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Rotational Diffusivity of Fractal Clusters Marco Lattuada, Hua Wu, and Massimo Morbidelli* Institut fu¨ r Chemie- und Bioingenieurwissenschaften, Swiss Federal Institute of Technology Zurich, ETHZ, ETH-Ho¨ nggerberg/HCI, CH-8093 Zu¨ rich, Switzerland Received February 23, 2004. In Final Form: April 21, 2004
The rotational diffusion behavior of fractal clusters generated through an off-lattice cluster-cluster aggregation algorithm in both diffusion-limited cluster aggregation and reaction-limited cluster aggregation conditions is investigated. The extended Kirkwood-Riseman theory (Garcia de la Torre et al., Macromolecules, 1987) is used to estimate the cluster rotational diffusion tensor. The three eigenvalues of this tensor, which correspond to the three main rotational diffusivity values of the cluster, have been computed for each generated cluster. Once the eigenvalues have been sorted in ascending order, each of them has been averaged over several thousands of clusters. It is found that one of the three main average rotational diffusivities is substantially larger than the other two, indicating significant anisotropy of fractal clusters. Moreover, a rotational hydrodynamic radius Rh,r has been determined on the basis of the mean value of the three average rotational diffusivities, which is about 25% larger than the mean translational hydrodynamic radius Rh calculated through the same Kirkwood-Riseman theory. Finally, the obtained Rh,r values have been applied to interpret dynamic light scattering data from aggregating colloidal systems and to investigate the reliability of the assumption, Rh ) Rh,r, typically made in the literature.
Introduction The quantitative investigation of the aggregation behavior of colloidal dispersions has been substantially boosted by the discovery that the structure of colloidal clusters can be rationalized using the concepts of fractal geometry.1-5 Since the first experimental observation of the fractal nature of colloidal clusters, many efforts have been devoted to the development of experimental techniques that can be used to characterize the structure of colloidal clusters.5,6 Among the many possible techniques, light scattering is perhaps the most used one, because it is noninvasive and allows recovering information about both the size and the fractal dimension of the clusters.5 In particular, static light scattering (SLS) experiments can be used to determine the average scattering structure factor of the clusters, from which the cluster average radius of gyration and fractal dimension can be estimated. Dynamic light scattering (DLS) measurements, on the other hand, give the possibility of estimating the average diffusion coefficient of the clusters, from which the average translational hydrodynamic radius can be obtained. For a better characterization of the aggregation kinetics of colloidal dispersions, it is convenient to combine DLS and SLS measurements, because the average radius of gyration and the average translational hydrodynamic radius correspond to different moments of the cluster mass distribution (CMD) and, therefore, provide together information * Corresponding author. E-mail:
[email protected]. Tel.: 0041-01-6323034. (1) Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman: New York, 1982. (2) Forrest, S. R.; Witten, T. A. J. Phys. A 1979, 12, L109. (3) Witten, T. A.; Sander, L. M. Phys. Rev. Lett. 1981, 47, 1400. (4) Family, F.; Landau, D. P. Kinetics of Aggregation and Gelation; North-Holland: Amsterdam, 1984. (5) Sorensen, C. M. Aerosol Sci. Technol. 2001, 35, 648. (6) Bushell, G. C.; Yan, Y. D.; Woodfield, D.; Raper, J.; Amal, R. Adv. Colloid Interface Sci. 2002, 95, 1.
about the broadness of the distribution.7 This is very useful because a direct determination of the CMD is typically a difficult task. However, to extract quantitative information about the CMD from the time evolution of the average translational hydrodynamic radius and the average radius of gyration, the structure properties of the individual fractal clusters are required. These may be obtained by analyzing the structure of clusters generated by Monte Carlo simulations. In particular, by simulating the aggregation processes under either diffusion- or reaction-limited conditions, we can obtain the particle-particle correlation function for a cluster of any mass and from this compute its radius of gyration Rg, scattering structure factor S(q), and translational hydrodynamic radius Rh.8,9 When these structure properties of the individual clusters are combined with the CMD calculated through population balance equations (PBEs), it was found that the time evolution of the average properties measured by light scattering can be correctly simulated in both reactionlimited and diffusion-limited aggregation conditions. One additional complication arises because the average translational hydrodynamic radius of fractal clusters has been shown to depend also on the rotational motion of the cluster.10,11 In other words, a conventional DLS measurement does not probe solely the translational motion of the cluster but receives a contribution also from the cluster rotational motion. To account for this complex phenomenon, a detailed theory and a simplified approach have been developed,10,11 both of which, however, require some (7) Lattuada, M.; Sandku¨hler, P.; Wu, H.; Sefcik, J.; Morbidelli, M. Adv. Colloid Interface Sci. 2003, 103, 33. (8) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 106. (9) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 96. (10) Lindsay, H. M.; Klein, R.; Weitz, D. A.; Lin, M. Y.; Meakin, P. Phys. Rev. A 1988, 38, 2614. (11) Lindsay, H. M.; Lin, M. Y.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. In Proceedings of the Photon Correlation Techniques and Applications; Abbiss, J. B., Smart, A. E., Eds.; Optical Society of America: Washinghton, DC, 1988; Vol. 1, p 122.
10.1021/la049530p CCC: $27.50 © 2004 American Chemical Society Published on Web 05/27/2004
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information about the rotational diffusion coefficient of an individual cluster. Although the translational diffusion coefficient and the corresponding translational hydrodynamic radius of fractal clusters have been widely investigated in the literature, little work has been done about their rotational diffusivity.10-12 Typically, it is assumed that the rotational diffusion coefficient Θr and the rotational hydrodynamic radius Rh,r are related through the Debye equation valid for spheres:13
Θr )
kT 8πηRh,r3
(1)
where k is the Boltzmann constant, T the absolute temperature, and η the solvent viscosity. However, neither Θr nor Rh,r can be easily estimated experimentally or theoretically. Actually, because fractal clusters are very anisotropic objects, the complete characterization of their diffusive behavior is a complex problem, which in the rigorous treatment requires the determination of a six-order tensor.14 Thus, it is generally assumed that Rh,r ) Rh. In this work, we investigate the rotational diffusivity of fractal clusters using the approach proposed by Kirkwood and Riseman,15 which has been extended by Garcia de la Torre et al.16,17 to compute the full diffusion matrix of an object. The Kirkwood-Riseman (KR) theory allows one to decouple the translational and the rotational motion of the cluster, thus reducing substantially the amount of calculations required to compute the rotational diffusion tensor. Once the rotational diffusion tensor of each cluster has been obtained, its three eigenvalues, corresponding to the three main rotational diffusion coefficients, can be computed, and their mean value is then used to interpret the DLS measurements. Finally, the effect of using the obtained rotational diffusion coefficient values, instead of the usual approximation Rh,r ) Rh, on the interpretation of DLS data is discussed. Theory Monte Carlo Algorithm. A Monte Carlo off-lattice cluster-cluster aggregation algorithm has been used to generate the clusters for obtaining the structural information related to this work. The details of the algorithm have been reported elsewhere.8,18,19 Each simulation starts from a configuration of N particles randomly positioned in a box, where N is chosen so as to reproduce a given value of the particle volume fraction. In all our simulations, we have chosen a box with a size equal to 50 particle diameters and started with a particle volume fraction of 1%, which corresponds to approximately N ) 3600. At each iteration loop, the algorithm selects one cluster (or particle), with a probability proportional to its diffusion coefficient that is assumed to be inversely proportional to the cluster size, and moves it over a distance of one particle (12) Torres, F. E.; Russel, B. R.; Schowalter, W. R. J. Colloid Interface Sci. 1991, 142, 554. (13) Dhont, J. K. G. An Introduction to Dynamics of Colloids; Elsevier: Amsterdam, 1996. (14) Favro, L. D. Phys. Rev. 1960, 119, 53. (15) Riseman, J.; Kirkwood, J. G. J. Chem. Phys. 1950, 18, 512. (16) Garcia de la Torre, J.; Lopez Martinez, M. C.; Garcia Molina, J. J. Macromolecules 1987, 20, 661. (17) Carrasco, B.; Garcia de la Torre, J. Biophys. J. 1999, 75, 3044. (18) Kolb, M.; Herrmann, H. J. J. Phys. A: Math. Gen. 1985, 18, L435. (19) Hasmy, A.; Foret, M.; Anglaret, E.; Pelous, J.; Vacher, R.; Jullien, R. J. Non-Cryst. Solids 1995, 186, 118.
diameter. The cluster size is related to its mass by the fractal scaling.18 If the movement produces a collision with another cluster, in diffusion-limited cluster aggregation (DLCA) conditions a new cluster is formed, with a mass equal to the sum of those of the colliding clusters. In reaction-limited cluster aggregation (RLCA) conditions, a sticking probability equal to 0.001 is introduced for the new cluster to be formed. Every time that a cluster with a mass equal to a selected value is formed, the coordinates of its primary particles are stored for subsequent analysis. In this work, clusters with masses in the range from 3 to 100 primary particles have been collected and analyzed. In particular, each class of clusters in the range of mass from 3 to 20 primary particles has been stored and analyzed in detail, while for clusters with masses larger than 20 primary particles, we have analyzed only those with masses that are multiples of 10 primary particles. To have a good statistical ensemble, more than 1000 simulations have been performed to have at least 1600 clusters of each mass. KR Theory. The full description of the diffusion behavior of anisotropic objects such as fractal clusters is a complex problem,14 which requires the determination of a six-order diffusion matrix D, proportional to the inverse of the six-order friction tensor X:17
D)
(
)
(
Dt DTtr X XTtr ) kTX-1 ) kT t Dtr Dr Xtr Xr
)
-1
(2)
where the superscript T stands for transpose. In eq 2, the diffusion matrix D can be decomposed into four threeorder tensors:16,17 the translational diffusion tensor Dt, the rotational diffusion tensor Dr, and two translationrotational coupling diffusion tensors Dtr. The friction tensor can also be decomposed into four three-order tensors: the translational friction tensor Xt, the rotational friction tensor Xr, and two translation-rotational friction coupling tensors Xtr. The coupling tensors exist as a consequence of the lack of symmetry of the fractal cluster, which results in the impossibility of decoupling its rotational and translational motions. Furthermore, it turns out that the diffusion matrix depends on the point chosen as the origin of the system of coordinates to which it refers. The general theory17 predicts the existence of a point, called the center of resistance, which for a symmetric object coincides with its center of symmetry, having the property that, when it is used as the center of the coordinates, the rotational diffusion tensor depends only upon the rotational part of the friction tensor. The difficulty is that the calculation of the center of resistance is usually rather cumbersome. In this work, we apply the simplified KR theory, which was devised for objects made of spherical particles15,20,21 and can reduce substantially the overall computational burden while maintaining a good accuracy. In particular, within the KR theory, it is assumed that the center of resistance coincides approximately with the center of mass if the object is made of equally sized spherical particles, as it is the case for the fractal clusters considered in this work. Once the position of the center of resistance is known, the translational and rotational motion of the cluster can be decoupled. The KR theory was used in our previous work9 to estimate the translational diffusion coefficient and, therefore, the translational hydrodynamic radius for (20) Bloomfield, V. A.; Dalton, W. O.; Van Holde, K. E. Biopolymers 1967, 5, 135. (21) Garcia de la Torre, J.; Bloomfield, V. A. Biopolymers 1977, 16, 1765.
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the clusters generated under both RLCA and DLCA conditions. Here, we will focus on the application of the KR theory to compute the rotational diffusion tensor of fractal clusters. When the cluster center of resistance can be approximated as its center of mass, the rotational diffusion coefficient tensor Dr depends only on the rotational friction tensor, as follows:16
Dr ) kTXCM,r-1
(3)
where XCM,r-1 is the inverse rotational friction tensor, computed with respect to the cluster center of mass rCM, given by the following relation:
rCM )
1
i
∑rj
(4)
i j)1
where rj is the position of the jth particle in the cluster. According to the KR theory, the drag resistance experienced by the cluster is determined as the sum of the resistances experienced by all its particles. The drag force acting on each particle is obtained by assuming that the movement of the particle in the fluid is perturbed by the presence of the other particles belonging to the cluster. Such perturbations are assumed to be additive and are estimated using the Oseen approach.20 Accordingly, Garcia de la Torre et al.16 have proven that the reverse rotational friction tensor can be well approximated by the following expression:
XCM,r-1 ) (A-1 + A-1BA-1)
(5)
where the tensors A and B are given by
∑ j)1
Uj‚Uj
(6)
∑ ∑ Uj‚TjmUm j)1 m)1*j
(7)
and i
i
2
B ) -fp
In eqs 6 and 7, I is the identity tensor and the particle friction factor fp is given by the Stokes formula:
fp ) 6πηRp
(8)
The tensor Tjm in eq 7 represents the effect of the perturbation produced at the location of particle j by a unit force acting at the position of the particle m. The expression proposed by Yamakawa for the tensor Tjm has been used:22
[(
)
(
-zj yj 0 -xj z Uj ) j 0 -yj xj 0
)
(10)
where xj, yj, and zj are the coordinates of the jth particle in the cluster with respect to the center of mass of the cluster. The procedure used for the determination of the rotational diffusion tensor of each Monte Carlo generated cluster is, therefore, the following. The Monte Carlo code provides the position of the centers of all the primary particles in a cluster, from which the position of the cluster center of mass is determined. Then, two tensors A and B are computed from eqs 6 and 7. Finally, the cluster rotational diffusion coefficient is evaluated using eq 3. To extract the main features of the rotational diffusivity, the eigenvalues of the diffusion coefficient tensor are computed for each generated cluster of a given mass. For an anisotropic object, the three eigenvalues are different from each other, and their difference is an indication of the cluster anisotropy. To test the reliability of the results obtained with the KR theory, we have also performed some calculations using the code HYDRO, developed by Garcia de la Torre et al.23 The code HYDRO was generated again on the basis of the Oseen method to account for the hydrodynamic perturbation experienced by a particle due to the presence of the other particles in the cluster but computes the full sixth order rotational diffusivity matrix without making the assumption that the center of resistance of a cluster coincides with the center of mass. We found that under the examined condition the eigenvalues of the rotational diffusion tensor extracted from the full diffusion matrix differ only by a few percent from the values estimated using the KR theory. Results and Discussion
i
A ) 8πηRp3iI - fp
the center of particle j with the center of particle m. The matrix Uj is instead given by
(
)]
rjmrjm 2Rp2 I rjmrjm 1 I+ + Tjm ) 8πηrjm rjm2 rjm2 3 rjm2
(9)
where rjm is the center-to-center distance between particles j and m inside the cluster, while rjm is the vector connecting (22) Yamakawa, H. J. Chem. Phys. 1970, 53, 436.
Rotational Hydrodynamic Radius. To obtain the main features of the rotational diffusion of fractal clusters, the eigenvalues of the cluster rotational diffusion tensors, computed as described in the previous section, have been averaged over several thousands of clusters with the same mass. Because for a typical fractal cluster the three eigenvalues of the rotational diffusion tensor are different from each other, they have been first sorted in ascending order and subsequently averaged over clusters with the same mass. Accordingly, for each cluster mass three main rotational diffusion coefficients have been determined. In the interpretation of DLS measurements, the mean of these three values is typically used to describe the relaxation behavior of the cluster.17 In the following, we extract from each average rotational diffusion coefficient the corresponding rotational hydrodynamic radius Rh,r using eq 1. With this, we can better analyze the rotational diffusivity of fractal clusters. The three rotational hydrodynamic radii, corresponding to the three eigenvalues, are represented by the open symbols in Figures 1 and 2, for DLCA and RLCA clusters, respectively, as a function of the number of particles per cluster. It is seen that two of the radii have rather similar values, while the third one is about 30% smaller. This indicates that the clusters have a characteristic direction along which the rotational diffusion coefficient is about 2.5 times larger than those along the other two charac(23) Garcia de la Torre, J.; Navarro, S.; Lopez Martinez, M. C.; Diaz, F. G.; Lopez Cascales, J. J. Biophys. J. 1994, 67, 530.
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Figure 1. Rotational hydrodynamic radii, normalized by the primary particle radius, calculated by means of eq 1 from the three average eigenvalues of the rotational diffusion tensor, in the case of DLCA clusters: (O) smallest eigenvalue, ()) intermediate eigenvalue, (0) largest eigenvalue. The continuous curve is the rotational hydrodynamic radius corresponding to the mean rotational diffusion coefficient. The dashed curve is the fractal scaling given by eq 11 with Df ) 1.87.
Figure 2. Rotational hydrodynamic radii, normalized by the primary particle radius, calculated by means of eq 1 from the three average eigenvalues of the rotational diffusion tensor, in the case of RLCA clusters: (O) smallest eigenvalue, ()) intermediate eigenvalue, (0) largest eigenvalue. The continuous curve is the rotational hydrodynamic radius corresponding to the mean rotational diffusion coefficient. The dashed curve is the fractal scaling given by eq 11 with Df ) 2.05.
teristic directions. This means that the average clusters are not spherically symmetric and encounter a smaller hydrodynamic hindrance when rotating along one of the main rotation axes rather than along the other two. This strong anisotropic structure of fractal aggregates generated by cluster-cluster aggregation algorithms is consistent with the work of other groups,24,25 which were focused on the eigenvalues of the radius of gyration tensor. In the same figures, the average rotational hydrodynamic radius computed from the average rotational diffusion coefficient is also shown by the continuous curve. It can be seen that all four radii plotted in Figures 1 and 2 follow a fractal scaling, given by6
Rh,r ) Ci1/Df Rp
(11)
Figure 3. Ratio between the rotational hydrodynamic radius, Rh,r, and the translational hydrodynamic radius, Rh, as a function of the number of particles per cluster: (0, dotted curve) RLCA, (O, solid curve) DLCA. The lines are computed by eq 12. Table 1. Values of the Parameters d, e, f, and m To Be Used in Equation 12 to Compute the Ratio between the Average Rotational Hydrodynamic Radius and the Translational Hydrodynamic Radius, Rh,r/Rh, for DLCA and RLCA Conditions aggregation mechanism
d
e
f
m
DLCA RLCA
1.2514 1.2561
0.1183 0.0259
0.7832 0.8578
2 2
where C is a constant of the order of unity and Df is the cluster fractal dimension. In particular, it is seen that eq 11 holds for clusters containing more than 20 particles, because the corresponding values exhibit a straight line behavior in the double logarithmic plot. The slopes correspond to fractal dimension values Df ) 1.87 for DLCA and 2.05 for RLCA clusters, identical to the values found in our previous work on translational hydrodynamic radii.9 To test the reliability of the typical assumption, Rh,r ) Rh, we have shown in Figure 3 the value of their ratio as a function of the number of particles per cluster, for both DLCA and RLCA clusters. In particular, we consider for the rotational hydrodynamic radius the average value shown in Figures 1 and 2 and for the translational hydrodynamic radius the values determined by applying the KR theory, as described in our previous work.9 It is seen that the Rh,r/Rh ratio increases with the cluster mass for a small cluster, and when the cluster mass is larger than 20 primary particles an asymptotic value has been reached. The values of the Rh,r/Rh ratio are comprised between 1.15 and 1.25, for RLCA and DLCA clusters. To obtain a parametrization of the ratio F(i) ) Rh,r/Rh, the results in Figure 3 have been fitted using the following empirical expression:
(i - e)m F(i) ) d (i - e)m + f
(12)
where d, e, f, and m are empirical parameters. The curves in Figure 3 are the results of such a fitting, and the corresponding values of the empirical parameters are reported in Table 1. (24) Family, F.; Vicsek, T.; Meakin, P. Phys. Rev. Lett. 1985, 55, 641. (25) Botet, R.; Jullien, R. J. Phys. A: Math. Gen. 1986, 19, L907.
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From these calculations, on the basis of the KR theory, it can be concluded that the assumption Rh,r ) Rh can lead to underestimating the actual Rh,r value by about 25%. Because the rotational diffusion coefficient is inversely proportional to the third power of Rh,r, the assumption of Rh,r ) Rh can overestimate the rotational diffusion coefficient by a factor of 2 for clusters containing more than 20 particles. Role of Rotational Diffusion in the Interpretation of DLS Data. DLS is a widely used technique to measure the translational diffusion coefficient of particles, from which the translational hydrodynamic radius can be estimated. In the case of fractal clusters, however, the rotational diffusive motion of the clusters affects the measured intensity autocorrelation function, and, therefore only a so-called effective translational hydrodynamic radius, Reff h , is obtained. To obtain the actual translational hydrodynamic radius, Rh, a proper correction factor has to be used. The decorrelation effect of the rotational motion on the scattered radiation can be computed using a rigorous theory,10 which involves a complex expansion of the cluster structure in spherical harmonics. On the other hand, a simplified but accurate expression for this correction factor, which requires the knowledge of only the cluster scattering structure factor, has been proposed by Ball et al.11 as follows:11,12
(
)
Rg2Rh 3 ∂ ln S(q) ) 1 + 1+ eff 3 Rh 2Rh,r ∂(qRg)2 Rh
Figure 4. Ratio between the actual translational hydrodynamic radius Rh and the measured translational hydrodynamic radius Reff h in light scattering experiments, as a function of the product qRg, computed using eq 13, in the case of a DLCA cluster made of 100 particles. The continuous line corresponds to the case where eq 12 is used for the rotational hydrodynamic radius, while the dotted line corresponds to the case where the assumption Rh,r ) Rh is used.
(13)
where Rg is the cluster radius of gyration, S(q) is the cluster scattering structure factor, and q is the wave vector. The scattering structure factor S(q) is the Fourier transform of the particle-particle correlation function g(r):
S(q) )
(
1 1 + 4π i
sin(qr) dr qr
∫0∞r2g(r)
)
(14)
Then, for a cluster with fixed Rg, the derivative of the logarithm of the scattering structure factor with respect to (qRg)2 in eq 13 can be written as
∂ ln S(q) ∂(qRg)2
)
qr ∞ 2 r g(r) 0
∫
4π 2qRg2S(q)i
cos(qr) - sin(qr) q2r
dr
(15)
The above integral can be easily computed numerically, using the piecewise expression of the g(r) function, as shown in a previous work.8 To obtain the value of the ratio Rh/Reff h from eq 13, however, one still needs to know the Rh,r value. In the literature, it is typically assumed that Rh ) Rh,r.10-12 Using the values of the average rotational hydrodynamic radius computed in the previous section, we are now in the position of better estimating the ratio Rh/Reff h and of assessing the inaccuracies deriving from the use of the approximation Rh,r ) Rh in eq 13. For this, we compare the Rh/Reff h values computed with the Rh,r values obtained from eq 12 with those obtained with the assumption of Rh ) Rh,r. The obtained results are shown as a function of qRg in Figures 4 and 5 for clusters containing 100 particles and generated under DLCA and RLCA conditions, respectively. The remaining quantities in eq 13, that is, Rg and Rh, have been calculated using literature relations.8,9 It is seen that the ratio Rh/Reff h is always larger than 1, indicating that the measured translational
Figure 5. Ratio between the actual translational hydrodynamic radius Rh and the measured translational hydrodynamic radius Reff h in light scattering experiments, as a function of the product qRg, computed using eq 13, in the case of a RLCA cluster made of 100 particles. The continuous line corresponds to the case where eq 12 is used for the rotational hydrodynamic radius, while the dotted line corresponds to the case where the assumption Rh,r ) Rh is used.
hydrodynamic radius is always smaller than the actual one. However, the assumption of Rh,r ) Rh overestimates this effect by a factor of about 2 at large wave vectors under both RLCA and DLCA conditions. This means that the influence of rotational diffusion on DLS measurements is weaker than what one would expect on the basis of the assumption Rh,r ) Rh. It is also clear that for sufficiently small clusters, that is, i < 10, because the ratio Rh,r/Rh approaches 1, the assumption Rh,r ) Rh leads to smaller errors. To appreciate the effect of the result reported above on a real system, one should consider that when fractal clusters are formed as a result of an aggregation process, the CMD is never monodisperse, but it can actually be quite broad. In this case, the DLS experiment allows one to determine an average translational hydrodynamic radius 〈Rh〉, where the average is a z average, typical of
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light scattering measurements, performed over the entire CMD: imax
〈Rh〉 )
Nii2Si(q) ∑ i)1
imaxN i2S (q) i i
∑ i)1
(16)
eff Rh,i
where Ni is the number of clusters with i primary particles, imax is the largest number of primary particles in a cluster, and all quantities with subscript i refer to clusters made of i primary particles. Therefore, by combining the CMD eff values estimated above, one can compute with the Rh,i with eq 16 the average translational hydrodynamic radius of the distribution at different values of the wave vector q, that is, at different scattering angles. Because the structure factor is usually a monotonically decreasing function of q, the average hydrodynamic radii measured at large scattering angles (i.e., large q) are smaller than those measured at small angles (i.e., small q). As an example, let us now generate a CMD, Ni, using the Smoluchowski PBE, and compute the 〈Rh〉 value from eff is estimated from eq 13 with the Rh,r,i eq 16, where Rh,i value computed either from eq 12 or from the assumption Rh,i ) Rh,r,i. The PBE used to simulate irreversible aggregation processes in their discrete form can be written as7,26
dNi(t) dt
∞
∑ j)1
)-
Ki,jNi(t) Nj(t) +
1 i-1
∑Ki-j,jNi-j(t) Nj(t)
2 j)1
Figure 6. Simulated ratio between the measured average translational hydrodynamic radius 〈Rh〉 and the average radius of gyration 〈Rg〉, as a function of q〈Rg〉, in the case of a DLCA process, for a colloidal system constituted of 70-nm particles and 0.01% volume fraction. The continuous curves correspond to the case where eq 12 has been used to compute the rotational hydrodynamic radius, while the dotted curves correspond to the case where the assumption Rh,r ) Rh has been used.
(17)
where Ki,j is the aggregation kernel between two clusters with masses i and j. The adopted kernel in the case of DLCA is the modified Smoluchowski kernel:7
Ki,j )
2kT 1/Df (i + j1/Df)(i-1/Df + j-1/Df) 3η
(18)
and in the case of RLCA is7
Ki,j )
2kT 1/Df (i + j1/Df)(i-1/Df + j-1/Df)(ij)λ 3ηW
(19)
where λ is a parameter equal to 0.1-0.4, whose precise value depends on the cluster structure, and W is the Fuchs stability ratio. The time evolution of the CMD has been generated starting from a monodisperse colloidal system with the primary particle radius, Rp ) 35 nm, and the particle volume fraction, φ ) 0.01%. For the RLCA simulation, we have used W ) 2 × 105 and λ ) 0.4 in eq 19. The obtained results are reported in the form of the ratio between the average translational hydrodynamic radius and the average radius of gyration, 〈Rh〉/〈Rg〉, where the latter is defined as imax
〈Rg〉2 )
Ni(t)i2Rg,i2 ∑ i)1 imax
(20)
Ni(t)i2 ∑ i)1
(26) Sandku¨hler, P.; Sefcik, J.; Lattuada, M.; Wu, H.; Morbidelli, M. AIChE J. 2003, 49, 1542.
Figure 7. Simulated ratio between the measured average translational hydrodynamic radius 〈Rh〉 and the average radius of gyration 〈Rg〉, as a function of q〈Rg〉, in the case of a RLCA process, for a colloidal system constituted of 70-nm particles and 0.01% volume fraction. The continuous curves correspond to the case where eq 12 has been used to compute the rotational hydrodynamic radius, while the dotted curves correspond to the case where the assumption Rh,r ) Rh has been used.
Figures 6 and 7 show the values of the ratio 〈Rh〉/〈Rg〉 as a function of q〈Rg〉 for DLCA and RLCA, respectively. In each case, the 〈Rh〉/〈Rg〉 values have been computed at two different q values, corresponding to scattering angles of 90 and 30°. It is seen that in the case of DLCA the 〈Rh〉/〈Rg〉 values computed using eq 12 to estimate the rotational hydrodynamic radius Rh,r (continuum curves) are substantially different from those obtained by assuming Rh,r ) Rh (broken curve). In the case of RLCA, however, the difference between the two approaches is relatively small. This can be explained by considering the Rh/Reff h values shown in Figures 4 and 5, which indicate that the effect of the rotational diffusivity on Reff h becomes increasingly more important as the product qRg increases (i.e., as Rg increases in the case of a fixed q value). If in an aggregation process the CMD is very broad as in the case of RLCA, the number of primary particles and small clusters is always significant, and for these the correction
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(eq 13) is negligible. In this case, the spread between 〈Rg〉 and 〈Rh〉 is dominated by the broadness of the distribution, while the effect of eq 13 is not significant, as shown in Figure 7. On the other hand, in the case of DLCA, because the distribution has a smaller polydispersity, the ratio 〈Rh〉/〈Rg〉 is more influenced by the effect of the correction (eq 13) as shown by Figure 6, particularly at large q〈Rg〉 values. In this case, the assumption Rh,r ) Rh tends to substantially overestimate the difference between 〈Rg〉 and 〈Rh〉 and, therefore, the broadness of the CMD. Concluding Remarks In this work, the rotational diffusion of fractal clusters formed under DLCA and RLCA conditions has been analyzed, using the clusters generated by a Monte Carlo off-lattice cluster-cluster aggregation algorithm. To estimate the cluster rotational diffusion tensor, the simplified KR theory, modified by Garcia de la Torre et al.,16 has been applied to decouple the cluster rotational and translational diffusive motions. First, the three eigenvalues of the rotational diffusion tensor of each cluster have been computed and sorted in ascending order, and then each eigenvalue has been averaged over several thousands of clusters with the same mass. It has been shown that among the three rotational hydrodynamic radii corresponding to the three main rotational diffusivities computed using eq 1, two have similar values, while the third one is about 30% smaller. This indicates that there is one main rotational axis in the cluster which offers a frictional resistance to rotation significantly smaller than
Lattuada et al.
the other two main axes. This gives an indication of significant anisotropy of the fractal clusters. The mean value of the three eigenvalues of the rotational diffusion tensor has been computed as a function of the cluster mass and is used to define the rotational hydrodynamic radius, Rh,r, based on the Debye equation (eq 1). The quantity Rh,r is often involved in the interpretation of DLS data, and it is typically assumed to be equal to the translational hydrodynamic radius, Rh. It has been found that the Rh,r values computed in this work are about 25% larger than the Rh values, at least for clusters containing more than twenty primary particles. Moreover, by using these Rh,r values, instead of assuming Rh,r ) Rh, we obtain values of the correction of the DLS data due to rotational diffusion, as given by eq 13, which are smaller by a factor of about 2 for significantly large clusters. When the present results are applied to predict the average translational hydrodynamic radius of the cluster populations generated under DLCA and RLCA conditions, it appears that under RLCA conditions the difference is small, while under DLCA conditions the assumption of Rh,r ) Rh significantly overpredicts the value of the ratio 〈Rg〉/〈Rh〉 and, therefore, the broadness of the CMD. Acknowledgment. This work was financially supported by the Swiss National Science Foundation (Grant 200020-101714). Many useful discussions with Dr. Jan Sefcik, Peter Sandku¨hler, and Andrea Vaccaro are gratefully acknowledged. LA049530P
