2784
S. P. WASIKAND N. M. ROSCHER
Diffusion Coefficients of Paraffin-Chain Salts and the Formation Energetics of Micelles by S. P. Wasik and Nina Matheny Roscher National Bureau of Standards, Washington, D.C. $0884
(Received January 14, 1970)
The diffusion coefficients of the colloidal electrolytes deoyl, dodecyl, and t tradecyltrimethylammonium bromide have been measured as a function of concentration in aquco 1s solutions at 25“. These values compared favorably with values obtained from the theoretical expression derived by Hartley. A method is presented for determining the standard Gibbs free energy change of micelle formation from diffusion coefficient vs. concentration plots.
In aqueous solutions of paraffin-chain salts there is a tendency for grouping of molecules to take place in which like is associated with like. The nonpolar paraffin chains tend to associate together in some form of fluid arrangement while the polar groups associate with the polar water molecules. These aggregates of molecules, called miceIIes, fluctuate under the influence of thermal motion and are in equilibrium with neighboring nonaggregated molecules. A simple form of micelle proposed by Hartleyl consists of a nearly spherical aggregation of paraffin-chain ions with the hydrophobic part of the ion in the center and the polar part of the outside, the electrical charge of the aggregate being partially neutralized by a number of oppositely charged ions adhering electrostatically to the primary groups. Below the critical micelle concentration, cmc, the paraffin-chain salt essentially behaves as a strong electrolyte. Experimental methods for investigating this phenomenon have varied widely; conductivity, solubilization, light scattering, viscosity, and other physical properties have been reported. Many of these measurements involve application of an external force, or an additive to the paraffin-chain salt solution. Such application of an external agent is liable to vitiate the results as far as calculations of size, shape, and structure of the micelles are concerned. Diffusion measurements by a free-boundary method provide a sensitive and accurate means of investigating the properties of micellar solutions without extraneous additives and without the limitations imposed by the use of glass disks or porous plates. Although little work has been reported2Iausing this technique it appears to be a good method for studying the physical properties of micellar solution. Hartley derived an expression fDr the diffusion of an aggregating electrolyte in the transition range from simple to colloidal solution4 which predicted that the diffusion coefficient should go through a very low minimum in the region where the paraffin-chain Salt goes The Journal of Physical Chemistry, Vol. 74, No. 14, 1870
from the simple to the colloidal form. Hartley lacked detailed knowledge of his compounds to treat his results adequately. Since then light scattering techniques have been developed for measuring the aggregation number and effective charge of the micelle, and it is now possible to evaluate the Hartley diffusion expression explicitly. I n this report are presented diffusion measurements on aqueous solutions of the decyltrimethylammonium bromide, CI,-,Br, dodecyltrimethylammonium bromide, CI2Br, and tetradecyltrimethylammonium bromide, CI4Br, by a free-boundary method. Hartley’s expressions for the diffusion coefficient concentration relationship is shown to explain adequately the experimental data. A method is presented for calculating the standard free-energy change from these measurements.
Experimental Section The alkyl bromides, obtained from the Fisher Scientific Co., were fractionally distilled through a PirasGlover spinning-band column, 60 cm in length. The center cuts having the same refractive index were used for the preparations. The n-alkyltrimethylammonium bromides were prepared by refluxing an excess (10%) of freshly distilled trimethylamine with the alkyl bromide in methyl alcohol for several hours. The excess alcohol and trimethylamine were removed by distillation at room temperature under reduced pressure, The quaternary ammonium bromides were then recrystallized four or five times from different solvents. Analyses for bromide content agree with the theoretical amount within experimental precision. (1) G. S. Hartley, “Aqueous Solutions of Paraffin-Chain Salts,” Hermann* 1936* (2) N. Bindney and L. Saunders, J . Pharm. Pharmacol., 7 , 1012 (1955). (3) R.Parker and s. Wasik, J . Phys. C h m . , 63,1921 (1959). (4) G. S. Hartley, Trans. Faradag SOC.,35,1109 (1939).
DIFFUSION COEFFICIENTS OF PARAFFIN-CHAIN SALTSFORMATION The cell used for the diffusion experiments and the experimental technique have been described in detail by Longsworth.6 A sharp, plane, horizontal boundary between two solutions of different concentrations was formed in a rectangular cell. The subsequent concentration distribution at the boundary was observed with a Rayleigh interference optical system. The temperature of the water bath in which the cell was immersed during the measurements was controlled at 25 f 0.005'. Photographs of the interference patterns were taken on glass plates. About forty measurements of the refractive index of the solution as a function of distance in the cell were made for each pattern with a two-dimensional comparator. The relation between the refractive index and the concentration of the solution was determined with a differential refractometer using dextrose (National Bureau of Standards Standard Sample No. 41) as the standard. From these data the concentration distribution was calculated. The calibration of the interferometer was checked by measuring the diffusion coefficient of dextrose in aqueous solution at 25', The experimental technique and calculation procedure were those used by Longsworth.6 The diffusion coefficient of dextrose in a solution of concentration 0.380 g/100 om3 obtained in six separate measurements was (6.729 i 0.006) X cm2/sec compared with the value of (6.728 f 0.006) X love cm2/sec measured by Longsworth.6 The zerotime correction in these experiments was less than 10 sec. The diffusion coefficients for the paraffin-chain salts were calculated from the general differential equation for a unidirected diffusion process
*
where C = concentration, t = time, x = distance in the direction of diffusion, D = diffusion coefficient, If diffusion is studied at a liquid junction between two solutions of different concentrations, the boundary conditions which are imposed upon the solution of the diffusion equation are that at t = 0, for x: < 0, C = G, and for x > 0, C = Czwhere CI < CZ. It is assumed that the concentrations a t large values of x: in the diffusion cell are unaltered during an experiment. With these restrictions, no general solution for eq 1 has been obtained. However, if it is assumed that x/dt = U, where u is a function of C only, eq 1 can be transformed and integrated at constant t to give
D
= -(2t
,> ac
-1
JcLx:dC c
D in this case is a differential diffusion coefficient corresponding to the definit,e concentration C rather than an integral coefficient appropriate to the range of concen-
2785
tration, C1 to C. This calculation is valid only if (x: dtc) is constant during the experimentaa The evaluation of the diffusion coefficient from eq 2 requires a simultaneous determination of C, 2 , and dC/dx at time t. This was done by measuring two of the quantities and calculating the third. Data for C as a function of x were differentiated to obtain dC/dx C
as a function of C and x:. The integral Jc, xdC in eq 2 was evaluated graphically for each of the experimental points. Since the free boundary experiments do not begin at zero time but effectively at a greater time, there is an error of about 1% in the calculated values of D.
Discussion and Results To describe the diffusion of an aggregating electrolyte in the transitional range from simple to colloidal solution, Hartley4derived the following expression
DF2 -= RT
x-0
z)+
dCM dC
- AM
- -(h+O
+ A+'
L'M
- - (x-' C
+ oh-' +
- 8xMo)
where Lo,X+O, and XMO are the equivalent conductance of the counterion, the paraffin-chain ion, and the micelle, respectively; 1 - e is the fraction of the charge of each micelle neutralized by the association of gegenions. Z is the charge of the micelle, F is the Faraday charge, C is the concentration of the whole electrolyte in equivalents per unit volume calculated without regard to association. The micelle concentration, CM,is expressed as CM =
c - cl
where C1 is the univalent concentration of the unassociated paraffin ion. This equation was derived on classical lines without introducing any thermodynamic or mobility corrections. It is valid only for dilute solutions. The quantities of dCM/dC and CM/Cwere calculated from the expression for the equilibrium between single ions and micelles
where KITis the equilibrium constant, m is the number of paraffin-chain ions per micelle, and p is the effective (5) L. G. Longsworth, J.Amer. Chem. Soc., 74,4155 (1952). (6) L. G. Longsworth, J.Phys. Chem., 58,770 (1954).
The Journal of Physical Chemistry, Vol. 7 4 , N o . 14, 1070
2786
S.
charge. Phillips’ justified using concentration instead of activities in eq 4 by substituting an effective charge for the real charge. The effective charge, p , may be calculated from the limiting slope of the HC/r vs. C plot, H being the Einstein constant, r the turbidity and C the micelle concentration.* Since p can be explained in terms of the Donnon concept and the fluctuation theory, it is not the real charge, 2,but rather an equivalent ideal charge. It may be interpreted as an activity coefficient. With this implication the double-layer interaction of a micelle ion of actual valence 2 is equivalent in behavior to an ideal ion of valence p. Calculations of CM/C and dCM/dC may be simplified by defining a concentration Coby the following relation KM = C02m-~-1 and the dimensionless quantities S = (C,/C), C/Co. Substituting S and if into eq 4
if
and CM = S€C,
From eq 5
(7)
+m - ( m - p ) S Combining eq 6 and 7 and rearranging terms dcM -dC
-
2m-p-1 m (m - P>2 1-S m-(m-p)S
-+-
1 S
+
+S
(8)
The quantity dCM/dC was calculated for different values of S using values of m and p reported by Debyes and Wasilr and Hubbardg from light scattering measurements on decyltrimethylammonium bromide, dodecyltrimethylammonium bromide, and tetradecyltrimethylammonium bromide solutions. The equivalent conductance of the paraffin-chain ion, X+O, was calculated from limiting conductance values reported by Scott and Tartarlo using a value of 78.2 (mol ohm cm)-l for the ion conductance of the bromide ion. Values for the micelle conductance were calculated from the electrophoretic mobility measurements of The Journal of Physical Chemistry, VoL 74,No.
14,1970
2
8.0
-
7.0
-
6.0 -
u
LD
0
x 5.0 n
-
4.0
-
3.0
-
2.0
20
.IO
.30
fl
c = cco
2m-p-1
-
9.0
=
Thus if may be calculated for given values of S, m, and p . Since
dS
P.WASIKAND N. M. ROSCHER
Figure 1. Experimental plots of diffusion Coefficient against concentration for decyltrimethylammonium bromide, CloBr, dodecyltrimethylammonium bromide, C1gBr, and tetradecyltrimethylammonium bromide, ClrBr, and the theoretical plots from eq 3 using Debye’s and Wasik and Hubbard’s data.
decyl-, dodecyl- and tetradecylamine hydrochloride micelles by Hoyer and Greenfie1d.l‘ These authors state that micelles of paraffin-chain salts of the same carbon chain have essentially the same mobility vs. concentration behavior. Although values for the mobility of the CloBr,C12Br,and C14Brmicelles obtained by this method are only an approximation, they do not seriously affect the shape of the theoretical diffusion-concentration plots. From eq 3 and 8 the theoretical diffusion coefficient I n order to comwas calculated as a function of &‘. pare the theoretical with experimental plots, a value for Co had to be estimated by assigning values of CO until the two curves gave a good fit. The value for Co in all cases was close to the concentration where dD/dl/C was a maximum. In Figure 1 are shown the experimental plots for CIOBr, C12Br, and C14Br along with the theoretical plots from the data given in Table I. The theoretical curves calculated from Debye’s* values for m and p gave better agreement than those of Wasik and Hubbard9 because apparently Debye’s values for p are the correct values. The minima of the theoretical curves were very sensitive t o p and fairly insensitive to m. J. N. Phillips, Trane. Faraday SOC.,51,561 (1955). P.Debye, Ann. N . Y . Acad. Sci., 51,575 (1949). (9) S. Waaik and W. Hubbard, J. Res. Nut. Bur. Stand., 68A, 539
(7)
(8)
(1964). (10) A. B. Scott and H. V. Tartar, J. Amer. Chem. SOC.,65, 692 (1943). (11) W.H.Hoyer and A. Greenfield, J. Phys. Chem., 61,735 (1957).
DIFFUSION COEFFICIENTS OF PARAFFIN-CHAIN SALTSFORMATION
where cmc is the critical micelle concentration. This equation was derived by defining the cmc as the point corresponding to the maximum change in gradient in an ideal property-concentration (4 against C) relationship, being given by
Table I: The Standard Gibbs Free Energy Change, AGO, per Molecule Associated with Micelle Formation for the Paraffin Salts Decyltrimethylammonium Bromide, CIOBr, Dodecyltrimethylammonium Bromide, C12Br,and Tetradecyltrimethylammonium Bromide, C14Br,Calculated from Eq 9 and 10 for Debye’s and Wasik, and Hubbard’s Values, for the Number of Paraffin-Chain Ions per Micelle, m, and the Effective Charge, p
CloBr ClzBr ClrBr
36 9 31 6 . 6 50 1 0 . 5 55 1 3 . 5 75 1 0 . 5
11.3 14.3 17.9
11.4 14.3 17.8
/du\
11.5 0.0767 1 4 . 1 0.0164 0.0036
a Debye.8 Ir Wasik and Hubbard.9 Equation 10, Debye’s data. Equation 9, Debye’s data. Equation 9, Wasik and Hubbard’s data.
The “valley” in the diffusion coefficient-concentration plots results from the fact that dCaI/dC increases much faster than CM/C. Since X+O is much bigger than XMO/Z, an increase in dCM/dC will cause a decrease in the second term of eq 3. I n the transition region the third term at first decreases when aggregation begins and subsequently increases. The overall effect is that in general the diffusion coefficient should decrease followed by a rise a t higher concentration. A mimimum in the D vs. l/c plots for associated colloids has previously been reported for sodium dodecyl sulfate2 and dodecyltrimethylammonium chloride. Standard Free Energy Change The standard Gibbs free energy change, A G O , per molecule associated with micelle formation may be calculated from the expression A G O
=
kT
(2m - p - 1) In CO m
(9)
The value of A G O depends on the standard states chosen. If the concentrations are expressed in mole fraction then A G O would be with reference to a state of mole fraction unity. In Table I are given A G O values calculated from eq 9 using values for COobtained from the diffusion experiments and values of m and p obtained by Debyes and Wasik and H ~ b b a r d . ~ Phillips’ has derived the following expression for the Gibbs free energy change
- --
A G O
kT
In3
+ 2 1 n m + 2m - P m
m
In crnc
2787
(10)
Phillips assumed that there exists an equilibrium between the single ions and the micelles and that the micelles are effectively monodispersed. In Table I are given A G O values for CloBr, G B r , and C14Br calculated by Phillips from light scattering data of Debye.8 The close agreement between the A G O values calculated from Phillip’s expression and those obtained from eq 9 lend further support to Hartley’s model of the micelle. Both expressions were derived from the assumption that there exists an equilibrium between the single ions and the micelle and that the micelles are effectively monodispersed. The method presented in this report for calculating AG” requires no knowledge of how the cmc should be defined. This is an advantage since the cmc is probably a range of concentration. The experimental measurements of D and theoretical values derived from light scattering and conductivity data agree well up to the concentration for the minimum value of D (see Figure 1). Above this value the agreement is poor. The likely explanation is a change in the character of the aggregates. Micelles are statistical in character; they are constantly being formed and broken up by thermal motion. They cannot be thought of as persistent entities having well defined geometrical shapes. Hartley’s’ model for the micelle as a spherical aggregate of amphiphilic molecules with the polar groups on its surface and the hydrocarbon chains forming a fluid core is probably valid at low concentrations. At higher concentrations it is replaced by more extended micellar forms. Equation 3 is valid for the diffusion of an aggregating electrolyte in the transition range from simple ion to a Hartley-like micelle. It does not take into consideration a more extended micellar form. This may be partly the reason why the agreement between the theoretical and experimental curves in Figure 1 is not as good a t higher concentrations as it is at concentrations near the cmc. Acknowledgment. We wish to thank Dr. Frederick Mies for his suggestions and his help with the mathematics in this paper.
The Journal of Physical Chemistry, VoZ. 7 4 , N o . 1.6, 1970
