3724
J . Phys. Chem. 1989, 93, 3724-3728
Ultrasonic Velocity, Apparent Molar Adiabatic Compressibility, and Solubilization Studies of Aqueous Solutions of Cetyltrimethylammonium Bromide as a Function of Surfactant and Soiubilizate Concentrations and Temperature M. Alauddin and Ronald E. Verrall* Department of Chemistry, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S 7 N 0 WO (Received: August 30, 1988; In Final Form: November 18, 1988)
Ultrasonic velocities and densities of aqueous solutions of cetyltrimethylammoniumbromide have been measured at concentrations below 0.40 mol kg-' at 25, 35, and 45 OC. Apparent molar adiabatic compressibilitiesand adiabatic compressibility coefficients of the aqueous surfactant solutions were derived from these data. Apparent molar compressibilities of 2,6-di-rert-butyl-4methylphenol and 2- or 3-tert-butyl-4-methoxyphenoldissolved in aqueous micellar solutions of cetyltrimethylammonium bromide were determined as a function of surfactant concentration and temperature. The partial molar compressibilities of solubilizates at infinite dilution in the aqueous micelle solutions were obtained from apparent molar compressibility data. The results were used to predict the possible structural changes of the micelle and to analyze the location of the solubilizates in the micellar aggregates. They show that cetyltrimethylammonium bromide micelles undergo a second transition in the postmicellar region at ca. 0.28 mol kg-'. The solubilization behavior of the micelles below and above this transition concentration is found to be significantly different. The results indicate that 2,6-di-tert-butyl-4-methylphenol is primarily solubilized in the hydrocarbon region of the micelle while 2- or 3-tert-butyl-4-methoxyphenol can be solubilized both at the micelle surface and in the hydrocarbon part of the micelle, depending upon the micelle and surfactant concentration and temperature. It appears that 2,6-di-tert-butyl-4-methylphenol solubilized in small ellipsoidal micelles initially occupies the outer part of the micelle interior and translocates into the inner part of the micelle as the number of solubilizate molecules per micelle increases. However, in rodlike micelles this hydrophobic molecule appears to be solubilized in a single type of site within the micelle.
Introduction A number of inve~tigationsI-~ have been carried out in recent years with the view of improving our understanding of the structural organization of amphiphilic molecules forming micelles or vesicles. The most important feature of micellar systems is their ability to solubilize various organic compounds that are ,~ experimental sparingly soluble or insoluble in ~ a t e r . ~Several techniques have been used in the study of micellar systems in both the presence and absence of additives to infer the environment of the solubilized molecule as well as the possible structural changes that occur in the micelle geometry as a result of solubilization. Such knowledge is of fundamental importance to improving our understanding of the nature of the solubilization process as well as the concomitant structural reorganization of micelles. Aqueous solutions of cetyltrimethylammonium bromide (CTAB) have shown complex behavior with respect to a number of micellar properties, especially when additives like electrolytes and different organic compounds are present.'-" The micellar properties are highly specific and depend on the associated counterions, chemical structure of the additives, and micelle concentrations. It has been reportedI2J3 that micellar aggregates of CTAB undergo transformation in size and shape as the surfactant concentration is increased, changing from a sphere to (1) Tanford, C. J . Phys. Chem. 1972, 76, 3020. (2) Menger, F. M. Acc. Chem. Res. 1979, 12, 11 1. (3) Ozeki, S.;Ikeda, S.J . Colloid Interface Sci. 1980, 77, 219. (4) Flamberg, A.; Pecora, R. J . Phys. Chem. 1984, 88, 3026. (5) Elworthy, P. H.; Florence, A. T.; McFarlane, C. B. Solubilization by Surface Acriue Agents; Chapman and Hall: London, 1968. (6) McBain, M. E. L.; Hutchinson, E. Solubilization and Related Phenomena; Academic Press: New York, 1955. (7) Erikson, J. C.; Gillberg, G. Acta Chem. Scand. 1966, 20, 2019. (8) Lindblom, G.: Lindman, B.; Mandell, L. J . Colloid Interface Sci. 1973, 42, 400. (9) Ulmious, J.; Lindman, B.; Drakenberg, T . J . Colloid Interface Sci. 1978, 65, 88. (10) Gratzel, M.; Kalyanasundaram, K.; Thomas, J. K. J . Am. Chem. Sot. 1974, 96, 7869. ( I 1 ) Alauddin, M.; Verrall, R. E. J . Phys. Chem. 1986, 90, 1947. (12) Ekwall, P.; Mandell, L.; Solyom, P. J . Colloid Interface Sci. 1971, 35, 519. (13) Backlund, S.; Hoiland, H.; Kvammen, 0.;Ljosland, E. Acta Chem. Scand. 1982, A36, 698.
0022-365418912093-3724$01.50/0
rodlike shape at -0.28 m CTAB in water.I3 However, it appears that the concentration at which this transition occurs in CTAB micelles is strongly dependent on the experimental technique being used. For example, Reiss-Husson and Luzzati14 observed the transition to occur at 0.15 m using small-angle X-ray scattering. More recently, Quirion and Desnoyersls observed transitions in enthalpy and heat capacity properties at the same CTAB concentration (0.15 m). It also has been reported that solubilization of different solutes in CTAB induces very marked changes in the micellar shape. Nuclear quadrupole studies8 of 81Brin aqueous micelle solutions of CTAB suggest that benzene, N,N-dimethylaniline, and hexanol are solubilized at or close to the micellar surface with the occurrence of a concomitant decrease in counterion binding. These solutes appear to promote a change in micellar shape. However, this was not the case with cyclohexane which appeared to be incorporated into the interior of the micelle. We have previously reported" that increasing amounts of 2,6-di-tert-butyl-4-(hydroxymethy1)phenol solubilized in aqueous solutions of CTAB result in an increase in aggregation number, thus reflecting a change in micelle shape and size. This paper describes the results of our continuing ~ t u d i e s " J ~ ~ ' ~ of micellar solutions that contain chemical compounds that have been reported to be either a potent inactivator of mammalian and bacterial viruses containing lipid structures18 or inhibitors of processes leading to cancer growth.lg We report adiabatic compressibility studies of aqueous micelle solutions of CTAB and (BHT) and 2- or 3-tert-b~of 2,6-di-tert-butyl-4-methylphenol tyl-4-methoxyphenol (BHA) solubilized in aqueous micelle solutions of CTAB. We also have examined the variation of ultrasonic velocity in these micellar systems, in both the presence and absence of additives. The results are interpreted qualitatively, in terms of the approximate location of the solubilizates in the (14) Reiss-Husson, F.; Luzzati, V. J . Phys. Chem. 1964, 68, 3504. (1 5 ) Quirion, F.; Desnoyers, J. E. J . Colloid Interface Sci. 1986, 112, 565. (16) Alauddin, M.; Verrall, R. E. J . Phys. Chem. 1984, 88, 5725. (17) Alauddin, M.; Verrall, R. E. J . Phys. Chem. 1987, 91, 1802. (18) Cupp, J.; Wanda, P.; Keith, A,; Snipes, W. Antimicrob. Agents Chemother. 1975, 8, 698. (19) Anderson, M. W.; Boroujendi, M.; Wilson, A. G. E. Cancer Res. 1981, 41, 4309.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3725
Aqueous Solutions of Cetyltrimethylammonium Bromide micellar aggregates, possible changes in the shape and size of the micelle, and interactions within the micelle between the solubilizate and surfactant species. Experimental Section Materials. Cetyltrimethylammonium bromide (Sigma) was recrystallized twice from an acetone-ethanol mixture (90/ 10, v/v) and dried under vacuum at 60 OC. Elemental analysis showed the purity to be >99%. All other chemicals used were purified as described previously.I6 Deionized water used in these experiments was obtained from a Millipore Super-Q-System. Apparatus and Procedure. Ultrasonic velocities were measured at a frequency of 4 MHz with a Nusonic (Model 6080, Mapco) single transducer velocimeter by using the sing-around technique.z0 The sound speed, u (in ms-'), was calculated from the average round-trip period of the ultrasonic wave in the fixed path length between the piezoelectric transducer and reflector. The period of the sound wave was measured with a frequency meter (Fluke 7261 Universal Counter). The following equation was used to calculate u at) -fA(1 . u = 7-fB where A is the sonic path length in meters, B is the electrical delay time in seconds, f is the frequency in hertz, a is the coefficient of thermal expansion of the transducer holder material, and t is the temperature of the solution in Celsius. The constants A and B were obtained at different temperatures by calibrating the transducer probe using the velocity data for waterz1 and aqueous solutions of NaC1.2z The sample container was an insulated double-walled glass cell which has a liquid volume capacity of ca. 80 mL. Accurate and rapid measurements of sound velocity as a function of solute concentration were made by successive additions of known weights of the solute to the container. The ultrasonic velocity measurements were estimated to be accurate to f0.005 ms-I. Adiabatic compressibility coefficients, 0(in units of Pa-]), were derived from the relation
I534 1531 I526 1525
-
1519
-E
1517
-I
Ln
3
(2)
where d is the solution density in kg m-3. The apparent molar volume (&) and apparent molar adiabatic compressibility (&) of liquid solutions were calculated from the following relations (3)
35°C 25OC
1497
(20) Garnsey, R.; Mahoney, R.; Litovitz, T. A. J . Chem. Phys. 1964,64, 2043. (21) Del Grosso, A.; Mader, C. W. J . Acoust. SOC.Am. 1972, 52, 1442. (22) Sakurai, M.; Nakajirna, T.; Kornatsu, T.; Nakagawa, T. Chem. Lett. 1975.91 1.
(23) Picker, P.;Tremblay, E.; Jolicoeur, C. J . Solution G e m . 1974,3, 377.
I
I
I
1
0
0.20
0 40
m ( m o l kg-') Figure 1. Ultrasonic velocities in aqueous solutions of CTAB as a function of molality at 25, 35, and 45 OC.
1 7 5 5 r
I
1735
0 E
16501
-
16.40
z
0 x
c
1630
0,"
l 5 301
0.10
~oo~o-o-o-o-Qo' I
I
I
0 30
0 50
0 70
L where m is the molality of the solution, M is the relative molar mass of the solute, d is the density of the solution, do is the density of the solvent, and Po and P are the compressibility coefficients of the solvent and solution, respectively. Density data were obtained with a high-precision flow digital densimeter (Model 02D, Sodev Inc.). Data for all systems are found in the supplementary material (see paragraph at the end of the text regarding supplementary material). The operation and calibration of the instrument have been previously d e s ~ r i b e d . ' ~The * ~ ~precision in the density data was found to be better than f1.5 ppm. The solution temperatures in both the velocimeter and densimeter were maintained to fO.OO1 OC by using a closed-loop temperature controller (Model CT-L, Sodev Inc.). One of the temperatures used in this study is within the range (24-26 OC) of critical micellization temperatures reported for pure CTAB in water. However, when the solutions were checked they did not show any signs of being metastable.
1
1498
15201,
(4)
1
1499
+
p = l/u2d
1515 1500
L
L
m 2 (mol2 kg-2)
Figure 2. Apparent molar compressibility of aqueous solutions of CTAB as a function of the square root of molality at 25, 35, and 45 OC.
Results The concentration dependence (0.05-0.40 m) of the ultrasonic velocities of aqueous solutions of CTAB at three temperatures is shown in Figure 1. At 25 OC a sharp break in sound velocity occurs at ca. 0.28 m CTAB. A less overt break also is seen at approximately the same concentration at 35 O C . However, the sound velocity data for aqueous solutions of CTAB at 45 OC show a smooth variation with increasing concentration of surfactant. A further important observation is the fact that the sound velocity increases with increasing CTAB concentration at the lowest temperature, 25 OC, whereas it decreases with increasing surfactant concentration at both 35 and 45 OC. These results are in good agreement with those previously reported13for this system in the temperature range 27.5-42.5 OC. Figure 2 shows the variation of & as a function of the square root of CTAB concentration in aqueous solutions. The magnitude of & increases with increasing CTAB concentration and increasing temperature. A sharp break in dkoccurs at ca. 0.28 m CTAB at 25 and 35 OC but is not observed at 45 OC. Less well defined breaks in the slopes
Alauddin and Verrall 4.480 -
98
7
7
t
4 4.450 -
1
4.350) 4.3501
,ado ' 45"c
I
T
O-O-o-
86-
25°C
0
'
1
4 250 I
I
I
I
0 IO
0 30
0 50
0 70
m i (mol
+
m x IO3 ( m o l k g - ' )
k9-i)
Figure 3. Adiabatic compressibility coefficient of aqueous solutions of CTAB as a function of square root of molality at 25, 35, and 45 "C. I7 0
I
Figure 5. Apparent molar compressibility of BHA as a function of molality in aqueous solutions of CTAB of different concentrations at 25, 35, and 45 "C: 0,0.10 rn; e, 0.30 rn. TABLE I: Partial Molar Compressibilities of Solubilizates at Infinite Dilution in Aqueous Solutions of CTAB at 25, 35, and 45 "C
[CTAB], mol kg-'
I
*W,450C
0.10
0.30
temp, "C BHT 25 35 45 25 35 45
&"(mic) x m3 mol-' Pa-' 12.05 13.45 13.78 16.54 14.58 16.40
BHA 0.10
0.30 13.51
It0
, 0
1
i
O
I
I
I
6 12 m x IO3 ( m o l k g - l )
I
I
18
Figure 4. Apparent molar compressibility of BHT as a function of molality in aqueous solutions of CTAB of different concentrations at 25, 35, and 45 "C: 0,0.10 rn; 0 , 0.30 m.
of /3 vs m1/2 of CTAB are observed (Figure 3) at m 0.25 and all temperatures. The slope of the /3 vs m112curve is negative at 25 "C whereas it is positive at both 35 and 45 "C. The results of the apparent molar adiabatic compressibilities of BHT and BHA solubilized in aqueous micelle solutions of CTAB at different concentrations and temperatures are shown in Figures 4 and 5 , respectively. It is apparent that there is considerable variation in this property for both solubilizates. For both BHT (Figure 4) and BHA (Figure 5) solubilized in 0.10 m CTAB at 25 "C, a sharp increase in & occurs at low solubilizate concentration, in the range (5-7) X m. At higher solubilizate concentrations @k decreases for BHT but is almost invariant in the case of BHA. In other cases the plots of & vs m of these solubilizates show a linear dependence with concentration at the temperatures studied except for BHA in 0.10 and 0.30 m CTAB
25 35 45 25 35 45
8.28 8.55 9.03 13.15 7.80 9.74
at 35 "C (Figure 5 ) . ,+#C of BHA solubilized in 0.10 m CTAB at 35 OC shows a decrease in magnitude at ca. 12 X m whereas, when it is solubilized in 0.30 m CTAB at the same temperature, a pronounced increase in $ J k occurs at BHA concentrations below 10 x m. The infinite dilution values of the apparent molar compressibility (& = $"(mic)) of the solubilizates in aqueous micelle solutions were determined, where possible, by using the weighted leastsquares method to fit the data to the assumed relation & = &" Skm. Derived values of the parameter &,"(mic) at different surfactant concentrations and temperatures are reported in Table I. The magnitude of R2"(mic) for the two solubilizates present in 0.10 m CTAB increases with increasing temperature. In the case of 0.30 m CTAB, the variation of R2O(mic) with temperature for both solubilizates appears to pass through a minimum. Adiabatic compressibility coefficients, 0, of BHT and BHA at a fixed CTAB concentration, generally, increase with increasing concentration of BHT and BHA at different temperatures studied. Typical plots of /3 vs m for BHT and BHA present in 0.30 m CTAB at 25 OC are shown in Figure 6.
+
Discussion Some surfactants in aqueous solutions form spherical micelles just above the critical micelle concentration (cmc) and some change shape at higher concentrations to form rodlike micelles.
Aqueous Solutions of Cetyltrimethylammonium Bromide 4 460
4.455
F
G 0
I
I
8
I
I
I
16
24
m x IO3 ( m o l k g - ' ) Figure 6. Adiabatic compressibility coefficients of aqueous solutions of CTAB containing solubilizates at 25 "C: 0, BHT in 0.30 m CTAB; 0 , BHA in 0.30 m CTAB.
Because of the rather high mean aggregation number of CTAB in water, at lower temperatures it may be argued that the micelles are not spherical even at the cmc. Ultrasonic measurements have been shown to be sensitive to transitions in micellar solutions. The results of our studies of sound velocity (Figure 1) and apparent molar adiabatic compressibility (Figure 2) properties both show that well-defined breaks occur at a CTAB concentration of ca. 0.28 mol kg-', in general agreement with previous sound velocity studies of this system." The break from linearity, however, disappears at 45 OC. The adiabatic compressibility coefficient (Figure 3) also shows a small but definite change in this concentration region. These results indicate that CTAB micelles apparently undergo a transition at a surfactant concentration of ca. 0.28 mol kg-'. This transition continues to be apparent in this work at 35 OC, in contrast to the previous results" which showed the transition to occur only below 32 OC. The observed compressibility of micellar solutions, generally, depends on two major fact01-s:~~ (i) the compressibility of the hydrocarbon core and (ii) the interaction between the head groups. This property may also be dependent on the variation of the counterion binding and the amount of water bound to the head groups of the surfactant monomer. Any hydrophobic solute added to an aqueous micelle solution may affect the solution compressibility by disrupting the water structure around the added molecule when it transfers from the bulk aqueous phase to the micellar pseudophase. It may also do so by occupying the free space between the surfactant monomers in the micelles. It is expected that the relative contribution of these factors would determine the sign and magnitude of the compressibility of the micellar solutions. The apparent molar compressibility, $k, of BHT solubilized in 0.10 m CTAB at 25 OC (Figure 4) shows a sharp increase in its value at a solubilizate concentration of -7 X mol kg-I. Further increase in BHT concentration leads to a decrease in &. This result suggests that a definite change is occurring in the solubilization process of BHT. It is possible that BHT is solubilized in the outer region of the hydrocarbon core of the micelle at low solubilizate concentrations and then moves into the inner mol region of the micelle as the concentration exceeds 7 X kg-'. If there is more free space in the interior of the micelle, then the decrease in f$k at BHT concentrations above 9 X mol kg-' may indicate the loss of this free space as the number of solubilizate molecules per micelle increases. It is observed that the values of & increase linearly with increasing BHT content over the concentration range studied when the temperature is increased to 35 and 45 OC (Figure 4). This suggests that, initially, (24)
Bloor, D. M.; Gormally, J.; Wyn-Jones, E. J . Chem. SOC.,Faraday
Trans. 1 1984, 80, 1915.
The Journal of Physical Chemistry, Vol. 93, No. 9, 1989 3121 BHT is solubilized in the outer region of the micelle interior and then begins to occupy the inner part of the micelle as the solubilizate concentration increases. The increase in f#)k at higher temperatures also may be attributed to the breakdown of water structure as the solubilizate transfers to the inner part of the micelle with increasing temperature. The apparent molar compressibility, &, of BHT solubilized in 0.30 m CTAB shows a linear decrease with BHT concentration at the highest and lowest temperatures studied while an increase in this property is observed for this additive dissolved in 0.10 m CTAB at the same temperatures. Such behavior may not be unexpected if the micelle changes its shape from ellipsoidal to rodlike aggregates as the CTAB concentration is increased from 0.10 to 0.30 mol kg-'. The rodlike micelles would have a larger surface area and hydrophobic core, and it is likely that the larger aggregates (rodlike micelles) may be able to solubilize all of the additive molecules in a similar site within the micelles. The decrease in with increasing BHT content may be attributed to the loss of free space within the micelles. Further, K20(mic) for BHT in 0.30 m CTAB is larger than Kzo(mic) in 0.10 m CTAB (Table I). This indicates that BHT in 0.30 m CTAB is probably more compressible,which is expected for the large rodlike micelles. BHT in 0.30 m CTAB at 35 "C, on the other hand, shows a slight increase in I& with BHT concentration over the concentration range studied. Reed et aLZ5reported that the outer methylene groups of CTAB are extensively exposed to water. An increase in temperature can cause desolvation of the micelles and their counterions, thus allowing the BHT molecules to be solubilized within the outer methylene groups of the micellar core of 0.30 m CTAB at 35 OC instead of penetrating into the inner part of the micelles. Further increase in temperature to 45 OC may cause the solubilizate to penetrate more into the hydrocarbon region of the aggregates. This is evident from the observation that &(45 " c ) > &(35 " c ) . Plots of qjk vs concentration of BHA a t different surfactant concentrations and temperatures show that the variation of the apparent molar adiabatic compressibility is very much dependent upon the concentration of both additive and surfactant as well as the temperature (Figure 5). An abrupt increase in & for BHA in 0.10 m CTAB at 25 "C is observed at a solubilizate concentration of (6-7) X mol kg-'. This result may reflect that the solubilizate is changing its location in the micellar aggregates with changing BHA concentration. It is speculated that BHA, which is relatively less hydrophobic than BHT, is solubilized at the micelle-water interfacial region and presumably undergoes iondipole or dipole4ipole interactions between its 4 C H 3 and - O H groups and the surfactant head groups but further penetrates into the inner part of the micelle as the solubilizate concentration increases. This argument is consistent with our previous results" obtained from volumetric and N M R studies for this additive solubilized in 0.10 m CTAB. Relatively higher compressibility values for BHA present in 0.10 m CTAB at 35 and 45 OC indicate that this molecule is favorably solubilized inside the micelle interior at higher temperatures. The solubilization behavior of BHA in 0.30 m CTAB is similar to that of BHT. On the basis of the same arguments given for BHT, it appears that BHA in 0.30 m CTAB is preferentially solubilized within the micelle and the extent of penetration depends on the additive concentration and temperature. It is observed that the compressibility of BHA in 0.30 m CTAB at 35 and 45 OC is lower than its value at 25 "C in the same surfactant solution. This can be understood by considering the decrease in hydration as well as the change in counterion binding with increasing temperature, both of which may enhance the intermicellar repulsion. Presumably, desolvation of micelles would favor the solubilization of BHA with its polar groups located at the surface between the surfactant head groups and with its hydrophobic part immersed in the hydrocarbon interior. This may reduce the repulsion between surfactant head groups and as a result the micelles become (25) Reed,W.; Politi, M. J.; Fendler, J. H.J . Am. Chem. SOC.1981, 103, 4591.
J . Phys. Chem. 1989, 93, 3728-3735
3728
more compact, thereby resulting in lower compressibility. The adiabatic compressibility coefficient of aqueous micelle solutions of CTAB containing these solubilizates is seen to increase with increasing solubilizate concentrations (Figure 6). This increase may arise because of the decrease in the degree of "structured" water as a result of transfer of the solubilizates from the aqueous to the micellar phase, this change being more than compensated by the loss of free space in the micelle interior upon solubilization. In conclusion, we find that CTAB micelles undergo a postmicellar transition that takes place around 0.28 mol kg-' CTAB in water. The solubilization behavior of CTAB micelles formed below and above the transition concentration is significantly different. It appears that both BHT and BHA are preferentially solubilized in the micellar aggregates. BHA solubilized in 0.10 m CTAB appears to be located at the micelle-water interfacial region at low solubilizate concentrations and penetrates into the inner part of the micelle with increasing surfactant and solubilizate concentration and temperature. BHT is more hydrophobic in nature than BHA and resides in the hydrocarbon-like environment
of the micelle. When solubilization in small ellipsoidal micelles (0.10 m CTAB) occurs, this hydrophobic molecule initially occupies the outer region of the nonpolar part of the micelle and moves into the inner part of the micelle with increasing solubilizate concentration. In 0.30 m CTAB it appears to be solubilized in a uniform site within the rod-shaped micelle. Acknowledgment. We are grateful to the Natural Sciences and Engineering Research Council of Canada for their financial support. The authors thank one of the reviewers for helpful suggestions concerning terminology. Registry No. CTAB, 57-09-0; 2,6-di-tert-butyl-4-methoxyphenol, 128-37-0;2-terr-butyl-4-methoxyphenol,121-00-6; 3-tert-butyl-4-methoxyphenol, 88-32-4.
Supplementary Material Available: Tables of ultrasonic velocity, density, apparent molar volume, apparent molar compressibility, and adiabatic compressibility coefficient data for CTAB with water, BHT, and BHA at 25,35, and 45 OC (8 pages). Ordering information is given on any current masthead page.
Influence of Surface Structure on the Kinetics of Diffusion-Controlled Reactive Processes on Molecular Organizates and Colloidal Catalysts Philip A. Politowicz, Research School of Chemistry, Australian National University, Canberra, Australia ACT 2601
Roberto A. Garza-Lbpez, David E. Hurtubise,+and John J. Kozak* Department of Chemistry, Franklin College of Arts and Sciences, University of Georgia, Athens, Georgia 30602, and Radiation Laboratory and Department of Chemistry,! University of Notre Dame, Notre Dame, Indiana 46556 (Received: September 6 , 1988)
At the molecular level, the surface of a colloidal catalyst particle or molecular organizate (such as a cell or vesicle) is not smooth and continuous, but rather differentiated by the geometry of the constituents and, if the surface composition is not homogeneous, often organized into clusters or domains. In this paper, we develop a model to study the influence of such structure on the efficiency of encounter-controlled, surface-mediated reactive processes. We focus on the surface structures defined by a number of classic figures, the five Platonic solids and (here) 16 Archimedean solids; in each case, the attendant polyhedral surface is characterized by dimension d = 2 and Euler characteristic x = 2, with N distinct locations (sites) on the surface organized into an array defined locally by the site valency v (or connectivity) of the consequent network. Given this structure, we consider a target molecule A anchored to the surface at one of the N sites and a coreactant B free to migrate among the N - 1 satellite sites and study the dynamics of the diffusion-controlled irreversible reaction A + B C by formulating a stochastic master equation for each surface geometry considered and solving this equation numerically for two classes of initial conditions. Specifically, we determine the survival probability p ( t ) versus time t of the diffusing coreactant B and calculate the first four moments of the underlying probability distribution function. From the consequent evolution curves we extract the relaxation times to, and from these and the associated moments we are able to disentangle the separate influences of the variables Nand v on the kinetics. In all cases, we find that, for fixed N, the time io decreases with increase in the (local or global) valency u. Secondly, for a given local symmetry (v fixed), we find that to increases with N in (almost) all cases; the single exception occurs when the number of domains of triangular symmetry begins to dominate the overall surface structure.
-
I. Introduction In many diffusion-controlled reactive processes in which the reactants are confined to the surface of a particle, the species are not free to diffuse freely over the surface. For example, in cellular systems, although it is known that lateral diffusion over the surface of the encompassing medium may be quite free, the diffusing species must nonetheless negotiate transmembrane (and other) 'To whom correspondence should be addressed at the University of Georgia. 'Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556. $This work was initiated at the University of Notre Dame, Notre Dame, Indiana. This is Document No. NDRL-3122 from the Notre Dame Radiation Laboratory.
0022-365418912093-3728$01.50/0
proteins whose presence bifurcates the surface into domains connected by channels. In colloidally dispersed catalytic systems, randomly dispersed "islands" of catalytic activity on the surface of a particle are believed to govern the turnover and overall kinetic response of the system. These and many other examples led two of us to investigate in a previous work' the consequences of developing a model wherein the presence of individual sites connected by "pathways" or "channels" to a reaction center (the consequence of having a surface broken up into domains or differentiated into clusters) was built into the formulation of the problem from the very outset. ( 1 ) Politowicz, P. A.; Kozak, J. J. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 8175.
0 1989 American Chemical Society
