Langmuir 2008, 24, 8245-8253
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Estimation of the Hamaker Coefficient for a Fuel-Cell-Supported Catalyst System Ram Subbaraman, Tom Zawodzinski, Jr., and J. A. Mann, Jr.* Department of Chemical Engineering, Case Western ReserVe UniVersity, CleVeland, Ohio 44106 ReceiVed January 9, 2008. ReVised Manuscript ReceiVed February 28, 2008 Hamaker coefficients are estimated for various nanoparticle-support systems commonly used in polymer electrolyte membrane fuel cells. The interaction energies, cohesion between nanoparticles, and adhesion of nanoparticles on the support are also estimated from the experimental data. Comparison between the bulk properties of platinum metal, calculated from the optical spectra available in the literature, and the nanoparticles are provided. Measurements to obtain the optical properties of the systems of interest are also reported. Implications of the van der Waals forces on the supported catalyst structure properties are discussed. The algorithm used in calculation of the Hamaker coefficient using Lifshitz theory was evaluated with known materials, using the spectral parameters available in the literature, and the results are presented.
Introduction Supported catalysts are a class of catalysts in which catalytic materials such as platinum, gold, etc. are distributed on the surface of an inexpensive support, such as carbon or alumina. Significant advantages of using supported catalysts over solid catalytic materials include increased surface to volume ratios for maximum available catalytic area and decreased cost of the catalysts. For electrocatalysts, the support material must be electrically conductive. Nanoparticles of platinum or platinum alloys supported on the surface of materials such as Vulcan XC-72 (carbon particles) are commonly used for electrocatalysis reactions in polymer electrolyte membrane fuel cells (PEMFCs). These systems catalyze both the hydrogen oxidation and oxygen reduction reactions. Figures of merit for a supported catalyst include high catalytic activity, durability under operating conditions, high dispersion, and, usually, fraction loading by weight of nanoparticles per unit weight of support. Catalytic activity depends strongly on the size and structure of these nanoparticles.1,2 Transmission electron microscopy shows both the microscopic and nanoscopic structures of these supported catalysts. Figure 1 shows the image of a catalyst cluster for a typical carbon-supported catalyst (Pt/ C). The dispersion of the nanoparticles on the support appears random, and the nanoparticles form clusters on the support. Figure 2 is in higher resolution, showing the pattern of the atom column of one nanoparticle. Nanoparticles are known to exist in icosahedron or cuboctahedron structures in their free state.3 However, TEM images show that the atoms are displaced from these wellordered shapes.4 A sample HRTEM image depicting the nanoparticle supported on a carbon substrate is shown in Figure 2. This suggests that the presence of a support influences the equilibrium structure of the catalyst particle. This view is consistent with molecular dynamic simulations done at Case.5 The simulations show the presence of strain on the nanoparticle (1) Kinoshita, K. J. Electrochem. Soc. 1990, 137, 845–8. (2) Sattler, M. L.; Ross, P. N. Ultramicroscopy 1986, 20, 21–8. (3) Yacaman, M. J.; Ascencio, J. A.; Liu, H. B.; Gardea-Torresdey, J. J. Vac. Sci. Technol., B 2001, 19, 1091–1103. (4) Markovic, N. M.; Radmilovic, V.; Ross, P. N. J. Catal. Electrocatal. Nanopart. Surf. 2003, 311–342. (5) Subbaraman, R.; Zawodzinski, T. A., Jr.; Mann, J. A., Jr. To be submitted for publication.
Figure 1. Transmission electron microscopy of a typical cluster of supported catalyst particles.
that is consistent with that estimated from the HRTEM image. The spatial deviation from the undistorted structure defines the strain field. Indeed, the strain exhibited by the catalyst particles will have a significant influence on the catalytic activity. An estimate of the strain energy created by the support can be used to identify the influence of the support on activity. This may also lead to a possibility of tailoring the supports to favorably influence the catalytic activity. One of the most commonly observed durability issues in supported catalyst systems used in fuel cells involves the detachment of the nanoparticles from the carbon support either during operation or during the processing of these catalysts to make fuel cell electrodes.6 This fact also motivates investigations aimed at understanding the nature of the support-nanoparticle interaction. No evidence of chemical bonding or any such strong interaction between the nanoparticles and support has been reported. The evidence suggests that the interaction between the (6) Xie, J.; Wood, David, L. I.; More, K. L.; Atanassov, P.; Borup, R. L. J. Electrochem. Soc. 2005, 152, A1011–A1020.
10.1021/la800064a CCC: $40.75 2008 American Chemical Society Published on Web 06/27/2008
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forces. Interactions between nanoparticles and the nanoparticles and support and the effect of intervening media are described in detail.
Experimental Section
Figure 2. High-resolution image of a Pt nanoparticle supported on a carbon substrate.
nanoparticle and support is weak and that the energy of adhesion is comparable with physisorption energies. As stated earlier, high dispersion of nanoparticles is desired to maximize the electrocatalyst area available for reaction. Poor dispersion leads to poor utilization of the catalyst due to loss in catalytic area from aggregation of particles. Aggregation of particles is a function of the energy of cohesion and is driven by a competition between the energy of adhesion of the particle with the support and energy of cohesion between nanoparticles. Loading of catalytic nanoparticles depends on the ability of the support to hold the particles. This again is dependent on the nature of the materials. Thus, the interaction forces between the nanoparticle and the support play a significant role in understanding the nature and functionality of supported catalysts. Interaction forces between two bodies in close proximity arise from various contributions such as double layer, structural, steric, depletion, hydration, and hydrophobic forces. One type of interaction which is ubiquitous is the van der Waals interaction. The Hamaker coefficient formalism represents a conventional and convenient way of assessing the magnitude of this interaction.7 The distance of separation between the bodies plays an important role in the strength of interaction. Hamaker7 constructed the distance dependence of the free energy change of macroscopic bodies by performing a pairwise summation over all the atoms in the bodies. This model is approximate and is replaced by using the Lifshitz formulation.8 The Hamaker coefficient is a material property that depends on the electron densities of the materials involved and the dielectric properties of the intervening media. In this analysis, the van der Waals force of interaction is treated as a result of fluctuations in the electromagnetic field between two objects, modified by the intervening media. Thus, the knowledge of the frequency-dependent dielectric properties of the interacting materials together with those of the intervening medium and the geometry of the bodies allows the calculation of the van der Waals forces and the resulting Hamaker coefficients, which is a function of separation, temperature, and the various dielectric properties. In this work we present an approach for estimating the van der Waals forces and the Hamaker coefficients for the nanoparticle systems of interest. Simple experimental methods are used to obtain the dielectric spectra for the materials. We present a description of these experiments and the calculation of interaction (7) Hamaker, H. C. Physica (The Hague) 1937, 4, 1058–72. (8) Lifshitz, E. M. Zh. Eksp. Teor. Fiz. 1955, 29, 94–110.
Measurement Protocol. Conventional measurements of optical properties for materials are performed on thin film samples under vacuum using ellipsometry and other dielectric spectroscopy measurements. Recently high-vacuum UV–vis-IR spectrometer measurements have been reported.9–11 One important observation is the negligible dependence of the dielectric spectra of conducting materials on very high energies. In our work, we have restricted the highest energy of measurement to the visible range. The lowest energy of measurement is in the far-IR region. To measure the optical properties for the nanoparticle systems of interest, we used a dilute dispersion of the nanoparticles in water. The Beer–Lambert function provides the means for estimating the optical properties of the nanoparticles that are required for estimating the Hamaker coefficient. The concentrations of the solutions were in the range of 0.05–0.2 vol % particles dispersed in water. This is equivalent to a number density in the range of 1015–1016 particles/mL for particles of average radius of 5 nm. This master dispersion was then diluted in varying proportions for the optical property measurements. The contribution of the scattered light component of the spectrum was then estimated from the absorbance and transmittance intensity of the spectrum. Typical scattering was found to be in the range of 5–8% of the intensity. Diluting the solution lowered the scattering. Scaling each spectrum with concentration showed that the spectra are consistent. This result also suggests that the effect of scattering is minimal in these dilute solutions. We studied several systems: stable suspensions of platinum (prepared in-house), platinum-cobalt (3:1, prepared by Sanjeev Mukerjee’sgroupatNortheasternUniversity)andplatinum-ruthenium (1:1, Johnson Mathey) nanoparticles, and Vulcan XC-72 (Etek inc.) carbon particles. Dilute suspensions of nanoparticles were prepared in water by sonication of the dispersion for 15 min. The suspension was observed to be stable during the experimental measurement period. The metal nanoparticles were observed to adsorb onto the carbon particles. To measure optical properties in the high-energy region, dualbeam UV–vis spectroscopy measurements were performed using a Perkin-Elmer Lambda 800 instrument. Two matched quartz cuvettes were used for this purpose. Pure water was used as the reference. The concentrations of the suspensions were varied to observe the log-linear dependence of absorption on the concentration of the particles. Typical concentrations ranged from 0.01 to 0.1 wt % for the particles. IR measurements were performed on the same suspensions in a quartz cuvette using a Perkin-Elmer Spectrum GX microscope referenced to a water sample for background correction. To confirm insignificant concentration effects on the absorption coefficient, samples of varying concentrations were compared; the Beer–Lambert function was followed accurately. Baseline correction was performed for the sample using Spectrum 5.3 software followed by the normalization of the absorption function. A typical combined UV-IR absorbance spectrum is shown in Figure 3. One important point of emphasis is that in spite of the simplicity of the experimental procedure and the limitation on the frequency range covered experimentally, it is powerful enough to obtain useful estimates of the Hamaker coefficients of the nanoparticles. Estimation of Dielectric Functions: Methodology and Approximations. The solvent is treated as a continuum, and the effect of solvent interactions with the particles is neglected. Since the solutions are dilute, the shielded, interparticle interactions are neglected. The effect of changing concentrations in the range of measurement is (9) Arakawa, E. T.; Inagaki, T.; Williams, M. W. Surf. Sci. 1980, 96, 248–74. (10) Ordal, M. A.; Long, L. L.; Bell, R. J.; Bell, S. E.; Bell, R. R.; Alexander, R. W., Jr.; Ward, C. A. Appl. Opt. 1983, 22, 1099–119. (11) Tan, G. L.; Lemon, M. F.; Jones, D. J.; French, R. H. Phys. ReV. B: Condens. Matter Mater. Phys. 2005, 72, 205117/1–205117/10.
Hamaker Coefficient Estimation for a Catalyst System
Langmuir, Vol. 24, No. 15, 2008 8247 ∞
∫
ωε′′(ω) 2 ε(iξ) ) 1 + P dω π 0 ω2 + ξ2
(4)
The Hamaker coefficient calculation requires computing the dielectric function along the imaginary axis. This function is a smoothly decreasing function of ξ. The data representing this function are then fit to the form of a harmonic oscillator function, as demonstrated by Parsegian.12 The typical form of the fitting function is given as
∑
ε(iξ) ) 1 +
j)1-3
Figure 3. Typical spectrum for the UV region and the IR region of absorbance for Pt nanoparticles. The patch between the UV and IR regions was provided by a spline fit of the interval around 0.9 eV. Spectra for other materials are similar up until 0 < energy j 0.2 eV, where there are features characteristic of the materials.
found to be minimal, supporting this assumption. The absorption coefficient, k, is estimated from the absorption spectra using the Beer–Lambert relationship, as in eq 1,
()
I1 λ k) ln 4πlc I0
(1)
where l is the path length given by the cuvette lateral dimension, c is the weight fraction of the particles, λ is the wavelength, and I1 and I0 are the measured intensity and the maximum intensity of the absorption spectra. The spectra were scaled with respect to the weight fractions of the dispersions. Typically absorption coefficients for particles are expressed in units of cross-section. The absorption coefficient (eq 1) was nondimensionalized using the weight fraction of the particles. The volume fraction of the particles can also be used for this purpose. The absorption coefficient is the imaginary part of the refractive index variable. Knowing the imaginary part of the complex function, we can estimate the real part of the refractive index function using a Kramers-Kronig (K-K) transformation. A Matlab program was developed for performing the K-K transformation of the optical data. The transformation function is given as
∫
(2)
The principal part of the integral is evaluated over the full range of frequency. The finite frequency domain of the experiment induces an error in the tail portion of the frequency spectrum. However, the nature of the absorption spectra ensures the tail function reaches 0 asymptotically. To match the data obtained from two different measurement techniques, the K-K transformation was performed over the whole range of data from both measurements. The bridging portion of the data is obtained by using a cubic spline to fit for the missing frequencies. The final task in the estimation of dielectric spectra includes obtaining the dielectric function along the imaginary axis. For this purpose we transform the refractive index function to the imaginary part of the dielectric function, εˆ ) ε′ + iε′′, using the following relationship:
ε′′(ω) ) 2nk
(3)
Knowing ε′′, one can then obtain the dielectric function along the imaginary axis using the K-K transformation. This is given by the following expression:
(5)
ωj2 + ξgj + ξ2
where fj, gj, and ωj are fitting coefficients for the dielectric spectra in the form of a series of harmonic oscillators. Knowing the experimental or theoretical values of the set of resonance frequencies (ωj) comes in handy in the fitting of this function. However, due to the absence of this information for the nanoparticles of interest, we used a nonlinear least-squares fitting routine of the Igor graphing software to fit for all parameters, including ωi. The fit parameters were then used in the calculation of the Hamaker coefficient using the Lifshitz relationship. This best fit function is used to estimate the Hamaker coefficient for interaction between two semi-infinite planes from the following equation:13 ∞
∞
∞
-
-
∑ ∑ ∫ p[(∆Am∆Bm)
1 3 AAmB(l) ) - kT ′rn2 2 n)0 q)1 q
q
+
1
]
(∆Am∆Bm)q e-rnpq dp (6) where the prime in the summation indicates the zero term is multiplied by a factor of 1/2. For convenience define the function si, i ) A, B, or m, by
si ) Then
¯ ij ) ∆
p2 - 1 +
siεj - sjεi siεj + sjεi
In eq 6
rn )
∞
k(ω′) 1 n)1+ P dω′ π -∞ ω′ - ω
fj
and
() εi εm
∆ij )
(7)
si - sj si + sj
( )
2lε1⁄2 ξn c
(8)
(9)
and l is the distance of separation between the planes. ξn are the eigenfrequencies at which the dielectric function ε′′ is evaluated for the materials A, m, and B:
ξn )
2πkBT n (n ) 0, 1, 2, ..., ∞) p
(10)
kB is Boltzmann’s constant, p ) h/2π, where h is Plank’s constant, c is the speed of light in a vacuum, and T is the temperature of interest. The frequency domain of the dielectric spectrum is limited experimentally and does not include either the zero frequency term or the infinite frequency term. The observed dielectric function in the accessible spectral range is a uniformly decreasing function at high frequencies approaching zero as shown in Figure 4. Hence, the decaying nature of the harmonic oscillator function reasonably extrapolates the tailing portion at very high frequencies. To estimate the zero frequency term, we use a Drude-type approximation of the (12) Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1981, 81, 285–9. (13) Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: New York, 2006.
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Figure 4. Dielectric functions computed from eq 4. Such functions were used for the calculation of Hamaker coefficients, for example, through eqs 6 or 17 or after a least-squares fit to eq 5.
dielectric function going to zero frequency. Note that, for ideal metals, the dielectric function is unbound at zero frequency. However, for real materials we can assume that the zero frequency dielectric coefficient is determined by the dc conductivity of the materials. Since conductivity data are not available for nanoparticles, we have used the literature value for the bulk material.14 For alloys we have assumed a linear mixing rule, and hence, the conductivities are averaged for the alloys on the basis of composition. Sensitivity analysis of these parameters shows that the magnitude of the Hamaker coefficient does not change significantly for a wide range of Drude parameters. The Drude term and the expressions for the parameters are given as
ε(iξ) ) 1 +
ωp2 ωτ2 - ξ2 ωp )
+
∑
j)1-3
e2n m/e ε0
ωj + ξgj + ξ
2
(11)
-ANmS [G1 + G2 + G3] + 6 (4 ! )ANmS σ 4 10! RN
( )( )
4RNRS (RN + RS + l) - 4RNRS 2
(13)
(
G3 ) ln
(12)
where e is the electronic charge, n is the density of the conduction electrons, me/ is the effective electron mass, ε0 is the permittivity of free space, and
ωτ ) Fε0ωp2
UNmS )
G1 )
fj 2
standard systems and the samples of interest are presented in the following sections. Work of Adhesion of Nanoparticles to the Support. One of the main objectives of this work is to estimate the work of adhesion between the nanoparticles and the support surface. Particle shapes are approximated as spheres. From the TEM image of the catalyst particles in Figure 1, this seems to be a reasonable assumption. Hamaker coefficients are computed for parallel plates; however, the energy of interaction between particles is a strong function of the geometry of the particles. The geometric factor in eq 14 must be modified for different shapes of particles. To calculate the Hamaker coefficient in the presence of an intervening medium and the support, a hybrid Hamaker coefficient, ANmS, between the nanoparticle (N) and support (S) in the presence of a medium (m) is estimated. The interaction energy between the nanoparticle and the support can then be calculated from the following procedure.7 Forces are calculated from the knowledge of the Hamaker coefficient and the geometric factor. The expression was derived by Hamaker for two interacting spheres. Using the same formulation, the Born repulsion component was calculated for the r12 repulsive term.15 The attractive and repulsive components for the forces are included in the geometric factor. The interaction energy can be separated as a function of two terms. The material properties, such as dielectric properties, are separated from the geometry of the interacting components. The Hamaker coefficient accounts for the material properties in the energy expression. A functional form of the energy of interaction between two spheres is presented in the following equation:
f(λ, R) )
[
G2 )
6
1 (f (λ, R)) (14) R 4RNRS
(RN + RS + l)2 (15)
(RN + RS + l)2 - 4RNRS (RN + RS + l)2
)
(16)
-R2 - 7(λ - 1)R - 6(λ2 - 7λ + 1) + (R - 1 + λ)7 - R2 + 7(λ - 1)R - 6(λ2 - 7λ + 1) + (R + 1 - λ)7 R2 + 7(λ + 1)R + 6(λ2 + 7λ + 1) + (R + 1 + λ)7 R2 - 7(λ - 1)R + 6(λ2 + 7λ + 1) (17) (R - 1 - λ)7
where F is the resistivity of the material of interest. Typical values for these parameters were found to be in the range of 3–10 for ωτ and 100–200 for ωp. A Fortran90 program was written to perform the calculation of the Hamaker coefficient. To perform the finite integral given in eq 6, an algorithm based on Gaussian quadratures was used. A standard IMSL subroutine, QDAGI, was used with accuracy restriction of 10-5. The quadrature approach was found to be much more efficient and computationally fast compared to other numerical integration schemes. The code was run using an Intel Xeon processor with 2 GB of RAM. The summation on the n index was varied between 1000 and 10 000 terms. The values of the Hamaker coefficient obtained were found to vary very little with increasing number of summation terms on the n index beyond 1000. Similarly the maximum value for the q index was found to be 500. The execution time for the code with a maximum n value of 1000 and q value of 500 was found to be 45 min. The code was tested by using known dielectric functions to estimate well-known Hamaker coefficients for materials such as gold, mica, and water. These systems are used as standards to test the code. The values of the Hamaker coefficients for the
Here λ is the ratio of the diameters of the two interacting particles and R is the center-to-center separation made dimensionless on the diameter of the smaller particle (RN). From the absorption spectra as a function of concentration, we obtain the imaginary part of the dielectric function as described above. A typical imaginary portion of the dielectric function is presented in Figure 4. As can be observed this function is a smoothly decreasing function. This function is fit as a sum of a series of harmonic oscillators as shown in eq 5. The slowly decreasing form of this last relation shows why incompleteness of spectral information need not always impede force computation. Limited data can still be used to obtain accurate measures for forces of interaction. The spectrum is assumed to reach a constant asymptotically as the frequency approaches zero on the basis of the Drude approximation
(14) Lide, D. R. CRC Handbook of Chemistry and Physics, 83rd ed.; CRC Press: Cleveland, OH, 2002.
(15) Feke, D. L.; Prabhu, N. D.; Mann, J. A., Jr.; Mann, J. A., III J. Phys. Chem. 1984, 88, 5735–9.
]
Hamaker Coefficient Estimation for a Catalyst System given by eq 11. Ideally we would like to obtain an extrapolation function based on Drude parameters for the function to satisfy the sum rule and to match the spectra for the lowest available frequency.16 This work treats using a constant Drude term as a first pass toward an elaborate extrapolation approach. The next step after obtaining the dielectric function fit parameters is to estimate the Hamaker coefficient using the Lifshitz relationship (eq 6). Summary of the Algorithm for Calculation of the Hamaker Coefficient. (1) Measure optical spectra with UV–vis and FTIR spectrometers for various concentrations. (2) Estimate the absorption coefficient as a function of frequency from the spectra using the Beer–Lambert law, eq 1. (3) Perform K-K transformation to get the real portion of the refractive index, eq 2. (4) Use the refractive index function, eq 2, and the absorption coefficient data to obtain the dielectric coefficient’s imaginary part, eq 3. (5) Calculate the imaginary part of the dielectric coefficient along the imaginary axis from K-K transformation of the imaginary part, eq 4. (6) Apply the Drude term estimation for the materials using the electrical resistivity information, eq 11. (7) Use the Lifshitz relationship to estimate the Hamaker coefficient, eq 6. (8) Estimate the interaction energies by multiplying by the corresponding geometric factors.
Results and Discussion Hamaker coefficients and energies of interaction were calculated for various systems of interest. The temperature dependence of the Hamaker coefficients has not been considered for this work. Standard systems such as water, gold, and mica were used to verify the algorithm for Hamaker coefficient calculations. Calculations for the nanoparticle systems of interest reveal the comparable natures of interaction energies between the nanoparticles themselves and the nanoparticles with the support. The effect of solvent on these interaction energies gives us insight into some durability issues observed with these supported catalyst systems. We represent the Hamaker coefficient and the energies as ratioed to kT, where T is always 300 K and k is the Boltzmann constant. This was done so that the strength of particle–particle and particle-substrate adhesion is obvious. This also gives an idea of the interaction energy scaled with respect to the thermal energy for the particles. For example, if the value of the Hamaker coefficient for nanoparticle-substrate interaction is ∼1 kT, the binding energy for the particle is not sufficient to keep the nanoparticle adsorbed on the substrate. All the values of the Hamaker coefficients and interaction energies reported in this paper are scaled with kT for a temperature T ) 300 K. Evaluation of the Algorithm: Comparison of Gold and Water Data from the Literature. Water, gold, and mica are well-studied systems for Hamaker coefficient estimation. Complete spectral fit parameters for these systems are compiled in the literature.13 To test the algorithm, we used the published fit parameters to estimate the Hamaker coefficients for these systems in a vacuum. The Hamaker coefficient function is shown in Figure 5. The values obtained for separation distances tending to 0 have been reported in the literature.17 The algorithm provides estimates close to the values reported. The nature of the profiles for the Hamaker coefficient follows the expected trend of reaching an asymptotic value as the separation distance approaches 0. Values of the Hamaker coefficient at this plateau are reported in Table 1. Table 1 shows the values reported in the literature and the values obtained from our algorithm. The agreement between the calculated and literature values of the (16) Seitz, F. The Modern Theory of Solids; Dover Publications: Mineola, NY, 1987. (17) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991.
Langmuir, Vol. 24, No. 15, 2008 8249
Figure 5. Comparison of Hamaker coefficients with the separation, l, of reference materials of gold, mica, and water across a vacuum. Literature spectra using the parameters given by Parsegian13 were used in eqs 5 and 6 to test our software and procedure. Note that here and in other figures 1 H/kT ) 4.05 zJ. Table 1. Validation of the Computation Procedurea material
obsd value (×10-20 J)
lit. value12 (×10-20 J)
water-vacuum-water gold-vacuum-gold mica-vacuum-mica
3.66 33.21 9.83
3.7–4.0 30–50 10
a Comparison between the estimated Hamaker constants and the values reported in the literature. Spectral data for the systems were obtained from the literature.
Figure 6. Comparison of the Hamaker coefficient and energy of interaction with separation as estimated from our experimental spectra for platinum and alloy nanoparticles. A vacuum is the intervening medium.
Hamaker constants justifies the use of the algorithm for calculation of Hamaker coefficients of nanoparticle samples. Hamaker Coefficient for Nanoparticle Suspensions. The work of cohesion for each of the colloidal nanoparticle systems was calculated. The work of cohesion is defined as the free energy of interaction between nanoparticles. It is directly scaled by the Hamaker coefficient for the particles and the intervening media as shown in eq 6. Hamaker coefficients for platinum and platinum alloy nanoparticles are presented in Figures 6 and 7 for a vacuum and water as media, respectively. A spherical geometry closely describes the shape of the nanoparticles. To calculate the Hamaker coefficients and energies of interaction for a spherical system, a modified Lifshitz relationship is used.13 This is given as
8250 Langmuir, Vol. 24, No. 15, 2008 ∞
∞
∞
-
Subbaraman et al. -
∑ ∑ ∫ p[(∆Am∆Bm)
1 3 AAmB(l) ) - kT ′rn 2 n)0 q)1 q2
q
+
1
]
(∆Am∆Bm)q e-rnpq dp (18) The interaction energies are then calculated for two spheres of equal radii as given by eq 14 with RS and RN as 5 × 10-9 m. The energies closely resemble the functional form of the LennardJones 6–12 potential form with an energy well at a separation of about 1 nm. The stability of a colloidal suspension is related to the energy of cohesion between the particles. As we can expect the energy of cohesion is lower in a vacuum than that observed in water. This could be attributed to the effect of the intervening media on the interaction forces. Both the Hamaker coefficients and the energy of interaction are nondimensionalized using kT energy units. Here k is the Boltzmann constant and T is the temperature () 300 K). The Hamaker coefficient of a pure platinum nanoparticle is found to be larger than that of the alloy nanoparticles. The Pt-Co nanoparticles are of the core–shell form, whereas the Pt-Ru black particles are bulk alloy nanoparticles. These values in the presence of water as the intervening medium is presented in Figure 7. The values are smaller compared to the case of a vacuum due to the effect of the solvent in shielding the interaction between the particles. The effect of the structure of the nanoparticle and the nature of the alloying elements seems to be intermixed in the difference of the Hamaker coefficient from that of the unalloyed nanoparticle. Detailed analysis is necessary to separate these effects. Work is under way to synthesize Pt-Co nanoparticles of different types such as core–shell (Co-Pt), shell–core (Co-Pt), and mixed alloy nanoparticles and study their properties. Two important clarifications are made regarding the nanoparticles. The surfaces of the nanoparticles are bare and do not contain any capping agents or surfactants. If surfactants are used during the synthesis of the nanoparticles, indeed corrections need to be made in the above equation to account for the adlayers. Also, the optical spectra measured for the core–shell nanoparticles provide an average value of the dielectric function for the nanoparticle. As a result, the Hamaker coefficients are calculated on the basis of the overall material, similarly to that for a homogeneous alloy nanoparticle. Knowledge of the dielectric response for the individual layers would help us to calculate the Hamaker coefficient using the multilayer approach presented by Parsegian.13
Figure 7. Comparison of the Hamaker coefficient and energy of interaction with separation as estimated from our experimental spectra for platinum and alloy nanoparticles. Water is the intervening medium.
Figure 8. Comparison of Hamaker coefficients with separation, l, for bulk platinum and platinum nanoparticles.
Comparison with Bulk Platinum Behavior. To understand the difference between the bulk behavior of these catalyst materials and the nanoparticle behavior, we need to compare the Hamaker coefficients of the bulk and nanoparticle systems. We have not found estimates of Hamaker coefficients reported previously in the literature for platinum nanoparticles. A dielectric spectrum for bulk platinum was estimated using the optical spectral data obtained from the literature.18 The Hamaker coefficient calculated for bulk platinum is compared with that for the nanoparticle system in Figure 8. As shown, the value for the bulk platinum is much larger than that observed for the platinum nanoparticle. The modification of the density of states for the surface atoms for the nanoparticles compared to the bulk provides a possible explanation for such behavior. Also, the geometry of the particle provides a surface structure which can lead to possible fluctuations in the electromagnetic response of the nanoparticles. From these data, it is apparent that the energy of cohesion between nanoparticle systems with Hamaker coefficients corresponding to the bulk will be much higher compared to that of the system with real Hamaker coefficients. Thus, the use of the bulk values of platinum in our calculations would lead to an inaccurate estimation of the nanoparticle-support interaction properties. Also, the knowledge of nanoparticle Hamaker coefficients helps us in determining the effects of interaction of the support on nanoparticle catalyst particles more accurately. Particle Size Effects on the Energy of Interaction. The dispersion and particle size distribution of catalyst nanoparticles are two significant properties for supported catalyst systems. One common denominator for both issues is that of particle agglomeration. Higher agglomeration of particles leads to lower utilization of the catalyst surface area, thereby lowering the performance per unit weight of the catalysts. To understand these phenomena, we need to understand the various driving forces for the growth/agglomeration of the particles. The work of cohesion between nanoparticles provides a good starting point to estimate the tendency toward particle agglomeration. Knowledge of the interaction energy helps us to calculate cohesion energies for particles of different sizes. This gives us an idea of the stability of a colloidal solution depending on the particle size. It also allows us to directly consider the impact of particle size effects on nanoparticle stability. For an ideal supported catalyst system, the work of cohesion between the nanoparticles must be smaller than the work of adhesion with the support. This would ensure (18) Palik, E. D. Handbook of Optical Constants of Solids; Academic Press: Orlando, FL, 1985.
Hamaker Coefficient Estimation for a Catalyst System
Figure 9. Effect of the particle size for colloidal particles on the interaction energy for platinum nanoparticles.
better dispersion of the nanoparticles on the support surface. Precious metal particle sizes were varied, and the energy of interaction between nanoparticles was estimated. The energies as a function of particle size are presented in Figure 9. As we would expect, the energies scale directly with the particle sizes. The higher energy of interaction between smaller particles indicates the decreased stability of the colloidal suspensions of smaller particles compared to those made of larger particles. The smaller the colloidal particle, the more destabilized it is. The minimum energy shifts well toward a higher distance of separation for larger particles. This is a direct consequence of the increased particle size leading to an increased collision cross section for the particles. Particle-Support Interaction. Fuel-cell-supported catalysts consist of catalyst nanoparticles dispersed on an electronconducting support such as carbon. The choice of the support mainly depends on the ability to adhere catalyst particles on it, the dispersability of the nanoparticles on the surface, its conductive properties and stability, and the absence of impurities that inhibit the catalytic activity of the metal nanoparticle. The nanoparticle-support interaction should play a major role in the selection of a support material. Strong adhesion of the nanoparticle on the support is desired. This keeps the electronic contact between the reactive catalytic site and current collector/potential surface of the support. Comparable values of nanoparticle-support and nanoparticle-nanoparticle interaction energies are desired to be able to disperse the catalysts uniformly on the support surface without significant agglomeration of the catalyst particles. Catalytic activity is a strong function of the catalyst structure and is influenced strongly by the presence of edges, grain boundaries, and defects. The strained structure observed from the HRTEM image in Figure 2 is mainly due to the interaction between the support and nanoparticle. Hence, the support choice should not significantly decrease the activity of the metal nanoparticle. A simple stochastic Monte Carlo model was developed to simulate the formation of supported catalyst particles from a colloidal suspension of nanoparticles.19 This model describes the evolution of the supported catalyst structure on the basis of the notions of nanoparticle-support interactions. Preliminary results19 were based on using bulk properties of the metal and graphite. Knowledge of the actual values of the Hamaker coefficients for the nanoparticle and support allows us (19) Subbaraman, R.; Zawodzinski, T. A., Jr.; Mann, J. A., Jr. Proc.sElectrochem. Soc. 2004, 21, 72–84.
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Figure 10. Hamaker coefficients and energy of interaction between platinum nanoparticles and the carbon support with a vacuum as the intervening medium.
Figure 11. Hamaker coefficients and energy of interaction between platinum nanoparticles and the carbon support with water as the intervening medium.
to refine the model to obtain more realistic structures. Results for catalyst–support interaction are presented in Figures 10 and 11. Hamaker coefficients are estimated for the case of nanoparticle-support interaction in the presence of a vacuum and water using eq 6. The values of the interaction energies for the corresponding cases are presented in the figures for a vacuum and water, respectively. The energies of interaction are calculated using typical particle size values of RN ) 5 nm and RS ) 150 nm, respectively. Since we are interested in obtaining the Hamaker coefficient function as a function of large to small separations, we have not used the commonly applied Derjaguin approximation for our calculations. The Derjaguin approximation is typically used17 for a sphere-sphere interaction to estimate the force of interaction between two spheres in terms of the energy per unit area for two flat surfaces. When the radius of curvature of one particle is significantly higher than the other, this approximation simply leads to the limiting case of sphere-plane interaction. The significant difference between the radii of the nanoparticle and the support validates this approximation. The energy well is observed in the range of 1–3 nm, which is of the same order of magnitude as that between the nanoparticles themselves. This further suggests the need for the use of the full form of sphere-sphere interaction instead of the sphere-plane approximation. Results are presented for two media of interest. Water is the medium commonly used during the preparation of the supported catalysts. A typical procedure involves mixing a
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slurry of carbon support with a nanoparticle suspension. Supported catalyst particles are separated by gravity filtration and dried for use in fuel cells. The driving force for the absorption of nanoparticles on the carbon support is that of the interaction energy between the support and nanoparticles. Energies are compared between three different nanoparticle materials. Pure Pt, Pt-Co, and Pt-Ru are commonly used nanoparticle catalysts in fuel cell electrodes. The nature of the nanoparticles plays a major role in determining the binding with the support. A comparison of the energies suggests that the platinum particle interacts the strongest compared to the alloy nanoparticles. This suggests an alloy of platinum with cobalt or ruthenium has a weaker interaction with the support compared to that of platinum. From Figure 7, we observe that the nanoparticles have comparable Hamaker coefficient values. This suggests that the energies of interaction between the nanoparticles are similar for Pt, Pt-Co, and Pt-Ru nanoparticles. Formation of clusters of nanoparticles depends on both the nanoparticle-support and nanoparticle-nanoparticle interactions. Thus, for systems with comparable nanoparticle-nanoparticle interactions, the nanoparticle-support system decides the extent of clustering or agglomeration of nanoparticles on the support. Thus, by comparing the support interactions of Pt, Pt-Co, and Pt-Ru, we can conclude that the agglomerate sizes will tend to be larger for the alloy cases compared to that of pure platinum. Effect of Intervening Media. The medium of interaction plays an equally important role in the structure of the supported catalyst. As we have observed, the intervening medium plays an important role in attenuation or enhancement of the interaction energies. By tuning the nature of the solvents, we can destabilize one interaction, such as the nanoparticle-support interaction, while keeping the other interaction, such as the nanoparticlenanoparticle interaction, constant, thereby modifying the energetics driving the local structure of sthe upported catalyst. Fuel cell electrodes are typically prepared by mixing the catalyst particles with a mixture of polymeric binder (typically Nafion) and other solvents (2-propanol, glycerol, etc.) to form an ink. This ink is then brushed onto a decal or the polymer membrane followed by drying. In the case of the decal the dried mixture is transferred onto a membrane by hot pressing the decal onto the membrane. The presence of other solvents in the preparation process leads to a need for studying the effect of the solvent on the various interaction energies of the supported catalyst system. A medium can have two types of impact on a supported catalyst particle. If the medium attenuates the interaction between the support and nanoparticle in such a way that the energy of adhesion between the nanoparticle and support is less favorable compared to the energy of cohesion of the particles in the solution, then the particles will tend to form higher order agglomerates on the surface and/or dislodge the particles off the support surface. The simplest nonaqueous solvent that is commonly used in the preparation of fuel cell catalyst inks is isopropyl alcohol (IPA). The dielectric fit parameters for IPA were obtained from the literature.20 Figure 12 shows a comparison between the works of adhesion of a nanoparticle on the carbon support in the presence of IPA and water. The energy of adhesion is lower in the presence of alcohol, indicating the destabilization of the particle-support interaction. Figure 13 shows a comparison of the energy of cohesion between platinum nanoparticles in the IPA medium and the energy of adhesion between platinum and the support in the IPA medium. The higher energy of cohesion suggests that the particle-support interaction will be destabilized in favor of the particles being in solution. This calculation is very significant (20) Bell, N.; Dimos, D. Mater. Res. Soc. Symp. Proc. 2001, 624, 275–280.
Subbaraman et al.
Figure 12. Comparison of the interaction energy of platinum nanoparticles with the carbon support in various media.
Figure 13. Competition between the energy of cohesion between nanoparticles and the energy of adhesion between a nanoparticle and the carbon support.
from the point of view of understanding the durability of catalyst particles in fuel cells. Knowledge of the effect of the solvent on the interaction will help us to predict the modification of the structure of the supported catalyst during the preparation of the catalyst ink for electrodes. The nature of solvents suitable for the process of making the catalyst ink could be modified suitably such that the “free” catalyst structure is not affected significantly during the process. Implications for PEMFC Catalysts. Molecular quantum mechanics models have been developed for studying the catalytic properties of fuel cell catalyst materials.21 Ordered structures such as (111), (100), and (110) are typically used for this purpose.21 Recently reports of structures with edges and kinks on the surface have been considered for these calculations. The actual nanoparticle structure of the catalyst particle has been very difficult to predict. Typical shapes of unsupported nanoparticles have been reported in the literature as icosahedrons and cuboctahedrons.3 However, these structures are seldom observed in the real supported catalyst systems. The presence of the support material affects the structure of the nanoparticle catalysts, with concomitant effects on catalyst performance. Knowledge of the energy of interaction between the support and nanoparticles helps us to identify/predict this distorted structure. Using the interaction energy as a boundary energy value for a molecular dynamics (21) Koper, M. T. M.; Van Santen, R. A.; Neurock, M. Catal. Electrocatal. Nanopart. Surf. 2003, 1–34.
Hamaker Coefficient Estimation for a Catalyst System
box, one can predict the nanoparticle structure due to the presence of the boundary. Preliminary results indicate the onset of distortion of the symmetric, platonic structure due to the introduction of “boundary energy”. Thus, knowledge of the interaction energies aids us in the prediction of the actual catalyst nanoparticle structure existent in the supported catalyst system. Dissociation of particles from the support and agglomeration of catalyst particles on the support are some of the commonly observed issues with supported catalysts during the electrode preparation process. Calculations show that the nature of the medium used for this process has a significant impact on the dispersion and agglomeration of catalyst nanoparticles on the support. Recent results suggesting alloy nanoparticles are more durable compared to pure nanoparticles have been reported.22 The difference in energies of interaction of the alloy and pure nanoparticles with the carbon support suggests a possible explanation for this phenomenon. Thus, understanding support-nanoparticle interactions helps us to answer two important questions regarding the supported catalyst system.
Conclusions A simple method for the estimation of Hamaker coefficients for materials has been demonstrated. The ability of the method to obtain information from materials in their native state (nanoparticle suspension) provides an ease of use and more realistic values. Verification of the algorithm with standard materials confirms the validity of the code used for calculation. (22) Ball, S.; Hudson, S.; Theobald, B.; Thompsett, D. ECS Trans. 2006, 1, 141–152.
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Fuel-cell-supported catalysts, chosen as the material system of interest, have been characterized. Interaction energies of platinum nanoparticles are estimated and compared with their corresponding bulk values. The disparity between the bulk and nanoparticle values shows a net effect in variation of the geometry of the particles along with the modification of the density of states of the surface atoms. Further analysis of this behavior is necessary to quantify this effect. Properties of the alloy nanoparticles are compared with those of the unalloyed platinum nanoparticles. The effectiveness of the calculations to show the effect of the particle size and medium is presented. The relevance of the interaction energies for the fuel cell catalyst system has been explained. Knowledge of the Hamaker coefficient equips us with the ability to predict the “real” structure of catalyst particles and to choose/tailor support materials to obtain the desired properties for supported catalysts. Durability issues such as detachment of nanoparticles and agglomeration of the same can be addressed with the help of the interaction force approach. This method will contribute significantly to progress in the understanding of supported catalysts for PEM fuel cells and in the design of stable, active, dispersed catalysts for PEMFCs. Acknowledgment. We thank the Army Research Office for their financial support of this work through MURI Grant No. DAAD19-03-1-0169. We also acknowledge Dr. Frank Ernst and Dr. Kui Du for their HRTEM image of platinum nanoparticles on the carbon support, Figure 2. We thank the reviewers for their careful reading of the manuscript and their useful remarks. LA800064A
