4466
J . Phys. Chem. 1988, 92,4466-4474
Thermodynamic Analysis of Polydispersity in Ionic Micellar Systems and Its Effect on Small-Angle Neutron Scattering Data Treatment Eric Y. Shell*,+ and Sow-Hsin Chen Department of Nuclear Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: November 17, 1987: In Final Form: February 5, 1988)
We analyze small-angle neutron scattering (SANS) data of sodium dodecyl sulfate (SDS) and sodium bis(2-ethylhexyl) sulfosuccinate (AOT) ionic micellar solutions taking into account the effect of size polydispersity on the particle form factor. The intermicellar structure factor is computed by a generalized one-componentmacroion theory (GOCM) assuming a constant fractional charge. The model fittings to SANS data for different concentrationsallow us to extract the free energy parameters of micelle formation and growth, the size distribution function of micelles, and the minimum micelle size which is consistent with the fully stretched hydrocarbon tail lengths of the surfactant molecules. The critical micellar concentration (cmc) is predicted from the free energy parameters correctly. Combining the free energy of micelle formation with the double-layer free energy around the averaged micellar surface calculated by the nonlinear Poisson-Boltzmann equation, we obtain the hydrophobic free energy of micellization which is in quantitative agreement with the literature value. These analyses confirm the applicability of the ladder model of micellar growth in salt free ionic micellar solutions at moderate concentrations. The degree of polydispersity in size is about 11% for 2% SDS solution at 40 OC and 17% for 1% AOT at 22.6 'C.
I. Introduction Sodium dodecyl sulfate (SDS) and sodium bis(Zethylhexy1) sulfosuccinate (AOT) are two well-known anionic surfactants which form micelles in water as the surfactant concentrations exceed their respective critical micellar concentration (cmc) values, in the few millimole per liter range at room temperature. Previous investigations of SDS'q2 and analogous LDS (lithium dodecyl sulfate)24 micellar solutions, free of salt, generally concluded that small-angle neutron scattering (SANS) intensity distribution can be well-fitted by a model with an effectively monodisperse micellar form factor, P(Q),and a one-component macroion structure factor, S,(Q), using a suitably defined average size, a, as the hard core diameter. Thus, at each concentration one obtains essentially two model parameters: a weight-average aggregation number Nw and a fractional surface charge a of the m i ~ e l l e . ~In relating a to iVw,one needs to take into account the hydration number per polar head group, which is about 10 for LDS and SDS and 20 for AOT micelles. These numbers do not change significantly with con~entration.~,~ This effective monodisperse model, although capable of yielding very good fits to SANS data, must be of suspected from several physical grounds. First of all, it can be shown from a general statistical thermodynamical argument* that a rapid change of NW with concentration is an evidence for a large distribution of micellar sizes at a given concentration. For example, the RW of SDS at 1.25 g/dL is 90 at T = 23 "C, and as the concentration doubles it becomes 100, 115, 127, and finally 149 at 20 g/dL.* Kotlarchyk and Chen9 have attempted to estimate the size polydispersity of a 8 g/dL LDS micellar system at 37 OC using a Schultz distribution of micellar sizes. They obtained Aula of 13%. In the absence of a known distribution function of micellar sizes this rather low estimated polydispersity was used to justify the use of the effective monodisperse model. However, within the past 2 years new experimental and theoretical evidences were produced to indicate that the micellar size fluctuation in moderate dilute solutions may be appreciable. Cabane, Duplessix, and Zemblo have made an asymptotic analysis of large Q SANS data (Q is the magnitude of the scattering vector) from SDS micelles at 2 g/dL. Their results showed that, with choices of different size distributions, the root-mean-square deviation is generally about 30% of the mean size. In two separate computer simulations of micellar formation by Owenson and Prattl' and by Jonsson, Edholm, and Teleman,'* both groups 'Present address: Exxon Research & Engineering Co., Annandale, NJ 08801,
0022-3654/88/2092-4466$0 1.50/0
reported observations of considerable fluctuations in the aggregation sizes. These simulations, however, have not yet reached a completely realistic stage because of the computational limitations and also because of the uncertainty in the potential functions between surfactant monomers and between water molecules. Thus, there seems to be sufficient reason for the investigation of the polydispersity effects on SANS data. Our motivation for this paper was provided by an experience that polydispersity effects of a zwitterionic micellar system made from short-chain lecithins in aqueous solution can be well-described by a ladder model of micellar growth.I3J4 The ladder model was originally used to describe the size distribution of lecithin micelles for interpretation of a static light scattering data by Tausk and O ~ e r b e e k . ' ~The model was later elaborated and applied to analysis of dynamic light scattering data from SDS solutions with 0.8 M NaCl by Mazer et a1.16 and more extensitively by Missel et al." It needs to be cautioned here, however, that the size distribution of SDS in 0.8 M salt solution is drastically different from SDS in salt-free aqueous solutions regardless of the concentration of the surfactant. Therefore, model parameters deduced from ref 16 and 17 cannot be directly compared with the results of analyses obtained in this paper. One compelling reason for (1) Hayter, J. B.; Penfold, J. J. Chem. SOC.,Faraday Trans. I 1981, 77, 1851. (2) Sheu, E. Y.; Wu, C. F.; Chen, S. H.; Blum, L. Phys. Reu. A 1985, 32, 3807. (3) Bendedouch, D.; Chen, S. H.; Koehler, W. C. J . Phys. Chem. 1983, 87, 153. (4) Bendedouch, D.; Chen, S. H.; Koehler, W. C. J . Phys. Chem. 1983, 87, 2621. ( 5 ) Chao, Y. S.;Sheu, E. Y.; Chen, S.H. J . Phys. Chem. 1985,89,4862. (6) Sheu, E. Y.; Wu, C. F.; Chen, S. H. J . Phys. Chem. 1986,90, 4179. ( 7 ) Chen, S. H.:Dill, K . Nature (London) 1985, 314, 385. (8) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 2 1976, 72, 1525. (9) Kotlarchyk, M.; Chen, S. H. J. Chem. Phys. 1983, 79, 2461. (10) Cabane, B.; Duplessix, R.; Zemb, T. J . Phys. 1986, 46, 2161. (11) Owenson, B.;Pratt, R.J . Phys. Chem. 1984, 88, 2905. (12) Jonsson, B.; Edholm, 0.;Teleman, 0.J. Chem. Phys. 1986,85,2259. (13) Lin, T. L.; Chen, S. H.; Gabriel, N. E.; Roberts, M. F. J. Phys. Chem. 1987, 91, 406. (14) Lin, T. L.; Chen, S.H.; Roberts, M. F. J . Am. Chem. SOC.1987,109, 232. (15) Tausk, R.J. M.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1976, 2, 379. (16) Mazer, N. A.; Carey, M. C.; Benedek, G. B. In Micellization, Solubilization, and Microemulsion; Mittal, K. L., Ed.; Plenum: New York, 1977. (17) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980, 84, 1044.
0 1988 American Chemical Society
Polydispersity in Ionic Micellar Systems initiating the following analysis came from a recent realization that the effective monodisperse model described before, in which the fully stretched hydrocarbon tail length of SDS was taken to be 16.7 8, according to Tanford,18seems to @vea poor result when used to analyze the very low concentration data (99.8%. In preparing samples, we made the solution of highest concentration of each series first and then made the rest of that series by dilution of this stock solution. (18) Tanford, C. The Hydrophobic Effect, 2nd ed.; Wiley: New York, 1980. (19) Onsager, L. Ann. N.Y. Acad. Sci. 1949, 51, 621. (20) Ben-Shaul, A,; Gelbart, W. M. J . Phys. Chem. 1982, 86, 316. (21) Chen, S. H.; Lin, T. L.; Wu, C. F. Presented at the Conference on Physics of Amphiphilic Layer, Les Houches, France, Feb 9-19, 1987. (22) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. J. Chem. Phys. 1986, 85, 7268. (23) Linkowsky, G., American Cyanamid Co., Linden, NJ, private communication. Williams, E. F.; Woodberry, N. T.; Dixon, J. K. J . Colloid Interface Sei. 1957, 12, 452.
The Journal of Physical Chemistry, Vol. 92, No. 15, I988 4467
B. SANS Measurement. SANS experiments were performed at the Low Angle Biology spectrometer in Brookhaven National Laboratory with neutron wavelength X fixed at 5.32 8,. Neutron flux at this wavelength was 1.12 X lo6 neutrons/(cm2 s). The detector used was an area detector of 64 cm X 64 cm with 128 X 128 pixels. The sample-to-detector distan_cewas chosen to be such that the magnitude of scattering vector lQl = (4n/X)sin ( 8 / 2 ) (8 is the scattering angle) ranges from 0.02 to 0.3 8,-1.24 Duration of data collection varies from sample to sample depending on the surfactant concentration and micelle-solvent contrast. It was chosen such that the intensity distribution function, Z(Q) (differential cross section per unit volume of the sample in iinits of cm-I), has statistical errors less than 1% for all Q. 111. Ladder Model and the Particle Form Factor
The intensity distribution function, Z(Q), for SANS consists of, aside from a factor representing number density of particles in the system, a product of two-dependent factors: the particle structure factor, which is mainly determined by particle size and shape, and the interparticle structure factor, which is a function depending on the interparticle correlations. Since the micellar sizes are considered to be polydisperse with a size distribution according to a ladder model in this study, we shall describe the ladder model for micellar growth first and then formulaic the procedure for calculating the average particle form factor ( P ( Q ) ) which is to be used for SANS data analysis. A. Thermodynamic Theory of Micellization. Let p N and p l be the chemical potential (free energy per particle) of a micelle of an aggregation number N a n d of a free monomer, respectively. Then the thermodynamic condition for a chemical equilibrium between the micelle and the monomer requires that pN
= Np1
(1)
In general, both p N and p 1 can be written as the sum of the standard chemical potential, a term representing the entropy of dispersion, and an interaction term. If A ' , and XI denote the mole fraction of a N-mer and a monomer, respectively, thenI8 p N = pN0 kBT In XN interaction term (2)
+ + p1 = p l o + kBT In XI + interaction term
(3)
Since the micellar size distribution, which is to be used for calculation of ( P ( Q )), is insensitive to interparticle interactions at least to the first order in the excluded-volume interaction for cylindrical particles as argued by O n ~ a g e r ,Ben-Shaul,20 '~ Chen et a1.,21and Blankschtein et a1.,22the interaction terms in eq 2 and 3 can then be dropped. We therefore substitute eq 2 and 3 into eq 1 to get
(4) Equation 4 shows that the probability of forming a micelle of aggregation number N is proportional to which is the probability of having N monomers at the same spatial location, and a Boltzmann factor exp[(pNO- Np1')/kBTl which represents the net energy gain of assembling N free monomers in the solvent to form a micelle. Thus, ( p N o - NpIo)/NkBT is a sum of the hydrophobic free energy gain and the double-layer repulsive energy loss per monomer in forming a micelle of aggregation number N. B. Ladder Model. This model starts from eq 4 and makes two basic assumptions: (1) the aggregation number of a micelle can only be greater than or equal to a minimum number No, and (2) the free energy of micellization takes a form of a first step A followed with a constant ladder spacing 6 according to the relation NO - Nplo = A + ( N - NO)^, N 1 No (5) In eq 5 A = pLNoo- N,,pIois the free energy advantage for forming a minimum micelle of size Noand 6 denotes the energy advantage of inserting an additional monomer into a micelle of size equal (24) Schneider, D. K.; Scheonborn, B. P. In Neurron in Biology; Schoenborn, B . P., Ed.; Plenum: New York, 1984; p 119.
4468
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
to or greater than No. In the ladder model both A and 6 are assumed to be independent of the surfactant concentration for a given micellar system. Clearly, this is so for A, because formation of the minimum micelle depends only on the chemical structure of the surfactant molecule, the type of counterion it has, and the solvent. It is not obvious, however, that 6 needs to be independent of the concentration. As a phenomenologicalconstant, it can vary with cbricentration because as micelle size distribution broadens the miaellar shape changes. S is a function of the head area per monomer aH,which changes when a monomer is inserted into a nonstraight section of the micelle. It is therefore expected that 6 may vary slightly as a function of concentration. However, it is normally assumed to be a constadt in the ladder model for the simplicity of carrying out the sum in eq 7 . From these assumptions one can readily see that the necessary condition for micelles to grow is that 161 > lA/NoI. Applying eq 5 to eq 4, one obtains
where P = X, exp(-6/kBT) = X l / X Bhas a value between 0 and 1.I4 Equation 6 shows that the micellar sizes follow an exponential distribution with a decay constant 111.1 PI-'. Thus, @ is a parameter that characterizes the micellar size distribution. When P approaches unity, the size distribution broadens. In order to relate P to the total surfactant molar fraction X, we impose a material conservation condition
which can be summed to obtain
Sheu and Chen I
0.75
2
E n
I
I
I
VOLUME PMIDISPERSITY
DOT: N0.15
Eio
m.
-
I
I
I
INDEX
VS BETA
I
SOLID: M=25 DASH: N0.42
r
I
-
iI ,
0.5
> J 0 a id
3
0.25
$
.... ........... .. . ... ........ . .....
__--______----
0.0
0.5
0.75
1.0
LADDER MODEL PARAMETER BETA
Figure 1. Theoreticalcalculation of volume polydispersity as a function of the ladder model parameter B according to eq 12 for No = 15,25, and 42. One can see that the system remains fairly monodisperse until p is very close to unity, particularly, in the case of large No.
and XBdefines the scale of monomer concentration at different surfactant concentrations. We define the volume polydispersity by an index
- -
(12) One can see that P, 1 as P 1 and decreases rapidly as /3 decreases. This indicates that system remains fairly monodispersed until B is very close to unity. Figure 1 shows the theoretical calculations of P, as a function of /3 for No = 15, 25, and 42. According to Tanford,I8 the cmc can be conveniently defined by the following relation
Xcmc = 1.05X1,,,
(13)
where XCmcis the molar fraction of the surfactant in solution at cmc and X I , , is the molar fraction of the free monomers at cmc. Substituting eq 13 into eq 8 gives
or
Equation 8 is the central equation of the ladder model, from which all the useful relations to be used for SANS data treatment are derived. These relations include the weight-average aggregation number iVw
=No+-
@
1-P
[,+
No(1 - P) + P
the number-averaged aggregation number, iVn m
Nn= N = N o Nx,/
m
x XN = ( N )
N=No
and the relation between (Nw - No) and ( X -
( X - X1)1/2
-
= 2K1/2 P 1 (11) It is easy to see from eq 8-1 1 that in order to determine @, and hence the size distribution of the micelles, three basic parameters should be specified, namely, the minimum micelle size, No, and two Boltzmann factors, K and X B . No and K control the degree of polydispersity as concentration of surfactant molecules increases,
By solving the implicit equation, eq 14, together with the condition, eq 13, one can obtain the cmc value within the ladder model providing K , X B ,and No are known. We shall show, in section VI, that the cmc value can be operationally defined by locating the sharp bending point of a P = X , / X versus XfXe plot according to eq 8 and that the ratio of 1.05 in eq 13 used by Tanford can be approximately justified in terms of this more reasonable definition of cmc, with SDS and AOT as examples. C. The Particle Structure Factor in the Ladder Model. For the monodisperse system the intensity distribution function, Z(Q), obtained in a S A N S experiment can be expressed as I(Q) = (C - Cl)N(Cbi
- Umps)'PdQ) S d Q )
=A Q ") SNCQ) (15) In eq 15 C is the total surfactant concentration, C, is the free monomer concentration in units of number of monomers per unit volume. N is the aggregation number of micelles, 7 3 , is the total scattering lengths of a surfactant molecule, u, is the dry volume of a monomer, ps is_the scattering length density of the solvent (in this case D20), P d Q ) is the normalized (i.?., normalized to unity a t Q = 0) particle structure factor, and SN(Q) is the orientationally averaged interparticle structure factor (in cases where micelles are not spherical). If the system is polydisperse and the size distribution is describable by the ladder model, then
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4469
Polydispersity in Ionic Micellar Systems
where b is taken to be the fully extended tail length of the surfactant molecule. The volume of the hydrophilic layer, VSh, can be calculated by
V,, = (4r/3)[(a
+ t ) ( b + t ) z - abz]
(24)
where t is computed by assuming the hydrophilic head of a surfactant to be spherical and that t is equal to its diameter, i.e.
t
From V,, we can estimate the hydration number (number of water molecules associated with each surfactant polar head group)z5
I
where CY is the fractional charge of the micelle, u+ is the hydrated volume of the counterion calculated by assuming the hydrated diameter of Na+ to be 2.58 and V, is the volume of a solvent molecule cVs = 30.27 A3 for DzO).Using eq 29-26, we can calculate P d Q ) and all the relevant parameters such as t , a, and N, once u,, ut, and mware known. It should be noted that we calculate p N ( Q )according to a spheroidal model which was found to be the case for both SDS and AOT micelle~?3~*~' However, the ladder model was originally formulated for spherocylindrical micelles only, because the, 6 parameter is assumed to be a constant in the ladder model and the minimum size micelle is taken to be spherical in shape. This model depicts the micelle growing from spherical to spherocylindrical with the added monomer going into the straight section. In our adaptation of the ladder model for spheroidal micelles such as SDS and AOT micelles, we argue that all the formulas in the ladder model may still be applied and that the deviation from a nonspherocylinder may only result in a renormalization of the ladder model parameter 6. We shall show that this is indeed the case from our SANS data analysis.
and ( s ( Q )) is the size and orientationally averaged interparticle structure factor to be written down in the following section. Equation 18 is the equation to be used for computing the particle form factor in SANS data analysis. To compute FdQ), we model the micelle as a particle spheroidal in shape having two distinctive regions: a hydrophobic compact core containing the hydrocarbon tails of the surfactants and a hydrophilic outer layer containing polar heads of the surfactants a n t t h e associated water molecules (see Figure 2). To calculate J"(Q), we adapt a form given previously for sodium dodecyl o-xylenesulfonate micellesz5
where p is the directional cosine between the 0 vector and a vector defining the symmetry axis of the spheroid,j is the spherical Bessel function of the first kind, and
The parameters contained in eq 20-22 are a, the semimajor axis, and b, the semiminor axis of the hydrocarbon core; t , the thickness of the hydrophilic layer; ut the tail volume of a surfactant molecule in the core; pc, the scattering length density of the core; and ,oh, the scattering length density of the polar head layer. Since the micellar core is assumed to be dry and compact, we can calculate the core volume of a micelle of aggregation number N according to Vc = (4r/3)abz = Nu,
(23)
(25) Sheu, E. Y.; Chen, S. H.; Huang, J. S. J . Phys. Chem. 1987,91, 1535.
IV. Interparticle Structure Factor As stated in section I, the interparticle interaction has a negligible effect on particle size distribution at least to the first order in the interaction. Another even more plausible reason is that the hydrophobic energy of micellization per monomer for SDS is about 10kBTwhile the intermicellar double-layer interaction energy per monomer is much less than kBT. Thus, the chemical equilibrium process between the monomers and micelles in solution would not be affected to any appreciable degree by the presence of micellar-micellar interaction. Even with fairly strong interparticle interactions such as SDS and AOT systems treated here, we therefore expect that this interaction effect, together with the nonspherocylindrical effect, would not cause appreciable deviation of the particle size distribution from that of the ladder model. However, the intermicellar double-layer interaction, although having a negligible effect on the particle size distribution, has to be computed as facas the calculation of the average interparticle structure factor (S(Q)) is concerned. The scattering intensity distribution function shown in eq 17 consists of a known factor Io, the average particle form factor ( F d Q ) ) ,which can be calculated according to eq 18-26, and the average interparticle structure factor ( S ( Q ) ) . For a system of polydisperse nonspherical particles it has been shown by Chen and S h e P that the average interparticle structure factor defined in eq 17 has the following form
L '
I
i
(26) Triolo, R.; Grigera, J. R.; Blum, L. J . Phys. Chem. 1976, 80,1858. (27) Sheu, E. Y.; Chen, S . H.; Huang, J. S . J . Phys. Chem. 1987,91,3306. (28) Chen, S . H.; Sheu, E. Y. Presented at 10th Discussion Conference on Small Angle Scattering and Related Method, Praque, July 13-16, 1987.
4470 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
where Vis the total volume containing M particles and the particle size indexes i and j run from 1 to s. There are Mi particles of size i , so that the total number of particles M is
Sheu and Chen
Z = cosh ( k / 2 ) + V [ ( k / 2 ) cosh ( k / 2 ) - sinh ( k / 2 ) ]
v = V / ( k / a 3- Y/(k/2) L‘
S
M =
-
CM,
=
ij
-
F,(Q) and Fi(Q)2 are respectively the orientationally averaged particle form factor and square of particle form factor. Equation 27 is a slight extension of an expression given originally by F o ~ r n e for t ~ polydisperse ~ spherical particles. A rigorous calculation of ( S ( Q ) ) is very difficult except for spherical particles. Van Beurten and Vrij30 showed that for a multicomponent hard sphere system having Schultz distribution of sizes the average interparticle structure factor can be well-approximated by an effective one-component structure factor in the neighborhood of the first diffraction peak provided that the polydispersity is less than 10%. The effective onecomponent hard sphere fluid should have a hard sphere diameter u chosen as
Recently, Senatore and Blum31 solved the multicomponent Ornstein-Zernike equation for a charged hard sphere system having the Schultz size distribution together with their neutralizing counterions in a mean spherical approximation. Their calculation of_( S ( Q ) ) showed again that, near the first diffraction peak, ( S ( Q ) )is very close to that of an equivalent one-component system provided the size distribution does not exceed 15%; Thus, we consider it an adequate approximation to replace ( S ( Q ) ) in eq 17 by an equivalent one-component structure factor, denoted by SGmM(Q). This structure factor is to be calculated by a generalized one-component macroion theory to be described below. A micellar solution can be regarded as a two-component charge fluid consisting of macroions and counterions in a solvent having a dielectric constant e. A proper description of particle correlations in this fluid can be based on a set of two-component coupled Ornstein-Zernike (OZ) equations.32 This two-component description can be contracted by eliminating the counterion pair correlations functions from the set of equations and arrive at the following equation for an effective one-component m a c r ~ i o n ~ ~
+ ( 6 / ? i ) + J c ( x ) h ( l x - XI)d3x’
h ( x ) = c(x)
(30)
where x’ = ?/a and 4 is the volume fraction of micelles in the solution. It can be shown34v3S that under assumptions of pointlike counterions and the mean spherical approximation (multicomponent version), the direct correlation c ( x ) outside the core, u, is given by
4 ~=)- ( ~ / ~ B T ) V G D L V O x( ~>) ,1
(31)
In addition, a boundary condition inside the core
h ( x ) = -1,
x I 1
(32)
is imposed. The generalized DLVO potential defined by eq 3 1 has a form (33)
where k = K X and counterions.
K
is the Debye screening constant of the
(29) Guinier, A.; Fournet, G. Small Angle Scattering of X-Ray; Wiley: New York, 1955; p 25. (30) Van Beurten, P.; Vrij, A. J . Chem. Phys. 1981, 7 4 , 2744. (31) Senatore, G.; Blum, L. J . Phys. Chem. 1985, 89, 2726. (32) Ornstein, L. S.; Zernike, F. Proc. Ned. Akad. Sci. 1914, 17, 793. (33) Beraford-Smith, B.; Chan, D. Y. C. Chem. Phys. Lett. 1982, 92,474. (34) Belloni, L. J . Chem. Phys. 1986, 85, 519. (35) Sheu, E. Y. Thesis, Massachusetts Institute of Technology, 1987.
(34)
(35)
= 34/(1 - 4)
(36)
(ru+ V ) / ( i + ra + v)
(37)
and F a is the solution of the an implicit equation
4 ( r u ) 2 = ( k / 2 ) 2 + (24e2z024/u&BT)(1+ ru + v ) ~ ( 3 8 ) It is important to note that, in the limit can be shown to be, using eq 34-38
I$
-
0, the factor Z
ekl2
elimo 2 = 1 + k/2
(39)
which, when substituted into eq 3 1, reproduces the well-known asymptotic form of the DLVO potential given in ref 36. In the one-component macroion theory one interprets the DLVO potential as the direct correlation function outside of the core. This is consistent with the Verway and Overbeek view that the DLVO double-layer interaction potential is a free energy of bringing two macroions from infinity to a finite distance r, which is a statistical mechanical quantity, depending on temperature. Since one knows c ( x ) outside the core, and h ( x ) inside the core, and hence c(x) inside the core also, one can Fourier invert c(x) to obtain S G ~ M ( Qby ) the relation SGoCM(H)
1 H = Qu = 1 - (6/7r)@C(X)’
(40)
The essential improvement of the GOCM approach is that the volume fraction, 4, of the macroions is involved in VGD,vo. This inclusion comes from the fact that the effect of the surrounding macroions on the effective pair potential between the two designated macroions has been properly taken into account in the derivation of VGDLVO. Derivation of the conventional DLVO potential includes only the bare interaction of the two macroions and their counterions. Actually, both VGDLVo and VDLvo have Yukawa form and the only difference is the prefactor. Thus, when one uses GOCM to analyze SANS data, the quality of the fit would be about the same as that of OCM. However, the fractional charge of the micelle, a, extracted through the prefactor would be different. In general, a obtained by GOCM is less than that by OCM and the two coincide in the dilute limit. This difference comes from the neglect of the screening effect of the surrounding macroions on the given pair of macroions in the OCM theory, and it can be substantial ,at high concentration. For the calculation of ( S ( Q ) )three inputs are needed, namely, the effective micellar size u, the micellar charge zo (or the fractional micellar charge a ) , and the Debye screening constant K . V. Data Analysis In this section we summarize the SANS data analysis procedure in which_theladder model is adapted to calculate the particle form factor ( P N ( Q ) )and the GOCM is used for the calculation of SGOCM In the course of SANS data analysis eq 17 is used to compute I ( Q ) for comparison with the experimental data. To calculate Z(Q) from this equation, several parameters such as No,B, a , A, 6, u,, and ut have to be known. However, if one uses all these parameters as free parameters to fit a SANS intensity distribution, the extracted parameter values would be nonunique. In order to reduce the number of free parameters so to avoid the nonuniqueness, we used the following self-consistent fitting procedure which involves only two free parameters in fitting a SANS intensity distribution: (1) Choose a minimum micellar aggregation number No and calculate the corresponding hydrophobic micellar core radius, which is essentially the fully extended hydrocarbon tail length of
(ea).
(36) Verway, E. J.; Overbeek, J . Th. G. Theory of the Srabiliry of Lyophobic Colloids;Elsevier: New York, 1948.
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4411
Polydispersity in Ionic Micellar Systems
TABLE I: Extracted Parameters from SANS AMI~WS" concn, % j3 XI,M N, N, a, A
a -8/kBT SDS ( T = 40 "C)
0.5 1.0 2.0 4.0 8.0 16.0
0.8959 0.9217 0.9522 0.9680 0.9745 0.9795
0.00795 0.00794 0.00792 0.00785 0.00771 0.00731
52.2 56.6 68.7 85.4 98.8 115.6
50.6 53.8 61.9 72.3 80.2 89.7
40.97 42.04 43.64 45.52 46.99 48.63
0.4 0.5 0.6 0.8 1.0
0.8951 0.9068 0.9165 0.9302 0.9378
0.001 39 0.001 40 0.00140 0.001 40 0.00140
27.0 29.0 31.0 35.1 38.2
23.5 25.6 26.8 29.0 30.9
38.28 38.65 39.14 40.04 40.66
0.235 0.256 0.254 0.242 0.194 0.140
-AINkBT
8.7410 8.7596 8.7868 8.7979 8.7943 8.7890
&nono)
K(Y
P,
N,
8.50 8.50 8.50 8.50 8.50 8.50
54.1 536 64b 736 82b 926
1.276 1.485 1.700 2.29 2.81 3.37
0.21 0.23 0.33 0.42 0.48 0.54
10.75 10.60 9.85 9.20 8.97 8.76
9.63 9.63 9.63 9.63 9.63
24.3 25.6 26.8 29.0 30.9
0.402 0.480 0.534 0.622 0.707
0.38 0.41 0.44 0.49 0.52
21.0 20.3 19.8 19.2 18.8
AOT (T = 22.6 "C) 0.225 0.253 0.254 0.247 0.248
10.4721 10.4787 10.4863 10.4976 10.5015
'M,(SDS) = 288, V, = 350 A' (SDS), V, = 405 A3, No = 42.0, cmc = 8.01 mM (this work), cmc (ref 36, H20)= 8.6 mM. MJAOT) = 444.5, V,= 546 A' (SDS),V, = 612 A', No = 15.0. cmc = 1.40 mM (this work), cmc (ref 22, H20)= 1.61 mM. bThe tail length of SDS surfactant was taken to be 16.7 A.
the surfactant molecule, by assuming the minimum micelle to be spherical and the micellar core to be compact, Le. (4~/3)1: = NOV,
(41)
where 1, is the full extended length of the surfactant tail and ut is taken to be 350 for a SDS molecule3and 544 A3 for an AOT molecule.27 (2) Use j3 defined in eq 6 and the fractional charge CY as free parameters to calculate Z(Q) and fit it to the experimental data for the cases of concentrated micellar solutions in which the molar fraction of the free monomers is negligible when compared to the total molar fraction of the surfactant molecules. With this approximation, Z(Q) for concentrated cases can be computed from eq 15 because C is known, C1can be neglected when solution is concentrated, RW can be computed according eq 9, bi of SDS and AOT can be calculated, u, is taken to be 405 for SDS and 612 A3for AOT,3*27 and ps for D 2 0 has been known to be 6.38 X lo4 A-2.27 The particle form factor, (PAQ)),can then be calculated according 9 18 where FAQ) was computed by use of eq 19-22. As for (S(Q)),we computed it by using eq 40. (3) The extracted CY and 0values from (2) were then used to calculate the ladder model parameter K via eq 11 where X , is assumed to be negligible when surfactant concentration is high. This approximation should be fairly good when SDS concentration is greater than 2% because the cmc of SDS at T = 40 OC is about 8.4 mM.37 For AOT, the approximation is good when concentration is greater than 0.5%. (cmc of AOT was reported by Cyanamid Co. to be about 0.05-0.07%.23) (4) According to the ladder model K should be independent of solution concentration. Thus, if K values obtained from different concentrations differ within 3%, then go to the next step; otherwise, select another No and repeat steps 1 through 3 until the K values obtained for all the concentrated cases have a difference less than 3%. (5) Use the K value obtained in (3), No,and 0extracted from (2) to calculate XI via eq 8 for the concentrated cases. ( 6 ) Compute 6 by using /3 obtained in (2) and X , obtained in (5) according to the definition of /3 (eq 6 ) . (7) Calculate A by K obtained in (3) and 6 obtained in ( 6 ) by using the definition of K (eq 6 ) . (8) Fix A and No obtained from fitting concentrated cases and use CY and 6 as free parameters to fit the S A N S intensity distribution function Z(Q) of all concentrations. In this fitting @was calculated by use of eq 8 since A, No, and X are given and 6 is a fitting parameter. In these eight steps of data analysis procedure one can see that only two free parameters, a and 6, are used in fitting each SANS data set, but we obtained all the physical parameters that describe the system. These parameters include the basic parameters A, 6, /3, a,and No and the derived parameters Nw, Nn, and the index
K
-
1.
*
5g
5.5
5.5
E
(37) Goddard, E. D.; Jones, T.G . Res. Corresp. 1955, 8(8), A l .
SQUARE : 2 Z SDS IN 020 TRIANGLE : 1% SDS IN D20 DIAMOND : 0.5% SDS IN 020 SOLID : FITTED CURVES
10.
1
0 IAtt-11
I
'
'
~
(
I I I I I SQUARE : 16% SDS IN 020 TRIANGLE: 8% SDS IN 020 DIAMOND : LtX SDS IN 020 SOLID : FITTED CURVES
I
I
i
Q tPltt-1 I
Figure 3. SANS intensity distributions for SDS micellar solutions of various surfactant concentrations (symbols) and the corresponding fitted
curves (solid lines) using the ladder model. of polydispersity Pv (by eq 12).
VI. Results and Discussion Figure 3 shows the results of fitting of SANS data from SDS micellar solutions of various concentration by using the ladder model for calculation of the particle structure factor and the eight-step fitting procedure described in section V. As one can see, the fittings are excellent for all concentrations. The minimum micellar aggregation number was found to be 42, which corresponds to a hydrophobic tail length of 15.19 A. This value is consistent with the fully extended tail length obtained by summing up all the bond-to-bond distances (15.3 A)38with the C-C-C bond angle taken to be a tetrahedral angle, Le., 109.4'. The extracted micellar and thermodynamical parameters are tabulated in Table I. Figure 4a gives the best fit for 0.5% SDS micellar solution (38) March, J. Advanced Organic Chemistry; Reaction, Mechanisms, and Structure; McGraw-Hill: New York, 1968; p 22.
4472 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 0.5 % SDS IN 020 AT T = 40 C
0.2
a
QUUARE: DATA SOLID: POLYDISPERSE FIT (No = 921 DASH: MONODISPERSE FIT (No = 531
Sheu and Chen
-
MICELLAR SIZE DISTRIBUTIIONS SOLID: 2X SDS [SIZE POLYDISPERSITY = 119.1 DASH : 1% AOT [SIZE POLYDISPERSITY = 17x1
-
-
i i 3
0.0
AGGREGATION NUMBER 0.2
0.5 Z SDS IN D20 AT T
90 C
SQUARE: DATA SOLID: POLYDISPERSE F I T [No = 421 DASH: MONODISPERSE F I T [No = W1
-.-.
0.9
m W
-
*
-
;; F-
e.
6 -
Figure 6. The micellar size distribution function (in terms of aggregation number) for 2% SDS (solid line) and 1% AOT (dash line) micellar solutions.
0.1
2 W
I - / L
0.8
/
J
SDS
f
D
[L
c(
a
i:
0.7
Ll 0
0.0
W [L
0.1
0.2
3
0.6
-I
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Figure 4. (a) Comparison of the ladder model fitting (No = 42) for 0.5% SDS micellar solution at 7' = 40 OC to that of the monodisperse model with It taken to be 16.7 A (Le., No = 53) according to Tanford's empirical formula.I8 (b) Same as (a) but using I, = 15.3 (Le., No = 42) in the monodisperse analysis.
0.5 0.5
I.
1.5
2.
X/XB
AOT DIAMOND CROSS F-SQUARE SOLID
-x x
IT
--
: : : :
0.6% AOT IN D20 0.5% AOT IN 020 0.4% AOT IN 020
FITTED CURVES
rt W W
5w 2
0.7
0.25
0 H
X/XB
0.0
Q IAXt-I1
Figure 5. Ladder model fits for the case of AOT micellar solutions. using the monodisperse model with surfactant tail length taken to be 16.7 A (or No = 53) calculated according to Tanford's empirical formula.'* The polydispersity model fit with No = 42 is also shown. It is obvious that the monodispersity fit is too poor to be accepted. In Figure 4b we give a monodispersity fit of 0.5% SDS case but with No = 42. The quality of the fit becomes reasonable. This indicates that Tanford's empirical formula for estimating surfactant tail length has to be used with caution. For the 0.5% solution the polydispersity fit suggests that the system is only slightly polydispersed (Pv= 18%, or 6% in size), and a reasonable fit should be obtained when the monodisperse model is used. Figure 5 shows the polydis rsity fits of AOT series. No was found to be 15 (or 1, = 12.57 which is, again, consistent with the value obtained by summing the bond-to-bond distances. The extracted parameters are also tabulated in Table I. Figure 6 gicss the size distribution of 2% SDS solution at T = 40 O C and that of 1% AOT at T = 22.6 O C . The polydispersity of the micellar size, Ps Pv/3, was found to be 11% for 2%SDS solution at T = 40 OC and 17% for 1% AOT at T = 22.6 O C . In Figure
6,
Figure 7. (a) Plots of ladder model parameter p = X,/XBversus X/XB for selected SDS micellar solutions. The cmc value is taken to be the X value at which B makes the transition from one asymptotic behavior to the other. The cmc so estimated is quantitatively consistent with those obtained via conductivity measurement. (b) Same as (a) for the case of AOT.
7 we illustrate the determination of cmc by plotting @ vs x/xB according to eq 8 using the extracted 6, A, and No parameters from SANS data analyses. The cmc value can be directly read from this curve by taking the x,,,/xB value where the curve switches from one asymptotic form to another (see Figure 7 ) . In principle, the cmc values determined from parameters obtained from different concentrations should be the same if the ladder model describes the system. However, the systems we are dealing with here have two features which do not coincide with the ladder model assumptions, namely, a strong interparticle interaction and a nonspherocylindrical shape. We have allowed for these two effects by taking the ladder model parameter, 6, as a free parameter in the fitting process. The 6 so obtained would have a renormalized value and would not have the same value for all concentrations. Due to this renormalization of 6, the cmc values predicted through eq 8 are different for different concentrations.
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4473
Polydispersity in Ionic Micellar Systems The deviation is less than 5% for the SDS case and less than 2% for the AOT case. We thus use 8.01 m M taken from the 0.5% case for SDS at T = 40 O C and 1.4 m M from the 0.4%case for AOT at T = 22.6 OC. We can compare cmc values predicted from our model to that determined by a conductivity measurement, which is 8.6 mM for SDS in H 2 0 at T = 40 0C.37The difference which is about 10%. However, the solvent used in our case is DzO, would make the cmc lower than that in H 2 0 by about 6%,39Le., 8.1 mM, which is in fair agreement with our value 8.01 mM. As for AOT, the measured cmc was reported to be 1.61 m M in H 2 0 at room temperature which, when taking into account the D 2 0 solvent effect, would result in 1.52 mM. This value differs from our value by about 7%. This level of agreement suggests that the ladder model is in fact applicable to both SANS analysis and cmc prediction. It is also worth noting that the ratio between X,,, so obtained and X I is 1.08, which suggests that the empirical definition of cmc by Tanford, Xc,,/Xl = 1.05 (eq 13), is quite reasonable in these cases. The next thermodynamic quantity to be tested is the hydrophobic energy gain, AFHp, of a surfactant monomer upon micellization. This hydrophobic free energy gain per monomer is related to the free energy parameter A we extracted from SANS data analyses. For a single micelle in solvent near cmc, one can write
where A F D L is a positive quantity (per monomer) representing the free energy of formation of the double layer upon micellization. This energy can be computed approximately by solving the nonlinear Poisson-Boltzmann equation involving a micelle and its associated c o u n t e r i ~ n (see s ~ ~eq~ 7~ of ~ ~ref~ 5). On the basis of this calculation, we obtained AFHp/kBT = A/(NokBT)
- AFDL/(kBT) = -8.5 - 4.3 = -12.8kBT (43)
in the case of SDS. A recent theoretical paper by K ~ r y l a n dgives ~~ AFHp = -1 1.7kBTfor SDS. Besides, T a n f ~ r dalso ~ ~gives an empirical formula for computing A F H p for single-chained surfactant molecules in water AFHp
= -1400 - 700(N, - 1)
+ 25(uH - 21)
(cal/mol) (44)
where N, is the number of carbons in the surfactant tail and uH is the area per polar head group. uH can be calculated for a concentration near cmc by taking the micelle to be spherical UH
= [(36~~;)/No]'/~
(45)
In the case of SDS, uH = 68.6 A2, and the hydrophobic energy AFHp calculated by eq 43 gives -13.4kBT. As one can see, our value is in quantitative agreement with these two calculated values. In Figure 8 we show the average interparticle structure factors extracted from SDS micellar solutions studied here. It is clearly seen that the local structure is gradually built up at concentrations greater than 1% and the first diffraction p_eak becomes more and more prominent. This prominent peak of ( S ( Q ) )is directly related to the peak in Z(Q), from which (PN(Q)) is extracted. Since ( s ( Q ) ) is not too sensitive to polydispersity when the width of the size distribution is less than 15% (or aggregation number polydispersity of 45%), polydispersity effects on Z(Q) enter through (39) Mukerjee, P.; Mysels, K. J.; Kapauan, P. J . Phys. Chem. 1967, 71, 4166. (40) Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J . Phys. Chem. 1984, 88, 6344. (41) Ohshima, H.; Healy, T.; White, C. R. J . Colloid Interface Sci. 1982, 90,17. (42) Kuryland, D. I. Colloid J . USSR (Engl. Transl.) 1985, 47(4), 600. (43) Tanford, C. In Micelliration Solubilization and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977; Vol. 1, p 119.
'
EXTRACTED STRUCTURE FACTORS OF SDS MICELLAR SOLUTIONS
AbA8 0
t
G
/
'
+
A
A
A.
0
0
D
0.5
a
'
DASH: 0.5% SOLID: 1% DOT: 2% CROSS: 'tX TRIANGLE: 8% SQUARE: 16%
0 IAtt-lI
Figure 8. The extracted interparticle structure factor S(Q) for SDS micellar solutions investigated.
( p N ( Q ) )mostly. (pN(Q)) is a smoothly varying monotomic Gaussian-like function which can always be reproduced by a suitable choice of size and shape of the particles. Thus, analysis of an individual SANS intensity distribution cannot distinguish a polydisperse system from a monodisperse system. Strictly speaking, I(Q) for all concentrations have to be reproduced by a finite set of parameters (No,A, 6, and a)in order to distinguish a polydisperse micellar solution from a monodisperse micellar solution. This explains why the previous monodisperse model can also fit the SANS data. VII. Conclusion We have proposed a method for treating the polydispersity effects in SANS data analyses involving ionic micellar solution. In spite of the fact that the intermicellar interactions are strong for these systems, we have demonstrated the applicability of the ladder model in these cases. From our analysis we find that the degree of polydispersity, in both SDS and AOT systems, increases rapidly as the concentration increases. The extracted ladder model paraxpetes No, A, and 6 allow us to predict both the cmc value and the hydrophobic free energy of a surfactant molecule in the minimum size micelle. The cmc as well as the hydrophobic free energy values so predicted are in good agreement with the theoretical and experimental estimates in the literature in the case of the SDS system. The minimum micelle aggregation number for the SDS system, No = 42, implies that the length of the surfactant tail in the core of a minimum size micelle can be well-estimated by summing up all the bond-to-bond distances with the C-C-C angle taken to be 109.4' (tetrahedral angle). The N, in the empirical formula for the tail length given by Tanford'* is to be interpreted as the number of carbon atoms in the alkyl chain minus one. Our previous analyses showed that a monodisperse model can also fit the SANS data provided that No is suitably chosen and the aggregation number is treated as a free parameter. The apparent success of the monodisperse model in analyzing the micellar solutions is attributed to the fact that the intermicellar structure factor is insensitive to polydispersity effects when the intermicellar interaction is strong. On the other_hand, at dilute concentrations, where the prominent peak of (S(Q)) no longer exists, the systems were found to be only slightly polydisperse, and thus, the monodisperse analysis is also applicable. Although the monodisperse model can also fit SANS data in most cases, the polydisperse analysis used here is still favored for the following reasons: (1) the micellization energy A and the energy for inserting an additional monomer into the minimum size micelle, 6, can be determined; (2) the correct minimum micellar aggregation number, No,can be extracted; (3) the micellar size distribution and the degree of polydispersity can be specified at any concentrations; and (4) the cmc value and the hydrophobic free energy of a surfactant molecule in a minimum micelle can be correctly deduced from SANS data. Above all the ladder model allows one a simple and intuitive way of understanding the for-
4474
J. Phys. Chem. 1988, 92, 474-4478
mation of micelles and their subsequent growth as a function of surfactant concentration in an aqueous medium.
Acknowledgment. E.Y.S. thanks Dr. J. S. Huang of Exxon Research & Engineering Co. for using his computer facility to analyze part of the SANS data presented here. He also thanks
Mr. J. Sung for purification of AOT surfactant for this study. This research is supported by a National Science Foundation grant administered through MIT Center for Material Science and Engineering. Registry No. SDS, 151-21-3; GOT, 577-11-7; DDAO, 1643-20-5.
Isotopic Identification of Surface Site Transfer on Ni/AI,O, Catalysts Paul G. Glugla, Keith M. Bailey, and John L. Falconer* Department of Chemical Engineering, University of Colorado, Boulder, Colorado 80309-0424 (Received: November 23, 1987; In Final Form: January 26, 1988)
Isotope labeling with temperature-programmed reaction (TPR) for CO hydrogenation was used to separate two distinct CO adsorption sites on a Ni/A1203 catalyst. One site is Ni metal and has the higher activity for CO hydrogenation. The less-reactive CO is on the A1203support. Only the Ni metal is occupied at 300 K, and transfer of CO between the two sites occurs at higher temperatures. In the presence of adsorbed Hz, CO that was adsorbed on the Ni metal moved to the A1203support. This is an activated process, and the only pathway to occupy the A1203sites is by adsorption on the Ni. The reverse transfer from A1203to Ni occurs if some of the H2 is desorbed; surface hydrogen inhibits this reverse process. These results show that during a typical TPR experiment, transfer between sites competes with reaction.
Introduction Nickel supported on y-alumina exhibits adsorption and reaction properties different from nickel on a silica support. For C O hydrogenation to methane, temperature-programmed reaction (TPR) has been used previously to observe two distinct types of sites on Ni/Al2O3 catalysts.',2 Only one type of site exists on Ni/Si02 for C O methanation. The sites on Ni/Si02 and one type of site on Ni/AlZO3are due to reduced Ni. Kester and Falconer' proposed that adsorbed CO could move between the reduced Ni sites (called A sites) and the second type of site (called B sites), and they proposed that this process was assisted by surface hydrogen. Recent TPR studies on Ni/AlZO3indicate that the B sites ~ studies for Ni,3,4R U , Rh,6 ,~ are on the A1203~ u p p o r t . ~Infrared and Pd7 on A1203are consistent with the T P R studies, and they show that C O and Hz are adsorbed on the A1203support as a formate or a methoxy species. Some of these studies also concluded that surface species move between the metal and the A1203.3*677 The purpose of this investigation was to examine the A1203sites on a Ni/A1203catalyst and study the transfer between sites. Temperature-programmed reaction, in combination with isotopic labeling, was found to be an effective way to study the properties of the A1203sites. Carbon monoxide on the Ni sites could be reacted away without reacting away the A1203sites. This was done by interrupting the TPR temperature ramp at the minimum point between the A and B peaks. A major innovation in this study was to cool the sample after interruption and then adsorb 13C0on the Ni sites while leaving the A 1 2 0 3 sites occupied by ITO. This technique allowed communication between the two reaction sites to be studied. This technique also showed that the sites were not saturated under normal experimental conditions, and adsorption at elevated temperatures, in the presence of H2, was found to be necessary to saturate the sites. Thus in previous TPR experiments, only a fraction of the catalyst surface was covered with CO. Moreover, in previous TPR studies, transfer between sites was occurring in competition with reaction to methane. (1) Kester, K.B.; Falconer, J. L. J . Coral. 1984, 89, 380. (2) Kester, K.B.; Zagli, E.; Falconer, J. L. Appl. Catal. 1986, 22, 31 1. (3) Lu, Y.;Xue,J.; Li, X.;Fu, G.; Zhang, D. Cuihuu Xuebao (Chinese J. Catal.) 1985, 6, 116. (4) Mirodatos, C.; Praliaud, H.; Primet, M. J . Catal. 1987, 207, 275. (5) Dalla-Betta, R. A.: Shelef. M. J. Caral. 1977. 48. 111. (6) Solymosi, F.; Bansagi, T.; Erdohelyi, A. J . Catal. 1981, 72, 166. (7) Palazov, A.; Kadinov, G.; Boney, C.; Shapov, S . J. Catul. 1982, 74,44.
The procedures described for isotope separation are general ones that can be applied to significantly improve our understanding of adsorbed species on surfaces. In a subsequent paper,* we have used isotope labeling in combination with temperature-programmed desorption of coadsorbed C O and H2to relate the reaction sites to adsorption sites. Experimental Section The temperature-programmed reaction (TPR) system was similar to that described previo~sly.~A 100-mg catalyst sample was located on a frit in a 1-cm 0.d. quartz down-flow reactor. An electric furnace was used to heat the catalyst at 0.7 K/s for most of the TPR experiments. A constant rate of heating was maintained by feedback from a small thermocouple located in the catalyst bed. Hydrogen flowed over the catalyst at 1 atm of pressure, and the gas residence time in the bed was less than 0.07 s. The effluent from the reactor was analyzed with a quadrupole mass spectrometer. A computer system allowed multiplexing between mass peaks. Carbon monoxide (%O or I3CO) was adsorbed by injecting 0.2-mL pulses into a Hz or He carrier gas that flowed through the catalyst bed. Adsorption was done at 300 or 385 K by repeated injections; in most experiments pulses were continued until no additional adsorption was detected. Some Ni(C0)4 may form during CO exposure at 385 K, but not much Ni was removed since a series of repeated experiments yielded the same TPR spectra. A 10% C O in H e (UHP, Matheson) gas was used for I2CO adsorption. The 13C0was from Monsanto Research Corp. and was specified as 99.25% carbon monoxide, of which 99.6% was labeled with 13C. The Hz and He carrier gases were UHP grade (Matheson) and were further purified. A 5.1% Ni/Al2O3 catalyst was prepared with nickel nitrate on Kaiser A-201 y-alumina by the same wet impregnation as described in the literature.' The nickel nitrate was directly reduced in H2without first calcining. The Ni loading was measured by atomic absorption. Temperature-programmed reaction was carried out by adsorbing carbon monoxide on the surface and then heating the catalyst in Hz while detecting I2CH, (mass 15) or I3CH4(mass 17) and I2CO or 13C0with the mass spectrometer. The cracking fraction of 13CH4at mass 15 was subtracted from the mass 15 (8) Glugla, P. G.; Falconer, J. L.; Bailey, K. M. J . Cutal., accepted for publication. (9) Falconer, J. L.; Schwarz, J. A,, Catal. Reu. Sci. Eng. 1983, 25, 141.
0022-3654188 12092-4474%01.50/0 0 1988 American Chemical Societv
