J . Phys. Chem. 1985,89, 1467-1473
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Figure 3. Relative polarization (in percent) of the charge on the nearest hydrogen by the second molecule, vs. the shortest H--0distance (angstroms) in the dimer. The dashed lines show the polarization in the B (bifurcated), C (cyclic), and E (linear) models fitted with PD (net atomic) charges. The solid lines show similar results for the SC (site charge) models.
.
The largest estimated polarization energies for the PD model were a t configuration D29 (-34.7) and at configuration E37 for the SC model (-10.8). Both of these configurations have very short H.. -0distances. In some cases the estimated polarization energy obtained was positive. More of the PD values were positive than were the SC values, which was consistent with the idea that SC was a better model for estimation of polarization energy. In the SC model only the A4, A5, and E40 structures, and the first five B structures, showed apparently positive polarization energies. In general polarization should be a stabilizing energy component. The positive values obtained (which never exceeded 1.0 kl/mol) indicated that this model for the polarization energy was approximate.
1467
Figure 2 compares the magnitudes of the Coulombic component of the dimer energy for the D, MCY, PD, and SC models. The D model showed the largest Coulombic component, except for longer distances in the cyclic and linear dimers. The MCY model showed a Coulombic component closer to the D model in the linear and cyclic structures. For the bifurcated structures the MCY, PD, and SC Coulombic components were very nearly equal and were much less than the D values, especially at short distance. For the linear and cyclic structures the PD model deviated the most from the D model. Figure 3 shows a plot of the relative change in the charge of the nearest hydrogen atom as the second molecule approaches to form the dimer. The dashed curves show the PD model. The three curves shown are for the B (bifurcated), C (cyclic), and E (linear) cases. For a given He. .O distance, the B structure showed the largest site charge polarization, and the C structure the least. The linear E structure was intermediate in its polarization, but since this case was evaluated at the very short H..-O distance of 1.51 A, the polarization reached a maximum of 36%. The solid lines in Figure 3 show the site charge polarization in the SC model. The apparent polarization was smaller, reaching a maximum of 28% at the closest distance for the E structure. The B points no longer showed a larger polarization than the E points; in fact both the B and E points could be plotted on the same curve. In addition, the C points for the SC model are closer to the E and B points than they were for the PD model. The figure showed that the S C model site charge polarization had a more uniform trend vs. the H. SOdistance than the PD model.
Acknowledgment. This work was supported by research grant GM16260 from the National Institutes of Health. Registry No. Water, 7732-18-5.
Simulation Study of Rotational and Translational Dynamics of Diatomic Molecules wlth Quadrupolar Interactions Vinayak N. Kabadi and W. A. Steele* Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: August 27, 1984; In Final Form: December 10, 1984)
Molecular dynamics simulationsof two model diatomic fluids have been performed over a range of temperature which includes the solid as well as the liquid at fixed density. Details of the simulation and thermodynamic properties have been reported elsewhere; here, time-correlation functions are reported. Rotational and translational velocity correlations are given as well as a number of reorientational time-correlation functions. The molecular interactions studied were based on site-site models interacting with and without quadrupolar energy. Attempts were made to fit the time correlations for the liquids to simple theoretical results. It is concluded that reorientation in these highly hindered fluids can be described as torsional oscillation with a fluctuating frequency. With the exception of the long-time tails, the translational correlation functions are reasonably well reproduced if one assumes memory functions with Gaussian time dependence.
Introduction In the preceding paper,' a computer simulation study of the thermodynamics of two diatomic systems in their melting regions was reported. In that work, the pairwise potential energies were taken to be those for the well-known sitesite model, with reduced site separation distance L* = L/o held constant at 0.547. For one system, ideal quadrupolar interactions were added, with ) ~1.00. / * Reduced reduced quadrupole moment Q* = Q / ( ~ u ~ = temperatures were varied over a rather wide range (greater than a factor of 2) with reduced density held fmed at p* = p d = 0.625. (1)
V. N. Kabadi and W. A. Steele, J . Phys. Chem., in press.
Here, t and IJ are the well-depth and distance scaling parameters in the Lennard-Jones site-site interaction function. Since the method of simulation was molecular dynamics, time-dependent quantities could also be extracted from the position and velocity data that were generated. In the present paper, some time-correlation functions for single-molecule translation and rotation are reported together with associated quantities such as correlation times and self-diffusion constants. These results have been fitted to theoretical expressions. Before we present the details, it should be pointed out that the values of the mean squared torque and force reported in paper 1 indicate clearly that both rotational and translational motions in these systems are highly hindered, for the temperatures and
0022-3654/85/2089-1467$01.50/0 0 1985 American Chemical Society
1468 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 I.Oh
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CJt) 0.0
CJt)
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.oo
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0.2
0.3
0.4
0.5
0.6
time (red. units)
Rpre 1. Angular velocity time-correlation functions are shown for diatomic molecules with L* = 0.547, Q* = 0 at fixed p* = 0.625 and at values of T.indicated in the figure. Solid curves are simulations and dashed curves were calculated from the fluctuating librator model. (Model and simulation for T.= 1.397 coincide.) The two highest r* correspond to liquids, and Ts = 0.597 is solid.
physical states of this study. This is due to the rather high density chosen, although nonsphericity implied by the large L* also plays an important role in hindering the rotation. Over the past decade or two, many models have been proposed to describe translation in dense monatomic fluids2 There is no obvious reason why these models would not be valid for nonspherical molecules, but previous attempts to fit simulation data for such systems to the existing models are quite sparse. Here, we will test a model due to Berne et al.3 and Lovesey4 for the liquid-state translational velocity correlation functions and will show that it gives too much oscillatory structure to match well with simulations. It will be shown that an alternative model based on memory functions with Gaussian decay gives velocity correlations that are closer to the simulations than those for the exponential decays of Berne and of Lovesey. In the case of reorientation, less work has been done on hightorque systems such as those of this study.5 Nevertheless, a simple model of libration with a fluctuating frequency has recently been suggested.5 We will show that good agreement between the simulated and model angular velocity correlation functions can be obtained and that many of the orientational relaxation correlation functions can be calculated to moderate accuracy by using the angular velocity correlation functions in first-order cumulant theory. Thus, it will be shown that much of the single-molecule dynamical behavior for these systems can be represented by simple models; furthermore, it will be argued that values obtained for the fitting parameters are physically reasonable and may be amenable to independent theoretical calculation.
Rotational Dynamics All time-correlation functions reported here are presented as a function of reduced time t*. For consistency, the reducing factor (2) Much of this work is summarized in monographs; in particular, see: J. P. Hansen and I. R. MacDonald, ”Theory of Simple Liquids”, Academic Press, New York, 1976; N. H. March and M. P. Tosi, “Atomic Dynamics in Liquids”, Wiley, New York, 1976. (3) B. J. Berne, J. P. Boon, and S. Rice, J. Chem. Phys., 45, 1086 (1966). (4) S. W. Lovesey, J. Phys. Chem. 6,399 (1973). (5) Reference to previous work can be found in R. Lynden-Bell and W. A. Steele, J. Phys. Chem., 88, 6514 (1984).
r
d.1
012
015
013 014 time (red. units)
016
Figure 2. Same as Figure 1, but for Q* = 1.0. The two lowest r* correspond to solids, and the two highest to liquids.
is taken to be “translational” throughout; thus, t* = t ( t / m a 2 ) 1 / 2 . These are related to the “rotational” reduced times t* = t*(T*/rC)1f2where T* = kT/c, r* = L*2/4. Figures 1 and 2 show angular velocity time-correlation functions Cw(t*)for each system over a range of temperatures. In general, the functions for Q* = 1.O are evaluated at considerably higher T* than those for Q* = 0 because of the lower melting temperature for Q* = 0. However, solid-state simulations were run for both systems at T* = 0.597, and these results are plotted in Figures 1 and 2 together with two representative liquid-state curves for each system, and a second solid-state correlation function for Q* = 1 at T* near the melting point. The pronounced negative regions in these curves have often been observed in other simulations of high-torque molecular fluids. For the solids, these dips are followed by damped oscillations at longer times which would normally be interpreted as due to librational motions. It is reasonable to assume that a more strongly damped version of this motion persists in the liquids, and thus we have attempted to fit the solid- and liquid-state data to a fluctuating ~ noted librator model based on ideas due primarily to K ~ b o .He that a cumulant expansion of the equation of motion for an oscillator can be written down which, for angular velocity, takes the form ~ ( t =) ~ ( 0cos ) (not)exp[-K(t)] (1) where Qo is the mean librational frequency and the cumulant K ( t ) is K ( t ) = x ‘ ( t - 7)(6Q0(0)s Q O ( 7 ) ) d7
(2)
where AQo is the fluctuation in no.It is here assumed that this is a Gaussian random variable so that (6Qo(0) 6 Q o ( f ) ) = Az exp(-t/.r,)
(3)
if Az is the mean square fluctuation in noand 7c is the correlation time for this fluctuation. In this way, one obtains Cw(t)= cos (not)exp(-A2~,2[exp(-t/7,)
- 1 + f/7,])
(4)
To fit this expression to the simulation results, one first notes that Cw(t)passes through zero at not = 7r/2. Secondly, it is well-known (6) R. Kubo in “Fluctuations, Relaxation and Resonance in Magnetic Systems”, D. ter Haar, Ed., Plenum Press, New York, 1962.
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1469
Dynamics of Diatomic Molecules
time (red. units)
TABLE I: Parameters for the Fluctuating Librator Model for the Angular Velocity Correlation Functiona
a,
TL
A
1,
I
0.2
0.4
0.6
0.8
I
I
I
I
1.0
(P2)sirn
Q* = 0.0 0.597 0.750 0.897 1.206 1.397
19.2 19.2 21.5 21.5 21.5
0.595 1.30 1.386 1 .502b 1.598 1.692 1.918 2.090 2.308 2.530
24.5 24.5 24.5 24.5 22.9 22.9 22.9 22.9 22.9 22.9
12.8 15.0 17.6 19.6 21.6
0.038 0.038 0.054 0.054 0.054
47 66 103 153 194
0.032 0.032 0.032 0.042 0.050 0.050 0.050 0.050 0.050 0.050
63 166 180 203 252 272 323 368 421 48 1
Q* = 1.0 10.2 15.9 16.4 17.4 23.0 23.4 24.5 25.5 26.4 27.3
‘All variables are in reduced units; consequently, no(9-’) and A (s-l) have been divided by and iC (seconds) has been multiplied by the same factor. The reducing factor for torque is t. bPoint very close to the transition region. time (red. units)
Figure 4. Same as Figure 3, but for T* = 0.897,which corresponds to
a liquid.
C
0.c
0-
c
4 -0.06
--+
-\
1
0.2 I
time (red. units) 0.4 0.6 I
I
0.8
1.0
I
-0.2
2 -
\
\
-i
L
2 C
9
-0.4 A&, \
-0.101 I I Figure 3. Solid curves are orientational timecorrelation functions for the simulated solid with Q* = 0.0, 7‘ = 0.597. They are plotted on a logarithmicscale and compared with model calculations. Curves A and B were obtained from first-order cumulant theory, by using model and
simulated C,,,(t),respectively. The dot-dashed curves are calculated from memory functions whose time dependence is approximated by the simulated C,(t). Numbers denote values of the index 1. that the initial curvature of C,(t) is equal to -(M)/2ZkT where ( M ) is the mean square torque calculated and reported in paper 1. However, eq 4 yields a curvature of -(no2 Az). In this way, noand A can be fixed, leaving only rCto be adjusted to give a fit to the curves a t longer times. Figure 1 and 2 show that good agreement between model and simulation are obtained for the liquids. These results are typical for the systems studied. Values for the parameters of the model are listed in Table I. The fits to the solid-state simulations are less successful, especially for the very low temperature Q* = 1 calculation, where the damping of the model is too large. In fact, for such systems it is likely that a “libron” model in which the coupling between molecular oscillatory motions is explicitly included will be. necessary to account for the observed long-time oscillations in C,(t).
+
-0.E Figure 5. Same as Figure 3, but for a liquid with T* = 1.397.
Given a reasonably successful model for C,(t) (or adequate simulations of this function), one can then attempt a calculation of the reorientational time-correlation functions Cl(t)defined in the usual way: = (PLcos M t ) ) ) (5) where PI is a Legendre function and 68 is the reorientation angle of the molecule axis. Applying first-order cumulant theory, one finds In C l ( t ) / l ( l + 1) = - A ‘ ( t - T)C,(T)dT Comparisons of the simulated C,(t) with the first cumulant expression evaluated by using both the simulated and the fitted model C,(t) are shown in Figures 3-9. The lack of success in fitting the solid-state data is not as serious as it appears, since the ap-
1470 The Journal of Physical Chemistry, Vol. 89, No. 8,1985 0.2
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Figure 6. Same as Figure 3,but for the solid with Q* = 1 .O, Tz = 0.595.
o,og.
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time (red. units) 0;4 Oi6
Oi8
‘p
Figure 8. Same as Figure 3, but for the liquid with Q* = 1.0, T.= 1.598.
0.0
O.On
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= t
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Figure 7. Same as Figure 3,but for the solid with Q* = 1 .O,Tz = 1.386.
proach to a constant value for Cr(t)a t long times requires that the area under Cw(t)be precisely zero. Neither the simulation nor the model yields zero area and even a relatively small integrated value will yield a significant slope to In C/(r)/l(l+ 1). In the case of the solid with Q* = 0, T* = 0.597 where the simulations and the Cr(t) calculated from cumulant theory using simulated Cw(t)are in reasonable agreement, it is felt to be more coincidence than an accurate calculation. An additional difficulty arises from the fact that higher cumulants are present in principle but have been neglected in practice. In fact, these higher terms are needed if the calculation of In C/(r)/l(l+ 1) is to exhibit the I dependence found in the liquid-state simulations of Figures 4, 5 , 8, and 9 and at other T* not shown here. An alternative, but less rigorous, way to use the velocity correlation function in a calculation of the C&) is to invoke the Nee-Zwanzig’ ‘diffusion with memory” model in which M/(t) = 4 1 + l ) C J t ) ( k T / I ) (7)
-1.0 Figure 9. Same as Figure 3, but for the liquid with Q* = 1.0,T.= 2.308.
where MI is the memory function. This yields correlation functions that are often quite close to those obtained from the first-order cumulant. Indeed, it is easy to show that both approaches yield exponentially decaying cr(t)a t long times with decay constants T / given by
Consequently, it would seem that there is little point in further exploring this memory function approximation. Note that it yields a single slope for In Cl(t)/l(l+ l), at long times, in disagreement (7)
T.W. Nee and R. Zwanzig, J . Chem. Phys., 52, 6353 (1970).
The Journal of Physical Chemistry, Vol. 89, No. 8, I985
Dynamics of Diatomic Molecules with the simulations. The cumulant approach is also preferred because it leads to a specific recipe for inserting correction terms to the first-order theory, while no such possibility is apparent for the Nee-Zwanzig theory. Parameters used in fitting the fluctuating librator model to the simulated C,(t) curves are listed in Table I. These numbers show striking regularities and indicate the following: (a) Over a large range of temperature, the average librational frequency is independent of T* (at fixed density); indeed, the only significant change in nooccurs when the system melts. (b) It is an increase in the magnitude of the fluctuations in the vibrational frequency that accounts for most of the temperature dependence of C,(t); as temperature increases, this gives rise both to a more rapid initial decay and to a larger damping of the oscillatory part of C,(t). (c) The correlation time for the fluctuations also appears to be independent of T* a t fixed p*. (d) The increase in mean square torque with increasing T* is due to an increase in the mean square fluctuations about a constant average librational frequency.
Translational Dynamics We now consider the translational velocity correlation functions. Figure 10 shows the C,(t) for three of the solids studied. The decaying oscillations are typical for a classical ensemble of molecules undergoing nearly harmonic vibration at a finite temperature. The most notable feature is that the introduction of quadrupolar interactions (at fixed P,p * ) has a significant effect upon the vibrational motion. Specifically, the decay of the oscillations is slower with Q* = 1 than with Q* = 0, even though the time scale (Le., the frequency) has not changed much. The liquid-state translational motions show what might be a remnant of vibrational oscillation (or a "cage effect", as it is often called). It seemed to be of interest to attempt to fit these curves to a model and, if possible, to attempt to learn whether there are characteristic differences between the CU(t)for molecules and atoms. Of course, there is a considerable amount of prior work on this function for monatomic fluids. For example, if one assumes that the time evolution of this function is governed by an exponentially decaying memory function M,(t), it is easy to show thats COS
(6t)
1 +sin 6t 267tr
1471
'.Or-----l
-LOo
r 0.1
0.2
0.3
0.4
0.5
0.6
time (red. units)
Figure 10. Translational velocity correlation functions for three diatomic solids with Q*,T* values indicated in the figure.
(9)
where
M d t ) = M Oe x ~ ( - f / ~ t r )
(11)
Furthermore, Mo = ( P ) / 3 m k B T so , that there is one adjustable In its parameter (T,~)for systems such as ours with known (P). original form, this parameter was fixed by requiring that the self-diffusion constant D obtained from a plot of mean square displacement vs. time be correctly given by the time integral of c,(t).As a consequence, T~ = kBT/&&. However, it was later realized that the long-time tails in Cu(t)make a significant contribution to the areas under these curves and thus to D.9 Since this tail is not given by an exponentially decaying memory function, an inconsistency is introduced in this approach which is most easily removed by taking Ttr to be adjustable. As a consequence, one expects that a tail may appear in the simulated curves which is not accounted for by this simple assumption for the memory function. Attempts were made to fit eq 9 to the simulated C,(t) for the liquids only-evidently a theory treating the nearly harmonic (8) J. P. Hansen and I. R. McDonald, 'Theory of Simple Liquids", Academic Press, New York, 1976, Section 9.2. (9) See, among others: B. J. Alder and T. E. Wainwright, Phys. Rev. A , 1, 18 (1970); J. J. Erpenbeckand W. W. Wood,J.Srac. Phys., 24,455 (1981); R. Bedeaux and P. Mazur, Phys. Lert. A , 43,401 (1973); J. R. Dorfman and E. G. D. Cohen, Phys. Rev. Len., 25, 1257 (1970).
I
1
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a4 0.5 0.6 time (red. unit4 Figure 11. Simulated translational velocity correlation function for the liquid with Q* = 0, T* = 0.897 is shown by the solid line. It can be compared with curves calculated from Gaussian (dot-dashed) and exponential (dashed) approximations to the memory function for this time-correlation function.
-0.5i
0.1
0.2
0.3
oscillations in the solids will involve an approach quite different from that taken here. If one requires that the initial decays be accurately fitted, it emerges that these model correlation functions give much more oscillatory behavior than the simulations, in agreement with previous analyses of monatomic liquids. Thus, an alternative form for the memory function Mu@)was tried, namely a simple Gaussian decay: M,(t) = hfo exp(-yt2) (12) Although no analytic expression for c,(t)can be obtained from eq 12, numerical solution of the memory function equation is straightforward. Figures 11-14 are typical of the results obtained
1472 The Journal of Physical Chemistry, Vol. 89, No. 8,1985
Kabadi and Steele
t , t
-0.50
1
0.1
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0.3
0.4
time (red. units) Figure 12. Same as Figure 11, but for Q* = 0,
0.5
r* =
k - 0.50 0.1 0.2 0.3 0.4 0.5 0.6
0.6
1.397.
time (red. units) Figure 14. Same as Figure 11, but for Q* = 1.0, T* = 2.308.
i
time (red. units) Figure 15. Simulations of the mean square displacement (in units of
-0.50
0.1
0.2
0.3
0.4
0.5
time (red. units) Figure 13. Same as Figure 11, but for Q* = 1.0, r* = 1.598
and show that y can be adjusted to give adequate fits to the simulated velocity time correlations for moderately short times and that there is a distinct long-time tail remaining in all cases. (We hasten to add that the identification of these tails with the theoretical tails is not really possible in the absence of a detailed justification for the accuracy of Gaussian memory functions.) It is particularly interesting to note that the Gaussian memory functions yield curves that are significantly closer to the simulations than those obtained from the exponential M,(t). (Consequently, similar comparisons have been performed for a variety of other simulated systems, including monatomics. The results, which will be reported elsewhere, clearly show the superiority of the Gaussian assumption in giving the velocity timecorrelation function except for a long-time tail.) Values used for the memory function parameters are listed in Table 11. Finally, mean square displacements were also simulated for solids and liquids. A few representative curves are shown in Figure 15, where the distinction between solid and liquid is immediately
distance/u) are shown for several liquids and solids. In all cases, Q* = 0; T.values are indicated in the figure. The small and constant (at long time) displacements are obviously for the solids; the straight lines are characteristicof diffusional motion in the liquids, with slopes proportional to the self-diffusion constants.
TABLE II: Parameters for Translational Velocity Memory Functions and Diffusion Constants“ Y
D+sim
D*cxpo
D*Gawinn
0.022 0.03 1 0.038
0.028 0.036 0.042
0.042 0.045 0.053 0.055 0.061 0.066
0.046 0.049 0.057 0.060 0.067 0.072
Q* = 0 0.897 1.206 1.397
782 0.050 897 0.041 965 \ 0.036
541 641 772
0.013 0.022 0.029
Q* = 1.0 1.598 1.692 1.918 2.090 2.308 2.530
1018 1045 1106 1177 1222 1275
0.035 0.034 0.030 0.029 0.028 0.027
794 860 1001 1082 1181 1231
0.024 0.027 0.029 0.031 0.041 0.047
‘All variables are in reduced units; consequently, M0 (s-~) and y (s-~) have been multiplied by m u 2 / €and D (cm2/s) has been multiplied by (a2e/m)-’12.
J. Phys. Chem. 1985, 89, 1473-1477 apparent. Self-diffusion constants calculated from the slopes of the liquid-state curves are also listed in Table 11. It is interesting to note that the contribution to D due to the long-time tail which is not given by the model Gaussian memory function is quite significant, often amounting to more than 30% of the total.
Conclusions Both molecular translational and rotational motions are highly hindered in the fluids simulated in this study-translations, because of the relatively high density, and rotations, because of the high density and the nonspherical potentials chosen. Upon melting, the nature of the motions changes from what appears to be coupled oscillations to what is basically diffusive. However, translations and rotations in the liquids are far from simple random walk. In particular, it is necessary to take umemoryninto account and, in addition, to deal with “cage effects” which produce damped oscillatory features in the velocity correlation functions. The rotational librations are reproduced reasonably well by a model embodying these ideas. In the translational case, the memory function used does not explicitly include a description of the translational cage effect, not is it capable of reproducing the long-time tails observed in the Cu(t).Nevertheless, the transla-
1473
tional Gaussian memory function seems to give correlation functions significantly closer to the simulations than does the exponential form suggested earlier. (Additional comparisons of simulation results and the C,(t) obtained from the memory functions are under way.) As is often the case the reasons for the success (or lack of it) for this simple memory function are not clear. We note that Lucas and co-workers10have obtained reasonable agreement between theory and experimental or simulated viscosities and self-diffusion constants for liquid argon by using Gaussian memory functions with parameters numerically evaluated from expressions relating them to equilibrium averages of functions of the interatomic potential. The work reported here is similar in spirit but indicates that much of their observed differences between experiment and theory may be due to difficulties in handling the long-time tails expected in the relevant time-correlation functions for these systems.
Acknowledgment. Support from the N.S.F. for this work via grant CHE-8305735 is gratefully acknowledged. (10) K. Lucas and B. Moser, Mol. Phys., 37, 1849 (1979);38, 1855 (1979); M. Luckas and K. Lucas, Mol. Phys., 48,989 (1983).
Microtitration Calorimetric Study of the Micelllzatlon of Three Poly(oxyethylene) Glycol Dodecyl Ethers Gerd Olofsson Division of Thermochemistry, Chemical Center, University of Lund, S - 221 00 Lund, Sweden (Received: October 5, 1984)
A new microtitration calorimetric technique has been utilized to determine the enthalpy of micelle formation AH(mic) and cmc for three poly(oxyethy1ene) glycol dodecyl ethers, C12H25(OC2H4),0H(x = 5,6, and 8), in water. Measurements were made at 10 and 25 “C and for C&8 also at 40 “C. The values of cmc determined in the present study are considered to be at least as accurate as previously reported values. Titration calorimetry offers a useful method for the determination of cmc, particularly for nonionic surfactants with low cmc’s. Measurements were also made of enthalpies of solution of pure liquid C12E, in aqueous solution. Comparison between the micellization enthalpies and solution enthalpies makes it possible to see how changes in head-group size and temperature influence the amphiphiles in the monomer and micellar states separately. The enthalpies of micelle formation are endothermic and vary only little with the size of polar group, the values (in kJ mol-’) being at 25 “C 13.5 0.3 for C12E5, 14.8 0.4 for C&6, and 16.3 0.4 for C1&. The enthalpies of solution to give aqueous monomers are strongly exothermic and vary considerably with head-group size, the exothermic contribution being 7 kJ mol-’ per (-OCZH4-) group. The contribution to the solution enthalpy from the C12alkyl chain is close to zero at 25 OC. The.observed AH(mic) therefore corresponds to the enthalpy of dehydration of 2-2.5 ethylene oxide groups. The values for the entropy of micelle formation are large and positive while the heat capacity changes are large and negative. The values for both properties are nearly the same for the three C12E, studied. Contributions from the dehydration of the alkyl chain dominate M(mic) and ACJmic). The temperatures for the minima in the cmc do not correlate with the cloud-point temperatures.
*
*
Introduction Nonionic surfactants have critical micelle concentrations (cmc’s) in water that are substantially lower than ionic surfactants having the same size of nonpolar group. For instance, poly(oxyethy1ene) glycol dodecyl ethers have cmc’s of the order 5 X 10-5-104 mol kg-’ at room temperature while the cmc of sodium dodceyl sulfate mol kg-’. Calorimetric measurements are needed to is 8 X give reliable information about enthalpy, entropy, and heat capacity changes for micelle formation,’ but only few studies have been made on nonionic surfactants?+ The scarcity of calorimetric (1) D. G. Hall and B. A. Pethica in “Nonionic Surfactants”, Vol. 1, M. J. Schick, Ed., Marcel Dekker, New York, 1966,p 549. (2) L. Benjamin, J . Phys. Chem., 68, 3575 (1964). (3) J. M. Corkill, J. F. Goodman, and J. R. Tate, Trans. Faraday SOC., 60, 996 (1964).
(4)J. L. Woodhead, J. A. Lewis,G. N . Malcolm, and I. D. Watson, J. Colloid Interface Sci., 79, 454 (1981).
0022-3654/85/2089-1473$01.50/0
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data is due at least in part to their low cmc’s. A new microtitration calorimetric method has been developed that allows the direct determination of both the enthalpy of micelle formation PJl(mic) and the cmc. Results of measurements on the pentakis-, hexakis-, and octakis(oxyethy1ene) glycol dodecyl ethers (C12E5, C1&, and C12E8)at various temperatures are reported in the present paper. The study was made to see how changes in size of the polar group affect the micellization. In a previous paper calorimetric measurements on C12E5 were reported and the thermodynamics of micellization of C12E5in water were discussed? The present study gave more precise results for AH(mic), and in addition values of cmc have been determined. Measurements have also been made on the dissolution of pure liquid C12E,in aqueous solution. These results make it possible (5) J. E. Desnoyers, G. Caron, R.DeLisi, D. Roberts, A. Roux, and G. Perron, J . Phys. Chem., 87, 1397 (1983). (6)G. Olofsson, J . Phys. Chem., 87, 4000 (1983).
0 1985 American Chemical Society
