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Experimental Investigation of Colloidal Gel Structures Marco Lattuada, Hua Wu, and Massimo Morbidelli* Swiss Federal Institute of Technology Zurich, ETHZ, Institut fu¨ r Chemie- und Bioingenieurwissenschaften, ETH-Ho¨ nggerberg/HCI, CH-8093 Zu¨ rich, Switzerland Received October 20, 2003. In Final Form: March 12, 2004 We investigate experimentally the structural properties of colloidal gels, formed under both diffusionlimited and reaction-limited aggregation conditions, using light scattering measurements and compare the results with the literature Monte Carlo (MC) simulations. The scattering structure factors have been measured for the two classes of gels in the range of the particle volume fractions between 0.02 and 0.07. From these, the corresponding fractal dimension values have been estimated. These have been found to be in good agreement with those estimated from the structure factors computed from MC simulated gels. On the basis of our previous research (Lattuada et al. Langmuir 2003, 19, 6312), this confirms that the scattering structure factor of a gel provides erroneously a small fractal dimension value, which decreases as the particle volume fraction increases. Furthermore, it is observed that the average size of the fractal clusters is larger in real gels than in simulated gels.
Introduction A large number of industrial processes involve colloidal dispersions.1,2 In particular, colloidal dispersions are intermediate steps in polymer and material processing, as well as in food and pharmaceutical industries, and their coagulation is often required to obtain the final product. Under appropriate conditions, the coagulation of colloidal dispersions can lead to the formation of gels,3 which are per-se interesting systems because they can be used for the preparation of new materials.4 It has been shown by several studies that colloidal gels, at least for particle volume fractions smaller than 10%, possess a fractal structure over certain length scales.3,6-11 This is not surprising because a colloidal gel is usually the last step of an aggregation process that first leads to the formation of fractal clusters.3,6 As a result of their open structure, when fractal clusters grow large enough, they can fill all the available space and then interconnect to form the gel phase.3,10 Therefore, the characterization of the fractal nature of a colloidal gel can give a deep insight into both its formation process and its mechanical and structural properties. However, it is not straightforward to obtain structural information about a colloidal gel. Although several experimental techniques have been used for this pur* Corresponding author. E-mail:
[email protected]. Tel.: 0041-1-6323034. (1) Hunter, R. J. Introduction to Modern Colloid Science; Oxford Science Publications: Oxford, 1994. (2) Russel, W. B.; Saville, D. A. S.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: London, 1989. (3) Poon, W. C. K.; Haw, M. D. Adv. Colloid Interface Sci. 1997, 73, 71. (4) Brinker, C. J.; Scherer, G. W. Sol-Gel Science; Academic Press: San Diego, 1990. (5) Hasmy, A.; Anglaret, E.; Foret, M.; Pelous, J.; Jullien, R. Phys. Rev. B 1995, 50, 6006. (6) Hasmy, A.; Foret, M.; Anglaret, E.; Pelous, J.; Vacher, R.; Jullien, R. J. Non-Cryst. Solids 1995, 186, 118. (7) Hasmy, A.; Jullien, R. J. Non-Cryst. Solids 1995, 186, 342. (8) Gaboriaud, F.; Chaumont, D.; Nonat, A.; Craievich, A. J. Appl. Crystallogr. 2000, 33, 597. (9) Mellema, M.; Heesakkers, J. W. M.; van Opheusden, J. H. J.; van Vliet, T. Langmuir 2000, 16, 6847. (10) Sorensen, C. M.; Hageman, W. B.; Rush, T. J.; Huang, H.; Oh, C. Phys. Rev. Lett. 1998, 80, 1782. (11) Lattuada, M.; Wu, H.; Morbidelli, M. Phys. Rev. E 2001, 64, 061404.
pose,9,12,13 those based on scattering phenomena are the most used ones.14 Light, X-ray, and neutron scattering have the advantages of being noninvasive, not requiring any manipulation of the sample, and having solid and well-developed theories for interpreting the obtained measurements.14 Among them, light scattering is getting more and more attractive after the introduction of new, improved methods to eliminate multiple scattering,15 which is the biggest problem encountered with highly concentrated systems such as gels. Alternatively, structural information about gels can be obtained through Monte Carlo (MC) simulations.3,5-7,16 The MC simulations were previously introduced to reproduce in a simplified way the aggregation process leading to the formation of fractal clusters in dilute conditions.17-20 More recently, they have been also applied to investigate the structure of aerogels.5-7 It is, therefore, natural to apply them also to colloidal gels, formed in both reaction-limited conditions (RLCA) and diffusionlimited conditions (DLCA). The advantage of MC simulations is that they provide the position of each particle in the system, from which the structure of the simulated gels can be computed in detail. In particular, one can compute the particle-density correlation function g(r), from which the scattering structure factor S(q) can be derived. This last step is very important when we need to verify experimentally the computed gel structures because only the scattering structure factor is experimentally accessible. One surprising result from MC simulations, first reported by Hasmy and Jullien,7 is that the value of fractal dimension Df of simulated gels estimated from the scattering structure factor, S(q), decreases as the particle volume fraction φ increases, while the same quantity of (12) Wu, H.; Morbidelli, M. Langmuir 2001, 17, 1030. (13) Cai, J.; Lu, N.; Sorensen, C. M. Langmuir 1993, 9, 2861. (14) Linder, P.; Zemb, Th. In Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter; Elsevier Science B.V.: Amsterdam, 2002. (15) Nicolai, T.; Urban, C.; Schurtenberger, P. J. Colloid Interface Sci. 2001, 240, 419. (16) Gimel, J. C.; Durand, D.; Nicolai, T. Phys. Rev. B 1995, 51, 11348. (17) Meakin, P. Adv. Colloid Interface Sci. 1988, 28, 249. (18) Meakin, P. Croat. Chem. Acta 1992, 65, 237. (19) Meakin, P. Phys. Scr. 1992, 46, 295. (20) Jullien, R. Croat. Chem. Acta 1992, 65, 215.
10.1021/la035949o CCC: $27.50 © 2004 American Chemical Society Published on Web 04/27/2004
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the same simulated gels estimated directly from the particle-density correlation function, g(r), increases as φ increases. We have recently shown21 that this result is due to the presence of a nonfractal region in g(r), which upon the Fourier transformation alters the shape of the power-law region of S(q). Accordingly, the Df value estimated from the structure factor of a gel does not represent the “true” Df value of its clusters. This is a serious problem because only the structure factor is accessible experimentally. In this work, we investigate experimentally the fractal behavior of clusters in colloidal gels to verify the conclusions just mentioned based on MC simulated gels. In particular, we consider gels formed through the aggregation of fluorinated polymer (Hyflon MFA) particles, in both DLCA and RLCA conditions, in the range of particle volume fractions 0.02 e φ e 0.07. Although the preparation of RLCA gels is rather simple, a uniform DLCA gel cannot be obtained by simply mixing a latex with an electrolyte solution. For this, a new technique, referred to here as frontal gelation, has been developed for the preparation of DLCA gels in the given range of particle volume fractions. A series of light scattering measurements on these colloidal gels have been performed to determine their scattering structure factors. From these, the fractal dimension has been estimated and compared with those obtained through MC simulations, together with other features of the measured gel scattering structure factors. Experimental Procedure Materials and Light Scattering Measurements. A fluorinated-polymer latex (copoly-tetrafluoroethylene and perfluoromethylvinyl ether, Hyflon MFA), manufactured by Solexis SPA through emulsion polymerization, has been used for all the gelation experiments. The polymer particles are practically monodisperse, with the mean radius determined by dynamic light scattering, Rp ) 37.5 nm. The main characteristic of MFA is the very low refractive index (1.35) compared to that of water (1.33), which allows one to perform the light scattering measurements in systems with a relatively high volume fraction without significant effects of multiple scattering. We used, as a reference system, a stock solution with a particle volume fraction φ ) 0.14. The light scattering instrument that was used is a BI-200SM (Brookhaven), with an argon laser (Lexel 95-2) as the light source (wavelength λ ) 514.5 nm) and an angular range of the goniometer from 8 to 150. The inner diameter of the cuvette used to measure the scattering structure factor of gels was 0.8 cm. During static light scattering (SLS) measurements of the gels, the sample cuvette was rotated by a motor at a constant speed of 3 rpm so that an average intensity over the whole section of the sample was measured to smoothen local inhomogeneities of the gel phase. All the experiments have been carried out at temperature T ) 298 K. Preparation of DLCA Gels by Frontal Gelation. The preparation of DLCA gels deserves considerable attention. The general procedure, based on mixing a latex and a salt solution, cannot be used because the gelation of a fully destabilized suspension containing particles of Rp ) 37.5 nm at a particle volume fraction φ > 0.01 occurs on a time scale of few tens of milliseconds.23 This is a consequence of the very fast diffusionlimited aggregation process that leads to quick space filling and subsequent interconnection of the fractal clusters. Such a short gelation time is (much) smaller than the time necessary to mix and homogenize the latex and salt solutions. It follows that mixing becomes the controlling process, and one cannot obtain a uniform gel. (21) Lattuada, M.; Wu, H.; Hasmy, A.; Morbidelli, M. Langmuir 2003, 19, 6312. (22) Gonzalez, A. E. Phys. Rev. Lett. 1993, 14, 2248. (23) Sandku¨hler, P.; Sefcik, J.; Morbidelli, M. Adv. Colloid Interface Sci. 2004, 108-109, 133.
Lattuada et al. Accordingly, to prepare gels in DLCA conditions, we have applied a technique, originally developed by Chittofrati et al.,24 which is referred to here as “frontal gelation”. This technique consists of the following steps: 1. A stable latex with a desired particle volume fraction is poured in the cuvette where the scattering measurement will be performed. 2. A small amount of a highly concentrated electrolyte (acid or salt) solution is very slowly added on top of the latex. As a result of the high ionic strength of the electrolyte solution and the high particle volume fraction of the latex, a layer of gel is immediately formed at the interface between the two solutions. In this work, a 0.5 M solution of aluminum nitrate [Al(NO3)3] has been used to induce gelation in DLCA conditions. 3. More electrolyte solution is added and kept in place by the gel layer, which exhibits sufficient mechanical strength. At this point, the electrolyte solution diffuses through the gel layer leading to a gelation front that moves further inside the latex solution. It should be pointed out that the presence of a gel layer at the boundary between the two solutions is crucial for frontal gelation to occur. This is needed to prevent not only a convection-driven mixing of the two solutions but also the diffusion of polymer particles in the electrolyte solution. On the other hand, because the gel contains more than 90% water, leading to the diffusivity of ions in the gel similar to that in water, the electrolyte ions can freely diffuse through the gel layer up to the latex solution where they induce almost instantaneous gelation. The thickness of the gel layer then increases with time, leading to a gelation front that moves downward toward the bottom of the cuvette, which constitutes the process referred to as frontal gelation. There are two important issues that need to be considered during the frontal gelation experiments. First, the particle volume fraction should be high enough to form an initial gel layer of sufficient mechanical resistance. We found that this condition can be satisfied for a particle volume fraction φ g 0.02. Thus, all the experiments in this work have been carried out at φ g 0.02. Second, the diffusing front of the electrolyte becomes smoother as the thickness of the gel layer increases, leading to a progressive change in the gelation mechanism from DLCA to RLCA. Therefore, the light scattering measurement was performed at a distance of about 5 mm from the initial gel layer, to be sure that the measured gel can be considered as a DLCA gel. To better understand how fast the gelation front moves in time as the electrolyte diffuses in the latex, we have performed some controlled experiments and have developed a simple model to predict the position of the gel front as a function of time. These controlled experiments have been performed in larger cuvettes, with an inner diameter of 1.5 cm. The procedure for the preparation of the sample is the same as that just described. The cuvette was filled with latex at a volume fraction of 10% and height of 8 cm. Then, the thin gel layer was formed, and the electrolyte solution was added, so as to reach a total hight of 10 cm (including the latex and the electrolyte solution). Because the colloidal particles were negatively charged, their destabilization was achieved using two electrolytes of different cations: CuSO4 and Ca(NO3)2, both at a concentration of 0.5 M. It turns out that, as a result of the presence of large fractal clusters, the gel scatters more light than the original latex, and the position of the gel front can be easily followed visually as a function of time. Note that the salt CuSO4 can form a gel of a light blue color, which makes it easier to identify the gel boundary. To model the time evolution of the position of the gel boundary, additional information is required: the critical coagulant concentration (CCC), Cc. This is defined as the minimum concentration of electrolyte necessary to induce fast DLCA aggregation. When operating at 10% volume fraction and electrolyte concentration equal to Cc, the latex undergoes instantaneous gelation. Such a CCC value has been measured independently by dropping a drop of latex in a series of electrolyte solutions with increasing concentration and observing the time needed to gelate. The obtained values for each electrolyte are reported in Table 1. In a later section, we will develop a model for the frontal gelation (24) Chittofrati, A.; Lazzari, P.; Lenti, D. U.S. Patent 6,013,712, 2000.
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Table 1. Values of the Electrolyte Diffusion Coefficients, Computed Using Equation 11,27 and the CCC, Cc, for the Two Electrolytes Used in the Experiments electrolyte
Diff, cm2/s
Cc, mol/L
CuSO4 Ca(NO3)2
8.55 × 10-6 1.296 × 10-5
0.005 0.004
formed cluster spans the entire box. The density correlation function of the gel g(r) is computed at the end of a simulation by counting the number of pairs of particles dN located at a distance between r and r + dr and dividing this number by the volume of the spherical shell with radius r:
Table 2. Values of the Gel Particle Volume Fractions and the Corresponding NaCl Concentrations Used To Induce Gelation in the RLCA Experiments φ NaCl, mol/L φ NaCl, mol/L
0.01 0.021 0.06 0.013
0.02 0.012 0.07 0.013
0.03 0.018 0.08 0.013
0.04 0.017 0.09 0.013
0.05 0.016 0.1 0.012
process and compare its predictions with the corresponding experimental results. Preparation of RLCA Gels. The preparation of RLCA gels is much simpler than that of DLCA gels. In this case, it is sufficient to tune the ionic strength of the gelling suspension so that gelation occurs on a time scale (several minutes or hours) much longer than that needed to mix the latex and the electrolyte solution. This guarantees that the gelation process occurs homogeneously in the gelling solution. The RLCA experiments have been carried out at eight different particle volume fractions, 0.01 < φ < 0.08. The suspensions used for the RLCA gelation experiments were prepared by diluting the reference latex with a Milli-Q (Millipore) deionized water and salt solution to the desired particle volume fraction and ionic strength. Sodium chloride (NaCl) was used to tune the ionic strength, and the salt concentrations used for the different particle volume fractions are summarized in Table 2.
Theoretical Background For all the MC simulations performed in this work, we use an off-lattice cluster-cluster aggregation algorithm, which has been widely used in the literature to simulate diffusion-limited and reaction-limited aggregation or gelation processes.6,7,19,21,22 The algorithm starts by placing in a cubic box of given size, l, a given number of spherical particles, N, of radius, Rp, according to a chosen particle volume fraction. At each step, the probability Pi of a cluster with mass i to be selected and moved in a random direction is given by6,7
Pi )
i-(1/Df)
∑
where it is assumed that the probability Pi is proportional to the cluster diffusion coefficient, which is inversely proportional to the cluster size Rc, while the cluster mass i is related to its size through the usual fractal scaling:25
()
1 dN 24Nr2 dr
(3)
The result is then averaged over at least 50 simulations of the same system to achieve a good statistical description. For the DLCA case at all particle volume fractions, the size of the simulation box, l, was set at l/(2Rp) ) 100, that is, 100 diameters of the primary particle. In RLCA conditions, because the computational time for each simulation is much higher than in DLCA conditions and increases substantially with N, we have progressively decreased the box size as the particle volume fraction increases. In particular, we have used a box size l/(2Rp) ) 116, 87, 75, 64, 58, and 30 for φ ) 0.01, 0.02, 0.03, 0.04, 0.05-0.07, and 0.08, respectively. A smaller box size is acceptable at large particle volume fractions because the average cluster size in the gel decreases as the particle volume fraction increases, and the structure of the gel becomes uniform at smaller length scales. This is clearly indicated by the density correlation function, which reaches the asymptotic value of 1 at a smaller correlation length. From the gel particle-density correlation function, the gel scattering structure factor S(q) can be easily determined through the following Fourier transform:14
S(q) ) 1 + 24φ
sin(qr) dr qr
∫0∞r2[g(r) - 1]
q)
(2)
(25) Jullien, R.; Botet, R. Aggregation and Fractal Aggregates; World Scintific: Singapore, 1987.
(5)
where K is a constant, which depends on the details of the experimental setup used to perform the experiments, and P(q) is the primary particle form factor. The wave vector q is related to the scattering angle θ, that is, the angle between the direction of the incident radiation and the direction along which the scattered radiation is detected,14 as follows:
Df
In DLCA conditions, each collision of two clusters forms a new cluster with mass equal to the sum of the masses of the colliding clusters, while in RLCA conditions we consider that the formation of a new cluster occurs with a substantially small probability of 0.001 to closely approximate aggregation in the RLCA regime. Periodic boundary conditions are applied so that when a cluster exits the box from one side, it reenters immediately from the opposite side. The calculation ends when all the particles are connected to form a single cluster. The gelation point is defined as the time when the size of a
(4)
where q is the wave vector modulus. Experimentally, the scattering structure factor S(q) of the gels is obtained from the static light scattering measurements. The intensity of the scattered radiation for a system made of monodisperse particles is given by14
I(q) ) KP(q) S(q) (1)
i-(1/Df)
Rc i∼ Rp
g(r) )
θ 4πn sin λ0 2
()
(6)
where λ0 is the wave vector of the incident radiation (measured in a vacuum) and n is the refractive index of the disperse medium. In the case of small spherical primary particles of low optical contrast, as it is the case for the MFA particles used in this work, the following expression for the form factor P(q) can be used:14
P(q) )
{
}
3[sin(qRp) - qRp cos(qRp)] (qRp)3
2
(7)
Thus, on the basis of eq 5, the experimental structure factors are obtained by dividing the measured intensity by the particle form factor, P(q), given by eq 7 with Rp ) 37.5 nm.
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Results and Discussion Modeling the Frontal Gelation Process. Because the particle volume fraction of the gel phase is small (φ < 0.1), we can assume that the electrolyte diffuses in the gel with the same diffusion coefficient as in the liquid solution. Accordingly, the time evolution of the electrolyte concentration profile can be computed by assuming that the concentration changes only along the axis of the cylindrical cuvette while it remains uniform in any section of the cuvette perpendicular to the axis. In this case, the equation to be solved for the electrolyte concentration C(x, t) is the time-dependent, one-dimensional diffusion equation:26
∂C(x, t) ∂2C(x, t) ) Diff ∂t ∂x2
(8)
where t is time, x is the axial coordinate of the cylindrical cuvette, and Diff is the electrolyte diffusion coefficient. The initial condition for the electrolyte concentration profile is set as follows: it is equal to that of the initial electrolyte solution in the upper part above the latex, while it is 0 in the lower part where the latex is present. For the boundary conditions, it is evident that there is no diffusion flux at both ends of the system. Thus, we have
{
C(x, 0) ) C0 C(x, 0) ) 0 ∂C(x, t) )0 ∂x ∂C(x, t) )0 ∂x
for 0 < x < H for H < x < L for x ) 0
(9)
for x ) L
where H ) 2 cm is the length of the upper part and L ) 10 cm is the total length of the system, which is fixed for all the experiments. The solution of eq 8 satisfying the conditions given by eq 9 can be found analytically by means of the separation of variables method, and it is given by the following infinite series:26
C C0
)
H L
∞
+
2
∑ sin k)1πk
( ) ( ) [ ( )] πkH L
cos
πkx L
exp -Diff
πk L
2
t
(10)
The values of the diffusion coefficients of the electrolytes in water have been computed using the Nernst theory developed for dilute solutions:27 on the basis of this theory, the diffusivity of the salt Diff depends on the valence z and the diffusion coefficient D of both the cation and the anion, according to the following relation:27
Diff )
(z+ + |z-|)D+Dz+D+ + |z-|D-
(11)
where z+ and D+ and z- and D- are the valence and diffusion coefficient of the cation and the anion, respectively. The diffusion coefficients D( of cations and anions are related to their molar equivalent conductivities λ( through the relation:27
D( )
RTλ( F2|z(|
(12)
(26) Crank, J. The mathematics of Diffusion; Oxford Clarendon Press: Oxford, 1994. (27) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry 1; Plenum Press: New York, 1998.
Figure 1. Normalized CuSO4 concentration, C/C0, as a function of the normalized distance, x/L, for five time values. The boundary between the electrolyte solution and the latex is located at x/L ) 0.2. The horizontal thin solid line is the normalized CCC of CuSO4.
where R is the molar gas constant, T is the absolute temperature, and F is the Faraday constant. As discussed previously, because the gelation occurs instantaneously at the CCC, Cc, we can compute the position of the gel front, xf, as the location where the electrolyte concentration becomes C(xf, t) ) Cc. Then, the problem is reduced to solve numerically a single nonlinear algebraic equation, namely, to find the value of xf in eq 10 such that, at a given time t, C(xf, t) ) Cc. Figure 1 shows the CuSO4 concentration profiles calculated at four different times. The horizontal line indicates the value of the CCC so that the intersection between this line and the CuSO4 concentration profile calculated at a given time indicates the position of the gel front at that time. It is seen that in the portion of the cuvette occupied by the latex the electrolyte concentration increases with time, and the gelation front, that is, the location where C(xf, t) ) Cc, propagates downward along the cuvette. However, it is known that the aggregation rate of colloidal particles is very sensitive to small changes in the electrolyte concentration. For example, in the RLCA regime, that is, below the CCC, an increase in the electrolyte concentration by a factor of 2 may lead to an increase in the coagulation rate by several orders of magnitude.2 Thus, we can expect that, as the electrolyte concentration in the latex part increases with time, although below the CCC, pre-aggregation may occur and this certainly accelerates the propagation of the gel front. Accordingly, to make sure that gelation occurs entirely in the DLCA regime, we should limit our observations to the early stages of the front propagation. Finally, as the concentration of electrolyte in the latex becomes everywhere larger than the CCC, gelation occurs in the whole remaining latex. In Figures 2 (electrolyte: CuSO4) and 3 [electrolyte: Ca(NO3)2], the gel front position calculated by the model as a function of time is compared with the experimental data. In each figure, the curve corresponds to the position of the front given by the condition C(xf, t) ) Cc, with the salt diffusion coefficients computed by eqs 11 and 12, as listed in Table 1. It can be seen that in the early stages of the process the model predictions are in good agreement with the experimental data for both electrolytes. At later times, the model predictions tend to underestimate the position of the gel front, which also becomes more difficult to detect experimentally. This is consistent with the fact discussed previously that for larger times the increase in electrolyte concentration leads to a significant pre-
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Figure 2. Normalized position of the gel front xf/L as a function of time, for the gelation induced by the CuSO4 solution. The circles are the experimental data. The solid curve is the model prediction using eq 11 for computing the electrolyte diffusion coefficient (see Table 1).
Figure 3. Normalized position of the gel front xf/L as a function of time, for the gelation induced by the Ca(NO3)2 solution. The circles are the experimental data. The solid curve is the model prediction using eq 11 for computing the electrolyte diffusion coefficient (see Table 1).
aggregation of the particles in the latex, thus, accelerating the propagation of the gel front. It is worth noting that when pre-aggregation of the particles in the latex occurs, because this occurs in RLCA conditions, the gel formed at the gelation front may significantly deviate from a DLCA gel. This is the reason our light scattering measurements were performed at a distance of about 5 mm below the initial gel layer, consistent with the results shown in Figures 2 and 3. Structure of the DLCA and RLCA Gels. It is wellknown that, if a cluster has a fractal structure over a sufficiently large range of length scales, both its particledensity correlation function and its scattering structure factor show a power-law behavior, with an exponent that depends on the fractal dimension Df,25 that is,
g(r) ∼ r-Df-3 for 6Rp < r , ξ
(13)
S(q) ∼ q-Df for 1/ξ , q , 1/Rp
(14)
and
where ξ is the characteristic size of the cluster. For a large, isolated fractal cluster, the two Df values estimated from eqs 13 and 14 are identical. However, when eqs 13 and 14 are used to estimate the Df value of the clusters in gels, inconsistencies have been found.6,7,21 Typical examples
Figure 4. Particle-density correlation function g(r) as a function of the normalized distance r/2Rp, for the MC simulated RLCA gel at φ ) 0.03. The continuous straight line represents the best fit obtained using eq 13 with Df ) 2.25, the dashed line corresponds to Df ) 2.1, and the dotted one to Df ) 2.4.
Figure 5. Scattering structure factor S(q) as a function of 2qRp, for the MC simulated RLCA gel at φ ) 0.03. The continuous straight line represents the best fit obtained using eq 14 with Df ) 1.77, the dashed line corresponds to Df ) 1.6, and the dotted one to Df ) 1.94.
are shown in Figures 4 and 5, where g(r) and the corresponding S(q) are reported for the RLCA gel generated by MC simulations at φ ) 0.03. It is seen that, for the given regions indicated by eqs 13 and 14 (ξ in this case corresponds to the r value at the minimum indicated by the arrow in Figure 4), both g(r) and S(q) are straight lines in the log-log plane, whose slopes give the estimate of the fractal dimension. The obtained Df values are 2.25 ( 7% and 1.77 ( 10% from g(r) and S(q), respectively, which are significantly different. Such a problem has been addressed in our previous work21 on the basis of MC simulations. In general, the Df value in a gel estimated from S(q) is smaller than that estimated from g(r), and the latter is considered to be the “true” Df value.6,7 Moreover, on the contrary to the “true” Df value from g(r), which increases with the particle volume fraction φ, the Df value estimated from S(q) decreases as φ increases. This was attributed to the short-range, nonfractal regime of g(r), which induces a perturbation in S(q) during the Fourier transform and leads to a reduction in the slope of the linear part of S(q) in the log-log plane and consequently to a reduced apparent fractal dimension. This effect becomes so strong at high particle volume
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Figure 6. Scattering structure factors S(q) as a function of 2qRp, measured experimentally for the RLCA gels at three particle volume fractions φ. All the S(q) curves have been normalized to have the value of 1 at the smallest q value measured. The lines correspond to the fitting using eq 14.
Figure 8. Values of the fractal dimension Df determined from experimental S(q) of DLCA gels (b), as a function of the particle volume fraction φ, compared with the Df values from the MC simulated DLCA gels estimated from S(q) (0) and from g(r) (9).
Figure 7. Scattering structure factors S(q) as a function of 2qRp, measured experimentally for the DLCA gels at three particle volume fractions φ. All the S(q) curves have been normalized to have the value of 1 at the smallest q value measured. The lines correspond to the fitting using eq 14.
Figure 9. Values of the fractal dimension Df determined from experimental S(q) of RLCA gels (b) as a function of the particle volume fraction φ, compared with the Df values from the MC simulated RLCA gels estimated from S(q) (0) and from g(r) (9).
fractions that although the “true” Df value computed from g(r) increases with φ, the Df value estimated from S(q) decreases with φ. To verify these findings for real gels, we have determined the Df values from the experimentally measured S(q) and compared the obtained results with those extracted from S(q) calculated from the MC simulations. Figure 6 shows three experimental structure factors for the RLCA gels at the particle volume fractions, φ ) 0.01, 0.04, and 0.07, respectively. It is seen that the slope of the structure factor decreases progressively as the volume fraction increases. This indicates that the fractal dimension estimated from the experimental S(q) decreases as φ increases, in agreement with the results from the MC simulations. In Figure 7, three experimental structure factors of the DLCA gels obtained through the frontal gelation at φ ) 0.02, 0.04, and 0.07, respectively, are shown. Again, their slopes decrease as φ increases, consistent with the MC simulations. In Figures 8 and 9, the Df values obtained from the slope of the experimental S(q) are shown as a function of the particle volume fraction, φ, for DLCA and RLCA gels, respectively, and compared to the Df values obtained from the MC simulations. For the MC gels, the Df values have
been estimated from both S(q) and g(r), as given in our previous work.21 It can be seen that for the Df values estimated from S(q), the agreement between experiments and MC simulations is remarkably good in both DLCA and RLCA. This demonstrates that the gel structure factor generated by MC simulations contains important features of real gels. In particular, the slope of S(q) obtained from both the experiments and the MC simulations is affected by the presence of the nonfractal region in the particledensity correlation function, confirming that the S(q) curve of a gel does not provide the “true” fractal dimension of its clusters for φ > 0.01.21 We can, therefore, conclude that the structure of colloidal gels with a particle volume fraction φ > 0.01 is more compact than what the slope of their structure factor indicates. Although the particles in the gel structure are progressively more densely packed as their volume fraction increases and the “true” fractal dimension increases, the fractal regime in the particledensity correlation function becomes progressively shorter and its contribution to the scattering structure factor is overcome by that of the nonfractal regime. As a consequence, extreme care has to be used in the physical interpretation of scattering measurements in these systems.
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Figure 10. Comparison between the particle-density correlation function g(r) of RLCA and DLCA gels from the MC simulations at the particle volume fraction φ ) 0.09. The arrow indicates the position of the minimum in g(r). Note that the g(r) values for the RLCA gel have been multiplied by 2 to avoid any overlapping with the DLCA data.
It should be noticed that, for large particle volume fractions (φ > 0.06), the correct determination of the Df values from g(r) is difficult and progressively becomes impossible as φ increases because of the extremely small region that exhibits a linear behavior in the double logarithmic plot. An example is shown in Figure 10, where it is seen that the g(r) curves of the RLCA and the DLCA gels at φ ) 0.09 are almost flat after the nonfractal regime, indicating indeed very high compactness [compared, for example, to the g(r) curve at φ ) 0.03 in Figure 4]. Actually, at large particle volume fractions, as a result of the large number of clusters formed during the aggregation process, the interconnection among the clusters at the gel point occurs early enough that they cannot grow very large and develop a fractal structure over a large enough range of length scales. In this case, the application of the fractal scaling becomes unrealistic. This is the reason we did not report the Df values from g(r) and S(q) for these large particle volume fractions in Figures 8 and 9. On the other hand, Figure 10 shows that, for large particle volume fractions, although the Df values cannot be estimated from g(r), the positions of the minimum in g(r) for RLCA and DLCA gels is different, as indicated by the arrows. This can be even better visualized in Figure 11, where the S(q) curves computed from the two correlation functions in Figure 10 are shown. It is well-known that the minimum in g(r) corresponds to the peak in S(q) and gives an idea of the average cluster size in the gels. In Figure 11, it appears that the position of the maximum in S(q) of the DLCA gel occurs at a larger q value compared to the RLCA gel, thus, indicating that the average cluster size in RLCA gels is larger than that in DLCA gels. This is consistent with the fact that because clusters formed under RLCA conditions have a more compact structure, at the same particle volume fraction the clusters can grow larger in a RLCA gel than in a DLCA gel. Hasmy and Jullien7 have investigated the scaling of the position of the minimum in g(r) as a function of φ for DLCA gels. Their results indicate that, although the Df values estimated from g(r) increase with φ, the average cluster size ξ in the gels estimated from the position of the minimum in g(r) scales with φ as follows:7
ξ ∼ φ-[1/(3-Df)]
(15)
The scaling in eq 15 holds for φ < 0.08, and the obtained value of the fractal dimension is very close to that of the typical DLCA fractal cluster formed in diluted conditions,
Figure 11. Comparison between the structure factors S(q) of RLCA and DLCA gels from the MC simulations at the particle volume fraction φ ) 0.09. Note that the S(q) values for the RLCA gel have been multiplied by 2 to avoid any overlapping with the DLCA data.
Figure 12. Comparison between the positions of the minimum ξ in g(r) of RLCA and DLCA gels from the MC simulations as a function of the particle volume fraction φ. The lines are fittings using eq 15, with the Df values equal to 2.05 and 1.75 for RLCA and DLCA, respectively.
that is, Df ∼ 1.7. This suggests that a DLCA gel can be considered to be made of clusters, whose fractal dimension is equal to that of DLCA clusters formed in dilute conditions. In Figure 12, we have reported the position of the minimum in g(r) as a function of φ for both RLCA and DLCA gels from the MC simulations obtained in this work. The dotted lines represent the fit with eq 15, using the fractal dimension Df ) 1.75 and 2.1 for the DLCA and RLCA gels, respectively. This indicates that the same conclusions reached by Hasmy and Jullien7 and just reported for the case of DLCA gels applies also to RLCA gels. It is worth mentioning that the apparent contradiction between the Df values and their dependence on φ estimated from g(r) and eq 15 could indicate two aspects of the colloidal gelation: first, before the gel point the clusters have grown according to a DLCA mechanism, characterized by Df ∼ 1.7, and second, the increase in their fractal dimension in the gel phase is the result of the interpenetration among the different clusters constituting the gel. Finally, it should be noted that, although a good agreement has been obtained between the Df values estimated from the experimental and the MC structure factors, there are indeed some differences when comparing the experimental and the MC structure factors at the same
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Figure 13. RLCA gel structure factors S(q) as a function of 2qRp, obtained from the MC simulations at three particle volume fractions φ.
Figure 14. DLCA gel structure factors S(q) as a function of 2qRp, obtained from the MC simulations at three particle volume fractions φ.
particle volume fraction. To better visualize these differences, we have plotted in Figures 13 and 14 the structure factors of the RLCA and DLCA gels from the MC simulations at the three particle volume fractions corresponding to the experimental gels shown in Figures 6 and 7 and also in the same q range accessible for the light scattering instrument used to perform the measurements. It can be observed by comparing Figure 13 (14) with Figure 6 (7) that in both RLCA and DLCA conditions, the S(q) curves from the MC simulations exhibit a peak within the q range, while the corresponding experimental S(q) bends at a smaller q value and does not present a peak in the accessible q range. This indicates that the average size of the gel clusters is larger in the real gel than in the MC simulated gel. To better assess this point, small-angle light scattering experiments should be performed. Concluding Remarks In this work, a detailed comparison between experimental light scattering data on gels and predictions of MC simulations has been performed, with the aim of better understanding the fractal nature of colloidal gels obtained
Lattuada et al.
in both RLCA and DLCA conditions. To perform light scattering measurements on gels with particle volume fractions in the range of 0.01 e φ e 0.08, we have prepared the gels by adding a suitable amount of salt to a fluorinated polymer colloid (Hyflon MFA latex), which has a low optical contrast with respect to water. While the preparation of RLCA gels is straightforward, the preparation of DLCA gels requires a special technique, referred to as frontal gelation. With this technique, a certain amount of concentrated electrolyte is placed on the top of the latex so that a thin layer of gel is formed at the interface, preventing any further mixing of the two solutions. Then the salt diffuses through the gel layer, reaching the latex. When the salt concentration reaches the CCC, the latex gels instantaneously leading to a DLCA gel, and the gel front moves downward. We have shown that the position of the gel front can be predicted as a function of time with a simple model. The experimental structure factors S(q) of all the prepared RLCA and DLCA gels have been determined from the light scattering measurements, and the values of the apparent fractal dimension Df have been estimated from the slope of S(q). It is found that, for both RLCA and DLCA gels in the given range of φ, the experimental Df values are in good agreement with those estimated from the structure factors computed from MC simulations. This confirms the finding from MC simulations21 that the apparent fractal dimension Df estimated from the slope of S(q) decreases as φ increases and implies that the MC simulations provide realistic structures of colloidal gels. Such a decrease in the Df value with φ has been explained in our previous work,21 and it is attributed to the strong effect of the short-range, nonfractal region of the particledensity correlation function g(r) on the Fourier transformation needed to compute the structure factor. The “true” Df value estimated directly from g(r) instead increases as φ increases and is also substantially larger than that estimated from S(q). The consequences are that the slope of S(q) for gels formed at φ > 0.01 cannot be interpreted as the fractal dimension of the clusters composing the gel network and that the structure of the gels is more compact than what the slope of S(q) indicates. On the other hand, there is a general disagreement between experiments and MC simulations concerning the position of the peak of the structure factor. In particular, under both RLCA and DLCA conditions, the S(q) curves from the MC simulations generally exhibit a peak at a certain q value, which is not observed in the experimental S(q) in the accessible q range. Moreover, at the same particle volume fraction, the experimental S(q) bends at a smaller q value, indicating that the average size of the gel clusters is larger in real gels than in MC simulated gels. Acknowledgment. This work was financially supported by the Swiss National Science Foundation (Grant 2000-061883). We are grateful to Dr. A. Chittofrati for introducing us to the literature regarding frontal gelation. The Hyflon MFA latex has been kindly supplied by Solvay Solexis SPA (Italy). Many useful discussions with Dr. Jan Sefcik, Peter Sandku¨hler, and Andrea Vaccaro are also gratefully acknowledged. LA035949O
