2033
COMMUNICATIONS TO THE EDITOR
of these papers. The notation that Dr. Swarbrick used is the hypothetical standard free energy, entropy, and enthalpy, but the enthalpy and entropy of micellization discussed in these papers are the real enthalpy and entropy change accompanied at the micellization. The chemical potentials (partial molal Gibbs free energy) of any component in the two phases in equilibrium at constant T and P are equal. Namely, the Gibbs free energy change of surfactant at the micellization, AG,, is zero. Therefore, the thermodynamics described on these two papers2s3are correct. In this occasion, I would like to emphasize that it is identical, in principle, to treat micelle formation either as a mass action equilibrium or as a pseudophaseU4 If the aggregation number of micelle is small, the mass action model may be used, while if the aggregation number is large, the phase separation model may be a ~ p l i e d . Two ~ ~ ~approaches are not incompatible, but are compatible. DEPARTMENT OF CHEMISTRY YOKOHAMA NATIONAL UNIVERSITY 0 0 K A - 2 , MINAMIKU YOKOHAMA, JAPAN
Path 2 must describe precisely the same changes since the initial and final states are identical with those of path 1. Step I represents a change in concentration, in which case
K6z8 S H ~ N O D A
AG,I
=
R T In aomo
(4)
Under ideal conditions of mixing, eq 4 becomes
AG,'
=
R T In cmc
(5)
Additionally, ideal mixing requires that
AHml = 0
(6)
The entropy change, ASml, is given by
AS,'
=
- AGml)/T
(AH,'
(7)
On substituting eq 5 and 6 into eq 7, one obtains
AS,'
=
-R In cmc
(8)
Step I1 of path 2 is simply the difference between path 1 and step I of path 2. Thus
AG,"
=
- AG,'
AG,"
(9)
From eq 1 and 5
RECEIVED OCTOBER 8, 1969
AGmI'
=
0
(10)
Equation 10 is the criterion for a state of equilibrium, presumably that of a phase change. In a similar fashion, the enthalpy change for step I1 is given by Entropy Changes Associated with Micellization
SiT: Comments by Shinodal on a recent paper from this group2 are indicative of certain differences which exist with regard to the present,ation of thermodynamic data on micellization. One may regard micellization as taking place along either of two paths path 1
standard state monomer, af = 1
\t,
z micelle, a, = 1
path 2
monomer, af = cmc where af and a, refer to the activities of the surfactant in the free and micellized forms, respectively. Path 1 describes the conversion of monomer in its hypothetical standard 1 m state to micelle in its hypothetical "pure" hydrocarbon state. The association thermodynamics have been described as3z4 AG,"
AH,' AS,"
= =
=
RT In cmc
6(AGmo/T>/6(l/T) (AH," - A G m o ) / T
where AS," is defined as (molar entropy in - (molar entropy in the standard state.)
AH,II
=
AH," - AH,,,'
(11)
which, through eq 6, reduces to AH,II
=
AH,"
(12)
The enthalpy change is then given, by using either eq 3 and 8 or eq 10 and 12, as
AS,II
=
AH,"/T
(13)
where AS,I1 is defined as (molar entropy in the micelle) - (molar entropy at the cmc). The literature reveals that the entropy of micellization is calculated in one of two ways: either through the use of eq 3 or eq 13. While the latter is more definitive in describing the entropy changes occurring upon micellization, actual data on this phenomenon are scarce, due to the fact that calorimetric AH," values suffer from various uncertainties, especially when the alkyl chain contains more than about ten carbon atom^.^,^ To circumvent this problem, a number of workers derive a AH, according to eq 2, and immediately introduce this term into eq 13. The entropy values thus obtained are (1) K. Shinoda, J . Phys. Chem., to be published. (2) J. Swarbrick and J. Daruwala, ibid., 73, 2627 (1969). (3) P. Molyneux, C. T. Rhodes, and J. Swarbrick, Trans. Faraday Soc., 61, 1043 (1965). (4) D. G. Hall and B. A. Pethica, "Nonionic Surfactants," M. J. Schick, Ed., Marcel Delrker, Inc., New York, N . Y., 1967, Chapter 16. (5) L. Benjamin, J . Phys. Chem., 68, 3575 (1964). Volume 74, h'umber 9 April SO, 1070
COMMUNICATIONS TO THE EDITOR
2034 quite misleading, since they include a cratic term which is of little significance in describing the events taking place during the formation of a micelle. On the other hand, use of eq 3 provides entropy values which more closely resemble unitary functions, since it excludes the entropy of mixing. This procedure would appear to offer more advantage, at least until such time as precise calorimetric data become available. Finally, it should be pointed out that the compatibility of the mass action and phase separation approaches is not in dispute. As previous work showedJ2 the data for the N-alkyl betaines support the view, widely held, that the longer the alkyl chain of the amphiphile, the better the agreement between the two models. DIVISIONO F PHARMACEUTICS SCHOOL OF PHARMACY OF CONNECTICUT, UNIVERSITY STORRS,CONNECTICUT 06268
1 oo
------I
10-
IC
RICHARDE. LINDSTROM JAMES SWARBRICK 10-
RECEIVED NOVEMBER 5, 1969
10-
Molecular Mobility in Simple .Glasses
I n a recent paper, one of us’ proposed a picture of viscous flow and the glass transition that suggested that secondary relaxations in the glassy state could arise solely from intermolecular processes, thus leading to the prediction that such molecular relaxations could be a universal feature of the glassy state. Argon2 made a similar conjecture on the basis of a description of viscous flow due to O r ~ w a n ,whose ~ ideas were also used as the basis of the picture referred to earlier.’ To test this hypothesis we have studied, by dielectric relaxation measurements, molecular mobility in glasses formed from molecules lacking obvious internal degrees of freedom capable of giving relaxations due to intramolecular processes. We studied the fused-salt glasses Ca(N03)2.4H20 and a 45 mol % Ca(N03)2-KN03 mix’ soluture; simple molecular glasses of 10-15 mol % tions of several mono- and di-halogen substituted benzenes and naphthalenes in the nonpolar solvent decalin; 50-60 mol % mixtures of pyridine with several halogensubstituted benzenes and naphthalenes and several pure aromatic hydrocarbons and aliphatic alcohols. We used a General Radio 1615A capacitance bridge and a parallel plate guarded electrode assembly as the dielectric cell. Measurements were made in the frequency range 50 HZ to 200 kHz and a t several temperatures near and below the respective glass transition temperature, T,. We may summarize our results by stating that almost all of these glasses show a secondary relaxation below T,. Figure 1 shows the dielectric loss factor (tan 6) a t 1 kHz plotted on a logarithmic scale against temSir:
The Journal of Physical Chemistry
I
TO
10-
I 123
K I
113
223
2.13
313
Figure 1. Dielectric loss factor of (1) 63.2 mol % pyridine in 1-chloronaphthalene (scale has been shifted to the right by 100’); (2) 16.5 mol % chlorobenzene in decalin; (3) 45 mol % Ca(NO3)2-KNO8 mixture; (4)o-terphenyl; and ( 5 ) Ca(N03)2. 4Hz0 (scale has been shifted downwards by two logarithmic cycles) a t 1 kHz as a function of temperature. Arrows indicate the literature values for T , when known.
perature for five of the systems studied. The Ca(N03)2-KWOa glass is atypical in that the secondary relaxation does not appear near T , (60”) but instead tan 6 shows a rise below 200”K, indicative of a maximum below the boiling point of liquid nitrogen. The behavior of most other systems we studied is more like that of the other four shown in the figure in that: (1) A secondary transition (peak in tan 6) appears some 30-50” below T , at a measuring frequency of lo3 Hz, (2) The “activation energy” of the main transition is much greater than that of the secondary peak, so that the two relaxations must either cross in a temperature-frequency diagram or else merge into a single loss region above T , a t higher frequencies. (3) Neither relaxation is described by a single relaxation (1) M. Goldstein, J . Chem. Phys., 51, 3728 (1969). (2) A. S.Argon, J . A p p l . Phys., 39, 4080 (1968). (3) E. Orowan, “Creep in Metallic and Non-Metallic Materials,” “Proceedings of the First National Congress of Applied Mechanics,” American Society of Mechanical Engineers, New York, N. Y., 1952, p p 453-472.
