Langmuir 1998, 14, 49-54
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On the Density and Structure Formation in Gels and Clusters of Colloidal Rods and Fibers Albert P. Philipse* and Anieke M. Wierenga Van’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute, Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands Received April 14, 1997. In Final Form: October 27, 1997X Structures of gels and clusters of disordered rods and fibers are described by scaling relations for a homogeneous random fiber network and a heterogeneous (that is, fractal) fiber structure. Both models relate the gel density to the particle shape. It is shown that gel densities for attractive colloid rods (iron hydroxide, clay minerals) are often too low to fill space with a homogeneous microstructure. Space filling with heterogeneous rod clusters allows for a very low solid content due to a combination of high particle aspect ratios and low fractal dimensions. This view is supported by reanalysis of early simulations of Vold (J. Phys. Chem. 1959, 63, 1608) of ballistic rod deposition, and yield stress measurements on boehmite-rod suspensions in an accompanying paper (Wierenga et al. Langmuir 1998, 14, 0000). Attention is also given to rapid rod coagulation and retarding factors, such as polydispersity in attraction strength, which may strongly affect gelation kinetics.
1. Introduction Gels and pastes of anisotropic colloids such as clay particles have been processed ever since the invention of ceramics in the form of pottery more than 8000 years ago.1 The anisotropic particle shape is important for the plasticity and permeability of clays in a casting or moulding process for ceramic shapes. (Our household, and possibly far more than that, would look very different if colloids were always spherical.) Nevertheless, many aspects of the influence of particle shape on suspension properties are poorly understood. One such property is the density of microstructures as encountered in packings or gels of nonspherical particles. The aim of this paper is to model the density and structure formation of gels of attractive rodlike particles. Sedimentation and gelation in a suspension of long particles such as V2O5 fibrils,2 cellulose fibers,2-4 attapulgite clay,5 and colloidal rods6,7 usually leads to structures with a low solid content. A striking example is a dispersion of imogolite rods,7,8 which may form space filling gels with volume fractions as low as Φ ≈ 0.002. These transparent gels do not collapse and may be stored over long periods of time. A similar observation is reported in our study of boehmite rod dispersions, presented in an accompanying paper,9 hereafter referred to as paper II. In these dispersions a yield stress is observed at very low concentrations. This phenomenon appears to be related to the high aspect ratio of the colloids. * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, December 15, 1997. (1) Kingery, W.; Bowen, H.; Uhlmann, D. An Introduction to Ceramics; Wiley: New York, 1976. (2) Hermans, P. H. Gels. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1949; Vol. 2. (3) Hermans, P. H. Physics and Chemistry of Cellulose Fibres; Elsevier: New York, 1949. (4) Dodson, C. T. J. Tappi J. 1996, 79, 212. (5) Buscall, R. Colloids Surf. 1982, 5, 269. (6) Philipse, A. P.; Nechifor, A.-M.; Pathmamanoharan, C. Langmuir 1994, 10, 4451. (7) Liz Marzan, L.; Philipse, A. P. Colloids Surf. A 1994, 90, 95. (8) Wijting, W.; Kluijtmans, S. G. J. M. Internal Report; Utrecht University, 1995. (9) Wierenga, A. M.; Philipse, A. P.; Lekkerkerker, H. N. W.; Boger, D. V. Langmuir 1998, 14, 55.
The open structure of (fiber) gels and its consequence for gel properties have been discussed extensively,2,3 but the role of the particle shape has not been specifically addressed. (Though Hermans already mentioned that low densities may be related to strongly anisotropic structural units.2) Monographs on gelation and aggregation10-12 usually focus on spherical particles. Here we formulate two models which may be useful in interpreting gel densities for rigid rods, namely the homogeneous random rod structure in section 2 and the heterogeneous fiber fractal in section 3. (In this paper “heterogeneous” and “fractal” structure are synonymous.) Apart from the density also the formation kinetics of rod clusters or gels is often puzzling. Gelation and coagulation in colloidal rod dispersions of comparable volume fractions occur on time scales ranging from seconds to months. In section 4 we estimate the (fast) coagulation kinetics for two rods in a dilute solution, using a simple extension of Smoluchowski’s coagulation theory for spheres, and identify retarding factors, such as “attraction polydispersity”. Experimental and simulation data from the (scarce) literature on rod gels are compared with our models in sections 5 and 6. The case of boehmite rods is discussed in detail in paper II. It should be noted that we do not consider (polymer) gels with chemical links, but only “physical particle gels” of rigid inorganic rods with high aspect ratios which stick together as a result of noncovalent forces, such as van der Waals attractions. Structures and packings of rods without attractions are treated elsewhere.13 2. The Homogeneous Random Fiber Network A suitable reference model for the structure of a collection of disordered rods is a homogeneous random (10) Aggregation Processes in Solution; Wyn-Jones, E., Gormally, J., Eds.; Elsevier: Amsterdam, 1983. (11) Family, F.; Landau, D. P. Kinetics of Aggregation and Gelation; North-Holland: Amsterdam, 1984. (12) Sonntag, H.; Strenge, K. Coagulation Kinetics and Structure Formation; Plenum: New York, 1987. (13) Philipse, A. P. Langmuir 1996, 12, 1127. Corrigendum: Langmuir 1996, 12, 5971.
S0743-7463(97)00375-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/06/1998
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fiber network. This network is a random collection of evenly distributed, thin rigid rods with effectively uncorrelated contacts. Within this mean field approach the rod number density ν is given by the mechanical equation of state13,14
νVex ) c
(1)
Here Vex is the orientational-averaged excluded volume, and c is the contact number, i.e., the average number of neighbors which make “physical” contact with a certain particle. For a thin cylinder with diameter D and length L . D, the volume fraction Φ and the average excluded volume are15
Φ)
πDL2 πνD2L L ; Vex ) for r ) . 1 4 2 D
Figure 1. Snapshots of a homogeneous random fiber network (a) and variations of it with a similar microstructure but composed of clusters with a smaller (b) and larger (c) aspect ratio. All structures have a Euclidean dimension d ) 3.
(2)
with r the aspect ratio. So eq 1 becomes
Φr ) c/2;
for r . 1
(3)
(An equation of this type was also derived by Dodson4 for the fiber density in paper.) The scaling Φr ) constant is also (experimentally) observed for random dense packings of rigid rods.13,16 The contact number in a network or gel, however, is lower than the value of c in a dense packing. This results in a lower network density in accordance with eq 3. What is a plausible value for c in the case of a gel, modeled by a homogeneous fiber network? In a spacefilling gel there will be (many) percolating routes from one vessel wall to another. Balberg et al.17,18 simulated homogeneous percolation in a system of randomly oriented rods with uncorrelated contacts. Their result for the threshold number density is
νtVex ) 1.4
(4)
which corresponds to a contact number of c ) 1.4. From eqs 4 and 3 we expect the inequality
Φgelr > 0.7
(5)
to apply for the density of gel which can be described by a homogeneous random fiber network. The maximum concentration of randomly oriented thin rods is the random dense packing value Φ ≈ 5.4/r.13,16 The homogeneous gel density in eqs 3 and 5 will change when the building blocks are rod clusters instead of single rods. For linear clusters two extremes can be evaluated by a redefinition of the aspect ratio (Figure 1). First, clusters of length L may be present, containing n parallel rods. The cluster diameter is proportional to Dxn. The relevant aspect ratio is now r/xn. Hence the gel density increases as
Φgelr ∝ (c/2)xn
(6)
The other extreme concerns n rods which form strings of (14) Philipse, A. P.; Verberkmoes, A. Physica A 1997, 235, 186. (15) Vroege, G. J.; Lekkerkerker, H. N. W. Rep. Prog. Phys. 1992, 55, 1241. (16) Nardin, M.; Papirer, E.; Schultz, J. J. Powder Technol. 1985, 44, 131. (17) Balberg, I.; Binenbaum, N.; Wagner, N. Phys. Rev. Lett. 1984, 52, 1465. (18) Balberg, I.; Binenbaum, N. Phys. Rev. A 1987, 35, 5174.
Figure 2. Snapshots of heterogeneous fiber networks built from the same units as in Figure 1. Structure c has a dimension d ) 1; (a) and (b) have intermediate fractal dimensions 1 < d < 3.
length nL and diameter D. Then we have the proportionality
Φgelr ∝ c/2n
(7)
indicating that stringlike clusters considerably reduce the homogeneous gel density. It should be emphasized that a random fiber network, whatever its constituents, is a homogeneous microstructure. Particle centers are evenly distributed in space and the density is independent of the radius Rc of the snapshots in Figure 1. (Of course Rc should be large enough such that the volume fraction is a meaningful quantity.) The random microstructure in Figure 1a is similar to the timeaveraged microstructure of a noncoagulated, isotropic suspension of Brownian rods or fibers. To form spacefilling gels with a microstructure as in Figure 1a, one needs a critical concentration of rods. In view of eq 5 it is unlikely that such a homogeneous gelation takes place below a suspension concentration of Φ ≈ 0.7/r, being the percolation threshold density. Below this suspension density homogeneous space filling seems only possible for elongated clusters as in Figure 1c in which building units have effectively a high aspect ratio. Another possibility to fill space at very low particle concentrations is the growth of a heterogeneous structure of highly ramified clusters of attractive fibers. The structure of such a low density network is the subject of the next section. 3. The Heterogeneous Fiber Fractal Consider a spherical cluster of particles (not necessarily rods) with a typical cluster radius Rc. For a homogeneous microstructure (Figure 1) the number of particles, Np, in the cluster scales as
NpVp ∝ Rc3
(8)
with Vp the particle volume. For ramified clusters as shown in Figure 2 the dimensionality d will be lower than d ) 3. (Such a heterogeneous structure of rods or fibers may be called a “fiber fractal”.) The number of particles in such a cluster is
Colloidal Rod Structures
Np ∝
Langmuir, Vol. 14, No. 1, 1998 51
( ) Rc
d
de3
1/3
Vp
(9)
corresponding to an average number density in the cluster of
νVp ∝
( ) Rc
d-3
(10)
Vp1/3
( ) Rc
Φr ∝ (c/2)
Vp
Vp1/3
(14)
∝ (c/2)ξd-3
(12)
For fractals composed of clusters of parallel rods (Figure 2b) the aspect ratio r in eq 12 may be rescaled as in eq 6 or eq 7. It should be noted that the cluster density depends on a dimensionless size ξ, rather than an absolute cluster size. The identification ξ ) Rc/Vp1/3 in eq 12 is somewhat arbitrary. One could, for example, choose another minimum length scale than Vp1/3, such as a length a below which no fractal geometry is present. In principle, scattering data may be employed to determine the length scales for fractal behavior. Following the approach for sphere gels,19,20 we assume that the fiber clusters grow until they are large enough to collectively fill space: the gel is a packing of “blobs” of the type in Figure 2. Then the condition for gelation is that the average volume fraction of the fiber fractals equals the overall volume fraction of the dispersion
( ) R hc
Vp1/3
(15)
and its value for thin rods in eq 12. The pair correlation function
g(x) ) Φ(x)/Φ
(16)
therefore equals
r.1
Φgelr ∝ (c/2)
Vp
c
d g(x) ) (x/Rc)d-3 3 g(x) ) 1
( )
d-3
1/3
∫0R Φ(x)4πx2 dx Φ) ∫0R 4πx2 dx
(11)
1/3
d-3
x
which follows from the definition of the average volume fraction
d-3
This general result for uncorrelated particle contacts may now be specified for thin rods to yield, in analogy with eq 3
Rc
( )
Φ(x)r ∝ d(c/6)
c
So far we essentially followed the theory11,19-21 for fractal sphere clusters in a gel of attractive spheres. Note, however, that eqs 8-10 are general and do not specify particle shape. In view of eq 1 we can replace eq 10 by the proportionality
νVex ∝ c
The local volume density in a spherical fiber cluster at a distance x from its center of mass is
(13)
(17)
x g Rc
with a cutoff at x ≈ Rc, because at distances beyond the correlation length Rc, g(x) equals unity. The Fourier transform of eq 17 leads to a structure factor of the wellknown form11,19-21
S(K) ∝ (KVp1/3)-d
(18)
with K the wavevector. Note that eqs 18 and 17 are also valid for spheres,19-21 whereas the density in eq 14 applies to rods. 4. Rapid Rod Coagulation: Initial Stage, Retarding Factors The fractal dimension will reflect the formation dynamics of the rod clusters (see also section 6). To find out whether the flocculation time for rods is very different from that for spheres, we first consider the initial stage of rapid coagulation in a very dilute dispersion. The rate of disappearance of primary particles due to binary collisions is22
-
d-3
x < Rc
dν ) kν2 dt
(19)
where the Smoluchowski rate constant22
r.1
k ) 8πD0R
Here R h c is an average cluster size or, alternatively, a typical correlation length in the gel. (Note that in this description the gel is homogeneous on a scale .R h c; the gel density is independent of the size of the vessel.) Equation 13 predicts very low gel densities for fibers with very high aspect ratios, in particular if the cluster-dimensionality is low. Another consequence of the non-Euclidian dimension is the development of a structure factor in light scattering, which in a random network equals unity (by definition).
is the collision frequency of particles with one test particle, which only depends on the diffusion coefficient D0 and the center-to-center distance R at which two particles collide. For spheres with radius a, R ≈ 2a so
(19) Bremer, L.; van Vliet, T.; Walstra, P. J. Chem. Soc., Faraday Trans. 1 1989, 85, 3359. (20) Bremer, L. Fractal Aggregation in Relation to Formation and Properties of Particle Gels. PhD thesis, Wageningen Agricultural University, 1992. (21) Dickinson, E. An Introduction to Food Colloids; Oxford University Press: Oxford, 1992.
k ) 8kT/3η0
(20)
(21)
where we use the diffusion coefficient
D0 ) kT/6πη0a
(22)
of a free sphere in a solvent of viscosity η0 (kT is the thermal energy). When applied to fast coagulation, eq 21 describes the kinetics of the initial stage during which only sphere (22) von Schmoluchowski, M. Z. Phys. Chem. 1917, 92, 129.
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Philipse and Wierenga
doublets are formed.22,12 For rods both D0 and R depend on the orientation of the particles during an encounter. We assume that the encounter does not change rod orientations. (Evans and Luner23 conclude in their study of cellulose fibers that van der Waals attractions (torques) hardly affect the relative orientation during pairwise collisions in the first stage of rapid coagulation.) Then orientational averages may be used to estimate the collision frequency. For the translational diffusion coefficient of a free rod this average is
D0 )
kT ln r 3πη0L
for r . 1
(23)
Here we neglect any (hydrodynamic) interaction effects on the diffusion; the same assumption underlies eq 21. A freely rotating rod sweeps a spherical volume with diameter L. Collisions may occur as soon as these “rotation volumes” overlap. Thus the average center-to-center distance when a collision occurs will be or order L. Substitution of eq 23 and R ≈ L in eq 20 yields
k≈
8kT ln r 3η0
for r . 1
(24)
The particle size cancels, just as for spheres, but the aspect ratio modestly increases the collision probability in a dilute dispersion in comparison to spheres. Equation 24 is virtually the same as the result of Mu¨ller’s24 extensive treatment of rapid coagulation of nonspherical colloids (see eqs 8 and 19 in ref 24). It seems, incidentally, that Mu¨ller’s theory from 192824 is the most recent one for monodisperse rods. (Booth25 calculated flocculation rates for two ellipsoids which differ appreciably in size. He also concluded that deviations from the spherical form at constant particle volume accelerate the process.) The estimate in (24) indicates that Smoluchowski’s result for spheres in eq 21 also gives the correct order of magnitude for the fast(est) coagulation rate in a very dilute suspension of rodlike colloids. The rapid coagulation kinetics may even be accelerated at higher densities where rotational volumes continuously overlap. In rod suspensions, however, one frequently encounters flocculation or gelation on a time scale of minutes to weeks. Clearly, strongly retarding factors must be present, such as a double layer repulsion. Another factor is the presence of only a small fraction of sufficiently attractive rods in a sea of rods which are stable against aggregation. (The origin of this “interaction polydispersity” is discussed in paper II.) Here the stable rods retard coagulation because the attractive rods can only encounter each other via strongly hindered self-diffusion over large distances. The effect of a small fraction of unstable colloids can often be observed in practice as a slow settlement of some flocs in a suspension with a long-term stability. Also surface heterogeneity may retard coagulation. In ref 26 the possibility has been mentioned that rods only stick (permanently) when certain attractive sites collide. (Such sites may be due to surface irregularities.) Suppose the area of sites on a rod is σ2, then the probability that attractive sites stick during collision of two rods is of order (σ2/πDL)2. Consequently eq 24 modifies to (23) Evans, R.; Luner, P. J. Colloid Interface Sci. 1989, 128. (24) Mu¨ller, H. Kolloidchem. Beih. 1928, 27, 223. (25) Booth, F. Discuss. Faraday Soc. 1954, 18, 104. (26) Wierenga, A. M.; Philipse, A. P. J. Colloid Interface Sci. 1996, 180, 360.
k≈
8kT 2 (σ /πDL)2 ln r 3η0
(25)
It is obvious that a small attractive site on a large “inert” rod is very inefficient to accelerate the coagulation rate. This reminds one of reaction limited aggregation, whereas eq 24 refers to diffusion limited kinetics. It is conceivable that stirring of a suspension may compensate for these retarding factors. However, a simple calculation shows that agitation has only an effect for large fiber clusters. For particles that are propelled by a shear flow, without any Brownian motion, the rate constant k for the disappearance of singlets (eq 19) is22,27
ks )
4 3 R γ˘ 3
(26)
in which γ˘ is the shear rate. For randomly oriented rods we again take R ≈ L. Consequently the ratio of shearinduced and Brownian collision rate (eq 24) in dilute dispersions is of order
ks η0L3γ˘ ) k 2kT ln r
(27)
For boehmite rods in paper II with length L ) 180 nm and aspect ratio r ) 22.5 in water at room temperature, the ratio ks/k is unity for γ˘ ≈ 300 s-1. So for rods of this length a high shear ratio is needed to achieve a significant contribution of shear-induced flocculation. (At these high shear rates, incidentally, rods will be (partly) oriented, which will reduces R and thus the flocculation rate.) However, in ref 28 we observe that γ˘ ≈ 1 s-1 may already enhance flocculation of boehmite rods. This is very likely due to a small minority of clusters, which hardly affect the viscosity at rest, but “nucleate” the flocculation in a shear field due to their large effective diameter or excluded volume. This flocculation may stop after some time if only a fraction of rods is sufficiently attractive.28 5. Experimental Densities and Structures A few studies report densities for structures of fairly well-defined, attractive colloidal rods which allow a comparison with the models in previous sections. Gel densities are summarised in Table 1. We only comment upon densities and structures; coagulation kinetics for the various colloids has, to our knowledge, not been studied yet. Iron Hydroxide Gels. Gelation of rodlike iron hydroxide colloids confirms the description of rod-gel microstructures in sections 2 and 3. Haas et al.29 present electron micrographs of iron hydroxide needles which have been “frozen” in a suspension by the polymerization of vinyl acetate. For stable colloids the micrographs show a homogeneous distribution of individual rods resembling our Figure 1a and parallel packages of a few rods as in Figure 1b. However, when the rods aggregate upon salt addition, homogeneity is lost and heterogeneous structures are formed which resemble the drawings in parts a and b of Figure 2. A structure as in Figure 2a is found after (27) Overbeek, J. T. G. Kinetics of flocculation. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1952; Vol. 1. (28) Wierenga, A. M.; Philipse, A. P. Langmuir 1997, 13, 4574. (29) Haas, W.; Zrinyi, M.; Kilian, H. G.; B. Heise. Colloid Polym. Sci. 1993, 271, 1024.
Colloidal Rod Structures
Langmuir, Vol. 14, No. 1, 1998 53 Table 1. Densities of Gels and Sediments
system attapulgite rods polystyrene spheres imogolite fibers silica-boehmite rodsc boehmite rodsd Vold sediments iron hydroxide rodsf d
ref
L (µm)
5 5 7, 8
∼2b
26 9 33 31
Φ (%)a
r ∼40b
0.57 0.3
111
0.29 0.21 28 0.025
22 25 e 10
1.87 5 0.19 0.76 0.17 0 0.2-0.6
Φgel (%) ∼2 ∼12 0.19 0.76 0.17 3.2 0.3 e 0.2-0.6
r*Φgel ∼0.8 0.2 0.8 0.2 0.7 0.07 0.3 0.02-0.06
a Initial volume fraction of sticky particles in suspension. b Rough estimate from Figure 10 in ref 5. c Uncharged silica rods in cyclohexane.26 Typical value for attractive boehmite rods. e Figure 3. f Range of gel densities, depending on electrolyte concentration.31
rapid aggregation of iron hydroxide, whereas slow flocculation favors the formation of bundles30 leading to a denser system as in Figure 2b. The heterogeneity of the structures leads, as expected from eq 12, indeed to low space filling gel densities31 (see Table 1). They are low enough to exclude the possibility of a homogeneous network: a fit of the densities to eq 3 leads to unphysical values for the contact number c. The heterogeneity of the iron hydroxide gels is also confirmed by the fractal dimension of d ) 1.95, measured by Brunner et al.30 Gel densities for charged rods depend on the ionic strength. At high ionic strength the net attraction between the rods is large which at first sight promotes low gel densities. However, the iron hydroxide rods show the reverse trend,31 and also for the boehmite rods in paper II we find that an increase in salt concentration increases the gel density. This may be due to an enhanced parallel clustering of rods which maximises the van der Waals attractions. The formation of such parallel clusters decreases the aspect ratio of the units forming the network. Consequently the gel density increases as indicated by eqs 6 and 12. Clay Particles. Buscall5 slowly sedimented rodlike clay particles (attapulgite) in an aqueous 10-2 M NaCl solution to study sediment volumes as a function of time and centrifugal field strength. The compression modulus of these volumes steeply increased above a volume fraction of about Φ ≈ 0.02, which was identified as a gel-point.5 (A similar gel-point was found for flocculated latex spheres at about Φ ≈ 0.125). No size distribution of the polydisperse attapulgite colloids was reported. From Figure 10 in ref 5, we estimate an average aspect ratio of roughly r ≈ 40, corresponding to a rescaled gel density of Φr ≈ 0.8 (Table 1). This estimate does not exclude the possibility of a (partly) homogeneous microstructure (cf. eq 5). Imogolite Rods. Imogolite is an aluminum silicate fiber with a diameter of about D ) 2.7 ( 0.4 nm and a length in the range 200-1000 nm.7,8 Equations 3 and 14 predict that colloids with such a high aspect ratio should form gels at low densities. This is indeed the case. In a typical experiment7,8 an aqueous imogolite dispersion (dialyzed against demineralized water) with a volume fraction of Φ ) 0.0019 is brought to its isoelectric point pH ) 9-10 by addition of ammonia or sodium hydroxide. Transparent gels are formed which generally fill the suspension volume (Table 1). Gelation is instantaneous and could be an instance of the rapid rod coagulation of section 4, even enhanced by the strong rod “entanglement” which must be present in these imogolite suspensions. (30) Brunner, R.; Gall, S.; Wilke, W.; Zrinyi, M. Physica A 1995, 214, 153. (31) Zrinyi, M.; Kabai-Fax, M.; Fuhos, S.; Horkay, F. Langmuir 1993, 9, 71.
The number density ν* at which rods start to “entangle”32 is of order
v* ≈ 1/R3
R≈L
(28)
corresponding to a volume fraction
rΦ* ≈ D/L
for r . 1
(29)
Table 1 shows that the imogolite density is an order of magnitude larger than that given in eq 29. If homogeneous gels are ever formed, it is probably in such a highly entangled system with very fast rod kinetics such that gelation preserves the initial isotropic particle distribution. The data for imogolite (Table 1) do not allow a firm conclusion. Note, however, that the Φr values for the iron hydroxide gels (Table 1), which gel below the onsetof-entanglement density in eq 29, are much smaller than those for imogolite. This may be an indication for the relative homogeneity of the imogolite gels. Boehmite and Silica Rods. Uncharged sterically stabilized silica rods in cyclohexane can be prepared as described elsewhere.6 Weak attractions may be present between these particles which may lead to an isotropic phase separation.26 We also observed a very slow gelation on a time scale of months. These transparent silica rod gels in cyclohexane are reversible in the sense that moderate shaking easily “liquefies” the system. The gel densities (Table 1) are about Φ ≈ 0.7/r. These densities are an order of magnitude higher than gel densities for attractive boehmite rods in water discussed in paper II, where gelation is much faster (minutes). The boehmite gel densities (Table 1) are too low for a homogeneous microstructure. Light scattering results in paper II indeed yield a fractal dimension of d ) 2.35. 6. Vold’s Simulations. Rod versus Sphere Gels Simulation results for Brownian rod aggregation for comparison with eq 12 are, to our knowledge, not available yet. Trends in the pioneering simulation results of Vold33 on ballistic rods can be interpreted using eq 12. Vold33 simulated sediment formation of randomly oriented, strongly attractive rods which cohere irreversibly on initial contact. Vold’s approach is as follows. Rods with aspect ratio r are modeled by a rigid chain of r spheres. The rods are initially positioned in random orientations and locations above a rectangular container. This starting suspension is infinitely dilute. Each particle in turn “drops” into the container by reduction of its z-coordinate until its (32) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1988. (33) Vold, M. J. J. Phys. Chem. 1959, 63, 1608.
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Langmuir, Vol. 14, No. 1, 1998
Figure 3. Sediment volume fraction percentage Φ after ballistic deposition of irreversibly sticking rods (rigid sphere chains) with aspect ratio r as simulated by Vold.33
Figure 4. Rescaling the data from Figure 3. (c ≈ 2 is the average number of contacts on a rod33) shows that the densities are significantly below the value of a random network: Φr ) c/2.
fall is arrested by contact with either a previously dropped particle or the bottom of the container. Figure 3 shows the considerable decrease in sediment volume fraction Φ for increasing aspect ratio. However, when we rescale the volume fraction to Φr, we observe only modest deviations from its average Φr ≈ 0.3 (see Figure 4). Vold also evaluated the average number of contacts on a particle, which slightly increases from c ) 1.99 at r ) 2 to c ) 2.17 at r ) 18. On average we find Φr ) 0.14c (see Figure 4). This result is significantly below the value of 0.5c in eq 3, expected for a homogeneous random network. In view of eq 7, the low density could be the consequence of rods forming predominantly long clusters by end-to-end contacts as in Figure 1c. Vold’s analysis, however, shows that crossed contacts occur much more frequently at higher aspect ratios than end-to-end contacts.33 So the low density of the “Vold network” is due to a heterogeneous structure with (regions of) a dimension d < 3. A heterogeneous microstructure is indeed likely to form, because settling rods will be intercepted by protruding branches of the cluster, which disfavors a homogeneous distribution of particle centers. The densities reported by Vold (Figure 3) exhibit no abrupt change going from a rod (string of beads) to a sphere (one bead), and the density Φr ≈ 0.3 (Figure 4) applies for nearly all aspect ratios. This may be a specific feature of the Vold model (infinite dilution, ballistic particles) which will not appear in a more extended simulation for Brownian particles at finite suspension densities. Nevertheless, it should be noted that in our model for
Philipse and Wierenga
structures with random contacts (e.g., eq 13), the particle shape is basically only a scaling factor for the concentration. For example, the g(x) in our model is actually independent of the shape; eq 17 only concerns the relative distribution of mass centers. This distribution of points (with a Euclidean or fractal dimension) is “dressed” with randomly oriented objects, which for thin rods leads to the average volume fraction in eq 12. Within this meanfield approach the fractal dimension primarily reflects differences in formation dynamics (e.g., diffusion or reaction limitation) rather than particle shapes. In paper II we find for the slowly forming boehmite rod gels a dimension d ) 2.35, whereas for the strongly attractive iron hydroxide rods30 the dimension of the (more ramified) gels is d ) 1.95. It is tempting to draw the analogy with reaction-limited and diffusion-limited sphere aggregation for which, respectively, d ) 2.1 and d ) 1.85.34 7. Conclusions The data in Table 1 and the scaling relations for the gel density demonstrate that structures of attractive (rigid) fibers and rods have much lower densities than the random rod packing value of Φr ) 5.4. Often, however, rod gels are not simply “diluted” versions of the homogeneous microstructure of a random rod packing. For low densities such that Φr , 1, a homogeneous rod structure has no mechanical coherence and space has to be filled with heterogeneous rod clusters. The resulting low solid content is, according to eq 13, due to a combination of a high fiber aspect ratio and a low fractal cluster dimension. (This equation also explains the trend in volume fractions in Vold’s simulations of rod deposition.33) The conclusion that fiber gels with very low densities have to be heterogeneous and therefore will have a structure factor as in eq 18 is a general one. In this respect particle interaction details such as nature and concentration of electrolytes, Hamaker constants, etc. are not relevant. These details, however, will influence the formation kinetics and eventually the fractal dimension of fiber clusters. Within our mean-field approach, the particle shape is only a scale factor for the average density, suggesting that well-known results for fractal spherecluster growth may also explain trends in the gelation of dispersions of rods (and other shapes). The initial (binary) aggregation kinetics for rods in very dilute suspensions differs little from rapid sphere coagulation. The slow gelation often observed in practice is therefore not due to the rod shape as such but to strongly retarding factors like surface heterogeneity and interaction polydispersity. The particle anisotropy is responsible for the low particle concentrations at which these factors become significant. Acknowledgment. H. N. W. Lekkerkerker is acknowledged for stimulating discussions. M. Uit de BultenWeerensteyn and H. de Vries are thanked for their help in the preparation of the manuscript. This work was supported by the Netherlands Organization for Chemical Research (SON) with financial aid from the Netherlands Organization for Scientific Research (NWO). LA9703757 (34) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Nature 1989, 339, 360.
