Self-Association of Dodecylammonlum Propionate
2609
Temperature-Dependent Self-Association of Dodecylammonium Propionate in Benzene and Cyclohexane’ Fred Yun-Fat Lo, Barnee M. Escott, Eleanor J. Fendler, E. T. Adams, Jr.,” Russell D. Larsen, and Philip W. Smith Departments of Chemistryand of Mathematics, Texas A&M lJn/vwsIty, CdEege Station, Texas 77843 (Received July 25, 1975) Publication costs assisted by The Robert A. Welch Foundation
A series of vapor pressure osmometry experiments at different temperatures was performed on solutions of dodecylammonium propionate (DAP) in benzene and cyclohexane. The increase in the apparent number average molecular weight (Mna)with increasing solute concentration ( c ) was characteristic of a self-association. In both solvents the association decreased with increasing temperature, and at the same temperatures the self-association was greater in cyclohexane than it was in benzene. B splines were found to be quite useful in obtaining smooth curves through the cM1/Mnevs. c data; M1 is the monomer molecular weight. From plots of MIIM,, vs. c it was possible to obtain apparent values of the weight average molecular weight (M,) and the natural logarithm of the weight fraction of monomer (In fa). The quantities M,,, M,, and In fa could be used to test for the type of association present and also to obtain values for the association equilibrium constant or constants, Ki, and the second virial coefficient, BM1. Several models were tested, and a sequential indefinite self-association seemed to describe the data best in both sets of experiments. The values of AHoobtained were .-83.4 kJ/mol in benzene and -57.4 kJ/mol in cyclohexane. These results are discussed in terms of micellar structure and catalysis in reversed micellar systems.
Introduction Surfactants have the general property of being soluble in water as well as in dipolar aprotic and nonpolar solvents. Although the physicochemical properties of these substances have been extensively investigated in aqueous solution, relatively little is known about their association behavior in nonaqueous solvents. In aqueous solution surfactants undergo a self-association forming micelles with a definite degree of aggregation. The structure of these micelles is generally such that the ionic or polar head groups are in contact with the bulk water, while the hydrophobic hydrocarbon moities comprise the interior. In nonpolar solvents, the aggregate structure is generally the converse with the ionic or polar groups surrounding the interior cavity or “solvent pool” and the hydrophobic hydrocarbon chains extending into and penetrated by the bulk nonpolar solvent.2 This type of aggregate has been referred to as a “reversed” or “inverted” mi~elle.~J Both types of micelles are known to exhibit considerable catalytic activity which depends on a variety of factors. However, it appears that catalyses in the latter systems are often more specific and of greater magnitude than those in aqueous micellar one^.^.^ Indeed, rate accelerations of up to a factor of 5 X lo6 have been observed for reactions of organic and inorganic ions in reversed micellar systems composed of alkylammonium carboxylates in benzene.2t415 Catalysis by alkylammonium carboxylates has been found to be dependent upon the nonpolar solvent as well as on the surfactant for a given reaction. For example, rate constants for the mutarotation of 2,3,4,6-tetramethyl-c~-D-glucose at 24.6OC in the presence of micellar dodecylammonium propionate (DAP) are enhanced by a factor of 380 in benzene and by a factor of 863 in cyclohexane relative to the rate constants in pure solvent^.^,^-^ Consequently, we have in-
* Address correspondence to Dr. E. T. Adams, Jr., Chemistry Department, Texas A&M University,College Station, Tex. 77843.
vestigated the state of aggregation of DAP in both solvents at various temperatures, since this information is essential to an understanding of the mechanisms, and hence the magnitudes, of reversed micellar interactions and catalyses. The monomer molecular weight (MI) of the DAP is 253.4 daltons. Because M1 is so small, one cannot do membrane osmometry. The low value of M1 tends to rule out sedimentation equilibrium experiments, as high speeds (30000 rpm or greater) may be required and the organic solvents would undergo compressibility effects, which would make the analysis of the data more complicated.‘ If M1 is small and the degree of aggregation is not large, then light scattering becomes technically difficuk8 It should be noted that light scattering has been used to study the self-association of ionic and nonionic detergents in aqueous solution, but here the degree of aggregation is usually quite large.2*9J0Hence, vapor pressure osmometry is the most advantageous and accurate technique for investigating the self-association of DAP in benzene and in cyclohexane. Although vapor pressure osmometry is a steady-state method, it has been used successfully to evaluate molecular weights (M) or number average molecular weights (Mn&and second virial coefficients (Bas) for nonassociating solutes.11J2 Elias has studied the self-association of nonionic detergents in aqueous solutions by light scattering, sedimentation equilibrium, and vapor pressure osmometry; he has obtained excellent agreement for the degree of aggregation as determined by the various methods.1° The self-association of cytidine in aqueous solutions has been studied by vapor pressure osmometry13 and by sedimentation equilibrium.14 Both methods gave the same model for the self-association (a sequential indefinite self-association with all molar equilibrium constants equal), and both methods gave excellent agreement for the value of the association equilibrium constant. Thus, there is considerable justification for using this technique. A method for analyzing self-associations using data from The Journal of Physical Chemistty, Vol. 79, No. 24, 1975
Adams et al.
2610
a series of osmotic pressure experiments at different solute concentrations (c) was developed by A d a m ~ . ' ~ ?Since ' ~ osmometry and vapor pressure osmometry both give Mn (or M,,, the apparent number average molecular weight, in nonideal solutions), the Adams method can be applied here. In this method M,,, the apparent weight average molecular weight, and In fa, the natural logarithm of the apparent weight fraction of monomer, can also be evaluated as ~ e l l . ~ The J ~ Jquantities ~ M,,, M,,, and In f a can be combined in various ways to test for the type of association present and to evaluate the equilibrium constant or constants (Ki) and the second virial coefficient (BM1).7J7-21 The method for doing this and its application to the selfassociation of DAP in the two solvents is presented below. The utility of spline functions in fitting smooth curves to raw data plots of cM1/Mna vs. c is introduced here for the first time.
The summation in eq 6 is carried out over all i solute components. With the aid of eq 3-5 and with the fact that V = (ATIS,, one notes that M,, is obtained in vapor pressure osmometry from the r e l a t i o n l l ~ ~ ~
--V ---1 --1 + Bo&+ . . .
(7) KVPC Mna Mn The quantity Bo, is the osmotic pressure second virial coefficient, and the quantity KVPis an instrumental constant. The value of KVPdepends on the temperature and also on the solvent used. In order to determine KVP, a solute of known molecular weight, M, is dissolved in the solvent at Theory temperature T. Values of V/c are recorded for several concentrations, c, of solute. The intercept a t c = 0 of a plot of When a nonvolatile solute is dissolved in a solvent, the Vlc vs. c gives KvpIM, (see plots a and b of Figure 1).If vapor pressure of the solvent is lowered; this lowering of the solute is homogeneous and does not associate, then M, the vapor pressure, po - p , is proportional to the mole fraction, x , of the solute, through Raoult's l a ~ ~ ~ 3 ~ ~becomes M, the molecular weight of the solute. For ideal, dilute solutions Bo, = 0, and the plot of V/c vs. c will be a (Po - P ) l P o = gx (1) horizontal line as in plot a of Figure 1, provided the solute does not self-associate. Once the calibration constant KVP At a temperature T, po is the vapor pressure of the pure has been determined, then measurements on the unknown solvent, p is the vapor pressure of the solvent in the solusolution are performed in the same manner. tion, and g is the osmotic coefficient, which is a measure of With nonassociating solutes in a nonideal solution, one the thermodynamic nonideality. For an ideal solution g = obtains M,, (see eq 7). From a series of vapor pressure os1. One of the most rapid and accurate ways of determining mometer experiments at different values of c, it is apparent vapor pressure lowering is to convert it to a temperature from eq 7 that a plot of l/Mna vs. c will have Bo, as a limitdifference (AT) which is the quantity measured in a vapor ing slope and 1/M, as an intercept. This situation is illuspressure osmometer. If an equilibrium state were present, trated in plot b of Figure 1. then A T would be related to gx by the relation22 For self-associating solutes in which the monomer is homogeneous, the situation becomes more complicated. SelfRT2 (AT)eq= -gx = K,,gx associations are reactions of the type7J5 mvap
where AHvapis the latent heat of vaporization and R is the universal gas constant. However, instead of an equilibrium state, a steady state is obtained in which heat losses are balanced by a continuous condensation of solvent onto the solution. Thus, the steady-state temperature difference can be expressed22as
(AT),, = Km@
(3)
where K,, is a proportionality constant between (AT),,and gx. In the vapor pressure osmometer, (AT),, is measured by a resistance change, Ar, or by the microvolts imbalance, V, in the Wheatstone bridge circuit of the instrument, Now gx is also related to the osmotic pressure through the relat i ~ n ~ ~ IIV1'IRT = gx
(5)
where c is the solute concentration in g/l. For ideal, dilute solutions M,, = M,, the number average molecular weight. M, is defined by The Journal of Physical Chemistry, Vol. 79, No. 24, 1975
nP1
P, qP2
n = 2,3,. .
+ mP3 + . . .
.
(8) (9)
and related equilibria. Here P represents the self-associating solute. Equilibria described by eq 8 are known as monomer-n -mer associations. Self-associations described by eq 9 can be discrete or they can continue without limit, in which case they are known as sequential, indefinite selfassociations. Whenever a self-association is present, Mn and the other average molecular weights are functions of c , the total solute c ~ n c e n t r a t i o n . ~This J ~ concentration dependence is indicated by writing M, as Mnc. Since M n c increases with increasing c, a plot of l/Mnc vs. c would resemble plot c in Figure 1. Note that the limiting slope of this plot is15J6
(4)
Here II is the osmotic pressure and V1O is the partial molar volume of the pure solvent a t temperature T . The quantity IIIRT is related to the apparent number average molecular weight, M,,, sincel1v2* 1000IIlRT = CIMna
nP1
(10) if dimer is present (K2 is the association constant for the dimerization). If dimer is absent, the limiting slope of a plot of l/Mncvs. c will be zero. If the solution is nonideal, then the concentration dependence of M,, denoted by M,,, is superimposed on the nonideal effect represented by the Bosc term in eq 7. Plot d in Figure 1 shows a possible result. Note that for a positive Bosc term, one can encounter a minimum in the plot of
Self-Assoclation of Dodecylammonium Propionate
2611
as one for which BM1 =-0.Now Mwaand Mna are interrelated, ~ i n c e ~ J ~ - ~ l
and
There is one other relation which can prove useful in the analysis of self-associations. The apparent weight fraction of monomer, fa, can be obtained from the relation7J5-21y25.26
C
Flgure 1. Simulated plots of l/Mn, vs. c for four different cases: (a) ideal, dilute solution, no self-association; (b) the corresponding plot
for a nonideal solution with no self-association; (c)self-association in an ideal, dilute solution. The Increase in Mnc with c is characteristic of a self-association, hence values of 1/M, decrease with increasing c. (d) The corresponding plot for a nonideal selfassoclatlon. A minimum In these plots is attributed to positive second virlal coefficient. l/Mna vs. c . Furthermore, the limiting slope of a plot of l/Mnavs. c is15J6
Here f l = c1/c is the weight fraction of monomer. Steiner27328showed for ideal, dilute solutions that In f l was given by (19)
Since K2 and Bo, are not known a priori, and since the limiting slope is not too reliable (large values of K2 may cause the initial portion of a plot of l/Mna vs. c to be very steep), one cannot use the same type of analysis that was used with nonassociating solutes. Clearly, a different type of analysis is required for a self-association. Fortunately, such a different treatment is a ~ a i l a b l e . ~ J ~We - ~ l can apply methods developed for the analysis of self-associations by sedimentation equilibrium to the analysis of osmotic pressure experiments, since Mnc and Mwc (the weight average molecular weight) or their apparent values, M,, and M,,, are interrelated.7J5 Thus, if one average or apparent average molecular weight is available as a function of associating solute concentration, then all the other average or apparent average molecular weights can be obtained. Previously it has been assumed that the natural logarithm of the activity coefficient for self-associating species i can be represented by7JS2l In yi = iBMlc
i = 2,3,
.. ,
(12)
where BM1 is the second virial coefficient or nonideal term. As a consequence of eq 12, the concentration of the n-mer ( c n ) can be expressed as Cn
Kncln
n
=:
2,3,. .
.
Adams and Williams25 showed that the replacement of MIIM,, in eq 19 by M1/Mwa(defined by eq 15) leads to eq 18. With strong self-associations one may not be able to evaluate In f a accurately. In order to evaluate In f a one must make a plot of [(MllM,,) - l]/c vs. c or a plot of [(MlIMna) - l ] /VS. ~ C . AS c 4 0'1~~ 1ia-O a : :
(
--
1) / c = -K2
+ BM1
and15
With strong associations, K2 (the dimerization constant) can be very large so that the plots required for the evaluation of In f a will be quite steep in the vicinity of zero solute concentration. The greatest contribution to the integral required for the evaluation of In f a comes from the region of lowest solute concentration (see Figure 4 of ref 7), and this is also the region where the experimental data have the greatest error. Additionally this is the region where it is most difficult to obtain data because of instrumental limitations. This problem can be overcome by modifying eq 18 to give19
(13)
where Kn is the association constant for the formation of n-mer. The quantity BM1 appears in the expressions for Mna and M,,, since they are defined by7JS2l (14) and
Here M,, = 2iCiMilC = MlBiicilc is the weight average molecular weight; the summation is carried out over the i associating species. For self-associations the molecular weight of the j-mer is related to M1 by the relation Mj = jM1 0' = 2,3, . . .I. An ideal, dilute solution will be defined
Here the asterisks refer to values a t a low concentration, c * ; the choice of c* is somewhat arbitrary. Equations 14, 16, and 18 can be combined in various ways so that the second virial coefficient (BM1) is eliminated, and the resulting functions can be used for the analysis of some types of self-association. Two useful relations The Journal of ~%yslcalChemistry, Vol. 7s. No. 24, 1975
2612 are17-21
and
The application of eq 20-22, as well as other relations to the analysis of self-associations, is demonstrated in the analysis of the results. Experimental Section Preparation of Dodecylammonium Propionate (DAP) Solutions. Reagent grade benzene ( 1 Negative equilibrium constant Generated negative values of MJM,, Generated values of f2 > 1 Negative equilibrium constant Generated negative values of MJM,,, Generated values of fi > 1 Negative equilibrium constant
As in Tables I and 11. These terms refer to data smoothed with ship curves (ship), free-spline (free),or forced-spline (forced) methods.
TABLE IV: Thermodynamic Functions for DAP in Benzene and Cyclohexane (Free Spline) Analyzeda as a Sequential Indefinite Self-Association with all Molar Equilibrium Constants Equal (Type I)* BM,7
K,
T ,K ml/g
ml/g
300 310
8.64 6.19
300 310 316 323
2.75 4.16 4.76 3.52
kJ mol''
AGO,
K, M"
A. In Benzene -13.3 207 70.4 -11.0 AH" = -83.4 kJ mol''
798 271
AS", J
deg" mol'' -2.34 -2.34
B. In Cyclohexane lo3 371 -14.8 -1.42 X lo3 296 -14.6 -1.38 125 -12.7 -1.41 77.3 -11.7 -1.41 AHO = -57.4 kJ mol-' * 13.4 (from van't Hoff plot, see Figure 7). 1.43 1.14 481 298
X
X
10'
X
lo2
X
10'' 10" 10'' 10''
X X X
As in Tables I and 11. * See comments, Tables I or IIB. k0.45 lo2 J deg-l mol-I.
a
X
C16l
AS' = (AH' - AGa)/T The values of the equilibrium constants, second virial coefficients, and other thermodynamic quantities obtained with free-spline data are listed in Table IVA for cyclohexane and in Table IVB for benzene. Discussion It is evident from the results reported here that the vapor pressure osmometer offers an excellent method for investigating the self-association of relatively small molecules in nonaqueous solvents. Other techniques, which are more amenable for determination of the critical micelle concentration (cmc), apparent aggregation numbers, assoThe Journal of Physical Chemistry, Vol. 79, No. 24, 1975
ciation-dissociation constants, and micellar structure, however, do not generally give information on the self-association behavior. Not only do we possess sensitive methods for testing for the type of association present, but we can also include nonideal effects when analyzing self-associations, since In (fJfa*) and M1IMwa can be calculated from the plots of MIIMna vs. c . Additionally, spline functions (see the Appendix) offer a very elegant and objective way of fitting the experimental cM1/Mna vs. c data. The free-spline method, which is a least-squares method, appears to be the best way of fitting the experimental data (see Table 111) Our observed curvilinear behavior as a function of surfactant concentration (Figures 2-4) is analogous to that observed using l H NMR chemical s h i f t ~and ~ l a~ variety ~~ of other physical properties (Table V). It should be noted that the cmc is generally taken as the surfactant concentration at which a discontinuity occurs in a plot of a physical property vs. stoichiometric surfactant concentration.2 In nonaqueous solvents, however, curvature is frequently encountered rather than a sharp break, especially for low molecular weight surfactants. In some series of similar surfactants, both types of behavior have been 0 b s e r ~ e d , 2 + as ~ ~well l~~-~~ as linear plots.2 In complete agreement with Markovits, Levy, and Kertes,47 this apparent anomalous behavior may be the consequence of the technique employed, the relative concentration of monomer with respect to the different oligomers, andlor the relative concentration of the different oligomers themselves. However, cmc values obtained by extrapolation of the high and low concentration data to the point of intersection in many cases of curvilinear behavior gives values which are in remarkable agreement, with those obtained from plots with relatively sharp breaks (see Table V and ref 39). The observation of relatively sharp breaks has generally been interpreted in terms of a monomer-nmer type of association, and this has been assumed in numerous cases in the calculation and interpretation of dataa2
2619
Self-Association of Dodecylammonium Propionate TABLE V: Association Parameters for Dodecylammonium Propionate (DAP) and Butanoate (DABu) in Benzene and Cyclohexane
-
Sur factant
DAP
crnc 2 x 5x 3-7 3 x
n
Method of determination Benzene Solubilization of water
10-3 M (26°C) 10-3 M (40°C) x 10-3 M (SOT) 10'5 M (37°C)
'H NMR spectroscopy 3+
DABu 1.2-1.8 X 10'' M (26°C) 2.0 x 1r2 M (40°C) 3.8 x 10-3 M ( 3 7 ~ )
Vapor pressure osmometry Solubilization of water
Comments Water alters cmc
29, 40, 41
Assumes monomerlz-mer association Sequential indefinite type of association Water alters cmc
31, 32, 42
Assumes monomer-n-mer association Freezing point depression Addition of up to 17 mole4.3 cules of water/molecule of surf act ant increases n to 34.5 Cyclohexane Assumes monomer-n-mer Uv spectrophotometry 7.1 x 10-4 M ( 2 0 ~ ) DAP association 10.1 (6.5"C) Freezing point depression Degree of association ( a ) is determined Probe may lower cmc; Uv-visible spectral 7.4-7.6 X M (20°C) assumes monomerlzchanges of solubilized mer association 7,7,8,8-tetracyanoquinodimethane Sequential indefinite type Vapor pressure osmom5+ of association etry DABu (6.3 0.2) X lom3M (20°C) 9 f 1 (20°C) Vapor pressure depression Assumes monomer-n-mer association for 6 1 (WC) 1.0 x 10-2 M (30°C) calculation of n 5.1 x lom3m (30°C) 3.7 (30°C) 2 1 x 10-3 M (200, SOT) Solubilization of water Water alters cmc 3 x lom3m (26°C) Uv-visible spectral Probe may lower cmc; 7.2-7.3 X lo'* M (20°C) changes of solubilized assumes monomer-n7,7,8,8-tetracyanomer association quinodimethane 5.3 x 1 0 ' 4 M ( 2 0 ~ ) Uv spectrophotometry Assumes monomer--mer association
*
'H NMR spectroscopy
*
*
It is obvious from our investigation and the calculations of Mullesg that such an a priori assumption is clearly unfounded (vide infra). The surprising result in this investigation is the apparent failure of a monomer-n-mer association to describe the observed self-association of DAP in either solvent. It is also remarkable that a type I indefinite self-association describes the association in either solvent so excellently. Additionally, the sequential indefinite self-association of DAP in benzene and cyclohexane is in complete accord with kinetic data for DAP catalysis in these solvent^.^^^^ It should be noted that Kertes and Markovits?O who studied the self-association of tridodecylammonium salts in CCL,benzene, and cyclohexane, had to invoke models that were different from a monomer-n-mer association to describe their observations. Their choices were restricted to discrete selfassociations, such as a monomer-dimer-hexamer or other models; a sequential, indefinite (type I) self-association or other indefinite self-associations were not reported among their choices. In a later paper, Kertes, Levy, and Marko-
Ref
This work 29, 40, 41 6, 42
40, 43
6, 42
44
45
This work 46 41 46 41 45
6, 42
vitsl5 studied the aggregation of alkylammonium tetrahaloferrates in benzene by vapor pressure osmometry, but in this case they also had to invoke models other than monomer-n-mer associations to describe their data. In both investigations the self-association was considered to be ideal, no plots of M1/Mna vs. c were made, and a dipole-dipole interaction was felt to play an important role in their observed self-associations. Their solutes formed ion pairs (as does DAP) which were not appreciably dissociated by organic solvents. It was found that the self-association was greater at low t e m p e r a t u r e ~ , 5although ~ # ~ ~ the temperature effect was smaller for the alkylammonium tetrahaloferrates.51 Studies with related surfactants in nonaqueous solvents indicate that this temperature effect appears to be a general trend.40*41 A very elegant method was developed by Steiner for analyzing ideal, self-associations from osmometry.28 Unfortunately, Kertes and coworkers have overlooked Steiner's method, as well as Adams' methods which allow for the inclusion of nonideal behavior.7~15-21 It is apparent from the thermodynamic functions listed The Journal of Physlcal Chemistry, Vol. 79, No. 24, 1975
Adams et al.
2620
in Table IV that the association is exothermic since A H o is negative. The major contributor to the AGO values is the AHo term. Negative values of ASo reflect, in part, the increase in order due to the self-association. The fact that the observed self-association of DAP decreases with increasing temperature tends to rule out hydrophobic interactions as the predominant cause of the self-association, since hydrophobic interactions generally increase with increasing temp e r a t ~ r e however, ; ~ ~ ~ ~the ~ temperature effect on the selfassociation is consistent with a dipole-dipole type of interaction. Indeed l H NMR spectroscopic data and that obtained via a variety of other techniques indicates that “reversed” micelles of alkylammonium carboxylates are formed with the ionic moieties surrounding a relatively rigid interior cavity or “solvent pool” while the hydrocarbon chains protrude into the bulk nonaqueous solvent.2 The mobility of the hydrocarbon chains has been demonstrated both by the solvent penetration observed using l H NMR s p e c t r o ~ c o p yand ~ ~ by the 13C (FT-NMR) relaxation times, TI, of the carbon atoms of the dodecylammonium ion of DAP (those for C-1 are very short, those for the central methylenes C-4 to C-9are intermediate, and those for the methyl tail, C-10 to (2-12, are relatively quite long54). These data indicate that a dipole-dipole type of interaction, rather than hydrophobic ones, is the primary contributor to the self-association in the systems. Investigations of the association behavior of other small molecules and surfactants in nonaqueous solvents are currently being carried out in our laboratories in order to determine the generality of our conclusions. An alternative way of analyzing our experimental data is in terms of a nonspecific nonideality in the system, which is expressed through the activity coefficients of the solute via the Gibbs-Duhem equation. We have not pursued this alternative method, but Kertes and his coworkers have described this alternative procedure very lucidly and elegantly.5031
Acknowledgments. We are very grateful to Professor Kurt J. Irgolic for letting us use his vapor pressure osmometer. Many thanks are due to T. J. Vonderhaar and E. S. Flora for their computational help and to J. L. Armstrong, J. L. Sarquis, and Peter J. Wan for their interest and comments in this work. This work was supported by grants from the Robert A. Welch Foundation (A 485) and the National Science Foundation (GB 32242-Al). Eleanor J. Fendler is a Research Career Development Awardee of the National Institutes of Health, US. Public Health Service.
of subjectivity is, of course, introduced into the data in this way. Several quantitative methods of approximating the desired curve are available which eliminate this subjectivity and central among these is, perhaps, a mathematical spline function representation. Several types of spline approximants for the Adams function as well as for molecular weight distribution curves are possible. Our initial efforts were concerned with a natural smoothing spline having N knots placed at the N data points. Although thE representation so obtained is accurate to a high degree it is, in fact, too faithful in the sense that the resulting spline is “ripply” in nature reflecting slight experimental deviations from expected monotonic dependence on concentration. Moreover, the intrinsic smoothing of a natural spline basis is not able to readily remove these features which are clearly manifested in the derivatives of the function. Considerable success has been achieved, however, through the use of the following variable knot spline (VKS) technique. Variable Knot Splines. The considerable recent interest, in spline functions mentioned above is a direct result of their flexibility. One of the salient features of (piecewise polynomial) spline approximation is the ease with which one may compute a good least-squares approximant. Work by de BOO^^^^^^ has given rise to a numerically stable basis for computing and evaluating splines, this being a normalized B spline basis. In this work we have used the B-spline algorithms suggested by de Boor along with a descent algorithm to locate the (local) best knot positions in order to obtain a good least-squares fit of the empirical data. As minimization in the knots is a nonlinear problem we can guarantee only that a local minimum has been found as opposed to a “global best’’ knot placement. In the case of the Adams function, however, we are quite sure that a global minimum has been found as well. We say that a spline of order k (degree k - 1)is a piecewise polynomial of degree less than or equal to k - 1. An empirical function defined over an interval [a,b] may be approximated by a spline function which is specified by a sequence of real numbers, called knots, (ti].The sequence (ti) over the interval [a,b] is said to be a partition, n, of [a,b].Consider the partition n = (tik”’+lk for which the knots satisfy the following conditions: 1,. , N + k - 1
ti I ti+l
i=
ti+k>ti
k=l,...,N
(ii)
tk‘a
tN+1= b
(iii)
e
.
(i)
where a and b are fixed numbers. The [ t i ] are given for Appendix fixed k, the order of the spline function. The set of all such Spline Representation of the Adarns Function (cM1I partitions satisfying (i)-(iii) is denoted by P. Given a fixed Mna). Spline functions are currently of considerable intern E P the normalized B-spline basis for the piecewise est in the area of mathematical approximation t h e ~ r y . ~ ~polynomials ?~ of order k on the partition n is defined as the Spline representations are also beginning to attract interest set of functions INi,k},?= lwhere in the physical sciences as is apparent by the introduction of several important germinal papers in different Ni,k(t)= (ti+k - t i ) [ t i , .. . > ti+k]s(S - t ) : (AI) area~.56-5~ Ultracentrifugal and vapor pressure osmometry data The notation [ti, . . . , ti+k]s denotes a kth divided difference operator in the variable s. That is, for a given k and analyses involve the key intermediate step of piecing toknots (ti),the B spline is the kth divided difference of the gether and curve fitting empirical data obtained over a cerfunction (s - t)? - where tain range of concentration (see eq 7, 14, and 15). It has been common practice to draw a smooth curve through the data points with the aid of “ship curves”, one of a variety of commercially available draftmen’s splines. A certain degree
’,
The Journal of Physical Chemistry, Vol, 79, No. 24, 1975
2621
Self-Association of Dodecylammonium Propionate
For this specific set of spline functions, Ni,k(t), and given empirical data, yi, the spline representation sought is the solution of the following least-squares minimization problem: min
I2
j=l
[~(II;A; x j )
- yj12]
(A31
The spline function, s, depends on the partition of N coefficients Aj, and the data points x j ; i.e.
n, the set
N
s(n;2; I ) = j=l AjNj,k(.%) A
(A4)
( A i , . , . ,A N )
The solution of this problem is characterized by normal equations; however, the coefficient matrix is especially simple in that it is banded with bandwidth of 2k - 1;i.e., there are less than k - 1nonzero subdiagonals and k 1nonzero superdiagonals. These equations are solved very efficiently by a banded Choleski decomposition.61 The problem solved in order to spline approximate the Adams function is a minimization problem in the knots (ti E n) as well as in the linear parameters (A,]?= 1; i.e.
(9) P. J. Debye. J. Phys. ColloM Chem., 51, 18 (1974); Ann. N.Y. Aced. Sci., 51, 575 (1949); J. Phys. ColloMChem.,53, 1 (1949). (10) H.Q. Ellas, J. Macromol. Sci., Chem., A7, 601 (1973). (1 1) F. W. Blllmeyer, Jr., “Textbook of Polymer Science”. 2nd ed, Wlley-lnterscience, New York, N.Y., 1971, Chapter 38. (12) J. L. Armstrong, Appl. Polym. Symp., No. 8, 17 (1969). (13) P. 0.P. T’so. 1. S. Melvin, and A. C. Olson, J. Am. Chem. SOC.,85, 1289 (1963). (14) K. E. Van H o b , G. P. Rossettl, and R. D. Dyson, Ann. N. Y. Aced. Sci., 184, 279 (1969). (15) E. T. Adams. Jr.. Bhhemistfy, 4, 1655 (1965). (16) M. P. Tombs and A. R. Peacocke. “The Osmotic Pressure of Biological Macromolecules”, Clarendon Press, Oxford, 1974. pp 55-62. (17) P. W. Chun, S. J. Kim, J. D. Williams, W. T. Cope, L.H. Tang, and E. T. Adams, Jr., Biopolymers, 11, 197 (1972). (18) L. H. Tang and E. 1.Adams, Jr., Arch. Blochem. Biophys., 157, 520 (1973). (19) . . W. E. Ferauson, C. M. Smith, E. T. Adams. Jr., and G. H. Bariow, Biophy. Chem., 1,325 (1974). (20) J. L. Sarquls and E. T. Adams, Jr., Arch. Biochem. Biophys., 163, 442 I6 1974). . - . .I.
H. Fujlta, “Foundations of Ultracentrifugal Analysis”. Wlley-lnterscience. New York, N.Y.. 1975, pp 377-418. J. Van Dam in “Characterlzatlon of Macromolecular Structure”. Publication No. 1573, National Academy of Sciences, Washington, D.C., 1968, pp 336-342. J. G. Klrkwood and I. Oppenheim, “Chemlcal Thermodynamics”. McGraw-Hill, New York, N.Y.. 1961, pp 178-187. E. A. Collins. J. Bard, and F. W. Blllmeyer, Jr., “Experiments in Polymer Sclence”, Wiley-lntersclence, New York, N.Y., 1973. pp 131-135 and
-
374-379.
E. 1.Adams, Jr.. and J. W. Wllllams, J. Am. Chem. Soc., 86, 3454 (1964).
min ljI1
[s(II;A;x j ) - yj12:
E P,A E R N ] (A5)
The knots were repositioned -by a steepest descent method until a relative minimum for eq 5 with respect to the knots ti was reached.62 Thus, in this application a polynomial spline of order k (kth order B-spline basis) was fitted by least-squares to the experimental data points of an Adams function plot of A ( c ) vs. c . The order of the spline, k , was generally 3 or 4. The general form of A ( c ) is given by
A single knot was found to be sufficient because of the slowly varying nature of A ( c ) . This knot was moved throughout the range of the independent variable, c , according to the aforementioned steepest descent iterative search (the knot usually did not coincide with a data point). A statistical analysis accompanied each spline fit for a given knot position. A resulting minimum sum of squares deviation was typically found to be less than o(10-5).
References and Notes (1) Part of this material was presented by F. Y.-F. Lo, E. J. Fendler. and E. T. Adams. Jr., at the 29th Southwest Regional Meeting, Amerlcan Chemical Society, El Paso, Tex., Dec 5-7, 1973, Abstract No. 284. (2) J. H. Fendler and E. J. Fendler, ”Catalysis in Mlcellar and Macromolecular Systems”. Academic Press, New York. N.Y., 1975. (3) E. J. Fendler. S. A. Chang, J. H. Fendler, R. 1.Medary, 0. A. El Seoud.
and V. A. Woods In “Reaction Kinetics in Micelles”, E. H. Cordes, Ed., Plenum Press, New York. N.Y.. 1973, pp 127-145. (4) C. J. O’Connor, E. J. Fendler, and J. H. Fendler, J. Am. Chem. Soc., 95, 3273 (1973). (5) C. J. O’Connor, E. J. Fendler. and J. H. Fendler. J. Chem. Soc.. Da/ton Trans., 625 (1974). (6) J. H. Fendler, E. J. Fendler, R. 1.Medary. and V. A. Woods. J. Am. Chem. Soc.. 94.7288 11972\. (7) E. T. Adams, Jr.: 6acthns.No. 3(1967). (8) The self-association of some low molecular weight antihistaminic pyri-
dine derivatives In aqueous solution has recently been studied by light scattering. See D. Attwood and 0. K. Udeala, J. Phys. Chem., 79, 889 (1975).
i39i (40) (41) (42) (43)
An equation similar to eq 18 was developed by Kreuzer (see eq 12 of J. Kreuzer, 2.Phys. Chem., B53, 213 (1943)). R. F. Steiner, Arch. Blochem. Biophys., 39,333 (1952); 44, 120 (1953). R. F. Steiner, Arch. Biochem. Biophys., 40, 400 (1954). A. Kltahara, Bull. Chem. Sac. Jpn., 28, 234 (1955). 0. A. El Seoud, E. J. Fendier, and J. H. Fendler, J. Chem. SOC.,Faraday Trans. 1,70,450 (1974). J. H. Fendler, E. J. Fendler, R. T. Medary, and 0. A. El Seoud. J. Chem. Soc., Faradey Trans. 1, 69, 280 (1973). E. J. Fendler, J. H. Fendler. and R. T. Medary, unpublished results. R. C. Weast, Ed., ”Handbook of Chgmlstry and Physics”, The Chemical Rubber Co., Cleveland, Ohio, 1974. C. de Boor, J. Approx. Theory, 8, 50 (1972). W. J. Hemmerle. “Statistical Computations on a Digital Computer”, Blalsdell PublishingCo., Waltham. Mass., 1967, Chapter 2. L.-H. Tang, Ph.D. Dissertation, Illinois Institute of Technology, Chicago, Ill., Dec 1971. H.Q. Ellas and R. Bareiss, Chimk, 21, 53 (1967). K. E. Van Hokle and G. P. Rossetti. Biochem/stfy, 6,2189 (1967). N. Muller, J. RIYS. Chem.. 79. 287 (1975). F. M. Fowkes in “Solvent Properties of Surfactant Solutions”, K. Shlnoda. Ed.. Marcel Dekker, New York, N.Y., 1967. pp 65-115. A. Kitahara in “Cationlc Surfactants”, E. Jungermann, Ed., Marcel Dekker. New York, N.Y., 1970, pp 289-310. E. J. Fendler, J. H. Fendler, R. 1.Medary, and 0. A. El Seoud, J. Phys. Chem., 77, 1432 (1973). S. R. Pallt and V. Venkateswarlu, Roc. R. Soc. (London),Ser. A, 208,
542 (1951). (44) S. R. Palit and V. Venkateswarlu. J. Chem. Soc., 2129 (1954). (45) S. Muto, Y. Shlmazaki, and K. Meguro, J. ColloMlnterface Sci., 49, 173 (1974). (46) A. Kltahara, Bull. Chem. SOC.Jpn., 31, 288 (1958). (47) 0. Y. Markovlts, 0. Levy, and A. S. Kertes, J. ColloMlnterlace Sci., 47, 424 (1974). (48) 0. A. El Seoud, E. J. Fendler. J. H. Fendler. and R. 1.Medary, J. Phys. Chem., 77, 1876 (1973). (49) E. J. Fendler, V. 0. Consteln. and J. H. Fendler, J. Phys. Chem., 79, 917 (1975). (50) A. S. Kertes and 0. Markovits. J. Phys. Chem., 72, 4202 (1968). (51) A. S. Kertes, 0. Levy, and 0. Markovlts. J. Phys. Chem., 74, 3568 (1970). (52) H. A. Scheraga in “The Proteins”, 2nd ed. Vol. 1. H. Neurath, Ed.. Academic Press, New York, N.Y.. 1963. pp 515-53.541.578-579. (53) S. N. Tlmasheff In “Protldes of the Biological Flulds”, H. Peeters, Ed., Pergamon Press, Oxford, 1972, p 511. (54) E. J. Fendler and S. N. Rosenthal, Unpublished results.
(55) G. G. Lorentz, Ed., “Approximation Theory”, Academic Press, New York. N.Y.. 1973. (56) B. W. Shore, J. Chem. Phys.. 58. 3855 (1973). (57) T. L. Gilbert and P.J. Bertonclnl, J. Chem. Phys., 81, 3026 (1974). (58) S. Wold, J. Phys. Chem., 76, 369 (1972). (59) R. L. Klaus and H. C. Van Ness, AlCMJ., 13, 1132 (1967). (60) C. de Boor, Los Alamos Scientific Laboratory Report No. LA-4728-MS, Aug 1971. (61) T. J. Vonderhaar, M.S. Thesis, Texas A&M University, May 1975. (62) E. S. Flora, M.S. Thesis, Texas ABM Universlty, May 1975.
The Journal of Physical Chemistry, Vol. 79. No. 24, 1975
