Anal. Chem. 1999, 71, 4069-4074
Taylor Dispersion Measurements of Monolayer Protected Clusters: A Physicochemical Determination of Nanoparticle Size W. Peter Wuelfing, Allen C. Templeton, Jocelyn F. Hicks, and Royce W. Murray*
Kenan Laboratories of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290
We describe the use of the Taylor dispersion method to determine the diffusion coefficients of monolayer protected clusters (MPCs). The MPCs have nanometer Au cores (∼1-5 nm diameter) coated with a passivating, selfassembled monolayer of thiolate ligands (∼40-600). The results are found to agree with electrochemically measured diffusion coefficients, validating both approaches for measuring nanoparticle diffusivities. Taylor diffusion coefficients are used to calculate MPC hydrodynamic diameters under the sticking and the slip boundary conditions of the Stokes-Einstein equation, which are compared to the overall MPC diametersestimated as the core diameter (measured by TEM) plus the length (2L) of two fully extended ligands. Better agreement between calculated and experimental hydrodynamic diameters of alkanethiolate-coated MPCs in organic solvents are obtained using slip boundary conditions, implying free draining of solvent through the outer portion of the extended chains. The hard-sphere (sticking) version of the Stokes-Einstein relation predicts unrealistically small hydrodynamic diameters. Similar but less clear-cut results are obtained for tiopronin-coated MPC ions in aqueous media. Nanometer-sized metal clusters, particularly those with selfassembled monolayer shells1 that serve to protect the metal cores from aggregation in dried samples, are now recognized for the wide range of chemical and physical attributes that their cores and monolayers provide. The cores give access to study of a range of different metals and alloys,2 of quantized double-layer charging in electrolyte solutions,3 and of electronic conductivity.4 The monolayer shellsswhich can be chemically functionalized in many (1) Brust, M.; Walker, M.; Bethel, D.; Schiffrin, D. J.; Whyman, R. J. J. Chem. Commun. 1994, 801. (2) Hostetler, M. J.; Zhong, C.-J.; Yen, B. K. H.; Anderegg, J.; Gross, S. M.; Evans, N. D.; Porter, M.; Murray, R. W. J. Am. Chem. Soc. 1998, 120, 9396. (3) (a) Chen, S.; Murray, R. W.; Feldberg, S. W. J. Phys. Chem. B 1998, 102, 9898. (b) Ingram, R. S.; Hostetler, M. J.; Murray, R. W.; Schaff, T. G.; Khoury, J. T.; Whetten, R. L.; Bigioni, T. P.; Guthrie, D. K.; First, P. N. J. Am. Chem. Soc. 1997, 119, 9279. (c) Hicks, J. F.; Templeton, A. C.; Chen, S.; Sheran, K. M.; Jasti, R.; Murray, R. W. Anal. Chem., in press. (d) Chen, S.; Ingram, R. S.; Hostetler, M. J.; Pietron, J. J.; Murray, R. W.; Schaaff, T. G.; Khoury, J. T.; Alvarez, M. M.; Whetten, R. L. Science 1998, 280, 2098. (4) Terrill, R. H.; Postlethwaite, T. A.; Chen, C.-C.; Poon, C.-D.; Terzis, A.; Chen, A.; Hutchison, J. E.; Clark, M. R.; Wignall, G. Londono, J. D.; Superfine, R.; Falvo, M.; Johnson, C. S.; Samulski, E. T.; Murray, R. W. J. Am. Chem. Soc. 1995, 117, 12537. 10.1021/ac990429c CCC: $18.00 Published on Web 08/17/1999
© 1999 American Chemical Society
waysslead to materials with ionic conductivity,5 provide for polylabeling with fluorophores and with electron donors and acceptors,6,7c and allow study of monolayer structure and dynamics by NMR and other methods.7 These attributes, and others, have led to investigations of these materials as potential nanoscale electronic devices,8a,b biosensors,9 and ordered 3-D metallic arrays.10 There is a premium on good characterization methods, given the current interest in monolayer protected clusters (which we dub MPCs). This paper concerns their physical dimensions, specifically their hydrodynamic diameters (dH) in solutions. The dimensions of MPC cores have been imaged in dry films by transmission electron microscopy (TEM)7a and in solutions by small-angle X-ray scattering (SAXS).4,7 The core mass has been determined by laser desorption/ionization mass spectrometry.11 The situation regarding overall (i.e., core plus monolayer) MPC dimensions in solutions is less satisfactory. Glimpses of the structure of the monolayer in the solid state have been inferred from TEM images, where the notion12 of interdigitation of monolayer (alkanethiolate) chains has been supported by theory predicting interweaving of chains in bundles.13 Vibrational spectroscopy indicates7d that alkanethiolate monolayers are more liquidlike in solutions than in solid form (where substantial crystallinity and trans-trans ordering7a,d is clearly present), but (5) Wuelfing, W. P.; Gross, S. M.; Miles, D. T.; Murray, R. W. J. Am. Chem. Soc. 1998, 120, 12696. (6) Templeton, A. C.; Chen, S.; Gross, S. M.; Murray, R. W. Langmuir 1999, 15, 66. (b) Templeton, A. C.; Cliffel, D. E.; Murray, R. W. J. Am. Chem. Soc., in press. (7) (a) Hostetler, M. J.; Wingate, J. E.; Zhong, C.-J.; Harris, J. E.; Vachet, R. W.; Clark, M. R.; Londono, J. D.; Green, S. J.; Stokes, J. J.; Wignall, G. D.; Glish, G. L.; Porter, M. D.; Evans, N. D.; Murray, R. W. Langmuir 1998, 14, 17. (b) Ingram, R. S.; Murray, R. W. Langmuir 1998, 14, 4115. (c) Templeton, A. C.; Hostetler, M. J.; Kraft, C. T.; Murray, R. W. J. Am. Chem. Soc. 1998, 120, 4845. (d) Hostetler, M. J.; Stokes, J. J.; Murray, R. W. Langmuir 1996, 12, 3604. (e) Hostetler, M. J.; Templeton, A. C.; Murray, R. W. Langmuir 1999, 15, 3782-3789. (8) (a) Sato, T.; Ahmed, H.; Varnsten, T.; Brown, D.; Johnson, B. F. G. J. Appl. Phys. 1997, 2, 69. (b) Feldheim, D. L.; Grabar, K. C.; Natan, M. J.; Mallouk, T. E. J. Am. Chem. Soc. 1996, 118, 7640. (9) Elghanian, R.; Storhoff, J. J.; Mucic, R. C.; Letsinger, R. L.; Mirkin, C. A. Science 1997, 277, 1078. (10) Sato, T.; Brown, D.; Johnson, B. F. G. Chem. Commun. 1997, 1007. (11) Whetten, R. L.; Khoury, J. T.; Alvarez, M. M.; Murthy, S.; Vezmar, I.; Wang, Z. L.; Stephens, P. W.; Cleveland, C. L.; Leudtke, W. D.; Landman, U. Adv. Mater. 1996, 8, 428. (12) Bethell, D.; Brust, M.; Schiffrin, D. J.; Kiely, C. J. Electrochem. Soc. 1996, 409, 137. (13) (a) Luedtke, W. D.; Landman, U. J. Phys. Chem. B 1998, 102, 6566. (b) Luedtke, W. D.; Landman, U. J. Phys. Chem. 1996, 100, 13323.
Analytical Chemistry, Vol. 71, No. 18, September 15, 1999 4069
this says little about the location of the hydrodynamic “outer edge” of MPCs in solutions. Exploring this outer edge is important for a variety of reasons, including capacitance and chemical reactivity characteristics. Electrochemical measurements14,15 of diffusion coefficients of MPCs with redox-labeled monolayers have been used to calculate (using the hard-sphere Stokes-Einstein relation) hydrodynamic radii (rH) (or hydrodynamic diameters dH). The values obtained sometimes seem unrealistically small, raising the question15 of whether the electrochemical measurements might contain artifacts. The present study was begun with the objective of adopting an alternative approach to diffusion coefficients, to corroborate or correct the electrochemical data. The method employed is an established hydrodynamic flow technique called the Taylor dispersion16 method. We will show that the electrochemical data appear to be generally correct and that the conversion of diffusion coefficients to hydrodynamic diameters requires use of the socalled slip condition form of the Stokes-Einstein equation, rather than the hard-sphere (“sticking condition”) form. The Taylor dispersion method is based on the Gaussian spreading of a plug of solute (MPC) solution injected into a slowly moving solvent stream in an open tubular column. The solute “plug” is spread, or dispersed, by a combination of radial diffusion of the solute in the column and the variation in cross-sectional velocity in an open tubular column. From this situation a diffusion coefficient can be calculated according to the equation16b,c,17
D)
r2t ln 2 0.23 1r2t ) 3W1/22 W1/22
(1)
where D is the diffusion coefficient of the solute, r the radius of the open tubular column, t the retention time of the solute in the column, and W1/2 the width at half-height of the eluted peak. This technique can be quite reproducible and has been used to measure D for molecular species in gases and in solutions, including nanoparticle-micelle mixtures.16b-d,g,h Experimental parameters that must be met were put forth by Taylor as16a
4L/r . Ur/D . 6.9
(2)
where L is the length of the open tubular column, r the radius of the column, U the velocity of the mobile phase, and in general, . means larger by a factor of g10. These conditions ensure that band spreading by longitudinal molecular diffusion of the solute is negligible relative to the dispersive effect of solvent flow. They are satisfied in the experiments reported here. (14) (a) Ingram, R. S.; Hostetler, M. J.; Murray, R. W. J. Am. Chem. Soc. 1997, 119, 9175. (b) Green, S. J.; Stokes, J. J.; Hostetler, M. J.; Pietron, J. J.; Murray, R. W. J. Phys. Chem. B 1997, 101, 2663. (15) Green, S. J.; Pietron, J. J.; Stokes, J. J.; Hostetler, M. J.; Vu, H.; Wuelfing, W. P.; Murray, R. W. Langmuir 1998, 14, 5612. (16) (a) Taylor, G. I. Proc. R. Soc. London, Ser. A 1954, 225, 473. (b) Yonezawa, T.; Tominaga, T.; Toshima, N. Langmuir 1995, 11, 4601. (c) Grushka, E.; Kitka, E. J. J. Phys. Chem. 1974, 78, 2297. (d) Madras, G.; Hamilton, B. L.; Mathews, M. A. Int. J. Thermophys. 1996, 17, 373. (e) Hao, L.; Lu, R.; Leaist, D. G. J. Solution Chem. 1996, 25, 231. (f) Quantitative Analysis using Chromatographic Techniques; Katz, E., Ed.; Wiley: New York, 1987; pp 359410. (g) Giddings, J. C.; Seager, S. L. J. Phys. Chem. 1960, 33, 1519. (h) Giddings, J. C.; Seager, S. L. Ind. Eng. Chem. Fundam. 1963, 1, 277. (17) Grushka, E.; Maynard, V. J. Chem. Educ. 1972, 49, 565.
4070 Analytical Chemistry, Vol. 71, No. 18, September 15, 1999
Conversion of diffusion coefficients into hydrodynamic diameters, dH, yields a measure of the overall (core and monolayer) MPC size, in a hydrodynamic sense. A general equation18a,d relating the D of a spherical object to its hydrodynamic diameter, dH
dH )
2kT 1 + 3η/βr 6πDη 1 + 2η/βr′
(
)
(3)
where k is the Boltzmann constant, T is absolute temperature, η is solvent viscosity, β is the coefficient of sliding (kinematic) friction (which varies with the nature of the solute), and r and r′ are solute and solvent radius, respectively. The solvent viscosity is taken as unmodified by the solute. As described by Chang and others,18a,d β, the friction between the solvent and solute, has a value between zero and infinity. In the limiting cases
dH ) kT/3πηD, where β . 0
(4)
for large solute molecules, with diameters larger than the solvent,
dH ) kT/2πηD, where β ≈ 0
(5)
the β . 0 case is appropriate and is termed no-slip, or sticking boundary conditions.18a-c,e This limit is the well-known hard-sphere Stokes-Einstein equation (4), which is commonly used for estimating dH from diffusion coefficients. For solute molecules with diameters similar to, or smaller than, the solvent, the slip boundary conditions and the limit of β ≈ 0 is more appropriate for relating dH to D. Equation 5 has been used with experimental data18d,f but is not as well-known. Equations 4 and 5 also result from a treatment by Landau18i that accounts for the diffusant’s viscosity. This approach yields the limiting conditions of eqs 4 and 5 by modeling a solute sphere’s viscosity as large (e.g., a hard sphere) and/or very small (e.g., a gas bubble). The MPC objects studied here are somewhat unique. MPCs have large hard cores (the Au cluster, 0.8-5 nm) surrounded by a solvent-penetrable organic monolayer (a low-viscosity shell) and thus are candidates for both the sticking and slip conditions. Herein, MPC dH values, calculated by eqs 4 and 5 from Taylor dispersion diffusion coefficients, will show that the slip conditions are more appropriate in the MPC case, yielding hydrodynamic diameters larger than the TEM-determined metal core diameter. That is, the use of eq 5 to calculate dH values that favorably compare to summation of the core diameter and extended chain lengths shows that the “outer edge” of the MPC is a soft surface freely draining of solvent, and with only small friction values (slip conditions). (18) (a) Li, J. C. M.; Chang, P. J. Chem. Phys. 1955, 23, 518. (b) Lamanna, R.; Delmelle, Cannistraro, M. Phys. Rev. E 1994, 49, 5878. (c) Zwanig, R. J. Chem. Phys. 1983, 79, 4507. (d) McLaughlin, E. Trans. Faraday Soc. 1959, 55, 28. (e) Parkhurst, H. J.; Jonas, J. J. Chem. Phys. 1975, 63, 2705. (f) Kivelson, D.; Jensen, S. J. K.; Ahn, M.-K. J. Chem. Phys. 1972, 58, 428. (g) Lammo, O.; Sjostedt, G. Trans. Faraday Soc. 1938, 34, 1158. (h) Einstein, A., Furth, R., Eds. Investigation on the Theory of Brownian Movement; Dover: New York, 1956. (i) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon: New York, 1987.
Figure 1. Cartoon of MPC core and monolayer and monolayer structures.
The Taylor measurements described are of MPCs with alkanethiolate monolayers, dissolved in toluene and dichloromethane, and of MPCs with tiopronin6 monolayers, dissolved in water. The alkanethiolate MPCs are of several different core sizes (as produced by the synthetic conditions and as characterized elsewhere7a) and of two different chain lengths. The tiopronin MPC studies encompassed four different core-sized species. EXPERIMENTAL SECTION The Taylor apparatus consisted of a Waters model U6K injector in-line with a Waters model 1050 HPLC pump, a 12-m silica capillary (from Micron) with an inner diameter of 125 µm, and a Waters model 484 UV-vis detector. All runs were performed at a mobile phase flow rate of 0.1 mL/min at 25 °C. These dimensions satisfy the Taylor criteria (eq 2: 4 × 105 . 4 × 104 . 6.9). The UV-vis detector wavelength was typically set to 250 nm, but was adjusted to 500 nm for larger MPCs whose absorbance at 250 nm overloaded the detector. The data were collected on a Waters model 745 B data module (strip chart recorder). W1/2 was measured manually. MPCs with alkanethiolate and tiopronin monolayers (Figure 1) were synthesized and characterized as to core size (determined by TEM) and monolayer composition as described previously.6,7a The core sizes reported are number-average values from TEMsize histograms. These numbers match well with those from previous SAXS studies,7a which gave the same value for core diameter generally within 10% of the TEM analysis. All solvents were degassed before use. The mobile-phase solvents used were toluene, dichloromethane, hexane, tetrahydrofuran, and filtered water (Nanopure by Barnstead). MPC concentrations ranged from 1 to 200 µM. Each reported diffusion coefficient was measured at least four times at several different concentrations and injection volumes (1-20 µL) to ensure that these components were not factors in the measured diffusion coefficients. Measurements with four different toluene flow rates (0.1, 0.2, 0.3, and 0.4 mL/min) were made with a C12 (3×, 0 °C, FD; see Table 2 for details) MPC giving the same value within error limits (2.9 ( 0.3, 2.8 ( 0.3, 2.9 ( 0.2, and 3.0 ( 0.1 × 10-6 cm2/s) ensuring no solvent velocity artifacts in the measurements. Column overloading, brought on by large-volume injections of the most concentrated samples, caused artificially large W1/2 values and low D values, which were discarded. Experiments with hexane and tetrahydrofuran solutions gave elution peak profiles that were substantially “tailed”, suggestive of adsorption to the capillary walls. These data were also not used. For aqueous
Figure 2. (s) C12 (1/6 X, 0 °C, FD) MPC elution peak from a capillary column of 12-m length and 125-µm radius. The peak was detected by absorbance at 500 nm, [MPC] ) 1.1 × 10-6 M in toluene; (O) is a fitted Gaussian line.
tiopronin MPC solutions, the column was preconditioned with 0.1 M NaOH for 1 h at a flow rate of 0.1 mL/min. The pH of the tiopronin MPC sample was adjusted to 6.9 (the pH of the Nanopure H2O (Barnstead) used as the mobile phase) by addition of 0.1 M NaOH. Buffer solutions were tried but lead to sharper, Lorentzian-type, or oddly bent elution curves. Occurrence of such effects is the main limitation of the Taylor method. RESULTS AND DISCUSSION A representative Taylor dispersion peak for a dodecanethiolate MPC (C12 MPC) is shown in Figure 2. The peak (solid line) is well-represented by a fitted Gaussian (open circles with dotted line), and the peak width from the fitted curve allows calculation of the MPC diffusion coefficient, D, from eq 1. This amount of tailing is typical and is not thought to be of consequence in the measurement of W1/2. The experimental setup is quite simple, and once the flow parameters have been established and an adequately Gaussian response ascertained, results such as Figure 2 were obtained very reproducibly. The method was tested on species with known, electrochemically determined (microelectrode voltammetry19) diffusion coefficients: ferrocene20a (Fc) in acetonitrile with tetra-n-butylammonium perchlorate and [(trimethylammonio)methyl]ferrocene20b (FcTMA) in water with NaNO3. The agreement20 of the Taylor numbers with the previous results (8% for Fc and 2% for FcTMA) was excellent. A second test of the Taylor D values was a comparison to previously determined14,15 (with microelectrodes) diffusion coefficients for several redox group-modified MPCs. Table 1 shows comparisons of electrochemical data for anthraquinone, ferrocene, phenothiazine, and viologen-labeled MPCs with Taylor results for (19) (a) Wightman, R. M. Science 1988, 240, 415. (b) Ewing, A. G.; Dayton, M. A.; Wightman, R. M. Anal. Chem. 1981, 53, 1842. (20) Fc was reported in acetonitrile in ref 20a as D ) 2.6 × 10-5 cm2/s. The Taylor result for Fc (1 mM/0.1 M tetra-n-butylammonium perchlorate/ acetonitrile) was D ) 2.4 ( 0.2 × 10-5 cm2/s. A 2 mM FcTMA solution (2 M NaNO3) in water was reported in ref 20b as D ) 5.0 × 10-6 cm2/s. The Taylor value (same concentrations of FcTMA and electrolyte) was D ) 5.1 ( 0.3 × 10-6 cm2/s. (a) Wooster, T. T.; Longmire, M. L.; Zhang, H.; Watanabe, M.; Murray, R. W. Anal. Chem. 1992, 64, 1132. (b) Fan, F. F.R.; Bard, A. J. Science 1995, 267, 871.
Analytical Chemistry, Vol. 71, No. 18, September 15, 1999
4071
Table 1. Comparison of Taylor Dispersion and Voltammetry Diffusion Coefficients MPC speciesa
av diam of MPC coreb (nm, dTEM)
electrochem diffusion coeff (10-6 cm2/s)
Au314[(FcC8)10(AQSC3)13(C8)60] Au314[(FcC8)6(C8)84] Au314[(C10)90] Au140[(PTZC12)9.7(C12)44] Au140[(PTZC12)3.2(C12)51] Au140[(C12)53] Au140[(PTZC12)9.7(C12)44] Au140[(PTZC12)3.2(C12)51] Au140[(C12)53] Au201[(MV)24(tiopronin)61] Au201[(tiopronin)85]
2.0 2.0 2.1 1.6 1.6 1.6 1.6 1.6 1.6 1.8 1.8
2.7 2.6
Taylor diffusion coeff (10-6 cm2/s)
solvent 2:1 toluene/CH3CN 2:1 toluene/CH3CN toluene 2:1 toluene/CH3CN 2:1 toluene/CH3CN toluene CH2Cl2 CH2Cl2 CH2Cl2 water water
2.2 2.2 2.6 2.9 2.5 2.8 2.6 3.1 1.8
a Nomenclature for the MPC: For example, the first species has a 314 Au atom core and the monolayer is a mixture of ferrocene (Fc) attached to octanethiolate chains (C8), anthraquinone attached to propanethiolate chains (C3), and octanethiolate chains (C8). Other abbreviations: C12 ) dodecanethiolate; PTZ ) phenothiazine attached to a dodecanethiolate chain via an ester bond; MV ) methyl viologen attached to tiopronin via an amide bond. b Determined by TEM. The value is an average; the standard deviation is typically about a third of the average value. c Calculated from limiting currents in microelectrode voltammograms (ref 19).
Table 2. Taylor Dispersion Results for Alkanethiol-MPCs (dH ) 2rH)
sample no.a
combined diam, dTEM + 2L(nm)b
Ddichloromethane/ Dtoluene
Taylor diffusion coeff -6 (10 cm2/s)c toluened CH2Cl2d
1
C12 (3X, 0 °C, FD)
1.6 + 2 × 1.5 ) 4.6
1.3
2.9 ( 0.2
3.7 ( 0.1
2
C10 (3X 0 °C, FD)
2.1 + 2 × 1.4 ) 4.9
1.5
2.2 ( 0.1
3.3 ( 0.1
3
C10 (3X, 0 °C, FD) cut Df
1.6 + 2 × 1.4 ) 4.4
1.4
2.5 ( 0.1
3.6 ( 0.1
4
1.6 + 2 × 1.5 ) 4.6 4.8 + 2 × 1.5 ) 7.8 (1:1 molar mix) 4.4 + 2 × 1.5 ) 7.4
1.7
2.0 ( 0.1
3.4 ( 0.1
5
C12 (3X 0 °C, FD) + C12 (1/10) X, 0 °C, FD) C12 (1/6X, 0 °C, FD)
1.5
1.9 ( 0.0
2.9 ( 0.2
6
C12 (1/10X, RT, FD)
4.8 + 2 × 1.5 ) 7.8
1.3
1.9 ( 0.1
2.5 ( 0.1
7
C12 (1/12X, RT, FD)
5.2 + 2 × 1.5 ) 8.2
2.1
1.7 ( 0.2
3.5 ( 0.0
dH (nm)c sticking conditions/slip conditions toluene CH2Cl2 2.5 ( 0.2 3.8 ( 0.3 3.4 ( 0.1 5.1 ( 0.2 3.0 ( 0.1 4.4 ( 0.2 3.7 ( 0.2 5.5 ( 0.3 3.9 ( 0.0 5.8 ( 0.0 3.9 ( 0.2 5.8 ( 0.3 4.4 ( 0.5 6.5 ( 0.8
3.0 ( 0.1 4.5 ( 0.1 3.4 ( 0.1 5.1 ( 0.2 3.1 ( 0.1 4.7 ( 0.1 3.3 ( 0.1 4.9 ( 0.1 3.9 ( 0.3 5.8 ( 0.4 4.5 ( 0.2 6.7 ( 0.3 3.2 ( 0.0 4.8 ( 0.0
% ligand detected,e sticking conditions/ slip conditions toluene CH2Cl2 30 ( 2 73 ( 5 46 ( 2 104 ( 5 50 ( 2 100 ( 4
47 ( 1 97 ( 3 46 ( 1 107 ( 3 54 ( 2 110 ( 3
50 ( 0
47 ( 3
33 ( 2
64 ( 3
43 ( 5
a Sample coding describes conditions for MPC synthesis (ref 7a): for example, sample 1 C12 (thiol chain length), 3X (molar excess thiol/Au) 0 °C (reaction temp), FD (BH4- reductant added rapidly). b Combined diameter modeled as diameter determined by TEM (dTEM) plus twice extended chain length (2L) as calculated from Hyperchem molecular modeling software. c Calculated by eq 4 (sticking conditions) or 5 (slip conditions). d η ) 0.50 and 0.39 cP for toluene and dichloromethane, respectively. e Calculated by eq 6. f Cut D refers to a monodisperse, sizefractionated MPC core. For more details on the precipitation fractionation technique, see ref 3c.
unlabeled MPCs with analogous monolayers and similar core sizes. A goal of the present study was to corroborate the electrochemical determinations, which potentially were complicated14b by the unusual problem of correcting for the diffusioncontrolled double-layer charging currents of the MPCs. The Taylor results for the alkanethiolate MPCs in the organic solvents proved to be very close ((2-15%) to the microelectrode results. (Small differences are readily attributed to differences between an MPC with a pure alkanethiolate monolayer and one with some fraction of the chains in the monolayer being labeled.) This important comparison provides mutual validation of the electrochemical and the Taylor determinations for alkanethiolate-monolayer materials. The comparison also shows that MPC diffusion coefficients can be measured without electrochemical labels. The comparison for tiopronin and tiopronin/viologen MPCs, on the other hand, shows a 40% difference between the two measurements, which is less satisfying. A further discussion of the tiopronin MPCs is given later. Table 2 presents Taylor diffusion coefficient results for a series of alkanethiolate MPCs dissolved in toluene and dichloromethane. The ratios of D values in the two solvents should, by either eq 4 4072 Analytical Chemistry, Vol. 71, No. 18, September 15, 1999
or 5, be the inverse ratio of their viscosities21 (ηtoluene/ηdichloromethane), which is 1.5. Inspection of the Taylor results for D shows that (except for sample 7), they do indeed display a ratio of 1.5 ( 0.1 (Ddichloromethane/Dtoluene). The MPCs in Table 2 differ primarily in terms of their average core size, which by TEM ranges over a factor greater than 3-fold. The combined MPC diameter, modeled as the core diameter plus twice the length (L) of a fully extended monolayer chain, varies by a factor of ∼2-fold. The Taylor diffusion coefficients decrease with increasing core size, but span a range of values smaller than 2-fold. The rationale of these changes is given in the following section. The Taylor diffusion coefficients in the two solvents were converted to hydrodynamic diameters, for both sticking and slip conditions. Inspection of eqs 4 and 5 shows that the slip limit yields dH values 1.5-fold larger than the sticking limit. In the past, slip conditions have been used for small solute molecules, where friction between the solute and solvent is ideally small.18d,f The various results for dH are shown in Table 2. (21) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 72nd ed.; CRC Press: Boca Raton, 1992.
Starting with the smaller core MPCs (samples 1-3), we see that the sticking condition assumption produces values of dH that are larger than the necessary minimum of the MPC core diameter (dTEM) but that are definitely smaller than the combined MPC core plus monolayer dimension (dTEM + 2L). Assuming slip conditions for samples 1-3, on the other hand, produces values for dH that are in generally good agreement with the dTEM + 2L values. For the larger samples (5-7), the sticking model produces dH values less than the dTEM core diameter itself, which of course is a physical impossibility, showing that this model is inappropriate for these materials. Calculation of dH based on the slip model produces values larger than dTEM but about a half monolayer short of a dTEM + 2L overall MPC dimension. (We were unable to measure Taylor D’s for MPCs with chains shorter than decanethiolate owing to their tailed elution peaks, indicative of adsorption to the capillary walls.) The above results strongly suggest that conversion of diffusion coefficients to hydrodynamic diameters using the sticking condition of eq 4, as has been done in previous publications,14,16b underestimates those values by up to 1.5-fold. The sticking (hardcore) model is clearly incorrect for larger core MPCs and, by inference, probably also for smaller core versions. Because the core dimension (dTEM) plus the monolayer dimension (2L) is usually larger than dH, the results for the MPCs can be interpreted in two ways: (i) They can be regarded as spheres with diameters equal to some fraction of that predicted by extended alkanethiolate chain lengths, inferring folding of the extended chains back to form a “hard” hydrodynamic interface. This seems somewhat unlikely considering that solubility properties of MPCs are dominated by the nature of the monolayer and considering that, for alkanethiolate MPCs, the core is more polar than the monolayer. (ii) The MPCs could be viewed as having an outer boundary defined by the termini of extended MPC chains offering a minimal drag resistance to hydrodynamic motion. The inference then is that the outer portions of the alkanethiolate chains are extensively penetrated by and are “free draining” of solvent and have substantial disordersa picture consistent with spectroscopic7d and chemical reactivity7c studies indicating that the alkanethiolate chains are liquidlike. The inner portions of the monolayer are not necessarily free draining (a strong radial gradient of chain density exists), and the friction to solvent motion must increase within the monolayer (as shown in the cartoon of Figure 3). It is important to remember that the friction between monolayer and solvent deals with the drag force exerted on the MPC by solvent, which is not exactly equivalent to how far individual solvent molecules penetrate into the monolayer. Thus, it is a matter less of how far a few solvent molecules penetrate than of how much of the chains act as free draining of solvent. This is the more likely model. It is instructive to estimate the amount of “hard” interface ligand and “free-draining” ligand detected by eqs 4 and 5, respectively. This fraction, dubbed “% chain detected”, is calculated as
% chain detected ) 100
(
)
dH - dTEM 2L
(6)
where dTEM is the TEM measured core diameter and 2L is twice
Figure 3. Cartoon of MPCs, hydrodynamic radius (dH), TEM core diameter (dTEM), and solvent (S) for cases in which the apparent monolayer/solvent boundary at which viscous drag occurs is located at the termini of extended chains (1) or at some shorter dimension (2).
the fully extended monolayer ligand length. Results are shown in Table 2; blank entries indicate the obviously not meaningful cases where dH < dTEM. The pictures that emerge of the MPCs in solution are again illustrated by the Figure 3 cartoon, showing the MPC dimensions offering drag force to the solvent. Examination of Table 2 shows that small-core MPCs (samples 1-3) offer, according to the slip model, a hydrodynamic dimension equivalent to the lengths of fully extended alkanethiolate chains (Figure 3-1) and, according to the sticking model, a drag equivalent to ∼50% of the extended chain length value (Figure 3-2). For the larger core MPCs (samples 5-7), the slip model depicts ∼50% of the extended chain length value offering drag resistance to the solvent (Figure 3-2), and the sticking model is obviously untenable. In the context of the slip model, ligands on large cores offer less drag than those on small cores. This difference is a bit tenuous since the dimension (core and ligand diameter) of the largest MPC is only about twice that of the smallest. Possibilities for distortion of the dH results by the dispersity of core sizes in the MPC must be considered. The values for dTEM in Table 2 are number averages; histograms of core diameters show that as-prepared alkanethiolate monolayer protected clusters have a dispersity (variance) of about one-third of the average diameter. The diffusion coefficient measurement is also an average property. Table 2 contains two control experiments. Sample 4 of Table 2 is a deliberate 1:1 molar mixture of large- and small-core MPCs. As anticipated, the diffusion coefficient measured for this mixture is intermediate between, and near to the average of, the D values for its two constituent MPCs. Sample 3 is a monodisperse core size MPC prepared by precipitation-fractionation3c of sample 2. Table 2 shows that D and the dH measured for sample 3 increase and decrease, respectively, almost exactly as expected on the basis of the change in dTEM for the monodisperse MPC (1.6 nm) in comparison to the polydisperse (2.1 nm) MPC sample 2. We conclude from these two tests that, within experimental uncertainAnalytical Chemistry, Vol. 71, No. 18, September 15, 1999
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Table 3. Tiopronin-MPC Taylor Dispersion Data in Water tiopronin-MPC sample no.a
combined diam, dTEM + 2L (nm)b
Taylor diffusion coeffc (10-6 cm2/s)
DHd (nm) sticking conditions/slip conditions
% of ligand detected,e sticking conditions/slip conditions
1.8 (( 0.7) + 2 × 0.8 ) 3.4
1.7 ( 0.1
2.4 ( 0.1 3.6 ( 0.2 2.6 ( 0.3 3.9 ( 0.4 3.0 ( 0.4 4.5 ( 0.6 2.3 ( 0.1 3.5 ( 0.2 2.4 ( 0.3 3.6 ( 0.5
38 113 50 131 50 144
1
3X
2 3
3X w/ NaNO3 (4.3 mM) 1X
2.2 (( 1.0) + 2 × 0.8 ) 3.8
1.4 ( 0.2
4
(1/6) X
3.1 ((1.2) + 2 × 0.8 ) 4.7
1.9 ( 0.1
5
(1/12)X
3.9 ((1.7) + 2 × 0.8 ) 5.5
1.5 ( 0.2
1.7 ( 0.2
25
a Sample coding refers to mole ratio of tiopronin thiol and AuCl 1- used in synthesis of MPC. b Combined diameter modeled as average TEM 4 diameter (dTEM) plus twice the extended tiopronin chain length (2L). c Calculated by eq 2. d Calculated by eq 4 (sticking conditions) or 5 (slip e conditions). Calculated by eq 6.
ties, MPC core polydispersity translates to an equivalently averaged polydispersity in the measured values of dH. A series of Taylor measurements was also carried out in aqueous solutions of MPCs with tiopronin monolayers.6 The average core diameter (dTEM) was again varied on the basis of synthetic reaction stoichiometry. Results are given in Table 3. The diffusion coefficients are smaller than those in Table 2 since water is a more viscous solvent, but the changes are roughly those expected. For example, one expects Dtoluene ) 1.7(DH2O) and Ddichloromethane ) 2.6(DH2O), and D in similar core size samples, D (sample 1, Table 2)/D (sample 1, Table 3) ) 1.7 and 2.2 for toluene and dichloromethane, respectively. (This comparison obviously ignores differences in the monolayers.) The Taylor results for the smaller core tiopronin MPCs (samples 1-3) are much like those for smaller core alkanethiolateprotected MPCs; the dH calculated with the slip model agrees with the monolayer extended chain predicted (dTEM + 2L), and the sticking model gives a smaller dH that detects about half of the monolayer shell. The larger core tiopronin MPCs also give dH results from the sticking model that are physically unreasonable, and disconcertingly for the largest core example, that are unreasonable (i.e., less than dTEM) even with the slip model. Table 3 shows that the Taylor D values do not respond systematically to increasing MPC core size as do those in Table 2, further vexing a comparison with the alkanethiolate series. Table 1 shows a comparison between D values determined electrochemically for a viologen-labeled tiopronin MPC and values Taylor-determined for an unlabeled tiopronin MPC. Unlike the alkanethiolateprotected MPCs, the agreement between the electrochemical and Taylor results is not very good. The tiopronin-MPCs are more complex than the alkanethiolateprotected materials in that the former are ionic with a substantial effective charge. The experiments reported were done at pH 6.9 at which the tiopronin monolayer is ∼95% deprotonated.6 The ensuing potential complications include hydrogen-bonding interactions within the monolayer/solvent boundary, formation of a screening shell of counterions around the tiopronin MPCs, and their interaction with surface charges on the glass capillary. These complications were briefly challenged by increasing the solution ionic strength (sample 2); there was no change in the measured diffusion coefficient. Unfortunately, we were not successful in measuring D as a function of solution pH as the elution peaks 4074 Analytical Chemistry, Vol. 71, No. 18, September 15, 1999
exhibited tailing (adsorption on capillary walls) at both higher (12) and lower (3, 5) pH values. In summary, when adsorption effects can be avoided, the analysis of MPC diffusion coefficients by the Taylor dispersion method is facile and, in the case of alkanethiolate-based MPCs in organic solvents, yields diffusion coefficients in agreement with electrochemical determinations. The agreement between quite different methods is very encouraging, since the absolute accuracy of diffusion coefficient measurements by any single given technique is in general somewhat uncertain. Often (50% variability is encountered from laboratory to laboratory, a level of uncertainty that would make useless comparisons of MPC hydrodynamic diameters as conducted in Table 2. The Table 2 assessment of dH values, and their comparison to core plus monolayer dimensions, has shown that the conventional sticking boundary conditions of the Stokes-Einstein equation (4)is not appropriate for conversion of D values to dH valuessat least for MPCs with large cores, and probably for those with small cores as well. This finding, and the confirmation of the electrochemically based diffusion coefficients (Table 1), has cleared up most of the questions14 raised in previous publications in which eq 4 was employed and hydrodynamic dimensions approaching or even less than the core size were encountered. There remain, however, uncertainties regarding the diffusion measurements and their translation into dH values for ionic MPCs in aqueous media. These are illustrated by the discussion above regarding the data in Table 3. In addition, some diffusion results obtained14 for oxidized (cationic) forms of ferrocene-labeled MPCs remain puzzling and in need of further experimental probing. Use of different column materials or surface modification, to avoid adsorption effects, is an obvious direction for further work. ACKNOWLEDGMENT This research was supported in part by grants from the National Science Foundation and the Office of Naval Research. A.C.T. acknowledges support from the Dobbins fund (UNC) and the Lord Corp. (RTP) fellowship.
Received for review April 22, 1999. Accepted June 24, 1999. AC990429C
