634 3.276 3.806

(0.0 0.00249 0.00483 0.00728 0.01053 0.01573 0.01966 0.029 12 0.0362 1 0.047 18 0.06046 0.073 17

0.8904) 0.9386 0.9848 1.034 1.103 1.227 1.338 1.568 1.751 2.067 2.396 2.688

0.02479 0.0496 1 0.07437 0.11517 0.14973 0.24876 0.30037 0.3 5 329 0.44987 0.50660 0.54460 0.64324 0.68762 0.74571 0.84190 0.99 186

1.128 1.390 1.636 2.014 2.216 2.377 2.334 2.252 2.059 1.939 1.86 1 1.669 1.589 1.490 1.343 1.102

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2.0 - - - 1 0

- - 0 8 4 I-- h

\ \ -

><'

F - 0 6 &

x- \ \ \ \ - 0.4 \

0 A cu + - rn

- - - a 6 0 2 - - 0 2 s

_I-- _/-

I J

' ' ' ' ' ' ' ' -

the coefficients cti (i = 2-7) are 5.22641, -0.467 678, Water-TBA is often compared with water-2- - 2.046 97, - 1.229 45, 0.732 980 and 0.409 297, respectively. butoxyethanol (BE). This system does possess a lower critical The standard deviation was 0.008 mPa s. Good agreement solution temperature (LCST) at 49.43 "C and a BE mole frac- with the results of Soliman and Mar~chal l ,~ was obtained. tion of 0.0599.31 The combined diffusion results of Castillo et

2.0

Q 1.5 cs'

< c -

1.0

- c m

0.5 v

0.0

Discussion The mutual-diffusion coefficient may be written as

0 . 0 -

where Cl2 is the resistance coefficient, P is the total molar volume, T the temperature, and R the gas constant. Much of the composition dependence of D , , is contained in the activ- ity term, B, . D , , becomes zero at a consolute point where this term is also zero, whereas the resistance coefficient remains finite.28

The resistance coefficient is similar to the frictional coeffi- cient, [, introduced by Einstein29 in his original description of Brownian motion of large particles in a continuum and in his treatment it represents the opposing force met by a particle moving through a fluid with unit velocity. For simple Brown- ian motion, with no correlation of velocities,

0.0 0.00 0.05 0.1 0 0.1 5

Dsi = kT/(

The Onsager-Miller-Dunlop resistance coeffcient, given by eqn. (14), is more rigorously defined by irreversible ther- modynamics,' being a proportionality coefficient between the thermodynamic force causing diffusion, the chemical poten- tial gradient and the flux of material produced, but it is then less easily interpreted at the molecular level. Suffice to say that its practical use is that numerical values are independent of the frame of reference and, for non-electrolyte solutions, it varies with composition much less than does D,, .

To first order the excess diffusion coefficient DE varies as B , , that is as

For the alcohol-water systems, the excess Gibbs energy GE is a large positive quantity and the activity coefficient factor is therefore large and negative. Koga et have measured the vapour pressure of dilute solutions of water-TBA at 25 and 30°C at sufficiently closely spaced intervals to enable a precise calculation of B,. Fig. 2 shows the effect of B , on the mutual-diffusion coefficient at 25 "C. The small inflection in D , , at x = 0.035 seems to be due to the maximum in B,, and the thermodynamically corrected diffusion coefficient M , , ( = D, , /B , ) rises rapidly as very low values of B , are approached above x = 0.06.

aL3, (Taylor dispersion), K a t ~ ~ ~ (light scattering, uncorrected from 23 "C) and Bender and P e ~ o r a ~ ~ (light scattering) are shown in Fig. 3. M,, , calculated with the aid of a thermody- namic term again derived from the vapour pressure data of K ~ g a , ~ ' shows quite remarkable double maxima, much larger than the reported experimental scatter in the data (& 2-3% precision). (More experimental data are clearly desirable.) The first maximum corresponds to the region where Koga et ~ 2 1 . ~ ~ have found a maximum in the derivative of the partial molar enthalpy of BE, aHE(BE)/dn(BE), which is a measure of solute-solute interactions, and may represent 'a transition in the local or intermediate scale structure in these solutions'. The second maximum corresponds to that of the LCST composition.

Koga and c o - w o r k e r ~ ~ ~ , ~ ~ have also reported a maximum in aH:/dn for water-TBA at a composition of about mole fraction 0.075, which is consistent with the rise in MI, report- ed here, and again may represent a transition in the local structure in solution. Whether double maxima are present remains to be established. At this stage, we are unable to extend the water-TBA data at 25°C to higher mole fractions for a bettter comparison with the water-BE system, since the high viscosity of the more concentrated solutions lengthens the experimental flow times to the point where it is hard to ensure that the detector baselines are linear for such periods, making diffusion measurements very difficult. It is hoped to examine the system at the higher temperature for which activity data is available at a later date.

An alternative way of examining the diffusion data is to combine them with the viscosities, to obtain the Ornstein- Zernicke correlation lengths, [. This was the approach used in the light-scattering studies referred to in the intro- duction.6*'0 In the critical region, this quantity represents the range of concentration fluctuations. Near a consolute point, D,, and B , have a different dependence on temperature and c o m p o ~ i t i o n . ~ ~ The mutual-diffusion Coefficient is not then a simple integral of an averaged velocity correlation function3 but must be averaged over the concentration fluctuations as well.38 Classical theory yields

k being the Boltzmann constant. A simple interpretation of the correlation length was made

by Alder and Wair~wright .~~.~ ' Mutual diffusion slows near a

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consolute point as droplets of the two new phases begin to form on a microscopic scale, becoming larger and of longer lifetime as the consolute point is approached. Self-diffusion is essentially unaltered as the new phases have almost identical densities and experiment records only a long time average over the whole sample. The mutual-diffusion coefficient can be approximated by the Stokes-Einstein diffusion coefficient for the Brownian motion of a droplet of radius 5, eqn. (16). Modern mode-coupling t h e o r i e ~ ~ ' . ~ ' split D,, into two con- tributions which decay at different rates as the consolute tem- perature is approached. One is a Stokes-Einstein term governed by the divergence of the correlation length, the other a background term governed by the slower divergence of the thermodynamic factor, B , . The classical result is used to obtain the correlation lengths discussed here.

The composition dependence of M , , for water-TBA shows similar behaviour to that near a consolute point. Fig. 4 shows the correlation length at 25 "C for aqueous solutions of meth- anol (0,43 q27*44), ethanol LO', 9 (this work)], propan-1-01,, butan-1-o14' and TBA. (The reference numbers give the sources of the original data.) It is clear that there is a trend from water-methanol, where 5 varies little with composition, through to water-propan- l-ol and water-TBA, where there is strong composition dependence. 5 has also been calculated for a number of other systems, including two where there is no consolute point, namely tetrachloromethane- c y ~ l o h e x a n e ~ ~ and benzene-cy~lohexane,~~ one where the components differ hugely in size, tetrachloromethane- octamethylcyclotetrasiloxane (OCTMS)$8 one where the components associate, trichloromethane-ethanoic acid,49 and one showing a true consolute point, water-triethylamine." The results are shown in Fig. 5. Only the last-mentioned system shows any features, with 5 having the expected maximum: the data here are for 19"C, just above the LCST at 18.33 "C.

Water-propan- l-ol shows a maximum in the correlation length at about a mole fraction of 0.14-0.15 (corresponding to an approximate ratio of water to alcohol molecules of 1 : 7), which is similar to that shown by water-triethylamine. This is consistent with both the models (A and C) postulating the formation of aggregates and (B) postulating pseudo- critical behaviour. The trend with water-TBA seems likely to be similar, but the data need to be extended to higher con- centrations (i.e. higher temperatures) to confirm this. Fig. 6 magnifies the region up to a mole fraction of 0.2. 5 rises for both water-TBA and water-propan-1-01 near a mole fraction of 0.04, close to the composition where clusters in water-TBA were postulated to overlap on the basis of molecular dynamics simulations' and dielectric relaxation measure- ments.20.2 ' However, dielectric relaxation time measurementss1 have not found any discontinuity for water- propan-1-01 at this composition, so a connection between the

1 .o

0.8 I r . . . i '.

i '., 0.6

\ $ w

0.4

0.2

2.0

1 .6

0.4

0.0 0.2 0.4 0.6 0.8 1 .o x (ROH)

Fig. 4 Correlation lengths for water-alcohol mixtures at 25 "C

0.0 L ' ' I , , I

0.0 0.2 0.4 0.6 0.8 1 .o

*2 Fig. 5 Correlation lengths for CC14<-C6H12, C6H6-C-C6H12, OMCTS-CCl,, CH,COOH-CHC1, and H,O-N(C,H,), . x2 is the mole fraction of the second-named component. The data are all for the temperature 25 "C, except for H,O-N(C,H,), , where the tem- perature is 19 "C.

two observations remains tenuous. for water-butan- l-ol follows the water-TBA line to the solute solubility limit. 5 for water-TBA at 15 "C is similar to that at 25 "C for the concen- tration range examined.

Conclusions Mutual-diffusion coefficients and viscosities have been mea- sured by the Taylor dispersion and capillary methods, respec- tively, for water-2-methylpropan-2-01 (TBA) at 15 and 25 "C. It has been shown that the composition dependence of the thermodynamically corrected diffusion coefficient, M , , at low TBA concentrations is similar to that exhibited by systems near a consolute point (where large concentration fluctuations occur), i.e. it tends toward a maximum. Exami- nation of literature data for the water-2-butoxyethanol system reveals similar but more complex behaviour, with a double maximum apparent. Correlation lengths for water- lower alcohols show a trend from water-methanol and water-ethanol, which behave like other non-electrolyte mix- tures with little variation in with composition, to water- propan-1-01 which shows a distinct maximum at low alcohol mole fraction and water-TBA, which tends towards such a maximum in the range of compositions available to our experiments. These results lend support to the possibility of pseudo-critical behaviour in these systems due to a transition in local and intermediate structure at a composition where the alcohol molecules are no longer shielded from one another by surrounding water molecules.

Work is continuing using high-pressure self-diffusion mea- surements, which are often sensitive to solution structure in

fBuOH - . - .. nPrOH

EtOH MeOH nBuOH

----.

_._..-. --- -

0.4 O ' I

"." 0.00 0.04 0.08 0.1 2 0.1 6 0.20

x (ROH)

Fig. 6 Correlation lengths for water-alcohol mixtures dilute in alcohol at 25 "C

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aqueous systems, to investigate further water-alcohol inter- actions.

We are grateful to Dr. M. Sakurai of Hokkaido University for kindly making available raw density data for the alcohol systems, to Professor T. Kato for details of Dr. Ito’s light- scattering data, and to the Australian Research Council for a grant in support of this work. Part of the work reported here was presented at the 23rd International Solution Chemistry Conference held at the University of Leicester in August 1993.

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Paper 5/03360H; Received 25th May, 1995

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